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The Scaling Limit of High-Dimensional Online Independent Component Analysis Chuang Wang and Yue M. Lu John A. Paulson School of Engineering and Applied Sciences Harvard University 33 Oxford Street, Cambridge, MA 02138, USA {chuangwang,yuelu}@seas.harvard.edu Abstract We analyze the dynamics of an online algorithm for independent component analysis in the high-dimensional scaling limit. As the ambient dimension tends to infinity, and with proper time scaling, we show that the time-varying joint empirical measure of the target feature vector and the estimates provided by the algorithm will converge weakly to a deterministic measured-valued process that can be characterized as the unique solution of a nonlinear PDE. Numerical solutions of this PDE, which involves two spatial variables and one time variable, can be efficiently obtained. These solutions provide detailed information about the performance of the ICA algorithm, as many practical performance metrics are functionals of the joint empirical measures. Numerical simulations show that our asymptotic analysis is accurate even for moderate dimensions. In addition to providing a tool for understanding the performance of the algorithm, our PDE analysis also provides useful insight. In particular, in the high-dimensional limit, the original coupled dynamics associated with the algorithm will be asymptotically ?decoupled?, with each coordinate independently solving a 1-D effective minimization problem via stochastic gradient descent. Exploiting this insight to design new algorithms for achieving optimal trade-offs between computational and statistical efficiency may prove an interesting line of future research. 1 Introduction Online learning methods based on stochastic gradient descent are widely used in many learning and signal processing problems. Examples includes the classical least mean squares (LMS) algorithm [1] in adaptive filtering, principal component analysis [2, 3], independent component analysis (ICA) [4], and the training of shallow or deep artificial neural networks [5?7]. Analyzing the convergence rate of stochastic gradient descent has already been the subject of a vast literature (see, e.g., [8?11].) Unlike existing work that analyze the behaviors of the algorithms in finite dimensions, we present in this paper a framework for studying the exact dynamics of stochastic gradient algorithms in the high-dimensional limit, using online ICA as a concrete example. Instead of minimizing a generic function as considered in the optimization literature, the stochastic algorithm we analyze here is solving an estimation problem. The extra assumptions on the ground truth (e.g., the feature vector) and the generative models for the observations allow us to obtain the exact asymptotic dynamics of the algorithms. As the main result of this work, we show that, as the ambient dimension n ? ? and with proper time-scaling, the time-varying joint empirical measure of the true underlying independent component ? and its estimate x converges weakly to the unique solution of a nonlinear partial differential equation (PDE) [see (6).] Since many performance metrics, such as the correlation between ? and 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. x and the support recover rate, are functionals of the joint empirical measure, knowledge about the asymptotics of the latter allows us to easily compute the asymptotic limits of various performance metrics of the algorithm. This work is an extension of a recent analysis on the dynamics of online sparse PCA [12] to more general settings. The idea of studying the scaling limits of online learning algorithm first appeared in a series of work that mostly came from the statistical physics communities [3, 5, 13?16] in the 1990s. Similar to our setting, those early papers studied the dynamics of various online learning algorithms in high dimensions. In particular, they show that the mean-squared error (MSE) of the estimation, together with a few other ?order parameters?, can be characterized as the solution of a deterministic system of coupled ordinary differential equations (ODEs) in the large system limit. One limitation of such ODE-level analysis is that it cannot provide information about the distributions of the estimates. The latter are often needed when one wants to understand more general performance metrics beyond the MSE. Another limitation is that the ODE analysis cannot handle cases where the algorithms have non-quadratic regularization terms (e.g., the incorporation of `1 norms to promote sparsity.) In this paper, we show that both limitations can be eliminated by using our PDE-level analysis, which tracks the asymptotic evolution of the probability distributions of the estimates given by the algorithm. In a recent paper [10], the dynamics of an ICA algorithm was studied via a diffusion approximation. As an important distinction, the analysis in [10] keeps the ambient dimension n fixed and studies the scaling limit of the algorithm as the step size tends to 0. The resulting PDEs involve O(n) spatial variables. In contrast, our analysis studies the limit as the dimension n ? ?, with a constant step size. The resulting PDEs only involve 2 spatial variables. This low-dimensional characterization makes our limiting results more practical to use, especially when the dimension is large. The basic idea underlying our analysis can trace its root to the early work of McKean [17, 18], who studied the statistical mechanics of Markovian-type mean-field interactive particles. The mathematical foundation of this line of research has been further established in the 1980s (see, e.g., [19, 20]). This theoretical tool has been used in the analysis of high-dimensional MCMC algorithms [21]. In our work, we study algorithms through the lens of high-dimensional stochastic processes. Interestingly, the analysis does not explicitly depend on whether the underlying optimization problem is convex or nonconvex. This feature makes the presented analysis techniques a potentially very useful tool in understanding the effectiveness of using low-complexity iterative algorithms for solving highdimensional nonconvex estimation problems, a line of research that has recently attracted much attention (see, e.g., [22?25].) The rest of the paper is organized as follows. We first describe in Section 2 the observation model and the online ICA algorithm studied in this work. The main convergence results are given in Section 3, where we show that the time-varying joint empirical measure of the target independent component and its estimates given by the algorithm can be characterized, in the high-dimensional limit, by the solution of a deterministic PDE. Due to space constraint, we only provide in the appendix a formal derivation leading to the PDE, and leave the rigorous proof of the convergence to a follow-up paper. Finally, in Section 4 we present some insight obtained from our asymptotic analysis. In particular, in the high-dimensional limit, the original coupled dynamics associated with the algorithm will be asymptotically ?decoupled?, with each coordinate independently solving a 1-D effective minimization problem via stochastic gradient descent. Notations and Conventions: Throughout this paper, we use boldfaced lowercase letters, such as ? and xk , to represent n-dimensional vectors. The subscript k in xk denotes the discrete-time iteration step. The ith component of the vectors ? and xk are written as ?i and xk,i , respectively. 2 Data model and online ICA We consider a generative model where a stream of sample vectors y k ? Rn , k = 1, 2, . . . are generated according to y k = ?1n ?ck + ak , (1) where ? ? Rn is a unique feature vector we want to recover. (For simplicity, we consider the case of recovering a single feature vector, but our analysis technique can be generalized to study cases involving a finite number of feature vectors.) Here ck ? R is an i.i.d. random variable drawn from an unknown non-Gaussian distribution Pc with zero mean and unit variance. And ak ? N (0, I ? n1 ?? T ) 2 2 models background noise. We use the normalization k?k = n so that in the large n limit, all elements ?i of the vector are O(1)  quantities. The observation model (1) is equivalent to the standard sphered c data model y k = A k , where A ? Rn?n is an orthonormal matrix with the first column being sk ? ?/ n and sk is an i.i.d. (n ? 1)-dimensional standard Gaussian random vector. ToP establish the large n limit, we shall assume that the empirical measure of ? defined by ?(?) = n 1 ? i=1 ?(? ? ?i ) converges weakly to a deterministic measure ? (?) with finite moments. Note that n this assumption can be satisfied in a stochastic setting, where each element ? of ? is an i.i.d. random variable drawn from ?? (?), or in a deterministic setting [e.g., ? = 2(i mod 2), in which case i ? 1 1 ? ? (?) = 2 ?(?) + 2 ?(? ? 2).] We use an online learning algorithm to extract the non-Gaussian component ? from the data stream {y k }k?1 . Let xk be the estimate of ? at step k. Starting from an initial estimate x0 , the algorithm update xk by e k = xk + ??kn f ( ?1n y Tk xk )y k ? ?nk ?(xk ) x ? (2) ek, xk+1 = kexknk x where f (?) is a given twice differentiable function and ?(?) is an element-wise nonlinear mapping introduced to enforce prior information about ?, e.g., sparsity. The scaling factor ?1n in the above equations makes sure that each component xk,i of the estimate is of size O(1) in the large n limit. The above online learning scheme can be viewed as a projected stochastic gradient algorithm for solving an optimization problem min ? kxk=n where F (x) = R K n 1 X 1X F ( ?1n y Tk x) + ?(xi ), K n i=1 (3) k=1 f (x) dx and Z ?(x) = ?(x) dx (4) is a regularization function. In (2), we update xk using an instantaneous noisy estimation PK 1 T ?1 f ( ?1 y T xk )y , in place of the true gradient ? ?1 k k=1 f ( n y k xk )y k , once a new sample y k n n k K n is received. In practice, one can use f (x) = ?x3 or f (x) = ? tanh(x) to extract symmetric non-Gaussian signals (for which E c3k = 0 and E c4k 6= 3) and use f (x) = ?x2 to extract asymmetric non-Gaussian signals. The algorithm in (2) with f (x) = x3 can also be regarded as implementing a low-rank tensor decomposition related to the empirical kurtosis tensor of y k [10, 11]. For the nonlinear mapping ?(x), the choice of ?(x) = ?x for some ? > 0 corresponds to using an L2 norm in the regularization term ?(x). If the feature vector is known to be sparse, we can set ?(x) = ? sgn(x), which is equivalent to adding an L1 -regularization term. 3 Main convergence result We provide an exact characterization of the dynamics of the online learning algorithm (2) when the ambient dimension n goes to infinity. First, we define the joint empirical measure of the feature vector ? and its estimate xk as n 1X n ?t (?, x) = ?(? ? ?i , x ? xk,i ) (5) n i=1 with t defined by k = btnc. Here we rescale (i.e., ?accelerate?) the time by a factor of n. The joint empirical measure defined above carries a lot of information ? about the performance of the algorithm. For example, as both ? and xk have the same norm n by definition, the normalized correlation between ? and xk defined by 1 Qnt = ? T xk n 3 can be computed as Qnt = E?nt [?x], i.e., the expectation of ?x taken with respect to the empirical Pn measure. More generally, any separable performance metric Htn = n1 i=1 h(?i , xk,i ) with some function h(?, ?) can be expressed as an expectation with respect to the empirical measure ?nt , i.e., Htn = E?nt h(?, x). Directly computing Qnt via the expectation E?nt [?x] is challenging, as ?nt is a random probability measure. We bypass this difficulty by investigating the limiting behavior of the joint empirical measure ?nt defined in (5). Our main contribution is to show that, as n ? ?, the sequence of random probability measures {?nt }n converges weakly to a deterministic measure ?t . Note that the limiting value of Qnt can then be computed from the limiting measure ?nt via the identity limn?? Qnt = E?t [?x]. Let Pt (x, ?) be the density function of the limiting measure ?t (?, x) at time t. We show that it is characterized as the unique solution of the following nonlinear PDE: ? ?t Pt (?, x)   ? ?2 = ? ?x ?(x, ?, Qt , Rt )Pt (?, x) + 21 ?(Qt ) ?x 2 Pt (?, x) (6) with ZZ Qt = ?xPt (?, x) dx d? (7) x?(x)Pt (?, x) dx d? (8) R2 ZZ Rt = R2 where the two functions ?(Q) and ?(x, ?, Q, R) are defined as D p
E ?(Q) = ? 2 f 2 cQ + e 1 ? Q2   ?(x, ?, Q, R) = x QG(Q) + ? R ? 12 ?(Q) ? ?G(Q) ? ? ?(x) where D D p p
E
E G(Q) = ?? f cQ + e 1 ? Q2 c + ? Q f 0 cQ + e 1 ? Q2 . (9) (10) (11) In the above equations, e and c denote two independent random variables, with e ? N (0, 1) and c ? Pc , the non-Gaussian distribution of ck introduced in (2); the notation h?i denotes the expectation over e and c; and f (?) and ?(?) are the two functions used in the online learning algorithm (2). When ?(x) = 0 (and therefore Rt = 0), we can derive a simple ODE for Qt from (6) and (7): d 1 Qt = (Q2t ? 1)G(Qt ) ? Qt ?(Qt ). dt 2 Example 1 As a concrete example, we consider the case when ck is drawn from a symmetric non-Gaussian distribution. Due to symmetry, E c3k = 0. Write E c4k = m4 and E c6k = m6 . We use f (x) = x3 in (2) to detect the feature vector ?. Substituting this specific f (x) into (9) and (11), we obtain G(Q) = ? Q3 (m4 ? 3) h i ?(Q) = ? 2 15 + 15Q4 (1 ? Q2 )(m4 ? 3) + Q6 (m6 ? 15) (12) (13) and ?(x, ?, Q, R) can be computed by substituting (12) and (13) into (10). Moreover, for the case ?(x) = 0, we derive a simple ODE for qt = Q2t as h i dqt = ?2?t qt2 (1 ? qt )(m4 ? 3) ? ?t2 qt 15qt2 (1 ? qt )(m4 ? 3) + qt3 (m6 ? 15) + 15 . (14) dt Numerical verifications of the ODE results are shown in Figure 1(a). In our experiment, the ambient dimension is set to n = 5000 and we plot the averaged results as well as error bars (corresponding to one standard deviation) over 10 independent trials. Two different initial values of q0 = Q20 are used. In both cases, the asymptotic theoretical predictions match the numerical results very well. The ODE in (14) can be solved analytically. Next we briefly discuss its stability. The right-hand side of (14) is plotted in Figure 1(b) as a function of qt . It is clear that the ODE (14) always admits a 4 1 0.5 0.8 0.6 ODE Simulation 0.4 0 0.2 0 -0.5 0 50 100 150 200 0 (a) 0.5 1 (b) Figure 1: (a) Comparison between the analytical prediction given by the ODE in (14) with numerical simulations of the online ICA algorithm. We consider two different initial values for the algorithm. The top one, which starts from a better initial guess, converges to an informative estimation, whereas the bottom one, with a worse initial guess, converges to a non-informative solution. (b) The stability of the ODE in (14). We draw g(q) = ?1 dq dt for different value of ? = 0.02, 0.04, 0.06, 0.08 from top to bottom. solution qt = 0, which corresponding to a trivial, non-informative solution. Moreover, this trivial solution is always a stable fixed point. When the stepsize ? > ?c for some constant ?c , qt = 0 is also the unique stable fixed point. When ? < ?c however, two additional solutions of the ODE emerge. One is a stable fixed point denoted by qs? and the other is an unstable fixed point denoted by qu? , with qu? < qs? . Thus, in order to reach an informative solution, one must initialize the algorithm with Q20 > qu? . This insight agrees with a previous stability analysis done in [26], where the authors investigated the dynamics near qt = 0 via a small qt expansion. Example 2 In this experiment, we verify the accuracy of the asymptotic predictions given by the PDE (6). The settings are similar to those in Example 1. In addition, we assume that ? the feature vector ? is sparse, consisting of ?n nonzero elements, each of which is? equal to 1/ ?. Figure 2 shows the asymptotic conditional density Pt (x|?) for ? = 0 and ? = 1/ ? at two different times. These theoretical predictions are obtained by solving the PDE (6) numerically. Also shown in the figure are the empirical conditional densities associated with one realization of the ICA algorithm. Again, we observe that the theoretical predictions and numerical results have excellent agreement. To demonstrate the usefulness of the PDE analysis in providing detailed information about the performance of the algorithm, we show in Figure 3 the performance of sparse support recovery using a simple hard-thresholding scheme on the estimates provided by the algorithm. By changing the threshold values, one can have trade-offs between the true positive and false positive rates. As we can see from the figure, this precise trade-off can be accurately predicted by our PDE analysis. 4 Insights given by the PDE analysis In this section, we present some insights that can be gained from our high-dimensional analysis. To simplify the PDE in (6), we can assume that the two functions Qt and Rt in (7) and (8) are given to us in an oracle way. Under this assumption, the PDE (6) describes the limiting empirical measure of the following stochastic process q ?(Qk/n ) zk+1,i = zk,i + n1 ?(zk,i , ?i , Qk/n , Rk/n ) + wk,i , i = 1, 2, . . . n (15) n where wk,i is a sequence of independent standard Gaussian random variables. Unlike the original online learning update equation (2) where different coordinates of xk are coupled, the above process is uncoupled. Each component zk,i for i = 1, 2, . . . , n evolves independently when conditioned on Qt and Rt . The continuous-time limit of (15) is described by a stochastic differential equation (SDE) p dZt = ?(Zt , ?, Qt , Rt ) dt + ?(Qt ) dBt , 5 6 6 4 4 2 2 0 -1 0 1 2 3 0 -1 4 6 6 4 4 2 2 0 -1 0 1 2 3 0 -1 4 0 1 2 3 4 0 1 2 3 4 3 4 (a) 0.15 0.2 0.1 0.1 0.05 0 -1 0 1 2 3 0 -1 4 0 1 2 (b) Figure 2: (a) A demonstration of the accuracy of our PDE analysis. See the discussions in Example 2 for details. (b) Effective 1-D cost functions. 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Figure 3: Trade-offs between the true positive and false positive rates in sparse support recovery. In our experiment, n = 104 , and the sparsity level is set to ? = 0.3. The theoretical results obtained by our PDE analysis can accurately predict the actual performance at any run-time of the algorithm. where Bt is the standard Brownian motion. We next have a closer look at the equation (15). Given a scalar ?, Qt and Rt , we can define a time-varying 1-D regularized quadratic optimization problem minx?R Et (x, ?) with the effective potential Et (x, ?) = 12 dt (x ? bt ?)2 + ? ?(x), (16) where dt = Qt G(Qt ) ? 12 ?(Qt ) + ? Rt , bt = G(Qt )/dt and ?(x) is the regularization term defined in (4). Then, the stochastic process (15) can be viewed as a stochastic gradient descent 6 for solving this 1-D problem with a step-sizeq equal to 1/n. One can verify that the exact gradient k) of (16) is ??(x, ?, Qt , Rt ). The third term ?(Q n wk in (15) adds stochastic noise to the true gradient. Interestingly, although the original optimization problem (3) is non-convex, its 1-D effective optimization problem is always convex for convex regularizers ?(x) (e.g., ?(x) = ? |x|.) This provides an intuitive explanation for the practical success of online ICA. To visualize this 1-D effective optimization problem, we plot in Figure 2(b) the effective potential Et (x, ?) at t = 0 and t = 100, respectively. From Figure (2), we can see that the L1 norm always introduces a bias in the estimation for all non-zero ?i , as the minimum point in the effective 1-D cost function is always shifted towards the origin. It is hopeful that the insights gained from the 1-D effective optimization problem can guide the design of a better regularization function ?(x) to achieve smaller estimation errors without sacrificing the convergence speed. This may prove an interesting line of future work. This uncoupling phenomenon is a typical consequence of mean-field dynamics, e.g., the SherringtonKirkpatrick model [27] in statistical physics. Similar phenomena are observed or proved in other high dimensional algorithms especially those related to approximate message passing (AMP) [28?30]. However, for these algorithms using batch updating rules with the Onsager reaction term, the limiting densities of iterands are Gaussian. Thus the evolution of such densities can be characterized by tracking a few scalar parameters in discrete time. For our case, the limiting densities are typically non-Gaussian and they cannot be parametrized by finitely many scalars. Thus the PDE limit (6) is required. Appendix: A Formal derivation of the PDE In this appendix, we present a formal derivation of the PDE (6). We first note that (xk , ? k )k with ? k = ? forms an exchangeable Markov chain on R2n driven by the random variable ck ? Pc and the Gaussian random vector ak . The drift coefficient ?(x, ?, Q, R) and the diffusion coefficient ?(Q) in the PDE (6) are determined, respectively, by the conditional mean and variance of the increment xk+1,i ? xk,i , conditioned upon the previous state vector xk and ? k . Let the increment of the gradient-descent step in the learning rule (2) be e k,i = x ? ek,i ? xk,i = ?k ? f ( ?1n y Tk xk )yk,i n ? ?k n ?(xk,i ) (17) e k . Let Ek denote the conditional expectation with where x ek,i is the ith component of the output x respect to ck and ak given xk and ? k . h i h i e k,i and Ek ? e 2 . From (1) and (17) we have We first compute Ek ? k,i h i e k,i = Ek ? ?k ? E n k h f (Qnk ck + eek,i + ?1 ak,i xk,i )( ?1 ?i ck n n i + ak,i ) ? ?k n ?(xk,i ),
where Qnk = n1 ? T xk and eek,i = ?1n aTk xk ? ak,i xk,i . We use the Taylor expansion of f around Qnk ck + eek,i up to the first order and get h i Ek f (Qnk ck + eek,i + ?1n ak,i xk,i )( ?1n ?i ck + ak,i ) h i h i = Ek f (Qnk ck + eek,i )( ?1n ?i ck + ak,i ) + ?1n xk,i Ek f 0 (Qnk ck + eek,i )( ?1n ?i ck + ak,i )ak,i + ?k,i , where ?k,i includes all higher order terms. As n ? ?, the random variable Qnk converges to a deterministic quantity Qk . Moreover, eek,i " # and ak,i are both zero-mean Gaussian with the covariance 1 1 2 ? 1 ? Qk + O( n ) ? n ?k,i Qk matrix . We thus have ? ?1n ?k,i Qk 1 + O( n1 )  q h i  Ek f 0 (Qnk ck + eek,i )( ?1n ?i ck + ak,i )ak,i = f 0 (Qk c + 1 ? Q2k e) + o(1) 7 and i h Ek f (Qnk ck + eek,i )( ?1n ?i ck + ak,i )   q ?i 1 2 ? ? = f (Qk c + 1 ? Qk e ? n Qk a)( n ?i c + a) "   # q q 0 1 + o( ?1n ), = ?n ? i cf (Qk c + 1 ? Q2k e) ? Qk f (Qk c + 1 ? Q2k e) p where in the last line, we use the Taylor expansion again to expand f around Qk c + 1 ? Q2k e and the bracket h?i denotes the average over two independent random variables c ? Pc and e ? N (0, 1). Thus, we have " #   q h i 1 0 e k,i = Ek ? ??i G(Qk ) + ?k xk,i f (Qk c + 1 ? Q2k e) ? ?k ?(xk,i ) + o( n1 ), n where the function G(Q) is defined in (11). To compute the (conditional) variance, we have h i h i 2 e 2k,i = ?k Ek f 2 (Qnk + eek,i ) + o( 1 ) = Ek ? n n ?k2 n  2 f (Qk c + q 1? Q2k e)  + o( n1 ). Next, we deal with the normalization step. Again, we use the Taylor expansion for the term  1 ?1 1  ?1 e x up to the first order, which yields e = x + ? k k k n n   ek +? eT? e k + ?k , e k + 1? xk+1 = xk ? n1 xk xTk ? k 2 h i e k ? 1 Pn xk,i Ek ? eT? ek ? e k,i , 1 ? where ? k includes all higher order terms. Note that n1 xTk ? k i=1 n n i h P n 1 1 T n e2 i=1 Ek ?k,i and n xk ?(x) = Rk ? Rk , we have n   Ek xk+1,i ? xk,i = n1 ?(xk,i , ?i , Qk , Rk ) + o( n1 ). Finally, the normalization step does not change the variance term, and thus h h i
2 i e 2k,i + o( 1 ) = 1 ?(Qk ) + o( 1 ). Ek xk+1,i ? xk,i = Ek ? n n n The above computation of Ek (xk+1,i ? xk,i ) and Ek (xk+1,i ? xk,i )2 connects the dynamics (2) to (15). In fact, both (2) and (15) have the same limiting empirical measure described by (6). A rigorous proof of our asymptotic result is built on the weak convergence approach for measurevalued processes. Details will be presented in an upcoming paper. Here we only provide a sketch of the general proof strategy: First, we prove the tightness of the measure-valued stochastic process (?nt )0?t?T on D([0, T ], M(R2 )), where D denotes the space of c?dl?g processes taking values from the space of probability measures. This then implies that any sequence of the measure-valued process {(?nt )0?t?T }n (indexed by n) must have a weakly converging subsequence. Second, we prove any converging (sub)sequence must converge weakly to a solution of the weak form of the PDE (6). 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Approximation Algorithms for `0-Low Rank Approximation Karl Bringmann1 kbringma@mpi-inf.mpg.de 1 Pavel Kolev1? pkolev@mpi-inf.mpg.de David P. Woodruff2 dwoodruf@cs.cmu.edu Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbr?cken, Germany 2 Department of Computer Science, Carnegie Mellon University Abstract We study the `0 -Low Rank Approximation Problem, where the goal is, given an m ? n matrix A, to output a rank-k matrix A0 for which kA0 ? Ak0 is minimized. Here, for a matrix B, kBk0 denotes the number of its non-zero entries. This NPhard variant of low rank approximation is natural for problems with no underlying metric, and its goal is to minimize the number of disagreeing data positions. We provide approximation algorithms which significantly improve the running time and approximation factor of previous work. For k > 1, we show how to find, in poly(mn) time for every k, a rank O(k log(n/k)) matrix A0 for which kA0 ? Ak0 ? O(k 2 log(n/k)) OPT. To the best of our knowledge, this is the first algorithm with provable guarantees for the `0 -Low Rank Approximation Problem for k > 1, even for bicriteria algorithms. For the well-studied case when k = 1, we give a (2+)-approximation in sublinear time, which is impossible for other variants of low rank approximation such as for the Frobenius norm. We strengthen this for the well-studied case of binary matrices to obtain a (1 + O(?))-approximation in sublinear time, where ? = OPT /kAk0 . For small ?, our approximation factor is 1 + o(1). 1 Introduction Low rank approximation of an m ? n matrix A is an extremely well-studied problem, where the goal is to replace the matrix A with a rank-k matrix A0 which well-approximates A, in the sense that kA ? A0 k is small under some measure k ? k. Since any rank-k matrix A0 can be written as U ? V , where U is m ? k and V is k ? n, this allows for a significant parameter reduction. Namely, instead of storing A, which has mn entries, one can store U and V , which have only (m + n)k entries in total. Moreover, when computing Ax, one can first compute V x and then U (V x), which takes (m + n)k instead of mn time. We refer the reader to several surveys [19, 24, 40] for references to the many results on low rank approximation. We focus on approximation algorithms for the low-rank approximation problem, i.e. we seek to output a rank-k matrix A0 for which kA ? A0 k ? ?kA ? Ak k, where Ak = argminrank(B)=k kA ? Bk is the best rank-k approximation to A, and the approximation ratio ? is as P small P as possible. One of m n the most widely studied error measures is the Frobenius norm kAkF = ( i=1 j=1 A2i,j )1/2 , for which the optimal rank-k approximation can be obtained via the singular value decomposition (SVD). Using randomization and approximation, one can compute an ? = 1 + -approximation, for any  > 0, in time much faster than the min(mn2 , mn2 ) time required for computing the SVD, namely, in O(kAk0 + n ? poly(k/)) time [9, 26, 29], where kAk0 denotes the number of non-zero entries ? This work has been funded by the Cluster of Excellence ?Multimodal Computing and Interaction? within the Excellence Initiative of the German Federal Government. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. of A. For the Frobenius norm kAk0 time is also a lower bound, as any algorithm that does not read nearly all entries of A might not read a very large entry, and therefore cannot achieve a relative error approximation. The rank-k matrix Ak obtained by computing the SVD is also optimal with respect to any rotationally invariant norm, such as the operator and Schatten-p norms. Thus, such norms can also be solved exactly in polynomial time. Recently, however, there has been considerable interest [10, 3, 32] in obtaining low rank approximations for NP-hard error measures such as the entrywise `p -norm
P p 1/p kAkp = , where p ? 1 is a real number. Note that for p < 1, this is not a norm, i,j |Ai,j | though it is still a well-defined quantity. For p = ?, this corresponds to the max-norm or Chebyshev norm. It is known that one can achieve a poly(k log(mn))-approximation in poly(mn) time for the low-rank approximation problem with entrywise `p -norm for every p ? 1 [36, 8]. 1.1 `0 -Low Rank Approximation A natural variant of low rank approximation which the results above do not cover is that of `0 -low rank approximation, where the measure kAk0 is the number of non-zero entries. In other words, we 0 seek a rank-k matrix A0 for which Ai,j is as small as possible. P the number of 0entries (i, j) with Ai,j 6= Letting OPT = minrank(B)=k i,j ?(Ai,j 6= Ai,j ), where ?(Ai,j 6= A0i,j ) = 1 if Ai,j 6= A0i,j and 0 otherwise, we would like to output a rank-k matrix A0 for which there are at most ? OPT entries (i, j) with A0i,j 6= Ai,j . Approximation algorithms for this problem are essential since solving the problem exactly is NP-hard [12, 14], even when k = 1 and A is a binary matrix. The `0 -low rank approximation problem is quite natural for problems with no underlying metric, and its goal is to minimize the number of disagreeing data positions with a low rank matrix. Indeed, this error measure directly answers the following question: if we are allowed to ignore some data - outliers or anomalies - what is the best low-rank model we can get? One well-studied case is when A is binary, but A0 and its factors U and V need not necessarily be binary. This is called unconstrained Binary Matrix Factorization in [18], which has applications to association rule mining [20], biclustering structure identification [42, 43], pattern discovery for gene expression [34], digits reconstruction [25], mining high-dimensional discrete-attribute data [21, 22], market based clustering [23], and document clustering [43]. There is also a body of work on Boolean Matrix Factorization which restricts the factors to also be binary, which is referred to as constrained Binary Matrix Factorization in [18]. This is motivated in applications such as classifying text documents and there is a large body of work on this, see, e.g. [28, 31]. The `0 -low rank approximation problem coincides with a number of problems in different areas. It exactly coincides with the famous matrix rigidity problem over the reals, which asks for the minimal number OPT of entries of A that need to be changed in order to obtain a matrix of rank at most k. The matrix rigidity problem is well-studied in complexity theory [15, 16, 39] and parameterized complexity [13]. These works are not directly relevant here as they do not provide approximation algorithms. There are also other variants of `0 -low rank approximation, corresponding to cases such as when A is binary, A0 = U V is required to have binary factors U and V , and multiplication is either performed over a binary field [41, 17, 12, 30], or corresponds to an OR of ANDs. The latter is known as the Boolean model [4, 12, 27, 33, 35, 38]. These different notions of inner products lead to very different algorithms and results for the `0 -low rank approximation problem. However, all these models coincide in the special and important case in which A is binary and k = 1. This case was studied in [20, 34, 18], as their algorithm for k = 1 forms the basis for their successful heuristic for general k, e.g. the PROXIMUS technique [20]. Another related problem is robust PCA [6], in which there is an underlying matrix A that can be written as a low rank matrix L plus a sparse matrix S [7]. Cand?s et al. [7] argue that both components are of arbitrary magnitude, and we do not know the locations of the non-zeros in S nor how many there are. Moreover, grossly corrupted observations are common in image processing, web data analysis, and bioinformatics where some measurements are arbitrarily corrupted due to occlusions, malicious tampering, or sensor failures. Specific scenarios include video surveillance, face recognition, latent semantic indexing, and ranking of movies, books, etc. [7]. These problems have the common theme of being an arbitrary magnitude sparse perturbation to a low rank matrix with no natural underlying metric, and so the `0 -error measure (which is just the Hamming distance, or number of disagreements) is appropriate. In order to solve robust PCA in practice, Cand?s et al. [7] 2 relaxed the `0 -error measure to the `1 -norm. Understanding theoretical guarantees for solving the original `0 -problem is of fundamental importance, and we study this problem in this paper. Finally, interpreting 00 as 0, the `0 -low rank approximation problem coincides with the aforementioned notion of entrywise `p -approximation when p = 0. It is not hard to see that previous work [8] for general p ? 1 fails to give any approximation factor for p = 0. Indeed, critical to their analysis is the scale-invariance property of a norm, which does not hold for p = 0 since `0 is not a norm. 1.2 Our Results We provide approximation algorithms for the `0 -low rank approximation problem which significantly improve the running time or approximation factor of previous work. In some cases our algorithms even run in sublinear time, i.e., faster than reading all non-zero entries of the matrix. This is provably impossible for other measures such as the Frobenius norm and more generally, any `p -norm for p > 0. For k > 1, our approximation algorithms are, to the best of our knowledge, the first with provable guarantees for this problem. First, for k = 1, we significantly improve the polynomial running time of previous (2 + )approximations for this problem. The best previous algorithm due to Jiang et al. [18] was based on the observation that there exists a column u of A spanning a 2-approximation. Therefore, solving the problem minv kA ? uvk0 for each column u of A yields a 2-approximation, where for a matrix B the measure P kBk0 counts the number of non-zero entries. The problem minv kA ? uvk0 decomposes into i mini kA:,i ? vi uk0 , where A:,i is the i-th column of A, and vi the i-th entry of vector v. The optimal vi is the mode of the ratios Ai,j /uj , where j ranges over indices in {1, 2, . . . , m} with uj 6= 0. As a result, one can find a rank-1 matrix uv T providing a 2-approximation in O(kAk0 n) time, which was the best known running time. Somewhat surprisingly, we show that one can achieve sublinear time for solving this problem. Namely, we obtain a (2 + )-approximation in (m + n) poly(?1 ? ?1 log(mn)) time, for any  > 0, where ? = OPT /kAk0 . This significantly improves upon the earlier O(kAk0 n) time for not too small  and ?. Our result should be contrasted to Frobenius norm low rank approximation, for which ?(kAk0 ) time is required even for k = 1, as otherwise one might miss a very large entry in A. Since `0 -low rank approximation is insensitive to the magnitude of entries of A, we bypass this general impossibility result. Next, still considering the case of k = 1, we show that if the matrix A is binary, a well-studied case coinciding with the abovementioned GF (2) and Boolean models, we obtain an approximation algorithm parameterized in terms of the ratio ? = OPT /kAk0 , showing it is possible in time (m + n)? ?1 poly(log(mn)) to obtain a (1 + O(?))-approximation. Note that our algorithm is again sublinear, unlike all algorithms in previous work. Moreover, when A is itself very well approximated by a low rank matrix, then ? may actually be sub-constant, and we obtain a significantly better (1 + o(1))-approximation than the previous best known 2-approximations. Thus, we simultaneously improve the running time and approximation factor. We also show that the running time of our algorithm is optimal up to poly(log(mn)) factors by proving that any (1 + O(?))-approximation succeeding with constant probability must read ?((m + n)? ?1 ) entries of A in the worst case. Finally, for arbitrary k > 1, we first give an impractical algorithm that runs in time nO(k) and achieves an ? = poly(k)-approximation. To the best of our knowledge this is the first approximation algorithm for the `0 -low rank approximation problem with any non-trivial approximation factor. To make our algorithm practical, we reduce the running time to poly(mn), with an exponent independent of k, if we allow for a bicriteria solution. In particular, we allow the algorithm to output a matrix A0 of somewhat larger rank O(k log(n/k)), for which kA ? A0 k0 ? O(k 2 log(n/k)) minrank(B)=k kA ? Bk0 . Although we do not obtain rank exactly k, many of the motivations for finding a low rank approximation, such as reducing the number of parameters and fast matrix-vector product, still hold if the output rank is O(k log(n/k)). We are not aware of any alternative algorithms which achieve poly(mn) time and any provable approximation factor, even for bicriteria solutions. 2 Preliminaries For an matrix A ? Am?n with entries Ai,j , we write Ai,: for its i-th row and A:,j for its j-th column. 3 Input Formats We always assume that we have random access to the entries of the given matrix A, i.e. we can read any entry Ai,j in constant time. For our sublinear time algorithms we need more efficient access to the matrix, specifically the following two variants: (1) We say that we are given A with column adjacency arrays if we are given arrays B1 , . . . , Bn and lengths `1 , . . . , `n such that for any 1 ? k ? `j the pair Bj [k] = (i, Ai,j ) stores the row i containing the k-th nonzero entry in column j as well as that entry Ai,j . This is a standard representation of matrices used in many applications. Note that given only these adjacency arrays B1 , . . . , Bn , in order to access any entry Ai,j we can perform a binary search over Bj , and hence random access to any matrix entry is in time O(log n). Moreover, we assume to have random access to matrix entries in constant time, and note that this is optimistic by at most a factor O(log n). (2) P We say that we are given P matrix A with row and column sums if we can access the numbers j Ai,j for i ? [m] and i Ai,j for j ? [n] in constant time (and, as always, access any entry Ai,j in constant time). Notice that storing the row and column sums takes O(n + m) space, and thus while this might not be standard information it is very cheap to store. We show that the first access type even allows to sample from the set of nonzero entries uniformly in constant time. Lemma 1. Given a matrix A ? Rm?n with column adjacency arrays, after O(n) time preprocessing we can sample a uniformly random nonzero entry (i, j) from A in time O(1). The proof of this lemma, as well as most other proofs in this extended abstract, can be found in the full version of the paper. 3 Algorithms for Real `0 -rank-k Given a matrix A ? Rm?n , the `0 -rank-k problem asks to find a matrix A0 with rank k such that the difference between A and A0 measured in `0 norm is minimized. We denote the optimum value by def OPT(k) = min rank(A0 )=k kA ? A0 k0 = min U ?Rm?k , V ?Rk?n kA ? U V k0 . (1) In this section, we establish several new results on the `0 -rank-k problem. In Subsection 3.1, we prove a structural lemma that shows the existence of k columns which provide a (k + 1)-approximation to OPT(k) , and we also give an ?(k)-approximation lower bound for any algorithm that selects k columns from the input matrix A. In Subsection 3.2, we give an approximation algorithm that runs in poly(nk , m) time and achieves an O(k 2 )-approximation. To the best of our knowledge, this is the first algorithm with provable non-trivial approximation guarantees. In Subsection 3.3, we design a practical algorithm that runs in poly(n, m) time with an exponent independent of k, if we allow for a bicriteria solution. 3.1 Structural Results We give a new structural result for `0 showing that any matrix A contains k columns which provide a (k + 1)-approximation for the `0 -rank-k problem (1). Lemma 2. Let A ? Rm?n be a matrix and k ? [n]. There is a subset J (k) ? [n] of size k and a matrix Z ? Rk?n such that kA ? A:,J (k) Zk0 ? (k + 1)OPT(k) . def def Proof. Let Q(0) be the set of columns j with U V:,j = 0, and let R(0) = [n] \ Q(0) . Let S (0) = [n], def def T (0) = ?. We split the value OPT(k) into OPT(S (0) , R(0) ) = kAS (0) ,R(0) ? U VS (0) ,R(0) k0 and def OPT(S (0) , Q(0) ) = kAS (0) ,Q(0) ? U VS (0) ,Q(0) k0 = kAS (0) ,Q(0) k0 . Suppose OPT(S (0) , R(0) ) ? |S (0) ||R(0) |/(k + 1). Then, for any subset J (k) it follows that minZ kA ? AS (0) ,J (k) Zk0 ? |S (0) ||R(0) | + kAS (0) ,Q(0) k0 ? (k + 1)OPT(k) . Otherwise, there is a column i(1) such that AS (0) ,i(1) ? (U V )S (0) ,i(1) 0 ? OPT(S (0) , R(0) )/|R(0) | ? OPT(k) /|R(0) |. 4 def Let T (1) be the set of indices on which (U V )S (0) ,i(1) and AS (0) ,i(1) disagree, and similarly S (1) = S (0) \T (1) on which they agree. Then we have |T (1) | ? OPT(k) /|R(0) |. Hence, in the submatrix T (1) ? R(0) the total error is at most |T (1) | ? |R(0) | ? OPT(k) . Let R(1) , D(1) be a partitioning of R(0) such that AS (1) ,j is linearly dependent on AS (1) ,i(1) iff j ? D(1) . Then by selecting column A:,i(1) the incurred cost on matrix S (1) ? D(1) is zero. For the remaining submatrix S (`) ? R(`) , we perform a recursive call of the algorithm. We make at most k recursive calls, on instances S (`) ? R(`) for ` ? {0, . . . , k ? 1}. In the `th iteration, either OPT(S (`) , R(`) ) ? |S (`) ||R(`) |/(k + 1 ? `) and we are done, or there is a column i(`+1) which partitions S (`) into S (`+1) , T (`+1) and R(`) into R(`+1) , D(`+1) such that Q` k?` 1 ) = k+1 ? m and for every j ? D(`) the column AS (`+1) ,j belongs |S (`+1) | ? m ? i=0 (1 ? k+1?i to the span of {AS (`+1) ,i(t) }`+1 t=1 . Suppose we performed k recursive calls. We show now that the incurred cost in submatrix S (k) ?R(k) is at most OPT(S (k) , R(k) ) ? OPT(k) . By construction, the sub-columns {AS (k) ,i }i?I (k) are linearly independent, where I (k) = {i(1) , . . . , i(k) } is the set of the selected columns, and AS (k) ,I (k) = (U V )S (k) ,I (k) . Since rank(AS (k) ,I (k) ) = k, it follows that rank(US (k) ,: ) = k, rank(V:,I (k) ) = k and the matrix V:,I (k) ? Rk?k is invertible. Hence, for matrix Z = (V:,I (k) )?1 V:,Rk we have OPT(S (k) , R(k) ) = kAS k ,Rk ? AS k ,I k Zk0 . The statement follows by noting that the recursive calls accumulate a total cost of at most k ? OPT(k) in the submatrices T (`+1) ? R(`) for ` ? {0, 1, . . . , k ? 1}, as well as cost at most OPT(k) in submatrix S (k) ? R(k) . We also show that any algorithm that selects k columns of a matrix A incurs at least an ?(k)approximation for the `0 -rank-k problem. Lemma 3. Let k ? n/2. Suppose A = (Gk?n ; In?n ) ? R(n+k)?n is a matrix composed of a Gaussian random matrix G ? Rk?n with Gi,j ? N (0, 1) and identity matrix In?n . Then for any subset J (k) ? [n] of size k, we have minZ?Rk?n kA ? A:,J (k) Zk0 = ?(k) ? OPT(k) . 3.2 Basic Algorithm We give an impractical algorithm that runs in poly(nk , m) time and achieves an O(k 2 )-approximation. To the best of our knowledge this is the first approximation algorithm for the `0 -rank-k problem with non-trivial approximation guarantees. Theorem 4. Given A ? Rm?n and k ? [n] we can compute in O(nk+1 m2 k ?+1 ) time a set of k indices J (k) ? [n] and a matrix Z ? Rk?n such that kA ? A:,J (k) Zk0 ? O(k 2 ) ? OPT(k) . Our result relies on a subroutine by Berman and Karpinski [5] (attributed also to Kannan in that paper) which given a matrix U and a vector b approximates minx kU x ? bk0 in polynomial time. Specifically, we invoke in our algorithm the following variant of this result established by Alon, Panigrahy, and Yekhanin [2]. Theorem 5. [2] There is an algorithm that given A ? Rm?k and b ? Rm outputs in O(m2 k ?+1 ) time a vector z ? Rk such that w.h.p. kAz ? bk0 ? k ? minx kAx ? bk0 . 3.3 Bicriteria Algorithm Our main contribution in this section is to design a practical algorithm that runs in poly(n, m) time with an exponent independent of k, if we allow for a bicriteria solution. Theorem 6. Given A ? Rm?n and k ? [1, n], there is an algorithm that in expected poly(m, n) time outputs a subset of indices J ? [n] with |J| = O(k log(n/k)) and a matrix Z ? R|J|?n such that kA ? A:,J Zk0 ? O(k 2 log(n/k)) ? OPT(k) . The structure of the proof follows a recent approximation algorithm [8, Algorithm 3] for the `p -low rank approximation problem, for any p ? 1. We note that the analysis of [8, Theorem 7] is missing an 5 O(log1/p n) approximation factor, and na?vely provides an O(k log1/p n)-approximation rather than the stated O(k)-approximation. Further, it might be possible to obtain an efficient algorithm yielding an O(k 2 log k)-approximation for Theorem 6 using unpublished techniques in [37]; we leave the study of obtaining the optimal approximation factor to future work. There are two critical differences with the proof of [8, Theorem 7]. We cannot use the earlier [8, Theorem 3] which shows that any matrix A contains k columns which provide an O(k)-approximation for the `p -low rank approximation problem, since that proof requires p ? 1 and critically uses scale-invariance, which does not hold for p = 0. Our combinatorial argument in Lemma 2 seems fundamentally different than the maximum volume submatrix argument in [8] for p ? 1. Second, unlike for `p -regression for p ? 1, the `0 -regression problem minx kU x ? bk0 given a matrix U and vector b is not efficiently solvable since it corresponds to a nearest codeword problem, which is NP-hard [1]. Thus, we resort to an approximation algorithm for `0 -regression, based on ideas for solving the nearest codeword problem in [2, 5]. Note that OPT(k) ? kAk0 . Since there are only mn + 1 possibilities of OPT(k) , we can assume we know OPT(k) and we can run the Algorithm 1 below for each such possibility, obtaining a rank-O(k log n) solution, and then outputting the solution found with the smallest cost. This can be further optimized by forming instead O(log(mn)) guesses of OPT(k) . One of these guesses is within a factor of 2 from the true value of OPT(k) , and we note that the following argument only needs to know OPT(k) up to a factor of 2. We start by defining the notion of approximate coverage, which is different than the corresponding notion in [8] for p ? 1, due to the fact that `0 -regression cannot be efficiently solved. Consequently, approximate coverage for p = 0 cannot be efficiently tested. Let Q ? [n] and M = A:,Q be an m ? |Q| submatrix of A. We say that a column M:,i is (S, Q)-approximately covered by a submatrix M:,S of M , if |S| = 2k and minx kM:,S x ? M:,i k0 ? 100(k+1)OPT(k) . |Q| Lemma 7. (Similar to [8, Lemma 6], but using Lemma 2) Let Q ? [n] and M = A:,Q be a submatrix of A. Suppose we select a subset R of 2k uniformly random columns of M . Then with probability at least 1/3, at least a 1/10 fraction of the columns of M are (R, Q)-approximately covered. Proof. To show this, as in [8], consider a uniformly random column index i not in the set R. Let def def T = R ? {i} and ? = minrank(B)=k kM:,T ? Bk0 . Since T is a uniformly random subset of 2k + 1 (k) (k) (2k+1)OPT M columns of M , ET ? ? ? (2k+1)OPT |Q| |Q| Then, by a Markov bound, Pr[E1 ] ? 9/10. . Let E1 be the event ? ? 10(2k+1)OPT(k) . |Q| Fix a configuration T = R ? {i} and let L(T ) ? T be the subset guaranteed by Lemma 2 such that |L(T )| = k and minX kM:,L(T ) X ? M:,T k0 ? (k + 1) minrank(B)=k kM:,T ? Bk0 . Notice that 1 Ei minx kM:,L(T ) x ? M:,i k0 | T = 2k+1 minX kM:,L(T ) X ? M:,T k0 , and thus by the law of total   probability we have ET minx kM:,L(T ) x ? M:,i k0 ? (k+1)? 2k+1 . Let E2 denote the event that minx kM:,L x ? M:,i k0 ? 10(k+1)? 2k+1 . By a Markov bound, Pr[E2 ] ? 9/10.
Further, as in [8], let E3 be the event that i ? / L. Observe that there are k+1 ways to choose a subset k
2k+1 0 0 0 R ? T such that |R | = 2k and L ? R . Since there are 2k ways to choose R0 , it follows that k+1 Pr[L ? R | T ] = 2k+1 > 1/2. Hence, by the law of total probability, we have Pr[E3 ] > 1/2. As in [8], Pr[E1 ? E2 ? E3 ] > 2/5, and conditioned on E1 ? E2 ? E3 , minx kM:,R x ? M:,i k0 ? (k) minx kM:,L x ? M:,i k0 ? 10(k+1)? ? 100(k+1)OPT , where the first inequality uses that L is a 2k+1 |Q| subset of R given E3 , and so the regression cost cannot decrease, while the second inequality uses the occurrence of E2 and the final inequality uses the occurrence of E1 . As in [8], if Zi is an indicator random variable indicating whether i is approximately covered P 3|Q| by R, and Z = i?Q Zi , then ER [Z] ? 2|Q| 5 and ER [|Q| ? Z] ? 5 . By a Markov bound, 2 Pr[|Q| ? Z ? 9|Q| 10 ] ? 3 . Thus, probability at least 1/3, at least a 1/10 fraction of the columns of M are (R, Q)-approximately covered. 6 Algorithm 1 Selecting O(k log(n/k)) columns of A. Require: An integer k, and a matrix A. Ensure: O(k log(n/k)) columns of A A PPROXIMATELY S ELECT C OLUMNS (k, A): if number of columns of A ? 2k then return all the columns of A else repeat Let R be a set of 2k uniformly random columns of A until at least (1/10)-fraction columns of A are nearly approximately covered Let AR be the columns of A not nearly approximately covered by R return R ? A PPROXIMATELY S ELECT C OLUMNS(k, AR ) end if Given Lemma 7, we are ready to prove Theorem 6. As noted above, a key difference with the corresponding [8, Algorithm 3] for `p and p ? 1, is that we cannot efficiently test if a column i is approximately covered by a set R. We will instead again make use of Theorem 5. Proof of Theorem 6. The computation of matrix Z force us to relax the notion of (R, Q)approximately covered to the notion of (R, Q)-nearly-approximately covered as follows: we say that a column M:,i is (R, Q)-nearly-approximately covered if, the algorithm in Theorem 5 returns a 2 (k) OPT vector z such that kM:,R z ? M:,i k0 ? 100(k+1) . By the guarantee of Theorem 5, if M:,i is |Q| (R, Q)-approximately covered then it is also w.h.p. (R, Q)-nearly-approximately covered. Suppose Algorithm 1 makes t iterations and let A:,?ti=1 Ri and Z be the resulting solution. We bound now its cost. Let B0 = [n], and consider the i-th iteration of Algorithm 1. We denote by Ri a set of 2k uniformly random columns of Bi?1 , by Gi a set of columns that is (Ri , Bi?1 )-nearly-approximately covered, and by Bi = Bi?1 \{Gi ? Ri } a set of the remaining columns. By construction, |Gi | ? 9 9 |Bi?1 |/10 and |Bi | ? 10 |Bi?1 | ? 2k < 10 |Bi?1 |. Since Algorithm 1 terminates when Bt+1 ? 2k, 1 t we have 2k < |Bt | < (1 ? 10 ) n, and thus the number of iterations t ? 10 log(n/2k). By Pt n i| construction, |Gi | = (1 ? ?i )|Bi?1 | for some ?i ? 9/10, and so i=1 |B|Gi?1 | ? t ? 10 log 2k . 2 Pt P OPT(k) Since minx(j) kA:,Ri x(j) ? A:,j k0 ? 100(k+1) , we have i=1 j?Gi kA:,Ri z (j) ? A:,j k0 ? |Bi?1 |
Pt P n (j) ? OPT(k) . ? A:,j k0 ? O k 2 ? log 2k i=1 j?Gi k ? minx(j) kA:,Ri x By Lemma 7, the expected number of iterations of selecting a set Ri such that |Gi | ? 1/10|Bi?1 | is O(1). Since the number of recursive calls t is bounded by O(log(n/k)), it follows by a Markov bound that Algorithm 1 chooses O(k log(n/k)) columns in total. Since the approximation algorithm of Theorem 5 runs in polynomial time, our entire algorithm has expected polynomial time. 4 Algorithm for Real `0 -rank-1 Given a matrix A ? Rm?n , the `0 -rank-1 problem asks to find a matrix A0 with rank 1 such that the difference between A and A0 measured in `0 norm is minimized. We denote the optimum value by def OPT(1) = min kA ? A0 k0 = rank(A0 )=1 min u?Rm , v?Rn kA ? uv T k0 . (2) In the trivial case when OPT(1) = 0, there is an optimal algorithm that runs in time O(kAk0 ) and finds the exact rank-1 decomposition uv T of a matrix A. In this work, we focus on the case when OPT(1) ? 1. We show that Algorithm 2 yields a (2 + )-approximation factor and runs in nearly linear time in kAk0 , for any constant  > 0. Furthermore, a variant of our algorithm even runs in def sublinear time, if kAk0 is large and ? = OPT(1) /kAk0 is not too small. In particular, we obtain time o(kAk0 ) when OPT(1) ? (?1 log(mn))4 and kAk0 ? n(?1 log(mn))4 . 7 Algorithm 2 Input: A ? Rm?n and  ? (0, 0.1). 1. Partition the columns of A into weight-classes S = {S (0) , . . . , S (log n+1) } such that S (0) contains all columns j with kA:,j k0 = 0 and S (i) contains all columns j with 2i?1 ? kA:,j k0 < 2i . 2. For each weight-class S (i) do: 2.1 Sample a set C (i) of ?(?2 log n) elements uniformly at random
from S (i) .  minv kA ? A:,j v T k0 , for 2.2 Find a vector z (j) ? Rn such that kA ? A:,j [z (j) ]T k0 ? 1 + 15 (i) each column A:,j ? C . S  3. Compute a (1 + 15 )-approximation Yj of kA ? A:,j [z (j) ]T k0 for every j ? i?[|S|] C (i) . Return: the pair (A:,j , z (j) ) corresponding to the minimal value Yj . Theorem 8. There is an algorithm that, given A ? Rm?n with column adjacency arrays and OPT(1) ? 1, and given  ? (0, 0.1], runs w.h.p. in time  
log2 n  n log m ?1 log n + min kAk0 , n + ? O and outputs a column A:,j and a vector z that 2 2 2 satisfy w.h.p. kA ? A:,j z T k0 ? (2 + )OPT(1) . The algorithm also computes a value Y satisfying w.h.p. (1 ? )OPT(1) ? Y ? (2 + 2)OPT(1) . The only steps for which the implementation details are not immediate are Steps 2.2 and 3. We will discuss them in Sections 4.1 and 4.2, respectively. Note that the algorithm from Theorem 8 selects a column A:,j and then finds a good vector z such that the product A:,j z T approximates A. We show that the approximation guarantee 2 +  is essentially tight for algorithms following this pattern. Lemma 9. There exist a matrix A ? Rn?n such that minz kA ? A:,j z T k0 ? 2(1 ? 1/n)OPT(1) , for every column A:,j . 4.1 Implementing Step 2.2 The Step 2.2 of Algorithm 2 uses the following sublinear procedure, given in Algorithm 3. Lemma 10. Given A ? Rm?n , u ? Rm and  ? (0, 1) we can compute in O(?2 n log m) time a vector z ? Rn such that w.h.p. kA:,i ? zi uk0 ? (1 + ) minvi kA:,i ? vi uk0 for every i ? [n]. Algorithm 3 Input: A ? Rm?n , u ? Rm and  ? (0, 1). def def def Let Z = ?(?2 log m), N = supp(u), and p = Z/|N |. 1. Select each index i ? N with probability p and let S be the resulting set. 2. Compute a vector z ? Rn such that zj = arg minr?R kAS,j ? r ? uS k0 for all j ? [n]. Return: vector z. 4.2 Implementing Step 3  In Step 3 of Algorithm 2 we want to compute a (1 + 15 )-approximation Yj of kA ? A:,j [z (j) ]T k0 S (i) for every j ? i?[|S|] C . We present two solutions, an exact algorithm (see Lemma 11) and a sublinear time sampling-based algorithm (see Lemma 13). Lemma 11. Suppose A, B ? Rm?n are represented by column adjacency arrays. Then, we can compute in O(kAk0 + n) time the measure kA ? Bk0 . For our second, sampling-based implementation of Step 3, we make use of an algorithm by Dagum et al. [11] for estimating the expected value of a random variable. We note that the runtime of their algorithm is a random variable, the magnitude of which is bounded w.h.p. within a certain range. def Theorem 12. [11] Let X be a random variable taking values in [0, 1] with ? = E[X] > 0. Let 0 < , ? < 1 and ?X = max{Var[X], ?}. There is an algorithm with sample access to X that computes an estimator ? ? in time t such that for a universal constant c we have: i) Pr[(1 ? )? ? ? ? ? (1 + )?] ? 1 ? ?, and ii) Pr[t ? c ?2 log(1/?)?X /?2 ] ? ?. 8 We state now our key technical insight, on which we build upon our sublinear algorithm. Lemma 13. There is an algorithm that, given A, B ? Rm?n with column adjacency arrays and kA ? Bk0 ? 1, and given  > 0, computes an estimator Z that satisfies w.h.p. (1 ? )kA ? Bk0 ? 0 +kBk0 Z ? (1 + )kA ? Bk0 . The algorithm runs w.h.p. in time O(n + ?2 kAk kA?Bk0 log n}). We present now our main result in this section. Theorem 14. There is an algorithm that, given A ? Rm?n with column adjacency arrays and OPT(1) ? 1, and given j ? [n], v ? Rm and  ? (0, 1), outputs an estimator Y that satisfies w.h.p. (1 ? )kA ? A:,j v T k0 ? Y ? (1 + )kA ? A:,j v T k0 . The algorithm runs w.h.p. in time O(min{kAk0 , n + ?2 ? ?1 log n}), where ? = OPT(1) /kAk0 . To implement Step 3 S of Algorithm 2, we simply apply Theorem 14 with A,  and v = z (j) to each sampled column j ? 0?i?log n+1 C (i) . 5 Algorithms for Boolean `0 -rank-1 Our goal is to compute an approximate solution of the Boolean `0 -rank-1 problem, defined by: def OPT = OPTA = min u?{0,1}m , v?{0,1}n kA ? uv T k0 , where A ? {0, 1}m?n . (3) In practice, approximating a matrix A by a rank-1 matrix uv T makes most sense if A is close to being rank-1. Hence, the above optimization problem is most relevant when OPT  kAk0 . In this section, we focus on the case OPT/kAk0 ? ? for sufficiently small ? > 0. We prove the following. Theorem 15. Given A ? {0, 1}m?n with row and column sums, and given ? ? (0, 1/80] with OPT/kAk0 ? ?, we can compute vectors u ?, v? with kA ? u ?v?T k0 ? (1 + 5?)OPT + 37?2 kAk0 in ?1 time O(min{kAk0 + m + n, ? (m + n) log(mn)}). In combination with Theorem 8 we obtain the following. Theorem 16. Given A ? {0, 1}m?n with column adjacency arrays and with row and column sums, for ? = OPT/kAk0 we can compute vectors u ?, v? with kA ? u ?v?T k0 ? (1 + 500?)OPT in time 3 ?1 w.h.p. O(min{kAk0 + m + n, ? (m + n)} ? log (mn)). A variant of the algorithm from Theorem 15 can also be used to solve the Boolean `0 -rank-1 problem exactly. This yields the following theorem,
which in particular shows that the problem is in polynomial p time when OPT ? O kAk0 log(mn) . Theorem 17. Given a matrix A ? {0, 1}m?n , if?OPTA /kAk0 ? 1/240 then we can exactly solve the Boolean `0 -rank-1 problem in time 2O(OPT/ kAk0 ) ? poly(mn). 6 Lower Bounds for Boolean `0 -rank-1 We give now a lower bound of ?(n/?) on the number of samples of any 1 + O(?)-approximation algorithm for the Boolean `0 -rank-1 problem, where OPT/kAk0 ? ? as before. Theorem 18. Let C ? 1. Given an n ? pn Boolean matrix A with column adjacency arrays and with row and column sums, and given log(n)/n  ? ? 1/100C such that OPTA /kAk0 ? ?, computing a (1 + C?)-approximation of OPTA requires to read ?(n/?) entries of A. The technical core of our argument is the following lemma. Lemma 19. Let ? ? (0, 1/2). Let X1 , . . . , Xk be Boolean random variables with expectations p1 , . . . , pk , where pi ? {1/2 ? ?, 1/2 + ?} for each i. 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The power of absolute discounting: all-dimensional distribution estimation Moein Falahatgar UCSD moein@ucsd.edu Mesrob Ohannessian TTIC mesrob@gmail.com Alon Orlitsky UCSD alon@ucsd.edu Venkatadheeraj Pichapati UCSD dheerajpv7@ucsd.edu Abstract Categorical models are a natural fit for many problems. When learning the distribution of categories from samples, high-dimensionality may dilute the data. Minimax optimality is too pessimistic to remedy this issue. A serendipitously discovered estimator, absolute discounting, corrects empirical frequencies by subtracting a constant from observed categories, which it then redistributes among the unobserved. It outperforms classical estimators empirically, and has been used extensively in natural language modeling. In this paper, we rigorously explain the prowess of this estimator using less pessimistic notions. We show that (1) absolute discounting recovers classical minimax KL-risk rates, (2) it is adaptive to an effective dimension rather than the true dimension, (3) it is strongly related to the Good?Turing estimator and inherits its competitive properties. We use powerlaw distributions as the cornerstone of these results. We validate the theory via synthetic data and an application to the Global Terrorism Database. 1 Introduction Many natural problems involve uncertainties about categorical objects. When modeling language, we reason about words, meanings, and queries. When inferring about mutations, we manipulate genes, SNPs, and phenotypes. It is sometimes possible to embed these discrete objects into continuous spaces, which allows us to use the arsenal of the latest machine learning tools that often (though admittedly not always) need numerically meaningful data. But why not operate in the discrete space directly? One of the main obstacles to this is the dilution of data due to the high-dimensional aspect of the problem, where dimension in this case refers to the number k of categories. The classical framework of categorical distribution estimation, studied at length by the information theory community, involves a fixed small k, [BS04]. Add-contant estimators are sufficient for this purpose. Some of the impetus to understanding the large k regime came from the neuroscience world, [Pan04]. But this extended the pessimistic worst-case perspective of the earlier framework, resulting in guarantees that left a lot to be desired. This is because high-dimension often also comes with additional structure. In particular, if a distribution produces only roughly d distinct categories in a sample of size n, then we ought to think of d (and not k) as the effective dimension of the problem. There are also some ubiquitous structures, like power-law distributions. Natural language is a flagship example of this, which was observed as early as by Zipf in [Zip35]. Species and genera, rainfall, terror incidents, to mention just a few all obey power-laws [SLE+ 03, CSN09, ADW13]. Are there estimators that mold to both dimension and structure? It turns out we don?t need to search far. In natural language processing (NLP) it was first discovered that an estimator proposed by Good and Turing worked very well [Goo53]. Only recently did we start understanding why and how [OSZ03, OD12, AJOS13, OS15]. And the best explanation thus far is that it implicitly competes with the best estimator in a very small neighborhood of the true distribution. But NLP researchers [NEK94, KN95, CG96] have long realized that another simpler estimator, absolute discounting, is equally good. But why and how this is the case was never properly determined, save some mention in [OD12] and in [FNT16], where the focus is primarily on form. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. In this paper, we first show that absolute discounting, defined in Section 3, recovers pessimistic minimax optimality in both the low- and high-dimensional regimes. This is an immediate consequence of an upper bound that we provide in Section 5. We then study lower bounds with classes defined by the number of distinct categories d and also power-law structure in Section 6. This reveals that absolute discounting in fact adapts to the family of these classes. We further unravel the relationship of absolute discounting with the Good?Turing estimator, for power-law distributions. Interestingly, this leads to a further refinement of this estimator?s performance in terms of competitivity. Lastly, we give some synthetic experiments in Section 8 and then explore forecasting global terror incidents on real data [LDMN16], which showcases very well the ?all-dimensional? learning power of absolute discounting. These contributions are summarized in more detail in Section 4. We start out in Section 2 with laying out what we mean by these notions of optimality. 2 Optimal distribution learning In this section we concretely formulate the optimal distribution learning framework and take the opportunity to point out related work. Problem setting Let p = (p1 , p2 , . . . , pk ) be a distribution over [k] := {1, 2, . . . , k} categories. Let [k]? be the set of finite sequences over [k]. An estimator q is a mapping that assigns to every sequence xn ? [k]? a distribution q(xn ) over [k]. We model p as being the underlying distribution over the categories. We have access to data consisting of n samples X n = X1 , X2 , ..., Xn generated i.i.d. from p. Intuitively, our goal is to find a choice of q that is guaranteed to be as close as any other estimator can be to p, in average. We first need to quantify how performance is measured. General notation: Let (?j : j = 1, ? ? ? , k) denote the empirical counts, i.e. the number of n n times symbol P j appears in X and let D be the number of distinct categories appearing in X , i.e. D = j 1{?j > 0}. We denote by d := E[D] its expectation. Let (?? : ? = 0, ? ? ? , n), P be the total number of categories appearing exactly ? times, ?? := j 1{?j = ?}. Note that P D = ?>0 ?? . Also let (S? : ? = 0, ? ? ? , n), be the total probability within each such group, P S? := j pj 1{?j = ?}. Lastly, denote the empirical distribution by qj+0 := ?j /n. KL-Risk We adopt the Kullback-Leibler (KL) divergence as a measure of loss between two distributions. When a distribution p is approximated by another q, the KL divergence is given by Pk p KL(p||q) := j=1 pj log qjj . We can then measure the performance of an estimator q that depends on data in terms of the KL-risk, the expectation of the divergence with respect to the samples. We use the following notation to express the KL-risk of q after observing n samples X n : rn (p, q) := nE n [KL(p||q(X n ))]. X ?p An estimator that is identical to p regardless of the data is unbeatable, since rn (p, q) = 0. Therefore it is important to model our ignorance of p and gauge the optimality of an estimator q accordingly. This can be done in various ways. We elaborate the three most relevant such perspectives: minimax, adaptive, and competitive distribution learning. Minimax In the minimax setting, p is only known to belong to some class of distributions P, but we don?t know which one. We would like to perform well, no matter which distribution it is. To each q corresponds a distribution p ? P (assuming the class is finite or closed) on which q has its worst performance: rn (P, q) := max rn (p, q). p?P The minimax risk is the least worst-case KL-risk achieved by any estimator q, rn (P) := min rn (P, q). q The minimax risk depends only on the class P. It is a lower bound: no estimator can beat it for all p, i.e. it?s not possible that rn (p, q) < rn (P) for all p ? P. An estimator q that satisfies an upper bound of the form rn (P, q) = (1 + o(1))rn (P) is said to be minimax optimal ?even to the constant? (an informal but informative expression that we adopt in this paper). If instead rn (P, q) = O(1)rn (P), we say that q is rate optimal. Near-optimality notions are also possible, but we don?t dwell on them. As an aside, note that universal compression is minimax optimality using cumulative risk. See [FJO+ 15] for such related work on universal compression for power laws. 2 Adaptive The minimax perspective captures our ignorance of p in a pessimistic fashion. This is because rn (P) may be large, but for a specific p ? P we may have a much smaller rn (p, q). How can we go beyond this pessimism? Observe that when a class is smaller, then rn (P) is smaller. This is because we?d be maximizing on a smaller set. In the extreme case noted earlier, when P contains only a single distribution, we have rn (P) = 0. The adaptive learning setting finds an intermediate ground where we have a family of distribution classes F = {Ps : s ? S} indexed by a (not necessarily countable) S index
set S. For each s, we have a corresponding rn (Ps ) which is often much smaller than rn s?S Ps , and we would like the estimator to achieve the risk bound corresponding to the smaller class. We say that an estimator q is adaptive to the family F if for all s ? S: rn (p, q) ? Os (1) rn (Ps ) ?p ? Ps ?? rn (Ps , q) ? Os (1) rn (Ps ) There often is a price to adaptivity, which is a function of the granularity of F and is paid in the form of varying/large leading constants per class. This framework has been particularly successful in density estimation with smoothness classes [Tsy09] and has been recently used in the discrete setting for universal compression [BGO15]. Competitive The adaptive perspective can be tightened by demanding that, rather than a multiplicative constant, the KL-risk tracks the risk up to a vanishingly small additive term: rn (p, q) = rn (Ps ) + n (Ps , q) ?p ? Ps . Ideally, we would like the competitive loss n (Ps , q) to be negligible compared to the risk of each class rn (Ps ). If n (Ps , q) = Os (1)rn (Ps ) for all s, then we recover adaptivity. And when n (Ps , q) = os (1)rn (Ps ) for all s ? S, we have minimax optimality even to the constant within each class, which is a much stronger form of adaptivity. We then say that the estimator is competitive with respect to the family F. We may also evaluate the worst-case competitive loss, over S. This formulation was recently introduced in [OS15] in the context of distribution learning. This work shows that the celebrated Good?Turing estimator [Goo53], combined with the empirical estimator, has small worst-case competitive loss over the family of classes defined by any given distribution and all its permutations. Most importantly, this loss was shown to stay bounded, even as the dimension increases. This provided a rigorous theoretical explanation for the performance of the Good?Turing estimator in high-dimensions. A similar framework is also studied for `1 -loss in [VV15]. 3 Absolute discounting One of the first things to observe is that the empirical distribution is particularly ill-suited to handle KL-risk. This is most easily seen by the fact that we?d have infinite blow-up when any ?j = 0, which will happen with positive probability. Instead, one could resort to an add-constant estimator, which for a positive ? is of the form qj+? := (?j + ?)/(n + k?). The most widely-studied class of distributions is P the one that includes all of them: the k?dimensional simplex, ?k := {(p1 , p2 , . . . , pk ), : pi = 1, pi ? 0 ?i ? [k]}. In the lowi dimensional scaling, when n/k ? ? (the ?dimension? here being the support size k), the minimax risk is k?1 rn (?k ) = (1 + o(1)) , 2n In [BS04], a variant of the add-constant estimator is shown to achieve this risk even to the constant. Furthermore, any add-constant estimator is rate optimal when k is fixed. But in the very highdimensional setting, when k/n ? ?, [Pan04] showed that the minimax risk behaves as k , n achieved by an add-constant estimator, but with a constant that depends on the ratio of k and n. rn (?k ) = (1 + o(1)) log Despite these classical results on minimax optimal estimators, in practice people often use other estimators that have better empirical performance. This was a long-running mystery in the language modeling community [CG96], where variants of the Good?Turing estimator were shown to perform the best [JM85, GS95]. The gap in performance was only understood recently, using the notion of competitivity [OS15]. In essence, the Good?Turing estimator works well in both low- and 3 high-dimensional regimes, and in-between. Another estimator, absolute discounting, unlike addconstant estimators, simply subtracts a positive constant from the empirical counts and redistributes the subtracted amount to unseen categories. For a discount parameter ? ? [0, 1), it is defined as: ( qj?? := ?j ?? n D? n(k?D) if ?j > 0, if ?j = 0. (1) Starting with the work of [NEK94], absolute discounting soon supplanted the Good?Turing estimator, due to both its simplicity and comparable performance. Kneser-Ney smoothing [KN95], which uses absolute discounting at its core was long held as the preferred way to train N -gram models. Even to this day, the state-of-the-art language models are combined systems where one usually interpolates between recurrent neural networks and Kneser-Ney smoothing [JVS+ 16]. Can this success be explained? Kneser-Ney is for the most part a principled implementation of the notion of back-off, which we only touch upon in the conclusion. The use of absolute discounting is critical however, as performance deteriorates if we back-off with care but use a more na??ve add-constant or even Katz-style smoothing [Kat87], which switches from the Good?Turing to the empirical distribution at a fixed frequency point. It is also important to mention the Bayesian approach of [Teh06] that performs similarly to Kneser-Ney, called the Hierarchical Pitman-Yor language model. The hierarchies in this model reprise the role of back-off, while the two-parameter Poisson-Dirichlet prior proposed by Pitman and Yor [PY97] results in estimators that are very similar to absolute discounting. The latter is not a surprise because this prior almost surely generates a power law distribution, which is intimately related to absolute discounting as we study in this paper. Though our theory applies more generally, it can in fact be straightforwardly adapted to give guarantees to estimators built upon this prior. 4 Contributions We investigate the reason behind the auspicious behavior of the absolute discounting estimator. We achieve this by demonstrating the adaptivity and competitivity of this estimator for many relevant families of distribution classes. In summary: ? We analyze the performance of the absolute discounting estimator by upper bounding the KLrisk for each class in a family of distribution classes defined by the expected number of distinct categories. [Section 5, Theorem 1] This result implies that absolute discounting achieves classical minimax rate-optimality in both the low- and high-dimensional regimes over the whole simplex ?k , as outlined in Section 2. ? We provide a generic lower bound to the minimax risk of classes defined by a single distribution and all of its permutations. We then show that if the defining distribution is a truncated (possibly perturbed) power-law, then this lower bound matches the upper bound of absolute discounting, up to a constant factor. [Section 6, Corollaries 3 and 4] ? This implies that absolute discounting is adaptive to the family of classes defined by a truncated power-law distribution and its permutations. Also, since classes defined by the expected number of distinct categories necessarily includes a power-law, absolute discounting is also adaptive to this family. This is a strict refinement of classical minimax rate-optimality. ? We give an equivalence between the absolute discounting and Good?Turing estimators in the high-dimensional setting, whenever the distribution is a truncated power-law. This is a finitesample guarantee, as compared to the asymptotic version of [OD12]. As a consequence, absolutediscounting becomes competitive with respect to the family of classes defined by permutations of power-laws, inheriting Good?Turing?s behavior [OS15]. [Section 7, Lemma 5 and Theorem 6] We corroborate the theoretical results with synthetic experiments that reproduce the theoretical minimax risk bounds. We also show that the prowess of absolute discounting on real data is not restricted only to language modeling. In particular, we explore a striking application to forecasting global terror incidents and show that, unlike naive estimators, absolute discounting gives accurate predictions simultaneously in all of low-, medium-, and high-activity zones. [Section 8] 4 5 Upper bound and classical minimax optimality We now give an upper bound for the risk of the absolute discounting estimator and show that it recovers classical minimax rates in the low- and high-dimensional regimes. Recall that d := E[D] is the expected number of distinct categories in the samples. The upper bound that we derive can be written as function of only d, k, and n, and is non-decreasing in d. For a given n and k, let Pd be the set of all distributions for which E[D] ? d. The upper bound is thus also a worst-case bound over Pd . Theorem 1 (Upper bound). Consider the absolute discounting estimator q = q ?? , defined in (1). Let p be such that E[D] = d. Given a discount 0 < ? < 1, there exists a constant c that may depend on ? and only ?, such that ? k? d ? d d ? ? log d 2 + c if d ? 10 log log k, n n rn (p, q) ? (2) 2 ? ? ? d log k + c d if d < 10 log log k. n n The same bound holds for rn (Pd , q). We defer the proof the theorem to the supplementary material. Here are the immediate implications. For the low-dimensional regime nk ? ? and the class ?k , the largest d can be once n > k is k. The risk of absolute discounting is thus bounded by c(1 + o(1)) nk = O(1) nk . This is minimax rate-optimal [BS04]. For the high-dimensional regime nk ? ? and the class ?k , the largest d can be when k > n is n. The risk of absolute discounting is thus dominated by the first term, which reduces to (1 + o(1)) log nk . This is the optimal risk for the class ?k [Pan04], even to the constant. Therefore on the two extreme ranges of k and n absolute discounting recovers the best performance, either as rate-optimal or optimal even to the constant. These results are for the entire k?dimensional simplex ?k . Furthermore, for smaller classes, it characterizes the worst-case risk of the class by the d, the expected number of distinct categories. Is this characterization tight? 6 Lower bounds and adaptivity In order to lower bound the minimax risk of a given class P, we use a finer granularity than the Pd classes described in Section 5. In particular, let Pp be the permutation class of distributions consisting of a single distribution p and all of its permutations. Note that the multiset of probabilities is the same for all distributions Pin Pp , and since the expected number of distinct categories only depends on the multiset (d = j [1 ? (1 ? pj )n ]) it follows that Pp ? Pd 1 . To find a good lower bound for Pd , we need a p that is ?worst case?. We first give the following generic lower bound. Theorem 2 (Generic lower bound). Let Pp be a permutation class defined by a distribution p and let ? > 1. Then for k > ?d, the minimax risk is bounded by: ? ?   k X 1 ?X ? k ? ?d pj log Pk rn (Pp ) ? 1 ? + pj log pj (3) ? j=?d pj j=?d i=?d Equation (3) can be used as a starting point for more concrete lower bounds on various distribution classes. We illustrate this for two cases. First, let us choose p to be a truncated power-law distribution with power ?: pj ? j ?? , for j = 1, ? ? ? , k. We always assume ? ? ?0 > 1. This leads to the following lower bound. Corollary 3. Let P be all permutations of a single power-law distribution with power ? truncated over k categories. Then there exists a constant c > 0 and large enough n0 such that when n > n0 1 1 and k > max{n, 1.2 ??1 n ? }, d k ? 2d rn (P) ? c log . n 2d Next, we use a different choice of p for Pp to provide a lower bound whenever d grows linearly with n. This essentially closes the gap of the previous corollary when ? approaches 1. 1 We abuse notation by distinguishing the classes by the letter used, while at the same time using the letters to denote actual quantities. From the context we understand that d is the expected number of distinct categories for p, at the given n. 5 Corollary 4. Let ? ? (1, 1.75) and let P be all permutations of a single uniform distribution over a subset k 0 = n? out of k categories. Then d ? (1 ? e?? )n/? and there exists a constant c > 0 and large enough n0 such that when n > n0 and k > n5 , d k ? 1.2d rn (P) ? c log . n d We defer the proofs of the theorem and its corollaries to the supplementary material. The upper bound of Theorem 1 and the lower bounds of Corollaries 3 and 4 are within constant factors of each other. The immediate consequence is that absolute discounting is adaptive with respect to the families of classes of the Corollaries. Furthermore, over the family of classes Pd where we can 1 write d as n ? for some ? > 1 or d ? n, we can select a distribution from the Corollaries among each class and use the corresponding lower bound to match the upper bound of Theorem 1 up to a constant factor. Therefore absolute discounting is adaptive to this family of classes. Intuitively, adaptivity to these classes establishes optimality in the intermediate range between low- and highdimensional settings in a distribution-dependent fashion and governed by the expected number of distinct categories d, which we may regard as the effective dimension of the problem. 7 Relationship to Good?Turing and competitivity We now establish a relationship between the absolute discounting and Good?Turing estimators and refine the adaptivity results of the previous section into competitivity results. When [OS15] introduced the notion of competitive optimality, they showed that a variation of the Good?Turing estimator is worst-case competitive with respect to the family of distribution classes defined by any given probability distribution and its permutations. In light of the results of Sections 5 and 6, it is natural to ask whether absolute discounting enjoys the same kind of competitive properties. Not only that, but it was observed empirically by [NEK94] and shown theoretically in [OD12] that asymptotically Good?Turing behaves exactly like absolute discounting, when the underlying distribution is a (possibly perturbed) power-law. We therefore choose this family of classes to prove competitivity for. We first make the aforementioned equivalence concrete by establishing a finite sample version. We use the following idealized version of the Good?Turing estimator [Goo53]: ? ? ?j +1 E[??j +1 ] if ? > 0, j n E[??j ] GT qj := (4) ? E[?1 ] if ?j = 0. n(k?D) Lemma 5. Let p be a power law with power ? truncated over k categories. Then for k > 1 max{n, n ??1 }, we have the equivalence:  1 3  ? ? 1 n o ?j ? ?1  1 j ? qjGT = 1 + O n? 2 2?+1 ? ?j ? 1, ? ? ? , n 2?+1 . ? n n An interesting outcome of the equivalence of Lemma 5 is that it suggests a choice of the discount ? in terms of the power, 1/?. To give a data-driven version of 1/?, we will use a robust version of the ratio ?1 /D proposed in [OD12, BBO17], which is a strongly consistent estimator when k = ?. Theorem 6. Let P be all permutations of a truncated power law n o p with power ?. Let q be the 1 ,1} absolute discounting estimator with ? = min max{? , ? max , for a suitable choice of ?max . D 1 Then for k > max{n, n ??1 }, the competitive loss is  2??1  n (Pp , q) = O n? 2?+1 . The implications are as follows. For the union of all such classes above a given ?, we find that we beat the n?1/3 rate of the worst-case competitive loss obtained for the estimator in [OS15]. Theorem 6 and the bounds of Sections 5 and 6, together imply that absolute discounting is not only worst-case competitive, but also class-by-class competitive with respect to the power-law permutation family. In other words, it in fact achieves minimax optimality even to the constant. One of the advantages of absolute discounting is that it gradually transitions between values that are close to the empirical distribution for abundant categories (since ? then dominates the discount ?), to a behavior that imitates the Good?Turing estimator for rare categories (as established by Lemma 5). In contrast, the estimator proposed in [OS15], and its antecedents starting from [Kat87], have to carefully choose a threshold where they switch abruptly from one estimator to the other. 6 8 Experiments We now illustrate the theory with some experimental results. Our purpose is to (1) validate the functional form of the risk as given by our lower and upper bounds and (2) compare absolute discounting on both synthetic and real data to estimators that have various optimality guarantees. In all synthetic experiments, we use 500 Monte Carlo iterations. Also, we set the discount value based on data, 1 ,1) ? = min{ max(? , 0.9}. This is as suggested in Section 7, assuming ?max = 0.9 is sufficient. D 0.025 0.5 0.4 1.6 n=500 n=1000 n=5000 n=10000 k=100 k=500 k=1000 k=3000 0.45 1.4 0.02 n=20 n=50 0.3 0.25 0.2 expected KL loss 1.2 expected KL loss expected KL loss 0.35 0.015 0.01 n=100 n=200 1 0.8 0.15 0.6 0.005 0.1 0.4 0.05 0 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5 5000 6 7 8 9 10 11 12 13 14 0.2 10 3 15 k n (a) k fixed 10 4 k (b) n fixed, k << n (c) n fixed, k >> n Figure 1: Risk of absolute discounting in different ranges of k and n for a power-law with ? = 2 Validation For our first goal, we consider absolute discounting in isolation. Figure 1(a) shows the decay of KL-risk with the number of samples n for a power-law distribution. The dependence of the risk on the number of categories k is captured in Figures 1(b) (linear x-axis) and 1(c) (logarithmic x-axis). Note the linear growth when k is small and the logarithmic growth when k is large. For the last plot we give 95% confidence intervals for the simulations, by performing 100 restarts. Synthetic data For our second goal, we start with synthetic data. In Figure 2, we pit absolute discounting against a number of distributions related to power-laws. The estimators used for our ? +? comparisons are: empirical q +0 (x) = ?nx , add-beta q +? (x) = x N ?x , and its two variants: ? Braess and Sauer, q BS [BS04] q +? with ?0 = 0.5, ?1 = 1, and ?i = 0.75 ?i ? 2 ? Paninski, q Pan [Pan04] q +? with ?i = nk log nk ?i, absolute discounting, q ?? , described in 1, Good?Turing + empirical q GT in [OS15], and an oracleaided estimator where S? is known. In Figures 2(a) and 2(b), samples are generated according to a power-law distribution with power ? = 2 over k = 1, 000 categories. However, the underlying distribution in Figure 2(c) is a piecewise power-law. It consists of three equal-length pieces, with powers 1.3, 2, and 1.5. Paninski?s estimator is not shown in Figures 2(b) and 2(c) since it is not well-defined in this range (it is designed for the case k > n only). Unsurprisingly, absolute discounting dominates these experiments. What is more interesting is that it does not seem to need a pure power-law (similar results hold for other kinds of perturbations, such as mixtures and noise). Also Good?Turing is a tight second. 10 1 Good-Turing + empirical Braess-Sauer absolute-discount oracle expected KL loss 10 0 expected KL loss expected KL loss Good-Turing + empirical Braess-Sauer Paninski absolute-discount oracle 10 0 10 -1 10 -1 10 -2 Good-Turing + empirical Braess-Sauer absolute-discount oracle 10 1 10 1 0 100 200 300 400 500 600 700 800 n (a) pure power-law 900 1000 10 -2 10 0 10 -1 10 -2 0 1000 2000 3000 4000 5000 6000 7000 8000 n (b) pure power-law 9000 10000 0 2000 4000 6000 8000 n (c) piece-wise power-law Figure 2: Comparing estimators for power-law variants with power ? = 2 and k = 1000. 7 10000 Real data One of the chief motivations to investigate absolute discounting is natural language modeling. But there have been such extensive empirical studies that have verified over and over the power of absolute discounting (see the classical survey of [CG96]) that we chose to use this space for something new. We use the START Global terrorism database from the University of Maryland [LDMN16] and explore how well we can forecast the number of terrorist incidents in different cities. The data contains the record of more than 50, 000 terror incidents between the years 1992 and 2010, in more than 12, 000 different cities around the world. First, we display in Figure 3(a) the frequency of incidents across the entire dataset versus the activity rank of the city in log-log scale, showing a striking adherence to a power-law (see [CSN09] for more on this). The forecasting problem that we solve is to estimate the number of total incidents in a subset of the cities over the coming year, using the current year?s data from all cities. In order to emulate the various dimension regimes, we look at three subsets: (1) low-activity cities with no incidents in the current year and less than 20 incidents in the whole data, (2) medium-activity cities, with some incidents in the current year and less than 20 incidents in the whole data, and (3) high-activity individual cities with a large number of overall incidents. The results for (1) are in Figure 3(b). The frequency estimator trivially estimates zero. Braess-Sauer does something meaningful. But absolute discounting and Good?Turing estimators, indistinguishable from each other, are remarkably on spot. And this, without having observed any of the cities! This nicely captures the importance of using structure when dimensionality is so high and data is so scarce. The results for (2) are in Figure 3(c). The frequency estimator markedly overestimates. But now absolute discounting, Good?Turing, and Braess-Sauer, perform similarly. This is a lower dimensional regime than in (1), but still not adequate for simply using frequencies. This changes in case (3), illustrated in Figure 4. To take advantage of the abundance of data, in this case at each time point we used the previous 2, 000 incidents for learning, and predicted the share of each city for the next 2, 000 incidents. In fact, incidents are so abundant that we can simply rely on the previous window?s count. Note how Braess-Sauer over-penalizes such abundant categories and suffers, whereas absolute discounting and Good?Turing continue to hold their own, mimicking the performance of the empirical counts. This is a very low-dimensional regime. The closeness of the Good?Turing estimator to absolute discounting in all of our experiments validates the equivalence result of Lemma 5. The robustness in various regimes and the improvement in performance over such minimax optimal estimators as Braess-Sauer?s and Paninski?s are evidence that absolute discounting truly molds to both the raw dimension and effective dimension / structure. 3000 4 2500 number of incidents number of incidents 10 3 10 2 10 2500 Good-Turing + empirical Braess-Sauer absolute-discount true value empirical 2000 2000 number of incidents 10 1500 1000 1 10 1 10 2 10 3 10 4 rank of the city (a) frequency vs rank 10 5 0 1992 1500 1000 500 500 10 0 10 0 Good-Turing + empirical Braess-Sauer absolute-discount true value empirical 1995 1997 1999 2002 2006 year (b) unobserved cities 2007 0 1992 1995 1997 1999 2002 2006 2007 year (c) observed cities Figure 3: (a) power-law behavior of frequency vs rank in terror incidents, (b), and (c) comparing forecasts of the number of incidents in unobserved cities and observed ones, respectively. 9 Conclusion In this paper, we offered a rigorous analysis of the absolute discounting estimator for categorical distributions. We showed that it recovers classical minimax optimality. The true reason for its success, however, is in adapting to distributions much more intimately, by recovering the right dependence on the distinct observed categories d, which can be regarded as an effective dimension, and optimally tracking structure such as power-laws. We also tightened its relationship with the celebrated Good?Turing estimator. 8 350 300 25 Good-Turing + empirical Braess-Sauer absolute-discount true value empirical 20 120 Good-Turing + empirical Braess-Sauer absolute-discount true value empirical Good-Turing + empirical Braess-Sauer absolute-discount true value empirical 100 200 150 number of incidents number of incidents number of incidents 250 15 10 80 60 40 100 5 20 50 0 1992 1995 1997 1999 2002 2006 2007 year (a) Baghdad 2008 2009 2010 0 1992 1995 1997 1999 2002 2006 2007 year (b) Fallujah 2008 2009 2010 0 1992 1995 1997 1999 2002 2006 2007 2008 2009 2010 year (c) Belfast Figure 4: Estimating the number of incidents based on previous data for different cities Some of our analysis could possibly be tightened, in particular in terms of the range of applicability over n, k, and d. Also, the limiting case of ? = 1 (very heavy tails, known as ?fast variation? [BBO17]) to which our results don?t directly apply, merits investigation. But more importantly, absolute discounting is often a module. For example, we already note how it is widely used in N -gram back-off models [KN95]. Also, recently, it has been successfully applied to smoothing low-rank probability matrices [FOO16]. Perhaps to further understand its power, it is worthwhile to study how it interacts with such larger systems. Acknowledgements We thank Vaishakh Ravindrakumar for very helpful suggestions, and NSF for supporting this work through grants CIF-1564355 and CIF-1619448. References [ADW13] Armen E. Allahverdyan, Weibing Deng, and Q. A. Wang. Explaining Zipf?s law via a mental lexicon. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 88(6), 2013. 1 [AJOS13] Jayadev Acharya, Ashkan Jafarpour, Alon Orlitsky, and Ananda Theertha Suresh. Optimal Probability Estimation with Applications to Prediction and Classification. In COLT, pages 764?796, 2013. 1 [BBO17] Anna Ben Hamou, St?ephane Boucheron, and Mesrob I Ohannessian. 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Few-Shot Adversarial Domain Adaptation Saeid Motiian, Quinn Jones, Seyed Mehdi Iranmanesh, Gianfranco Doretto Lane Department of Computer Science and Electrical Engineering West Virginia University {samotian, qjones1, seiranmanesh, gidoretto}@mix.wvu.edu Abstract This work provides a framework for addressing the problem of supervised domain adaptation with deep models. The main idea is to exploit adversarial learning to learn an embedded subspace that simultaneously maximizes the confusion between two domains while semantically aligning their embedding. The supervised setting becomes attractive especially when there are only a few target data samples that need to be labeled. In this few-shot learning scenario, alignment and separation of semantic probability distributions is difficult because of the lack of data. We found that by carefully designing a training scheme whereby the typical binary adversarial discriminator is augmented to distinguish between four different classes, it is possible to effectively address the supervised adaptation problem. In addition, the approach has a high ?speed? of adaptation, i.e. it requires an extremely low number of labeled target training samples, even one per category can be effective. We then extensively compare this approach to the state of the art in domain adaptation in two experiments: one using datasets for handwritten digit recognition, and one using datasets for visual object recognition. 1 Introduction As deep learning approaches have gained prominence in computer vision we have seen tasks that have large amounts of available labeled data flourish with improved results. There are still many problems worth solving where labeled data on an equally large scale is too expensive to collect, annotate, or both, and by extension a straightforward deep learning approach would not be feasible. Typically, in such a scenario, practitioners will train or reuse a model from a closely related dataset with a large amount of samples, here called the source domain, and then train with the much smaller dataset of interest, referred to as the target domain. This process is well-known under the name finetuning. Finetuning, while simple to implement, has been found to be sub-optimal when compared to later techniques such as domain adaptation [5]. Domain Adaptation can be supervised [58, 27], unsupervised [15, 34], or semi-supervised [16, 21, 63], depending on what data is available in a labeled format and how much can be collected. Unsupervised domain adaptation (UDA) algorithms do not need any target data labels, but they require large amounts of target training samples, which may not always be available. Conversely, supervised domain adaptation (SDA) algorithms do require labeled target data, and because labeling information is available, for the same quantity of target data, SDA outperforms UDA [38]. Therefore, if the available target data is scarce, SDA becomes attractive, even if the labeling process is expensive, because only few samples need to be processed. Most domain adaptation approaches try to find a feature space such that the confusion between source and target distributions in that space is maximum (domain confusion). Because of that, it is hard to say whether a sample in the feature space has come from the source distribution or the target distribution. Recently, generative adversarial networks [18] have been introduced for image generation which can also be used for domain adaptation. In [18], the goal is to learn a discriminator 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. G1 G3 G2 G4 Figure 1: Examples from MNIST [32] and SVHN [40] of grouped sample pairs. G1 is composed of samples of the same class from the source dataset in this case MNIST. G2 is composed of samples of the same class, but one is from the source dataset and the other is from the target dataset. In G3 the samples in each pair are from the source dataset but with differing class labels. Finally, pairs in G4 are composed of samples from the target and source datasets with differing class labels. to distinguish between real samples and generated (fake) samples and then to learn a generator which best confuses the discriminator. Domain adaptation can also be seen as a generative adversarial network with one difference, in domain adaptation there is no need to generate samples, instead, the generator network is replaced with an inference network. Since the discriminator cannot determine if a sample is from the source or the target distribution the inference becomes optimal in terms of creating a joint latent space. In this manner, generative adversarial learning has been successfully modified for UDA [33, 59, 49] and provided very promising results. Here instead, we are interested in adapting adversarial learning for SDA which we are calling few-shot adversarial domain adaptation (FADA) for cases when there are very few labeled target samples available in training. In this few-shot learning regime, our SDA method has proven capable of increasing a model?s performance at a very high rate with respect to the inclusion of additional samples. Indeed, even one additional sample can significantly increase performance. Our first contribution is to handle this scarce data while providing effective training. Our second contribution is to extend adversarial learning [18] to exploit the label information of target samples. We propose a novel way of creating pairs of samples using source and target samples to address the first challenge. We assign a group label to a pair according to the following procedure: 0 if samples of a pair come from the source distribution and the same class label, 1 if they come from the source and target distributions but the same class label, 2 if they come from the source distribution but different class labels, and 3 if they come from the source and target distributions and have different class labels. The second challenge is addressed by using adversarial learning [18] to train a deep inference function, which confuses a well-trained domain-class discriminator (DCD) while maintaining a high classification accuracy for the source samples. The DCD is a multi-class classifier that takes pairs of samples as input and classifies them into the above four groups. Confusing the DCD will encourage domain confusion, as well as the semantic alignment of classes. is an extensive 1 1 Our third contribution 1 validation of FADA against the state-of-the-art. Although our method is general, and can be used for all domain adaptation applications, we focus on visual recognition. 2 Related work Naively training a classifier on one dataset for testing on another is known to produce sub-optimal results, because an effect known as dataset bias [42, 57, 56], or covariate shift [51], occurs due to a difference in the distributions of the images between the datasets. Prior work in domain adaptation has minimized this shift largely in three ways. Some try to find a function which can map from the source domain to the target domain [47, 28, 19, 16, 11, 55, 52]. Others find a shared latent space that both domains can be mapped to before classification [35, 2, 39, 13, 14, 41, 37, 38]. Finally, some use regularization to improve the fit on the target domain [4, 1, 62, 10, 3, 8]. UDA can leverage the first two approaches while SDA uses the second, third, or a combination of the two approaches. In addition to these methods, [6, 36, 50] have addressed UDA when an auxiliary data view [31, 37], is available during training, but that is beyond the scope of this work. For this approach we are focused on finding a shared subspace for both the source and target distributions. Siamese networks [7] work well for subspace learning and have worked very well with deep convolutional neural networks [9, 53, 30, 61]. Siamese networks have also been useful in 2 1 G2 Source g ? h G2 h Loss (1) DCD Loss (1) Loss (3) Loss (4) Loss (1) (a) (c) (b) Figure 2: Few-shot adversarial domain adaptation. For simplicity we show our networks in the case of weight sharing (gs = gt = g). (a) In the first step, we initialized g and h using the source samples Ds . (b) We freeze g and train a DCD. The picture shows a pair from the second group G2 when the samples come from two different distributions but the same class label. (c) We freeze the DCD and update g and h. domain adaptation recently. In [58], which is a deep SDA approach, unlabeled and sparsely labeled target domain data are used to optimize for domain invariance to facilitate domain transfer while using a soft label distribution matching loss. In [54], which is a deep UDA approach, unlabeled target data is used to learn a nonlinear transformation that aligns correlations of layer activations in deep neural networks. Some approaches went beyond Siamese weight-sharing and used couple networks for DA. [27] uses two CNN streams, for source and target, fused at the classifier level. [45], which is a deep UDA approach and can be seen as an SDA after fine-tuning, also uses a two-streams architecture, for source and target, with related but not shared weights. [38], which is an SDA approach, creates positive and negative pairs using source and target data and then finds a shared feature space between source and target by bringing together the positive pairs and pushing apart the negative pairs. Recently, adversarial learning [18] has shown promising results in domain adaptation and can be seen as examples of the second category. [33] introduced a coupled generative adversarial network (CoGAN) for learning a joint distribution of multi-domain images for different applications including UDA. [59] has used the adversarial loss for discriminative UDA. [49] introduces an approach that leverages unlabeled data to bring the source and target distributions closer by inducing a symbiotic relationship between the learned embedding and a generative adversarial framework. Here we use adversarial learning to train inference networks such that samples from different distributions are not distinguishable. We consider the task where very few labeled target data are available in training. With this assumption, it is not possible to use the standard adversarial loss 1 used in [33, 59, 49], because the training target data would be insufficient. We address that problem 1 by modifying the usual pairing technique used in many applications such as learning similarity metrics [7, 23, 22]. Our pairing technique encodes domain labels as well as class labels of the training data (source and target samples), producing four groups of pairs. We then introduce a multi-class discriminator with four outputs and design an adversarial learning strategy to find a shared feature 1 other adversarial UDA space. Our method also encourages the semantic alignment of classes, while approaches do not. 1 1 3 Few-shot adversarial domain adaptation In this section we describe the model we propose to address supervised domain adaptation (SDA). We s are given a training dataset made of pairs Ds = {(xsi , yis )}N i=1 . The feature xi ? X is a realization s s from a random variable X , and the label yi ? Y is a realization from a random variable Y s . In t addition, we are also given the training data Dt = {(xti , yit )}M i=1 , where xi ? X is a realization from t t a random variable X , and the labels yi ? Y. We assume that there is a covariate shift [51] between X s and X t , i.e., there is a difference between the probability distributions p(X s ) and p(X t ). We say that X s represents the source domain and that X t represents the target domain. Under this settings the goal is to learn a prediction function f : X ? Y that during testing is going to perform well on data from the target domain. The problem formulated thus far is typically referred to as supervised domain adaptation. In this work we are especially concerned with the version of this problem where only very few target labeled 3 1 Algorithm 1 FADA algorithm 1: 2: 3: 4: 5: 6: 7: 8: Train g and h on Ds using (1). Uniformly sample G1 ,G3 from Ds xDs . Uniformly sample G2 ,G4 from Ds xDt . Train DCD w.r.t. gt = gs = g using (3). while not convergent do Update g and h by minimizing (5). Update DCD by minimizing (3). end while samples per class are available. We aim at handling cases where there is only one target labeled sample, and there can even be some classes with no target samples at all. In absence of covariate shift a visual classifier f is trained by minimizing a classification loss LC (f ) = E[`(f (X s ), Y )] , (1) where E[?] denotes statistical expectation and ` could be any appropriate loss function. When the distributions of X s and X t are different, a deep model fs trained with Ds will have reduced performance on the target domain. Increasing it would be trivial by simply training a new model ft with data Dt . However, Dt is small and deep models require large amounts of labeled data. In general, f could be modeled by the composition of two functions, i.e., f = h ? g. Here g : X ? Z would be an inference from the input space X to a feature or inference space Z, and h : Z ? Y would be a function for predicting from the feature space. With this notation we would have fs = hs ? gs and ft = ht ? gt , and the SDA problem would be about finding the best approximation for gt and ht , given the constraints on the available data. If gs and gt are able to embed source and target samples, respectively, to a domain invariant space, it is safe to assume from the feature to the label space that ht = hs = h. Therefore, domain adaptation paradigms are looking for such inference functions so that they can use the prediction function hs for target samples. Traditional unsupervised DA (UDA) paradigms try to align the distributions of the features in the feature space, mapped from the source and the target domains using a metric between distributions, Maximum Mean Discrepancy [20] being a popular one and other metrics like Kullback Leibler divergence [29] and Jensen?Shannon [18] divergence becoming popular when using adversarial learning. Once they are aligned, a classifier function would no longer be able to tell whether a sample is coming from the source or the target domain. Recent UDA paradigms try to find inference functions to satisfy this important goal using adversarial learning. Adversarial training looks for a domain discriminator D that is able to distinguish between samples of source and target distributions. In this case D is a binary classifier trained with the standard cross-entropy loss Ladv?D (Xs , Xt , gs , gt ) = ?E[log(D(gs (X s )))] ? E[log(1 ? D(gt (X t )))] . (2) Once the discriminator is learned, adversarial learning tries to update the target inference function gt in order to confuse the discriminator. In other words, the adversarial training is looking for an inference function gt that is able to map a target sample to a feature space such that the discriminator D will no longer distinguish it from a source sample. From the above discussion it is clear that in order to perform well, UDA needs to align the distributions effectively in order to be successful. This can happen only if distributions are represented by a sufficiently large dataset. Therefore, UDA approaches are in a position of weakness when we assume Dt to be small. Moreover, UDA approaches have also another intrinsic limitation; even with perfect confusion alignment, there is no guarantee that samples from different domains but with the same class label will map nearby in the feature space. This lack of semantic alignment is a major source of performance reduction. 3.1 Handling Scarce Target Data We are interested in the case where very few labeled target samples (as low as 1 sample per class) are available. We are facing two challenges in this setting. First, since the size of Dt is small, we need to find a way to augment it. Second, we need to somehow use the label information of Dt . Therefore, we create pairs of samples. In this way, we are able to alleviate the lack of training target samples by 4 pairing them with each training source sample. In [38], we have shown that creating positive and negative pairs using source and target data is very effective for SDA. Since the method proposed in [38] does not encode the domain information of the samples, it cannot be used in adversarial learning. Here we extend [38] by creating 4 groups of pairs (Gi , i = 1, 2, 3, 4) as follows: we break down the positive pairs into two groups (Groups 1 and 2), where pairs of the first group consist of samples from the source distribution with the same class labels, while pairs of the second group also have the same class label but come from different distributions (one from the source and one from the target distribution). This is important because we can encode both label and domain information of training samples. Similarly, we break down the negative pairs into two groups (Groups 3 and 4), where pairs of the third group consist of samples from the source distribution with different class labels, while pairs of the forth group come from different class labels and different distributions (one from the source and one from the target distributions). See Figure 1. In order to give each group the same amount of members we use all possible pairs from G2 , as it is the smallest, and then uniformly sample from the pairs in G1 , G3 , and G4 to match the size of G2 . Any reasonable amount of portions between the numbers of the pairs can also be used. In classical adversarial learning we would at this point learn a domain discriminator, but since we have semantic information to consider as well, we are interested in learning a multi-class discriminator (we call it domain-class discriminator (DCD)) in order to introduce semantic alignment of the source and target domains. By expanding the binary classifier to its multiclass equivalent, we can train a classifier that will evaluate which of the 4 groups a given sample pair belongs to. We model the DCD with 2 fully connected layers with a softmax activation in the last layer which we can train with the standard categorical cross-entropy loss 4 X LF ADA?D = ?E[ yGi log(D(?(Gi )))] , (3) i=1 where yGi is the label of Gi and D is the DCD function. ? is a symbolic function that takes a pair as input and outputs the concatenation of the results of the appropriate inference functions. The output of ? is passed to the DCD (Figure 2). In the second step, we are interested in updating gt in order to confuse the DCD in such a way that the DCD can no longer distinguish between groups 1 and 2, and also between groups 3 and 4 using the loss LF ADA?g = ?E[yG1 log(D(?(G2 ))) ? yG3 log(D(?(G4 )))] . (4) (4) is inspired by the non-saturating game [17] and will force the inference function gt to embed target samples in a space that DCD will no longer be able to distinguish between them. Connection with multi-class discriminators: Consider an image generation task where training samples come from k classes. Learning the image generator can be done by any standard kclass classifier and adding generated samples as a new class (generated class) and correspondingly increasing the dimension of the classifier output from k to k + 1. During the adversarial learning, only the generated class is confused. This has proven effective for image generation [48] and other tasks. However, this is different than the proposed DCD, where group 1 is confused with 2, and group 3 is confused with 4. Inspired by [48], we are able to create a k + 4 classifier to also guarantee a high classification accuracy. Therefore, we suggest that (4) needs to be minimized together with the main classifier loss LF ADA?g = ??E[yG1 log(D(g(G2 )))?yG3 log(D(g(G4 )))]+E[`(f (X s ), Y )]+E[`(f (X t ), Y )] , (5) where ? strikes the balance between classification and confusion. Misclassifying pairs from group 2 as group 1 and likewise for groups 4 and 3, means that the DCD is no longer able to distinguish positive or negative pairs of different distributions from positive or negative pairs of the source distribution, while the classifier is still able to discriminate positive pairs from negative pairs. This simultaneously satisfies the two main goals of SDA, domain confusion and class separability in the 5 Table 1: MNIST-USPS-SVHN datasets. Classification accuracy for domain adaptation over the MNIST, USPS, and SVHN datasets. M, U, and S stand for MNIST, USPS, and SVHN domain. LB is our base model without adaptation. FT and FADA stand for fine-tuning and our method, respectively. LB Traditional UDA [60] [45] [15] Adversarial UDA [33] [59] [49] M?U 65.4 47.8 60.7 91.8 91.2 89.4 92.5 U ?M 58.6 63.1 67.3 73.7 89.1 90.1 90.8 S?M 60.1 - - 82.0 76.0 - 84.7 M?S 20.3 - - 40.1 - - 36.4 S?U 66.0 - - - - - - U ?S 15.3 - - - - - - SDA FT [38] FADA FT [38] FADA FT FADA FT FADA FT FADA FT FADA 1 82.3 85.0 89.1 72.6 78.4 81.1 65.5 72.8 29.7 37.7 69.4 78.3 19.9 27.5 2 84.9 89.0 91.3 78.2 82.2 84.2 68.6 81.8 31.2 40.5 71.8 83.2 22.2 29.8 3 85.7 90.1 91.9 81.9 85.8 87.5 70.7 82.6 36.1 42.9 74.3 85.2 22.8 34.5 4 86.5 91.4 93.3 83.1 86.1 89.9 73.3 85.1 36.7 46.3 76.2 85.7 24.6 36.0 5 87.2 92.4 93.4 83.4 88.8 91.1 74.5 86.1 38.1 46.1 78.1 86.2 25.4 37.9 6 88.4 93.0 94.0 83.6 89.6 91.2 74.6 86.8 38.3 46.8 77.9 87.1 25.4 41.3 7 88.6 92.9 94.4 84.0 89.4 91.5 75.4 87.2 39.1 47.0 78.9 87.5 25.6 42.9 feature space. UDA only looks for domain confusion and does not address class separability, because of the lack of labeled target samples. Connection with conditional GANs: Concatenation of outputs of different inferences has been done before in conditional GANs. For example, [43, 44, 64] concatenate the input text to the penultimate layers of the discriminators. [25] concatenates positive and negative pairs before passing them to the discriminator. However, all of them use the vanilla binary discriminator. Relationship between gs and gt : There is no restriction for gs and gt and they can be constrained or unconstrained. An obvious choice of constraint is equality (weight-sharing) which makes the inference functions symmetric. This can be seen as a regularizer and will reduce overfitting [38]. Another approach would be learning an asymmetric inference function [45]. Since we have access to very few target samples, we use weight-sharing (gs = gt = g). Choice of gs , gt , and h: Since we are interested in visual recognition, the inference functions gs and gt are modeled by a convolutional neural network (CNN) with some initial convolutional layers, followed by some fully connected layers which are described specifically in the experiments section. In addition, the prediction function h is modeled by fully connected layers with a softmax activation function for the last layer. Training Process: Here we discuss the training process for the weight-sharing regularizer (gs = gt = g). Once the inference functions g and the prediction function h are chosen, FADA takes the following steps: First, g and h are initialized using the source dataset Ds . Then, the mentioned four groups of pairs should be created using Ds and Dt . The next step is training DCD using the four groups of pairs. This should be done by freezing g. In the next step, the inference function g and prediction function h should be updated in order to confuse DCD and maintain high classification accuracy. This should be done by freezing DCD. See Algorithm 1 and Figure 2. The training process for the non weight-sharing case can be derived similarly. 4 Experiments We present results using the Office dataset [47], the MNIST dataset [32], the USPS dataset [24], and the SVHN dataset [40]. 4.1 MNIST-USPS-SVHN Datasets The MNIST (M), USPS (U), and SVHN (S) datasets have recently been used for domain adaptation [12, 45, 59]. They contain images of digits from 0 to 9 in various different environments including in the wild in the case of SVHN [40]. We considered six cross-domain tasks. The first two tasks include M ? U, U ? M, and followed the experimental setting in [12, 45, 33, 59, 49], which involves randomly selecting 2000 images from MNIST and 1800 images from USPS. For the rest of 6 Table 2: Office dataset. Classification accuracy for domain adaptation over the 31 categories of the Office dataset. A, W, and D stand for Amazon, Webcam, and DSLR domain. LB is our base model without adaptation. Since we do not train any convolutional layers and only use pre-computed DeCaF-fc7 features as input, we expect a more challenging task compared to [58, 27]. A?W A?D W?A W?D D?A D?W Average LB 61.2 ? 0.9 62.3 ? 0.8 51.6 ? 0.9 95.6 ? 0.7 58.5 ? 0.8 80.1 ? 0.6 68.2 Unsupervised Methods [60] [34] [15] 61.8 ? 0.4 68.5 ? 0.4 68.7 ? 0.3 64.4 ? 0.3 67.0 ? 0.4 67.1 ? 0.3 52.2 ? 0.4 53.1 ? 0.3 54.09 ? 0.5 98.5 ? 0.4 99.0 ? 0.2 99.0 ? 0.2 52.1 ? 0.8 54.0 ? 0.4 56.0 ? 0.5 95.0 ? 0.5 96.0 ? 0.3 96.4 ? 0.3 70.6 72.9 73.6 [58] 82.7 ? 0.8 86.1 ? 1.2 65.0 ? 0.5 97.6 ? 0.2 66.2 ? 0.3 95.7 ? 0.5 82.2 Supervised Methods [27] [38] 84.5 ? 1.7 88.2 ? 1.0 86.3 ? 0.8 89.0 ? 1.2 65.7 ? 1.7 72.1 ? 1.0 97.5 ? 0.7 97.6 ? 0.4 66.5 ? 1.0 71.8 ? 0.5 95.5 ? 0.6 96.4 ? 0.8 82.6 85.8 FADA 88.1 ? 1.2 88.2 ? 1.0 71.1 ? 0.9 97.5 ? 0.6 68.1 ? 06 96.4 ? 0.8 84.9 the cross-domain tasks, M ? S, S ? M, U ? S, and S ? U, we used all training samples of the source domain for training and all testing samples of the target domain for testing. Since [12, 45, 33, 59, 49] introduced unsupervised methods, they used all samples of a target domain as unlabeled data in training. Here instead, we randomly selected n labeled samples per class from target domain data and used them in training. We evaluated our approach for n ranging from 1 to 4 and repeated each experiment 10 times (we only show the mean of the accuracies for this experiment because standard deviation is very small). Since the images of the USPS dataset have 16 ? 16 pixels, we resized the images of the MNIST and SVHN datasets to 16 ? 16 pixels. We assume gs and gt share weights (g = gs = gt ) for this experiment. Similar to [32], we used 2 convolutional layers with 6 and 16 filters of 5 ? 5 kernels followed by max-pooling layers and 2 fully connected layers with size 120 and 84 as the inference function g, and one fully connected layer with softmax activation as the prediction function h. Also, we used 2 fully connected layers with size 64 and 4 as DCD (4 groups classifier). Training for each stage was done using the Adam Optimizer [26]. We compare our method with 1 SDA method, under the same condition, and 6 recent UDA methods. UDA methods use all target samples in their training stage, while we only use very few labeled target samples per category in training. Table 1 shows the classification accuracies, where FADA - n stands for our method when we use n labeled target samples per category in training. FADA works well even when only one target sample per category (n = 1) is available in training. Also, we can see that by increasing n, the accuracy goes up. This is interesting because we can get comparable accuracies with the state-of-the-art using only 10 labeled target samples (one sample per class) instead of using more than thousands unlabeled target samples. We also report the lower bound (LB) of our model which corresponds to training the base model using only source samples. Moreover, we report the accuracies obtained by fine-tuning (FT) the base model on available target data. Although Table 1 shows that FT increases the accuracies over LB, it has reduce performance compared to SDA methods. Figure 3 shows how much improvement can be obtained with respect to the base model. The base model is the lower bound LB. This is simply obtained by training g and h with only the classification loss and source training data; so, no adaptation is performed. Weight-Sharing. As we discussed earlier, weight-sharing can be seen as a regularizer that prevents the target network gt from overfitting. This is important because gt can be easily overfitted since target data is scarce. We repeated the experiment for the U ? M with n = 5 without sharing weights. This provides an average accuracy of 84.1 over 10 repetitions, which is less than the weight-sharing case. 4.2 Office Dataset The office dataset is a standard benchmark dataset for visual domain adaptation. It contains 31 object classes for three domains: Amazon, Webcam, and DSLR, indicated as A, W, and D, for a total of 4,652 images. The first domain A, consists of images downloaded from online merchants, the second W, consists of low resolution images acquired by webcams, the third D, consists of high resolution images collected with digital SLRs. We consider four domain shifts using the three domains (A ? W, A ? D, W ? A, and D ? A). Since there is not a considerable domain shift between W and D, we exclude W ? D and D ? W. 7 50 70 LB FT 60 FADA n=1 n=2 n=3 n=4 n=5 n=6 n=7 Accuracy % 50 Accuracy % 80 (b) M ?S 40 30 LB FT 20 FADA n=1 n=2 n=3 n=4 n=5 n=6 n=7 40 30 LB 20 FT FADA 10n=1 n=2 n=3 n=4 n=5 n=6 n=7 100 100 (d) U ?M (e) S ?U 90 80 LB FT FADA 60 n=1 n=2 n=3 n=4 n=5 n=6 n=7 70 100 90 Accuracy % Accuracy % 90 (c) U ?S Accuracy % (a) M ?U (f) S ?M 90 80 70 LB FT 60 FADA 50 n=1 n=2 n=3 n=4 n=5 n=6 n=7 Accuracy % 100 80 70 LB FT FADA 50 n=1 n=2 n=3 n=4 n=5 n=6 n=7 60 Figure 3: MNIST-USPS-SVHN summary. The lower bar of each column represents the LB as reported in Table 1 for the corresponding domain pair. The middle bar is the improvement of finetuning FT the base model using the available target data reported in Table 1. The top bar is the improvement of FADA over FT, also reported in Table 1. We followed the setting described in [58]. All classes of the office dataset and 5 train-test splits are considered. For the source domain, 20 examples per category for the Amazon domain, and 8 examples per category for the DSLR and Webcam domains are randomly selected for training for each split. Also, 3 labeled examples are randomly selected for each category in the target domain for training for each split. The rest of the target samples are used for testing. Note that we used the same splits generated by [58]. In addition to the SDA algorithms, we report the results of some recent UDA algorithms. They follow a different experimental protocol compared to the SDA algorithms, and use all samples of the target domain in training as unlabeled data together with all samples of the source domain. So, we cannot make an exact comparison between results. However, since UDA algorithms use all samples of the target domain in training and we use only very few of them (3 per class), we think it is still worth looking at how they differ. Here we are interested in the case where gs and gt share weights (gs = gt = g). For the inference function g, we used the convolutional layers of the VGG-16 architecture [53] followed by 2 fully connected layers with output size of 1024 and 128, respectively. For the prediction function h, we used a fully connected layer with softmax activation. Similar to [58], we used the weights pre-trained on the ImageNet dataset [46] for the convolutional layers, and initialized the fully connected layers using all the source domain data. We model the DCD with 2 fully connected layers with a softmax activation in the last layer. Table 2 reports the classification accuracy over 31 classes for the Office dataset and shows that FADA has performance comparable to the state-of-the-art. 5 Conclusions We have introduced a deep model combining a classification and an adversarial loss to address SDA in few-shot learning regime. We have shown that adversarial learning can be augmented to address SDA. 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Spectral Mixture Kernels for Multi-Output Gaussian Processes Gabriel Parra Department of Mathematical Engineering Universidad de Chile gparra@dim.uchile.cl Felipe Tobar Center for Mathematical Modeling Universidad de Chile ftobar@dim.uchile.cl Abstract Early approaches to multiple-output Gaussian processes (MOGPs) relied on linear combinations of independent, latent, single-output Gaussian processes (GPs). This resulted in cross-covariance functions with limited parametric interpretation, thus conflicting with the ability of single-output GPs to understand lengthscales, frequencies and magnitudes to name a few. On the contrary, current approaches to MOGP are able to better interpret the relationship between different channels by directly modelling the cross-covariances as a spectral mixture kernel with a phase shift. We extend this rationale and propose a parametric family of complex-valued cross-spectral densities and then build on Cram?r?s Theorem (the multivariate version of Bochner?s Theorem) to provide a principled approach to design multivariate covariance functions. The so-constructed kernels are able to model delays among channels in addition to phase differences and are thus more expressive than previous methods, while also providing full parametric interpretation of the relationship across channels. The proposed method is first validated on synthetic data and then compared to existing MOGP methods on two real-world examples. 1 Introduction The extension of Gaussian processes (GPs [1]) to multiple outputs is referred to as multi-output Gaussian processes (MOGPs). MOGPs model temporal or spatial relationships among infinitelymany random variables, as scalar GPs, but also account for the statistical dependence across different sources of data (or channels). This is crucial in a number of real-world applications such as fault detection, data imputation and denoising. For any two input points x, x0 , the covariance function of an m-channel MOGP k(x, x0 ) is a symmetric positive-definite m ? m matrix of scalar covariance functions. The design of this matrix-valued kernel is challenging since we have to deal with the trade off between (i) choosing a broad class of m(m ? 1)/2 cross-covariances and m auto-covariances, while at the same time (ii) ensuring positive definiteness of the symmetric matrix containing these m(m+1)/2 covariance functions for any pair of inputs x, x0 . In particular, unlike the widely available families of auto-covariance functions (e.g., [2]), cross-covariances are not bound to be positive definite and therefore can be designed freely; the construction of these functions with interpretable functional form is the main focus of this article. A classical approach to define cross-covariances for a MOGP is to linearly combine independent latents GPs, this is the case of the Linear Model of Coregionalization (LMC [3]) and the Convolution Model (CONV, [4]). In these cases, the resulting kernel is a function of both the covariance functions of the latent GPs and the parameters of the linear operator considered; this results in symmetric and centred cross-covariances. While these approaches are simple, they lack interpretability of the dependencies learnt and force the auto-covariances to have similar behaviour across different channels. The LMC method has also inspired the Cross-Spectral Mixture (CSM) kernel [5], which uses the 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Spectral Mixture (SM) kernel in [6] within LMC and model phase differences across channels by manually introducing a shift between the cosine and exponential factors of the SM kernel. Despite exhibiting improved performance wrt previous approaches, the addition of the shift parameter in CSM poses the following question: Can the spectral design of multiouput covariance functions be even more flexible? We take a different approach to extend the spectral mixture concept to multiple outputs: Recall that for stationary scalar-valued GPs, [6] designs the power spectral density (PSD) of the process by a mixture of square exponential functions to then, supported by Bochner?s theorem [7], present the Spectral Mixture kernel via the inverse Fourier transform of the so-constructed PSD. Along the same lines, our main contribution is to propose an expressive family of complex-valued square-exponential cross-spectral densities, and then build on Cram?r?s theorem [8, 9], the multivariate extension of Bochner?s, to construct the Multi-Output Spectral Mixture kernel (MOSM). The proposed multivariate covariance function accounts for all the properties of the Cross-Spectral Mixture kernel in [5] plus a delay component across channels and variable parameters for auto-covariances of different channels. Additionally, the proposed MOSM provides clear interpretation of all the parameters in spectral terms. Our experimental contribution includes an illustrative example using a trivariate synthetic signal and validation against all the aforementioned literature using two real-world datasets. 2 Background Definition 1. A Gaussian process (GP) over the input set X is a real-valued stochastic process N (f (x))x?X such that for any finite subset of inputs {xi }N i=1 ? X , the random variables {f (xi )}i=1 n are jointly Gaussian. Without loss of generality we will choose X = R . A GP [1] defines a distribution over functions f (x) that is uniquely determined by its mean function m(x) := E(f (x)), typically assumed m(x) = 0, and its covariance function (also known as kernel) k(x, x0 ) := cov(f (x), f (x0 )), x, x0 ? X . We now equip the reader with the necessary background to follow our proposal: we first review a spectral-based approach to the design of scalar-valued covariance kernels and then present the definition of a multi-output GP. 2.1 The Spectral Mixture kernel To bypass the explicit construction of positive-definite functions within the design of stationary covariance kernels, it is possible to design the power spectral density (PSD) instead [6] and then transform it into a covariance function using the inverse Fourier transform. This is motivated by the fact that the strict positivity requirement of the PSD is much easier to achieve than the positive definiteness requirement of the covariance kernel. The theoretical support of this construction is given by the following theorem: Theorem 1. (Bochner?s theorem) An integrable1 function k : Rn ? C is the covariance function of a weakly-stationary mean-square-continuous stochastic process f : Rn ? C if and only if it admits the following representation Z > k(? ) = e?? ? S(?)d? (1) Rn where S(?) is a non-negative bounded function on Rn and ? denotes the imaginary unit. For a proof see [9]. The above theorem gives an explicit relationship between the spectral density S and the covariance function k of the stochastic process f . In this sense, [6] proposed to model the spectral density S as a weighted mixture of Q square-exponential functions, with weights wq , centres ?q and diagonal covariance matrices ?q , that is, S(?) = Q X q=1 wq 1 (2?)n/2 |? q |1/2
exp ? 12 (? ? ?q )> ??1 q (? ? ?q ) . (2) Relying on Theorem 1, the kernel associated to the spectral density S(?) in eq. (2) is given the spectral mixture kernel defined as follows. 1 A function g(x) is said to be integrable if R Rn |g(x)|dx < +? 2 Definition 2. A Spectral Mixture (SM) kernel is a positive-definite stationary kernel given by k(? ) = (q) Q X   1 wq exp ? ? > ?q ? cos(?> q ?) 2 q=1 (3) (q) where ?q ? Rn , ?q = diag(?1 , . . . , ?n ) and wq , ?q ? R+ . Due to the universal function approximation property of the mixtures of Gaussians (considered here in the frequency domain) and the relationship given by Theorem 1, the SM kernel is able to approximate continuous stationary kernels to an arbitrary precision given enough spectral components as is [10, 11]. This concept points in the direction of sidestepping the kernel selection problem in GPs and it will be extended to cater for multivariate GPs in Section 3. 2.2 Multi-Output Gaussian Processes A multivariate extension of GPs can be constructed by considering an ensemble of scalar-valued stochastic processes where any finite collection of values across all such processes are jointly Gaussian. We formalise this definition as follows. Definition 3. An m-channel multi-output Gaussian process f (x) := (f1 (x), . . . , fm (x)), x ? X , is an m-tuple of stochastic processes fp : X ? R ?p = 1, . . . , m, such that for any (finite) subset of N inputs {xi }N i=1 ? X , the random variables {fc(i) (xi )}i=1 are jointly Gaussian for any choice of indices c(i) ? {1, . . . , m}. Recall that the construction of scalar-valued GPs requires choosing a scalar-valued mean function and a scalar-valued covariance function. Conversely, an m-channel MOGP is defined by an Rm -valued mean function, whose ith element denotes the mean function of the ith channel, and an Rm ? Rm valued covariance function, whose (i, j)th element denotes the covariance between the ith and j th channels. The symmetry and positive-definiteness conditions of the MOGP kernel are defined as follows. Definition 4. A two-input matrix-valued function K(x, x0 ) : X ? X ? Rm?m defined element-wise by [K(x, x0 )]ij = kij (x, x0 ) is a multivariate kernel (covariance function) if it is: (i) Symmetric, i.e., K(x, x0 ) = K(x0 , x)> , ?x, x0 ? X , and (ii) Positive definite, i.e., ?N ? N, c ? RN ?m , x ? X N such that, [c]pi = cpi , [x]p = xp , we have m N X X i,j=1 p,q=1 cpi cqj kij (xp , xq ) ? 0. (4) Furthermore, we say that a multivariate kernel K(x, x0 ) is stationary if K(x, x0 ) = K(x ? x0 ) or equivalently kij (x, x0 ) = kij (x ? x0 ) ?i, j ? {1, . . . , m}, in this case, we denote ? = x ? x0 . The design of the MOGP covariance kernel involves jointly choosing functions that model the covariance of each channel (diagonal elements in K) and functions that model the cross-covariance between different channels at different input locations (off-diagonal elements in K). Choosing these m(m + 1)/2 covariance functions is challenging when we want to be as expressive as possible and include, for instance, delays, phase shifts, negative correlations or to enforce specific spectral content while at the same time maintaining positive definiteness of K. The reader is referred to [12, 13] for a comprehensive review of MOGP models. 3 Designing Multi-Output Gaussian Processes in the Fourier Domain We extend the spectral-mixture approach [6] to multi-output Gaussian processes relying on the multivariate version of Theorem 1 first proved by Cram?r and thus referred to as Cram?r?s Theorem [8, 9] given by Theorem 2. (Cram?r?s Theorem) A family {kij (? )}m i,j=1 of integrable functions are the covariance functions of a weakly-stationary multivariate stochastic process if and only if they (i) admit the 3 representation Z kij (? ) = e?? > ? Sij (?)d? Rn ?i, j ? {1, . . . , m} (5) where each Sij is an integrable complex-valued function Sij : Rn ? C known as the spectral density associated to the covariance function kij (? ), and (ii) fulfil the positive definiteness condition m X zi zj Sij (?) ? 0 ?{z1 , . . . , zm } ? C, ? ? Rn (6) i,j=1 where z denotes the complex conjugate of z ? C. Note that eq. (5) states that each covariance function kij is the inverse Fourier transform of a spectral density Sij , therefore, we will say that these functions are Fourier pairs. Accordingly, we refer to the set of arguments of the covariance function ? ? Rn as time or space Domain depending of the application considered, and to the set of arguments of the spectral densities ? ? Rn as Fourier or spectral domain. Furthermore, a direct consequence of the above theorem is that for any element m?m ? in the Fourier domain, the matrix defined by S(?) = [Sij (?)]m is Hermitian, i.e., i,j=1 ? R Sij (?) = S ji (?) ?i, j, ?. Theorem 2 gives the guidelines to construct covariance functions for MOGP by designing their corresponding spectral densities instead, i.e., the design is performed in the Fourier rather than the space domain. The simplicity of design in the Fourier domain stems from the positive-definiteness condition of the spectral densities in eq. (6), which is much easier to achieve than that of the covariance functions in eq. (4). This can be understood through an analogy with the univariate model: in the single-output case the positive-definiteness condition of the kernel only requires positivity of the spectral density, whereas in the multioutput case the positive-definiteness condition of the multivariate kernel only requires that the matrix S(?), ?? ? Rn , is positive definite but there are no constraints on each function Sij : ? 7? Sij (?). 3.1 The Multi-Output Spectral Mixture kernel We now propose a family of Hermitian positive-definite complex-valued functions {Sij (?)}m i,j=1 , thus fulfilling the requirements of Theorem 2, eq. (6), to use them as cross-spectral densities within MOGP. This family of functions is designed with the aim of providing physical parametric interpretation and closed-form covariance functions after applying the inverse Fourier transform. Recall that complex-valued positive-definite matrices can be decomposed in the form S(?) = H (?)R:j (?); RH (?)R(?), meaning that the (i, j)th entry of S(?) can be expressed as Sij (?) = R:i Q?m th H where R(?) ? C , R:i (?) is the i column of R(?), and (?) denotes the Hermitian (transpose and conjugate) operator. Note that this factor decomposition fulfils eq. (6) for any choice of R(?) ? CQ?m :  m 2 m  X X   2 H zi R:i (?) = ||R(?)z|| ? 0 ?z = [z1 , . . . , zm ]> ? Cm , ? ? Rn zi R:i (?)R:j (?)zj =    i,j=1 i=1 (7) We refer to Q as the rank of the decomposition, since by choosing Q < m the rank of S(?) = RH (?)R(?) can be at most Q. For ease of notation we choose2 Q = 1, where the columns of R(?) are complex-valued functions {Ri }m i=1 , and S(?) is modeled as a rank-one matrix according to Sij (?) = Ri (?)Rj (?). Since Fourier transforms and multiplications of square exponential (SE) functions are also SE, we model Ri (?) as a complex-valued SE function so as to ensure closed-form expression of its corresponding covariance kernel, that is,  
1 Ri (?) = wi exp ? (? ? ?i )> ??1 (? ? ? ) exp ??(?i> ? + ?i ) , i = 1, . . . , m (8) i i 4 2 2 where wi , ?i ? R, ?i , ?i ? Rn and ?i = diag([?i1 , . . . , ?in ]) ? Rn?n . With this choice of the m m functions {Ri }i=1 , the spectral densities {Sij }i,j=1 are given by  
1 > Sij (?) = wij exp ? (? ? ?ij )> ??1 (? ? ? ) + ? ? ? + ? , i, j = 1, . . . , m (9) ij ij ij ij 2 2 The extension to arbitrary Q will be presented at the end of this section. 4 meaning that the cross-spectral density between channels i and j is modeled as a complex-valued SE function with the following parameters: ? covariance: ?ij = 2?i (?i + ?j )?1 ?j ? mean: ?ij = (?i + ?j )?1 (?i ?j + ?j ?i ) ? magnitude: wij = wi wj exp ? 14 (?i ? ?j )> (?i + ?j )?1 (?i ? ?j )
? delay: ?ij = ?i ? ?j ? phase: ?ij = ?i ? ?j where the so-constructed magnitudes wij ensure positive definiteness and, in particular, the autospectral densities Sii are real-valued SE functions (since ?ii = ?ii = 0) as in the standard (scalarvalued) spectral mixture approach [6]. The power spectral density in eq. (9) corresponds to a complex-valued kernel and therefore to a complex-valued GP [14, 15] . In order to restrict this generative model only to real-valued GPs, the proposed power spectral density has to be symmetric with respect to ? [16], we then make Sij (?) symmetric simply by reassigning Sij (?) 7? 12 (Sij (?) + Sij (??)), this is equivalent to choosing Ri (?) to be a vector of two mirrored complex SE functions. The resulting (symmetric with respect to ?) cross-spectral density between the ith and j th channels Sij (?) and its corresponding real-valued kernel kij (? ) = F ?1 {Sij (?)}(? ) are the following Fourier pairs  

   ?1 ?1 > > wij (???ij )> ??1 (?+?ij )> ??1 ij (???ij )+? ?ij ?+?ij ij (?+?ij )+? ??ij ?+?ij +e 2 Sij (?) = e 2 2  
1 kij (? ) = ?ij exp ? (? + ?ij )> ?ij (? + ?ij ) cos (? + ?ij )> ?ij + ?ij (10) 2 n where the magnitude parameter ?ij = wij (2?) 2 |?ij |1/2 absorbs the constant resulting from the inverse Fourier transform. We can again confirm that the autocovariances (i = j) are real-valued and contain square-exponential and cosine factors as in the scalar SM approach since ?ii ? 0 and ?ii = ?ii = 0. Conversely, the proposed model for the cross-covariance between different channels (i 6= j) allows for (i) both negatively- and positively-correlated signals (?ij ? R), (ii) delayed channels through the delay parameter ?ij 6= 0 and (iii) out-of-phase channels where the covariance is not symmetric with respect to the delay for ?ij 6= 0. Fig. 1 shows cross-spectral densities and their corresponding kernel for a choice of different delay and phase parameters. Cross Covariances ?ij = 0 ?ij = 0 ?ij 6= 0 ?ij = 0 ?ij = 0 ?ij 6= 0 ?ij 6= 0 ?ij 6= 0 0 5 ?5 0 5 ?5 0 5 ?5 0 5 ?5 0 5 ?5 0 5 ?5 0 5 ?5 0 5 Cross-Spectral Densities ?5 Figure 1: Power spectral density and kernels generated by the proposed model in eq. (10) for different parameters. Bottom: Cross-spectral densities, real part in blue and imaginary part in green. Top: Cross-covariance functions in blue with reference SE envelope in dashed line. From left to right: zero delay and zero phase; zero delay and non-zero phase; non-zero delay and zero phase; and non-zero delay and non-zero phase. 5 The kernel in eq. (10) resulted from a low rank choice for the PSD matrix Sij , therefore, increasing the rank in the proposed model for Sij is equivalent to consider several kernel components. Arbitrarily choosing Q of these components yields the expression for the proposed multivariate kernel: Definition 5. The Multi-Output Spectral Mixture kernel (MOSM) has the form:   Q   X 1 (q) (q) (q) (q) (q) (q) (q) kij (? ) = ?ij exp ? (? + ?ij )> ?ij (? + ?ij ) cos (? + ?ij )> ?ij + ?ij (11) 2 q=1 (q) (q) n (q) where ?ij = wij (2?) 2 |?ij |1/2 and the superindex (?)(q) denotes the parameter of the q th component of the spectral mixture. This multivariate covariance function has spectral-mixture positive-definite kernels as autocovariances, while the cross-covariances are spectral mixture functions with different parameters for different output pairs, which can be (i) non-positive-definite, (ii) non-symmetric, and (iii) delayed with respect to one another. Therefore, the MOSM kernel is a multi-output generalisation of the spectral mixture approach [6] where the positive definiteness is guaranteed by the factor decomposition of Sij as shown in eq. (7). 3.2 Training the model and computing the predictive posterior Fitting the model to observed data follows the same rationale of standard GP, that is, maximising log-probability of the data. Recall that the observations in the multioutput case consist of (i) a location x ? X , (ii) a channel identifier i ? {1, . . . , m}, and (iii) an observed value y ? R; therefore, we denote N observations as the set of 3-tuples D = {(xc , ic , yc )}N c=1 . As all observations are jointly Gaussian, we concatenate the observations into the three vectors x = [x1 , . . . , xN ]> ? X N , i = [i1 , . . . , iN ]> ? {1, . . . , m}N , and y = [y1 , . . . , yN ]> ? RN , to express the negative loglikelihood (NLL) by N 1 1 ? log p(y|x, ?) = log 2? + log |Kxi | + y> K?1 (12) xi y 2 2 2 where all hyperparameters are denoted by ?, and Kxi is the covariance matrix of all observed samples, that is, the (r, s)th element [Kxi ]rs is the covariance between the process at (location: xr , channel: ir ) and the process at (location: xs , channel: is ). Recall that, under the proposed MOSM model, this covariance [Kxi ]rs is given by eq. (11), that is, kir is (xr ? xs ) + ?i2r ,noise ?ir is , where ?i2r ,noise is a diagonal term to cater for uncorrelated observation noise. The NLL is then minimised with respect to (q) (q) (q) (q) (q) 2 ? = {wi , ?i , ?i , ?i , ?i , ?i,noise }m,Q i=1,q=1 , that is, the original parameters chosen to construct R(?) in Section 3.1, plus the noise hyperparameters. Once the hyperparameters are optimised, computing the predictive posterior in the proposed MOSM follows the standard GP procedure with the joint covariances given by eq. (11). 3.3 Related work Generalising the scalar spectral mixture kernel to MOGPs can be achieved from the LMC framework as pointed out in [5] (denoted SM-LMC). As this formulation only considers real-valued cross spectral densities, the authors propose a multivariate covariance function by including a complex component to the cross spectral densities to cater for phase differences across channels, which they call the Cross Spectral Mixture kernel (denoted CSM). This multivariate covariance function can be seen as the proposed MOSM model with ?i = ?j , ?i = ?j , ?i = ?j ?i, j ? {1, . . . , m} and ?i = ?> i ?i for ?i ? Rn . As a consequence, the SM-LMC is a particular case of the proposed MOSM model, where the parameters ?i , ?i , ?i are restricted to be same for all channels and therefore no phase shifts and no delays are allowed?unlike the MOSM example in Fig. 1. Additionally, Cram?r?s theorem has also been used in a similar fashion in [17] but only with real-valued t-Student cross-spectral densities yielding cross-covariances that are either positive-definite or negative-definite. 4 Experiments We show two sets of experiments. First, we validated the ability of the proposed MOSM model in the identification of known auto- and cross-covariances of synthetic data. Second, we compared 6 MOSM against the spectral-mixture linear model of coregionalization (SM-LMC, [3, 6, 5]), the Gaussian convolution model (CONV, [4]), and the cross-spectral mixture model (CSM, [5]) in the estimation of missing real-world data in two different distributed settings: climate signals and metal concentrations. All models were implemented in Tensorflow [18] using GPflow [19] in order to make use of automatic differentiation to compute the gradients of the NLL. The performance of all the models in the experiments was measured by the mean absolute error given by MAE : N 1 X |yi ? y?i | N i=1 (13) where yi denotes the true value and y?i the MOGP estimate. 4.1 Synthetic example: Learning derivatives and delayed signals All models were implemented to recover the auto- and cross-covariances of a three-output GP with the following components: (i) a reference signal sampled from a GP f (x) ? GP(0, KSM ) with spectral mixture covariance kernel KSM and zero mean, (ii) its derivative f 0 (x), and (iii) a delayed version f? (x) = f (x ? ?). The motivation for this illustrative example is that the covariances and cross covariances of the aforementioned processes are known explicitly (see [1, Sec. 9.4]) and we can therefore compare our estimates to the true model. The derivative was computed numerically (first order through finite differences) and the training samples were generated as follows: We chose N1 = 500 samples from the reference function in the interval [-20, 20], N2 = 400 samples from the derivative signal in the interval [-20, 0], and N3 = 400 samples from the delayed signal in the interval [-20, 0]. All samples were randomly uniformly chosen in the intervals mentioned and Gaussian noised was added to yield realistic observations. The experiment then consisted in the reconstruction the reference signal in the interval [-20, 20], and the imputation of the derivative and delayed signals over the interval [0, 20]. Fig. 2 shows the ground truth and MOSM estimates for all three synthetic signals and the covariances (normalised), and Table 1 reports the MAE for all models over ten realisations of the experiment. Notice that the proposed model successfully learnt all cross-covariances cov(f (?), f 0 (x)) and cov(f (x), f (x ? ?)), and autocovariances without prior information about the delayed or the derivative relationship between the two channels. Furthermore, MOSM was the only model that successfully extrapolated the derivate signal and the delayed signal simultaneously, this is due the fact that the cross-covariances needed for this setting are not linear combinations of univariate kernels, hence models based on latent processes fail in this synthetic example. Synthetic Example: MOSM Reference Signal 4 k11 (? ) 2 0.5 0 0.0 ?2 ?0.5 f (x) ?20 5.0 Derivative Signal Cov: Reference 1.0 ?15 ?10 ?5 0 5 10 15 20 0.0 2.5 5.0 Cov: Reference and Derivative 1.0 1 k22 (? ) k21 (? ) 0.5 0.0 0 0.0 ?2.5 ?0.5 ?5.0 ?20 ?15 ?10 ?5 0 5 10 15 ?5.0 20 ?2.5 0.0 2.5 5.0 ?1 Cov: Delayed ?5.0 ?2 f (x ? ?) ?20 ?15 ?10 ?5 0 5 10 15 2.5 5.0 k31 (? ) 0.5 0.0 0.0 ?0.5 ?0.5 ?5.0 20 Input 0.0 1.0 k33 (? ) 0.5 0 ?2.5 Cov: Reference and Delayed 1.0 2 Delayed Signal ?2.5 Cov: Derivative f 0 (x) 2.5 ?5.0 ?2.5 0.0 2.5 5.0 ?5.0 ?2.5 0.0 2.5 5.0 Figure 2: MOSM learning of the covariance functions of a synthetic reference signal, its derivative and a delayed version. Left: synthetic signals, middle: autocovariances, right: cross-covariances. The dashed line is the ground truth, the solid colour lines are the MOSM estimates, and the shaded area is the 95% confidence interval. The training points are shown in green. 7 Table 1: Reconstruction of a synthetic signal, its derivative and delayed version: Mean absolute error for all four models with one-standard-deviation error bars over ten realisations. 4.2 Model Reference Derivative Delayed CONV SM-LMC CSM MOSM 0.211 ? 0.085 0.166 ? 0.009 0.148 ? 0.010 0.127 ? 0.011 0.759 ? 0.075 0.747 ? 0.101 0.262 ? 0.032 0.223 ? 0.015 0.524 ? 0.097 0.398 ? 0.042 0.368 ? 0.089 0.146 ? 0.017 Climate data The first real-world dataset contained measurements3 from a sensor network of four climate stations in the south on England: Cambermet, Chimet, Sotonmet and Bramblemet. We considered the normalised air temperature signal from 12 March, 2017 to 16 March, 2017, in 5-minute intervals (5692 samples), from where we randomly chose N = 1000 samples for training. Following [4], we simulated a sensor failure by removing the second half of the measurements for one sensor and leaving the remaining three sensors operating correctly; we reproduced the same setup across all four sensors thus producing four experiments. All models considered had five latent signals/spectral components. For all four models considered, Fig. 3 shows the estimates of missing data for the Cambermet-failure case. Table 2 shows the mean absolute error for all models and failure cases over the missing data region. Observe how all models were able to capture the behaviour of the signal in the missing range, this is because the considered climate signals are very similar to one another. This shows that the MOSM can also collapse to models that share parameters across pairs of outputs when required. Temperature?[?C] 3 MOSM 95%?CI 2 3 3 CONV 95%?CI 2 SM?LMC 95%?CI 2 3 1 1 1 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 0 1 2 3 4 0 1 2 3 4 CSM 95%?CI 2 1 0 1 2 3 4 3 0 1 2 3 4 Time?[Days]? Figure 3: Imputation of the Cambermet sensor measurements using the remaining sensors. The red points denote the observations, the dashed black line the true signal, and the solid colour lines the predictive means. From left to right: MOSM, CONV, SM-LMC and CSM. Table 2: Imputation of the climate sensor measurements using the remaining sensors. Mean absolute error for all four experiments with one-standard-deviation error bars over ten realisations. Model Cambermet Chimet Sotonmet Bramblemet CONV SM-LMC CSM MOSM 0.098 ? 0.008 0.084 ? 0.004 0.094 ? 0.003 0.097 ? 0.006 0.192 ? 0.015 0.176 ? 0.003 0.129 ? 0.004 0.137 ? 0.007 0.211 ? 0.038 0.273 ? 0.001 0.195 ? 0.011 0.162 ? 0.011 0.163 ? 0.009 0.134 ? 0.002 0.130 ? 0.004 0.129 ? 0.003 These results do not show a significant difference between the proposed model and the latent processes based models. In order to test for statistical significance, the Kolmogorov-Smirnov test [20, Ch. 7] was used with a significance level ? = 0.05, concluding that for the Sotonmet sensor we can assure that the MOSM model yields the best results. Conversely, for the Cambermet, Chimet and Bramblemet sensors, MOSM and CSM provided similar results, though we cannot confirm their difference is statistically significant. However, given the high correlation of these signals and the 3 The data can be obtained from www.cambermet.co.uk. and the sites therein. 8 similarity between the MOSM model and the CSM model, the close performance of these two models on this dataset is to be expected. 4.3 Heavy metal concentration The Jura dataset [3] contains, in addition to other geological data, the concentration of seven heavy metals in a region of 14.5 km2 of the Swiss Jura, and it is divided into a training set (259 locations) and a validation set (100 locations). We followed [3, 4], where the motivation was to aid the prediction of a variable that is expensive to measure by using abundant measurements of correlated variables which are less expensive to acquire. Specifically, we estimated Cadmium and Copper at the validation locations using measurements of related variables at the training and test locations: Nickel and Zinc for Cadmium; and Lead, Nickel and Zinc for Copper. The MAE?see eq. (13)?is shown in Table 3, where the results for the CONV model were obtained from [4] and all models considered five latent signals/spectral components, except for the independent Gaussian process (denoted IGP). Observe how the proposed MOSM model outperforms all other models over the Cadmium data, which is statistical significant with a significance level ? = 0.05. Conversely, we cannot guarantee a statistically-significant difference between the CSM model and the MOSM in the Copper case. In both cases, testing for statistical significance against the CONV model was not possible since those results were obtained from [4]. On the other hand, the higher variability and non-Gaussianity of the Copper data may be the reason of why the simplest MOGP model (SM-LMC) achieves the best results. Table 3: Mean absolute error for the estimation of Cadmium and Copper concentrations with onestandard-deviation error bars over ten repetitions of the experiment. 5 Model Cadmium Copper IGP CONV SM-LMC CSM MOSM 0.56 ? 0.005 0.443 ? 0.006 0.46 ? 0.01 0.47 ? 0.02 0.43 ? 0.01 16.5 ? 0.1 7.45 ? 0.2 7.0 ? 0.1 7.4 ? 0.3 7.3 ? 0.1 Discussion We have proposed the multioutput spectral mixture (MOSM) kernel to model rich relationships across multiple outputs within Gaussian processes regression models. This has been achieved by constructing a positive-definite matrix of complex-valued spectral densities, and then transforming them via the inverse Fourier transform according to Cram?r?s Theorem. The resulting kernel provides a clear interpretation from a spectral viewpoint, where each of its parameters can be identified with frequency, magnitude, phase and delay for a pair of channels. Furthermore, a key feature that is unique to the proposed kernel is the ability joint model delays and phase differences, this is possible due to the complex-valued model for the cross-spectral density considered and validated experimentally using a synthetic example?see Fig. 2. The MOSM kernel has also been compared against existing MOGP models on two real-world datasets, where the proposed model performed competitively in terms of the mean absolute error. Further research should point towards a sparse implementation of the proposed MOGP which can build on [4, 21] to design inducing variables that exploit the spectral content of the processes as in [22, 23]. Acknowledgements We thank Crist?bal Silva (Universidad de Chile) for useful recommendations about GPU implementation, Rasmus Bonnevie from the GPflow team for his assistance on the experimental MOGP module within GPflow, and the anonymous reviewers. This work was financially supported by Conicyt Basal-CMM. 9 References [1] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning. The MIT Press, 2006. [2] D. Duvenaud, ?Automatic model construction with Gaussian processes,? Ph.D. dissertation, University of Cambridge, 2014. [3] P. Goovaerts, Geostatistics for natural resources evaluation. Oxford University Press on Demand, 1997. [4] M. A. ?lvarez and N. D. Lawrence, ?Sparse convolved Gaussian processes for multi-output regression,? in Advances in Neural Information Processing Systems 21, 2008, pp. 57?64. [5] K. R. Ulrich, D. E. Carlson, K. Dzirasa, and L. Carin, ?GP kernels for cross-spectrum analysis,? in Advances in Neural Information Processing Systems 28, 2015, pp. 1999?2007. [6] A. G. Wilson and R. P. Adams, ?Gaussian process kernels for pattern discovery and extrapolation,? in Proceedings of the 30th International Conference on Machine Learning (ICML-13), 2013, pp. 1067?1075. [7] S. Bochner, M. Tenenbaum, and H. Pollard, Lectures on Fourier Integrals, ser. Annals of mathematics studies. Princeton University Press, 1959. [8] H. Cram?r, ?On the theory of stationary random processes,? Annals of Mathematics, pp. 215?230, 1940. [9] A. Yaglom, Correlation Theory of Stationary and Related Random Functions, ser. Correlation Theory of Stationary and Related Random Functions. Springer, 1987, no. v. 1. [10] F. Tobar, T. D. Bui, and R. E. Turner, ?Learning stationary time series using Gaussian processes with nonparametric kernels,? in Advances in Neural Information Processing Systems 28. Curran Associates, Inc., 2015, pp. 3501?3509. [11] F. Tobar and R. E. Turner, ?Modelling time series via automatic learning of basis functions,? in Proc. of IEEE SAM, 2016, pp. 2209?2213. [12] M. A. ?lvarez, L. Rosasco, and N. D. Lawrence, ?Kernels for vector-valued functions: A review,? Found. Trends Mach. Learn., vol. 4, no. 3, pp. 195?266, Mar. 2012. [13] M. G. Genton and W. Kleiber, ?Cross-covariance functions for multivariate geostatistics,? Institute of Mathematical Statistics, vol. 30, no. 2, 2015. [14] F. Tobar and R. E. Turner, ?Modelling of complex signals using Gaussian processes,? in Proc. of IEEE ICASSP, 2015, pp. 2209?2213. [15] R. Boloix-Tortosa, F. J. Pay?n-Somet, and J. J. Murillo-Fuentes, ?Gaussian processes regressors for complex proper signals in digital communications,? in Proc. of IEEE SAM, 2014, pp. 137?140. [16] S. M. Kay, Modern spectral estimation : Theory and application. 1988. Englewood Cliffs, N.J. : Prentice Hall, [17] T. Gneiting, W. Kleiber, and M. Schlather, ?Mat?rn cross-covariance functions for multivariate random fields,? Journal of the American Statistical Association, vol. 105, no. 491, pp. 1167?1177, 2010. [18] M. Abadi et al., ?TensorFlow: Large-scale machine learning on heterogeneous systems,? 2015, software available from tensorflow.org. [Online]. Available: http://tensorflow.org/ [19] A. G. d. G. Matthews, M. van der Wilk, T. Nickson, K. Fujii, A. Boukouvalas, P. Le?n-Villagr?, Z. Ghahramani, and J. Hensman, ?GPflow: A Gaussian process library using TensorFlow,? 2016. [20] J. W. Pratt and J. D. Gibbons, Concepts of nonparametric theory. Springer Science & Business Media, 2012. [21] M. A. ?lvarez, D. Luengo, M. K. Titsias, and N. D. Lawrence, ?Efficient multioutput Gaussian processes through variational inducing kernels.? in AISTATS, vol. 9, 2010, pp. 25?32. [22] J. Hensman, N. Durrande, and A. Solin, ?Variational Fourier features for Gaussian processes,? arXiv preprint arXiv:1611.06740, 2016. [23] F. Tobar, T. D. Bui, and R. E. Turner, ?Design of covariance functions using inter-domain inducing variables,? in NIPS 2015 - Time Series Workshop, December 2015. 10 

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Neural Expectation Maximization Klaus Greff? IDSIA klaus@idsia.ch Sjoerd van Steenkiste? IDSIA sjoerd@idsia.ch J?rgen Schmidhuber IDSIA juergen@idsia.ch Abstract Many real world tasks such as reasoning and physical interaction require identi?cation and manipulation of conceptual entities. A ?rst step towards solving these tasks is the automated discovery of distributed symbol-like representations. In this paper, we explicitly formalize this problem as inference in a spatial mixture model where each component is parametrized by a neural network. Based on the Expectation Maximization framework we then derive a differentiable clustering method that simultaneously learns how to group and represent individual entities. We evaluate our method on the (sequential) perceptual grouping task and ?nd that it is able to accurately recover the constituent objects. We demonstrate that the learned representations are useful for next-step prediction. 1 Introduction Learning useful representations is an important aspect of unsupervised learning, and one of the main open problems in machine learning. It has been argued that such representations should be distributed [13, 37] and disentangled [1, 31, 3]. The latter has recently received an increasing amount of attention, producing representations that can disentangle features like rotation and lighting [4, 12]. So far, these methods have mostly focused on the single object case whereas, for real world tasks such as reasoning and physical interaction, it is often necessary to identify and manipulate multiple entities and their relationships. In current systems this is dif?cult, since superimposing multiple distributed and disentangled representations can lead to ambiguities. This is known as the Binding Problem [21, 37, 13] and has been extensively discussed in neuroscience [33]. One solution to this problem involves learning a separate representation for each object. In order to allow these representations to be processed identically they must be described in terms of the same (disentangled) features. This would then avoid the binding problem, and facilitate a wide range of tasks that require knowledge about individual objects. This solution requires a process known as perceptual grouping: dynamically splitting (segmenting) each input into its constituent conceptual entities. In this work, we tackle this problem of learning how to group and ef?ciently represent individual entities, in an unsupervised manner, based solely on the statistical structure of the data. Our work follows a similar approach as the recently proposed Tagger [7] and aims to further develop the understanding, as well as build a theoretical framework, for the problem of symbol-like representation learning. We formalize this problem as inference in a spatial mixture model where each component is parametrized by a neural network. Based on the Expectation Maximization framework we then derive a differentiable clustering method, which we call Neural Expectation Maximization (N-EM). It can be trained in an unsupervised manner to perform perceptual grouping in order to learn an ef?cient representation for each group, and naturally extends to sequential data. ? Both authors contributed equally to this work. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. 2 Neural Expectation Maximization The goal of training a system that produces separate representations for the individual conceptual entities contained in a given input (here: image) depends on what notion of entity we use. Since we are interested in the case of unsupervised learning, this notion can only rely on statistical properties of the data. We therefore adopt the intuitive notion of a conceptual entity as being a common cause (the object) for multiple observations (the pixels that depict the object). This common cause induces a dependency-structure among the affected pixels, while the pixels that correspond to different entities remain (largely) independent. Intuitively this means that knowledge about some pixels of an object helps in predicting its remainder, whereas it does not improve the predictions for pixels of other objects. This is especially obvious for sequential data, where pixels belonging to a certain object share a common fate (e.g. move in the same direction), which makes this setting particularly appealing. We are interested in representing each entity (object) k with some vector ?k that captures all the structure of the affected pixels, but carries no information about the remainder of the image. This modularity is a powerful invariant, since it allows the same representation to be reused in different contexts, which enables generalization to novel combinations of known objects. Further, having all possible objects represented in the same format makes it easier to work with these representations. Finally, having a separate ?k for each object (as opposed to for the entire image) allows ?k to be distributed and disentangled without suffering from the binding problem. We treat each image as a composition of K objects, where each pixel is determined by exactly one object. Which objects are present, as well as the corresponding assignment of pixels, varies from input to input. Assuming that we have access to the family of distributions P (x|?k ) that corresponds to an object level representation as described above, we can model each image as a mixture model. Then Expectation Maximization (EM) can be used to simultaneously compute a Maximum Likelihood Estimate (MLE) for the individual ?k -s and the grouping that we are interested in. The central problem we consider in this work is therefore how to learn such a P (x|?k ) in a completely unsupervised fashion. We accomplish this by parametrizing this family of distributions by a differentiable function f? (?) (a neural network with weights ?). We show that in that case, the corresponding EM procedure becomes fully differentiable, which allows us to backpropagate an appropriate outer loss into the weights of the neural network. In the remainder of this section we formalize and derive this method which we call Neural Expectation Maximization (N-EM). 2.1 Parametrized Spatial Mixture Model We model each image x ? RD as a spatial mixture of K components parametrized by vectors ?1 , . . . , ?K ? RM . A differentiable non-linear function f? (a neural network) is used to transform these representations ?k into parameters ?i,k = f? (?k )i for separate pixel-wise distributions. These distributions are typically Bernoulli or Gaussian, in which case ?i,k would be a single probability or a mean and variance respectively. This parametrization assumes that given the representation, the pixels are independent but not identically distributed (unlike in standard mixture models). A set of binary latent variables Z ? [0, 1]D?K encodes the unknown true pixel assignments, such that ? zi,k = 1 iff pixel i was generated by component k, and k zi,k = 1. A graphical representation of this model can be seen in Figure 1, where ? = (?1 , . . . ?K ) are the mixing coef?cients (or prior for z). The full likelihood for x given ? = (?1 , . . . , ?K ) is given by: P (x|?) = D ? ? i=1 zi 2.2 P (xi , zi |?i ) = D ? K ? i=1 k=1 Expectation Maximization P (zi,k = 1) P (xi |?i,k , zi,k = 1). ?? ? ? (1) ?k Directly optimizing log P (x|?) with respect to ? is dif?cult due to marginalization over z, while for many distributions optimizing log P (x, z|?) is much easier. Expectation Maximization (EM; [6]) takes advantage of this and instead optimizes a lower bound given by the expected log likelihood: Q(?, ? old ) = ? P (z|x, ? old ) log P (x, z|?). z 2 (2) D Figure 1: left: The probabilistic graphical model that underlies N-EM. right: Illustration of the computations for two steps of N-EM. Iterative optimization of this bound alternates between two steps: in the E-step we compute a new estimate of the posterior probability distribution over the latent variables given ? old from the previous iteration, yielding a new soft-assignment of the pixels to the components (clusters): ?i,k := P (zi,k = 1|xi , ?iold ). (3) In the M-step we then aim to ?nd the con?guration of ? that would maximize the expected loglikelihood using the posteriors computed in the E-step. Due to the non-linearity of f? there exists no analytical solution to arg max? Q(?, ? old ). However, since f? is differentiable, we can improve Q(?, ? old ) by taking a gradient ascent step:2 ? new = ? old + ? ?Q ?? D ? ?Q ??i,k ? ?i,k (?i,k ? xi ) . ??k ??k i=1 where (4) The resulting algorithm belongs to the class of generalized EM algorithms and is guaranteed (for a suf?ciently small learning rate ?) to converge to a (local) optimum of the data log likelihood [42]. 2.3 Unrolling In our model the information about statistical regularities required for clustering the pixels into objects is encoded in the neural network f? with weights ?. So far we have considered f? to be ?xed and have shown how we can compute an MLE for ? alongside the appropriate clustering. We now observe that by unrolling the iterations of the presented generalized EM, we obtain an end-to-end differentiable clustering procedure based on the statistical model implemented by f? . We can therefore use (stochastic) gradient descent and ?t the statistical model to capture the regularities corresponding to objects for a given dataset. This is implemented by back-propagating an appropriate loss (see Section 2.4) through ?time? (BPTT; [39, 41]) into the weights ?. We refer to this trainable procedure as Neural Expectation Maximization (N-EM), an overview of which can be seen in Figure 1. Upon inspection of the structure of N-EM we ?nd that it resembles K copies of a recurrent neural network with hidden states ?k that, at each timestep, receive ?k ? (?k ? x) as their input. Each copy generates a new ?k , which is then used by the E-step to re-estimate the soft-assignments ?. In order to accurately mimic the M-Step (4) with an RNN, we must impose several restrictions on its weights and structure: the ?encoder? must correspond to the Jacobian ??k /??k , and the recurrent update must linearly combine the output of the encoder with ?k from the previous timestep. In- Figure 2: RNN-EM Illustration. Note the stead, we introduce a new algorithm named RNN-EM, changed encoder and recurrence compared when substituting that part of the computational graph to Figure 1. of N-EM with an actual RNN (without imposing any restrictions). Although RNN-EM can no longer guarantee 2 Here we assume that P (xi |zi,k = 1, ?i,k ) is given by N (xi ; ? = ?i,k , ? 2 ) for some ?xed ? 2 , yet a similar update arises for many typical parametrizations of pixel distributions. 3 convergence of the data log likelihood, its recurrent weights increase the ?exibility of the clustering procedure. Moreover, by using a fully parametrized recurrent weight matrix RNN-EM naturally extends to sequential data. Figure 2 presents the computational graph of a single RNN-EM timestep. 2.4 Training Objective N-EM is a differentiable clustering procedure, whose outcome relies on the statistical model f? . We are interested in a particular unsupervised clustering that corresponds to grouping entities based on the statistical regularities in the data. To train our system, we therefore require a loss function that teaches f? to map from representations ? to parameters ? that correspond to pixelwise distributions for such objects. We accomplish this with a two-term loss function that guides each of the K networks to model the structure of a single object independently of any other information in the image: L(x) = ? D ? K ? i=1 k=1 ?i,k log P (xi , zi,k |?i,k ) ? (1 ? ?i,k )DKL [P (xi )||P (xi |?i,k , zi,k )] . ?? ? ? ?? ? ? intra-cluster loss (5) inter-cluster loss The intra-cluster loss corresponds to the same expected data log-likelihood Q as is optimized by N-EM. It is analogous to a standard reconstruction loss used for training autoencoders, weighted by the cluster assignment. Similar to autoencoders, this objective is prone to trivial solutions in case of overcapacity, which prevent the network from modelling the statistical regularities that we are interested in. Standard techniques can be used to overcome this problem, such as making ? a bottleneck or using a noisy version of x to compute the inputs to the network. Furthermore, when RNN-EM is used on sequential data we can use a next-step prediction loss. Weighing the loss pixelwise is crucial, since it allows each network to specialize its predictions to an individual object. However, it also introduces a problem: the loss for out-of-cluster pixels (?i,k = 0) vanishes. This leaves the network free to predict anything and does not yield specialized representations. Therefore, we add a second term (inter-cluster loss) which penalizes the KL divergence between out-of-cluster predictions and the pixelwise prior of the data. Intuitively this tells each representation ?k to contain no information regarding non-assigned pixels xi : P (xi |?i,k , zi,k ) = P (xi ). A disadvantage of the interaction between ? and ? in (5) is that it may yield con?icting gradients. For any ?k the loss for a given pixel i can be reduced by better predicting xi , or by decreasing ?i,k (i.e. taking less responsibility) which is (due to the E-step) realized by being worse at predicting xi . A practical solution to this problem is obtained by stopping the ? gradients, i.e. by setting ?L/?? = 0 during backpropagation. 3 Related work The method most closely related to our approach is Tagger [7], which similarly learns perceptual grouping in an unsupervised fashion using K copies of a neural network that work together by reconstructing different parts of the input. Unlike in case of N-EM, these copies additionally learn to output the grouping, which gives Tagger more direct control over the segmentation and supports its use on complex texture segmentation tasks. Our work maintains a close connection to EM and relies on the posterior inference of the E-Step as a grouping mechanism. This facilitates theoretical analysis and simpli?es the task for the resulting networks, which we ?nd can be markedly smaller than in Tagger. Furthermore, Tagger does not include any recurrent connections on the level of the hidden states, precluding it from next step prediction on sequential tasks.3 The Binding problem was ?rst considered in the context of Neuroscience [21, 37] and has sparked some early work in oscillatory neural networks that use synchronization as a grouping mechanism [36, 38, 24]. Later, complex valued activations have been used to replace the explicit simulation of oscillation [25, 26]. By virtue of being general computers, any RNN can in principle learn a suitable mechanism. In practice however it seems hard to learn, and adding a suitable mechanism like competition [40], fast weights [29], or perceptual grouping as in N-EM seems necessary. 3 RTagger [15]: a recurrent extension of Tagger that does support sequential data was developed concurrent to this work. 4 ?????? ???? ?????? ? ? ? ? ? ? Figure 3: Groupings by RNN-EM (bottom row), NEM (middle row) for six input images (top row). Both methods recover the individual shapes accurately when they are separated (a, b, f), even when confronted with the same shape (b). RNN-EM is able to handle most occlusion (c, d) but sometimes fails (e). The exact assignments are permutation invariant and depend on ? initialization; compare (a) and (f). Unsupervised Segmentation has been studied in several different contexts [30], from random vectors [14] over texture segmentation [10] to images [18, 16]. Early work in unsupervised video segmentation [17] used generalized Expectation Maximization (EM) to infer how to split frames of moving sprites. More recently optical ?ow has been used to train convolutional networks to do ?gure/ground segmentation [23, 34]. A related line of work under the term of multi-causal modelling [28] has formalized perceptual grouping as inference in a generative compositional model of images. Masked RBMs [20] for example extend Restricted Boltzmann Machines with a latent mask inferred through Block-Gibbs sampling. Gradient backpropagation through inference updates has previously been addressed in the context of sparse coding with (Fast) Iterative Shrinkage/Tresholding Algorithms ((F)ISTA; [5, 27, 2]). Here the unrolled graph of a ?xed number of ISTA iterations is replaced by a recurrent neural network that parametrizes the gradient computations and is trained to predict the sparse codes directly [9]. We derive RNN-EM from N-EM in a similar fashion and likewise obtain a trainable procedure that has the structure of iterative pursuit built into the architecture, while leaving tunable degrees of freedom that can improve their modeling capabilities [32]. An alternative to further empower the network by untying its weights across iterations [11] was not considered for ?exibility reasons. 4 Experiments We evaluate our approach on a perceptual grouping task for generated static images and video. By composing images out of simple shapes we have control over the statistical structure of the data, as well as access to the ground-truth clustering. This allows us to verify that the proposed method indeed recovers the intended grouping and learns representations corresponding to these objects. In particular we are interested in studying the role of next-step prediction as a unsupervised objective for perceptual grouping, the effect of the hyperparameter K, and the usefulness of the learned representations. In all experiments we train the networks using ADAM [19] with default parameters, a batch size of 64 and 50 000 train + 10 000 validation + 10 000 test inputs. Consistent with earlier work [8, 7], we evaluate the quality of the learned groupings with respect to the ground truth while ignoring the background and overlap regions. This comparison is done using the Adjusted Mutual Information (AMI; [35]) score, which provides a measure of clustering similarity between 0 (random) and 1 (perfect match). We use early stopping when the validation loss has not improved for 10 epochs.4 A detailed overview of the experimental setup can be found in Appendix A. All reported results are averages computed over ?ve runs.5 4.1 Static Shapes To validate that our approach yields the intended behavior we consider a simple perceptual grouping task that involves grouping three randomly chosen regular shapes (???) located in random positions of 28 ? 28 binary images [26]. This simple setup serves as a test-bed for comparing N-EM and RNN-EM, before moving on to more complex scenarios. We implement f? by means of a single layer fully connected neural network with a sigmoid output ?i,k for each pixel that corresponds to the mean of a Bernoulli distribution. The representation ?k is 4 Note that we do not stop on the AMI score as this is not part of our objective function and only measured to evaluate the performance after training. 5 Code to reproduce all experiments is available at https://github.com/sjoerdvansteenkiste/ Neural-EM 5 ?????? ???? ???? ???? ???? ???? ?????? ?????? ?????? ?????? ?????? ?????? ?????? ?????? ?????? ?????? ??????? ??????? ??????? ??????? ??????? ??????? ??????? ??????? ??????? ??????? ??????? Figure 4: A sequence of 5 shapes ?ying along random trajectories (bottom row). The next-step prediction of each copy of the network (rows 2 to 5) and the soft-assignment of the pixels to each of the copies (top row). Observe that the network learns to separate the individual shapes as a means to ef?ciently solve next-step prediction. Even when many of the shapes are overlapping, as can be seen in time-steps 18-20, the network is still able to disentangle the individual shapes from the clutter. a real-valued 250-dimensional vector squashed to the (0, 1) range by a sigmoid function before being fed into the network. Similarly for RNN-EM we use a recurrent neural network with 250 sigmoidal hidden units and an equivalent output layer. Both networks are trained with K = 3 and unrolled for 15 EM steps. As shown in Figure 3, we observe that both approaches are able to recover the individual shapes as long as they are separated, even when confronted with identical shapes. N-EM performs worse if the image contains occlusion, and we ?nd that RNN-EM is in general more stable and produces considerably better groupings. This observation is in line with ?ndings for Sparse Coding [9]. Similarly we conclude that the tunable degrees of freedom in RNN-EM help speed-up the optimization process resulting in a more powerful approach that requires fewer iterations. The bene?t is re?ected in the large score difference between the two: 0.826 ? 0.005 AMI compared to 0.475 ? 0.043 AMI for N-EM. In comparison, Tagger achieves an AMI score of 0.79 ? 0.034 (and 0.97 ? 0.009 with layernorm), while using about twenty times more parameters [7]. 4.2 Flying Shapes We consider a sequential extension of the static shapes dataset in which the shapes (???) are ?oating along random trajectories and bounce off walls. An example sequence with 5 shapes can be seen in the bottom row of Figure 4. We use a convolutional encoder and decoder inspired by the discriminator and generator networks of infoGAN [4], with a recurrent neural network of 100 sigmoidal units (for (t?1) details see Section A.2). At each timestep t the network receives ?k (?k ?x ?(t) ) as input, where (t) x ? is the current frame corrupted with additional bit?ip noise (p = 0.2). The next-step prediction objective is implemented by replacing x with x(t+1) in (5), and is evaluated at each time-step. Table 1 summarizes the results on ?ying shapes, and an example of a sequence with 5 shapes when using K = 5 can be seen in Figure 4. For 3 shapes we observe that the produced groupings are close to perfect (AMI: 0.970 ? 0.005). Even in the very cluttered case of 5 shapes the network is able to separate the individual objects in almost all cases (AMI: 0.878 ? 0.003). These results demonstrate the adequacy of the next step prediction task for perceptual grouping. However, we ?nd that the converse also holds: the corresponding representations are useful for the prediction task. In Figure 5 we compare the next-step prediction error of RNN-EM with K = 1 (which reduces to a recurrent autoencoder that receives the difference between its previous prediction and the current frame as input) to RNN-EM with K = 5 on this task. To evaluate RNN-EM on next-step ? prediction we computed its loss using P (xi |?i ) = P (xi | maxk ?i,k ) as opposed to P (xi |?i ) = k ?i,k P (xi |?i,k ) to avoid including information from the next timestep. The reported BCE loss for RNN-EM is therefore an upperbound to the true BCE loss. From the ?gure we observe that RNN-EM produces signi?cantly lower errors, especially when the number of objects increases. 6 ??? ?????? ???????????? ??? ?? ??? ????????????? ??? ?? ??? ??? ?????? ??? ?? ??? ?? ? ? ? ????????? ??? ? Figure 5: Binomial Cross Entropy Error obtained by RNN-EM and a recurrent autoencoder (RNN-EM with K = 1) on the denoising and next-step prediction task. RNN-EM produces signi?cantly lower BCE across different numbers of objects. ? ?? ?? ????? ?? ?? ?? Figure 6: Average AMI score (blue line) measured for RNN-EM (trained for 20 steps) across the ?ying MNIST test-set and corresponding quartiles (shaded areas), computed for each of 50 time-steps. The learned grouping dynamics generalize to longer sequences and even further improve the AMI score. Train Test Test Generalization # obj. K AMI # obj. K AMI # obj. K AMI 3 3 5 5 3 5 3 5 0.969 ? 0.006 0.997 ? 0.001 0.614 ? 0.003 0.878 ? 0.003 3 3 5 5 3 5 3 5 0.970 ? 0.005 0.997 ? 0.002 0.614 ? 0.003 0.878 ? 0.003 3 3 3 3 5 3 3 5 0.972 ? 0.007 0.914 ? 0.015 0.886 ? 0.010 0.981 ? 0.003 Table 1: AMI scores obtained by RNN-EM on ?ying shapes when varying the number of objects and number of components K, during training and at test time. Finally, in Table 1 we also provide insight about the impact of choosing the hyper-parameter K, which is unknown for many real-world scenarios. Surprisingly we observe that training with too large K is in fact favourable, and that the network learns to leave the excess groups empty. When training with too few components we ?nd that the network still learns about the individual shapes and we observe only a slight drop in score when correctly setting the number of components at test time. We conclude that RNN-EM is robust towards different choices of K, and speci?cally that choosing K to be too high is not detrimental. 4.3 Flying MNIST In order to incorporate greater variability among the objects we consider a sequential extension of MNIST. Here each sequence consists of gray-scale 24 ? 24 images containing two down-sampled MNIST digits that start in random positions and ?oat along randomly sampled trajectories within the image for T timesteps. An example sequence can be seen in the bottom row of Figure 7. We deploy a slightly deeper version of the architecture used in ?ying shapes. Its details can be found in Appendix A.3. Since the images are gray-scale we now use a Gaussian distribution for each pixel with ?xed ? 2 = 0.25 and ? = ?i,k as computed by each copy of the network. The training procedure is identical to ?ying shapes except that we replace bit?ip noise with masked uniform noise: we ?rst sample a binary mask from a multi-variate Bernoulli distribution with p = 0.2 and then use this mask to interpolate between the original image and samples from a Uniform distribution between the minimum and maximum values of the data (0,1). We train with K = 2 and T = 20 on ?ying MNIST having two digits and obtain an AMI score of 0.819 ? 0.022 on the test set, measured across 5 runs. In early experiments we observed that, given the large variability among the 50 000 unique digits, we can boost the model performance by training in stages using 20, 500, 50 000 digits. Here we exploit the generalization capabilities of RNN-EM to quickly transfer knowledge from a less varying set of 7 ?????? ???? ???? ???? ?????? ?????? ?????? ?????? ?????? ?????? ?????? ?????? ?????? ?????? ??????? ??????? ??????? ??????? ??????? ??????? ??????? ??????? ??????? ??????? ??????? Figure 7: A sequence of 3 MNIST digits ?ying across random trajectories in the image (bottom row). The next-step prediction of each copy of the network (rows 2 to 4) and the soft-assignment of the pixels to each of the copies (top row). Although the network was trained (stage-wise) on sequences with two digits, it is accurately able to separate three digits. MNIST digits to unseen variations. We used the same hyper-parameter con?guration as before and obtain an AMI score of 0.917 ? 0.005 on the test set, measured across 5 runs. We study the generalization capabilities and robustness of these trained RNN-EM networks by means of three experiments. In the ?rst experiment we evaluate them on ?ying MNIST having three digits (one extra) and likewise set K = 3. Even without further training we are able to maintain a high AMI score of 0.729 ? 0.019 (stage-wise: 0.838 ? 0.008) on the test-set. A test example can be seen in Figure 7. In the second experiment we are interested in whether the grouping mechanism that has been learned can be transferred to static images. We ?nd that using 50 RNN-EM steps we are able to transfer a large part of the learned grouping dynamics and obtain an AMI score of 0.619 ? 0.023 (stage-wise: 0.772 ? 0.008) for two static digits. As a ?nal experiment we evaluate the directly trained network on the same dataset for a larger number of timesteps. Figure 6 displays the average AMI score across the test set as well as the range of the upper and lower quartile for each timestep. The results of these experiments con?rm our earlier observations for ?ying shapes, in that the learned grouping dynamics are robust and generalize across a wide range of variations. Moreover we ?nd that the AMI score further improves at test time when increasing the sequence length. 5 Discussion The experimental results indicate that the proposed Neural Expectation Maximization framework can indeed learn how to group pixels according to constituent objects. In doing so the network learns a useful and localized representation for individual entities, which encodes only the information relevant to it. Each entity is represented separately in the same space, which avoids the binding problem and makes the representations usable as ef?cient symbols for arbitrary entities in the dataset. We believe that this is useful for reasoning in particular, and a potentially wide range of other tasks that depend on interaction between multiple entities. Empirically we ?nd that the learned representations are already bene?cial in next-step prediction with multiple objects, a task in which overlapping objects are problematic for standard approaches, but can be handled ef?ciently when learning a separate representation for each object. As is typical in clustering methods, in N-EM there is no preferred assignment of objects to groups and so the grouping numbering is arbitrary and only depends on initialization. This property renders our results permutation invariant and naturally allows for instance segmentation, as opposed to semantic segmentation where groups correspond to pre-de?ned categories. RNN-EM learns to segment in an unsupervised fashion, which makes it applicable to settings with little or no labeled data. On the downside this lack of supervision means that the resulting segmentation may not always match the intended outcome. This problem is inherent to this task since in real world images the notion of an object is ill-de?ned and task dependent. We envision future work to alleviate this by extending unsupervised segmentation to hierarchical groupings, and by dynamically conditioning them on the task at hand using top-down feedback and attention. 8 6 Conclusion We have argued for the importance of separately representing conceptual entities contained in the input, and suggested clustering based on statistical regularities as an appropriate unsupervised approach for separating them. We formalized this notion and derived a novel framework that combines neural networks and generalized EM into a trainable clustering algorithm. We have shown how this method can be trained in a fully unsupervised fashion to segment its inputs into entities, and to represent them individually. Using synthetic images and video, we have empirically veri?ed that our method can recover the objects underlying the data, and represent them in a useful way. We believe that this work will help to develop a theoretical foundation for understanding this important problem of unsupervised learning, as well as providing a ?rst step towards building practical solutions that make use of these symbol-like representations. Acknowledgements The authors wish to thank Paulo Rauber and the anonymous reviewers for their constructive feedback. This research was supported by the Swiss National Science Foundation grant 200021_165675/1 and the EU project ?INPUT? (H2020-ICT-2015 grant no. 687795). We are grateful to NVIDIA Corporation for donating us a DGX-1 as part of the Pioneers of AI Research award, and to IBM for donating a ?Minsky? machine. References [1] H.B. Barlow, T.P. Kaushal, and G.J. Mitchison. Finding Minimum Entropy Codes. Neural Computation, 1(3):412?423, September 1989. 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Learning Linear Dynamical Systems via Spectral Filtering Elad Hazan, Karan Singh, Cyril Zhang Department of Computer Science Princeton University Princeton, NJ 08544 {ehazan,karans,cyril.zhang}@cs.princeton.edu Abstract We present an efficient and practical algorithm for the online prediction of discrete-time linear dynamical systems with a symmetric transition matrix. We circumvent the non-convex optimization problem using improper learning: carefully overparameterize the class of LDSs by a polylogarithmic factor, in exchange for convexity of the loss functions. From this arises a polynomial-time algorithm with a near-optimal regret guarantee, with an analogous sample complexity bound for agnostic learning. Our algorithm is based on a novel filtering technique, which may be of independent interest: we convolve the time series with the eigenvectors of a certain Hankel matrix. 1 Introduction Linear dynamical systems (LDSs) are a class of state space models which accurately model many phenomena in nature and engineering, and are applied ubiquitously in time-series analysis, robotics, econometrics, medicine, and meteorology. In this model, the time evolution of a system is explained by a linear map on a finite-dimensional hidden state, subject to disturbances from input and noise. Recent interest has focused on the effectiveness of recurrent neural networks (RNNs), a nonlinear variant of this idea, for modeling sequences such as audio signals and natural language. Central to this field of study is the problem of system identification: given some sample trajectories, output the parameters for an LDS which generalize to predict unseen future data. Viewed directly, this is a non-convex optimization problem, for which efficient algorithms with theoretical guarantees are very difficult to obtain. A standard heuristic for this problem is expectation-maximization (EM), which can find poor local optima in theory and practice. We consider a different approach: we formulate system identification as an online learning problem, in which neither the data nor predictions are assumed to arise from an LDS. Furthermore, we slightly overparameterize the class of predictors, yielding an online convex program amenable to efficient regret minimization. This carefully chosen relaxation, which is our main theoretical contribution, expands the dimension of the hypothesis class by only a polylogarithmic factor. This construction relies upon recent work on the spectral theory of Hankel matrices. The result is a simple and practical algorithm for time-series prediction, which deviates significantly from existing methods. We coin the term wave-filtering for our method, in reference to our relaxation?s use of convolution by wave-shaped eigenvectors. We present experimental evidence on both toy data and a physical simulation, showing our method to be competitive in terms of predictive performance, more stable, and significantly faster than existing algorithms. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. 1.1 Our contributions Consider a discrete-time linear dynamical system with inputs {xt }, outputs {yt }, and a latent state {ht }, which can all be multi-dimensional. With noise vectors {?t }, {?t }, the system?s time evolution is governed by the following equations: ht+1 = Aht + Bxt + ?t yt = Cht + Dxt + ?t . If the dynamics A, B, C, D are known, then the Kalman filter [Kal60] is known to estimate the hidden state optimally under Gaussian noise, thereby producing optimal predictions of the system?s response to any given input. However, this is rarely the case ? indeed, real-world systems are seldom purely linear, and rarely are their evolution matrices known. We henceforth give a provable, efficient algorithm for the prediction of sequences arising from an unknown dynamical system as above, in which the matrix A is symmetric. Our main theoretical contribution is a regret bound for this algorithm, giving nearly-optimal convergence to the lowest mean squared prediction error (MSE) realizable by a symmetric LDS model: Theorem 1 (Main regret bound; informal). On an arbitrary sequence {(xt , yt )}Tt=1 , Algorithm 1 makes predictions {? yt }Tt=1 which satisfy   m, d, log T ) ? poly(n, ? MSE(? y1 , . . . , y?T ) ? MSE(? y1? , . . . , y?T? ) ? O , T compared to the best predictions {yt? }Tt=1 by a symmetric LDS, while running in polynomial time. Note that the signal need not be generated by an LDS, and can even be adversarially chosen. In the less general batch (statistical) setting, we use the same techniques to obtain an analogous sample complexity bound for agnostic learning: Theorem 2 (Batch version; informal). For any choice of ? > 0, given access to an arbitrary distribution D over training sequences {(xt , yt )}Tt=1 , Algorithm 2, run on N i.i.d. sample trajectories ? such that from D, outputs a predictor ?   ? (poly(n, m, d, log T, log 1/?)) O ? ? ? E MSE(?) ? MSE(? ) ? ? + , D N compared to the best symmetric LDS predictor ?? , while running in polynomial time. Typical regression-based methods require the LDS to be strictly stable, and degrade on ill1 conditioned systems; they depend on a spectral radius parameter 1?kAk . Our proposed method of wave-filtering provably and empirically works even for the hardest case of kAk = 1. Our algorithm attains the first condition number-independent polynomial guarantees in terms of regret (equivalently, sample complexity) and running time for the MIMO setting. Interestingly, our algorithms never need to learn the hidden state, and our guarantees can be sharpened to handle the case when the dimensionality of ht is infinite. 1.2 Related work The modern setting for LDS arose in the seminal work of Kalman [Kal60], who introduced the Kalman filter as a recursive least-squares solution for maximum likelihood estimation (MLE) of Gaussian perturbations to the system. The framework and filtering algorithm have proven to be a mainstay in control theory and time-series analysis; indeed, the term Kalman filter model is often used interchangeably with LDS. We refer the reader to the classic survey [Lju98], and the extensive overview of recent literature in [HMR16]. Ghahramani and Roweis [RG99] suggest using the EM algorithm to learn the parameters of an LDS. This approach, which directly tackles the non-convex problem, is widely used in practice [Mar10a]. However, it remains a long-standing challenge to characterize the theoretical guarantees afforded by EM. We find that it is easy to produce cases where EM fails to identify the correct system. In a recent result of [HMR16], it is shown for the first time that for a restricted class of systems, gradient descent (also widely used in practice, perhaps better known in this setting as backpropagation) 2 guarantees polynomial convergence rates and sample complexity in the batch setting. Their result applies essentially only to the SISO case (vs. multi-dimensional for us), depends polynomially on the spectral gap (as opposed to no dependence for us), and requires the signal to be created by an LDS (vs. arbitrary for us). 2 2.1 Preliminaries Linear dynamical systems Many different settings have been considered, in which the definition of an LDS takes on many variants. We are interested in discrete time-invariant MIMO (multiple input, multiple output) systems with a finite-dimensional hidden state.1 Formally, our model is given as follows: Definition 2.1. A linear dynamical system (LDS) is a map from a sequence of input vectors x1 , . . . , xT ? Rn to output (response) vectors y1 , . . . , yT ? Rm of the form ht+1 = Aht + Bxt + ?t yt = Cht + Dxt + ?t , (1) (2) where h0 , . . . , hT ? Rd is a sequence of hidden states, A, B, C, D are matrices of appropriate dimension, and ?t ? Rd , ?t ? Rm are (possibly stochastic) noise vectors. Unrolling this recursive definition gives the impulse response function, which uniquely determines the LDS. For notational convenience, for invalid indices t ? 0, we define xt , ?t , and ?t to be the zero vector of appropriate dimension. Then, we have: yt = T ?1 X CAi (Bxt?i + ?t?i ) + CAt h0 + Dxt + ?t . (3) i=1 We will consider the (discrete) time derivative of the impulse response function, given by expanding yt?1 ? yt by Equation (3). For the rest of this paper, we focus our attention on systems subject to the following restrictions: (i) The LDS is Lyapunov stable: kAk2 ? 1, where k?k2 denotes the operator (a.k.a. spectral) norm. (ii) The transition matrix A is symmetric and positive semidefinite.2 The first assumption is standard: when the hidden state is allowed to blow up exponentially, finegrained prediction is futile. In fact, many algorithms only work when kAk is bounded away from 1, so that the effect of any particular xt on the hidden state (and thus the output) dissipates exponentially. We do not require this stronger assumption. We take a moment to justify assumption (ii), and why this class of systems is still expressive and useful. First, symmetric LDSs constitute a natural class of linearly-observable, linearly-controllable systems with dissipating hidden states (for example, physical systems with friction or heat diffusion). Second, this constraint has been used successfully for video classification and tactile recognition tasks [HSC+ 16]. Interestingly, though our theorems require symmetric A, our algorithms appear to tolerate some non-symmetric (and even nonlinear) transitions in practice. 2.2 Sequence prediction as online regret minimization A natural formulation of system identification is that of online sequence prediction. At each time step t, an online learner is given an input xt , and must return a predicted output y?t . Then, the true response yt is observed, and the predictor suffers a squared-norm loss of kyt ? y?t k2 . Over T rounds, the goal is to predict as accurately as the best LDS in hindsight. 1 We assume finite dimension for simplicity of presentation. However, it will be evident that hidden-state dimension has no role in our algorithm, and shows up as kBkF and kCkF in the regret bound. 2 The psd constraint on A can be removed by augmenting the inputs xt with extra coordinates (?1)t (xt ). We omit this for simplicity of presentation. 3 Note that the learner is permitted to access the history of observed responses {y1 , . . . , yt?1 }. Even in the presence of statistical (non-adversarial) noise, the fixed maximum-likelihood sequence produced by ? = (A, B, C, D, h0 ) will accumulate error linearly as T . Thus, we measure performance against a more powerful comparator, which fixes LDS parameters ?, and predicts yt by the previous response yt?1 plus the derivative of the impulse response function of ? at time t. We will exhibit an online algorithm that can compete against the best ? in this setting. Let y?1 , . . . , y?T be the predictions made by an online learner, and let y1? , . . . , yT? be the sequence of predictions, realized by a chosen setting of LDS parameters ?, which minimize total squared error. Then, we define regret by the difference of total squared-error losses: def Regret(T ) = T T X X kyt ? y?t k2 ? kyt ? yt? k2 . t=1 t=1 This setup fits into the standard setting of online convex optimization (in which a sublinear regret bound implies convergence towards optimal predictions), save for the fact that the loss functions are non-convex in the system parameters. Also, note that a randomized construction (set all xt = 0, and let yt be i.i.d. Bernoulli random variables) yields a lower bound3 for any online algorithm: ? E [Regret(T )] ? ?( T ). To quantify regret bounds, we must state our scaling assumptions on the (otherwise adversarial) input and output sequences. We assume that the inputs are bounded: kxt k2 ? Rx . Also, we assume that the output signal is Lipschitz in time: kyt ? yt?1 k2 ? Ly . The latter assumption exists to preclude pathological inputs where an online learner is forced to incur arbitrarily large regret. For a true noiseless LDS, Ly is not too large; see Lemma F.5 in the appendix. ? ? T ) regret bound can be trivially achieved in this setting by algorithms We note that an optimal O( such as Hedge [LW94], using an exponential-sized discretization of all possible LDS parameters; this is the online equivalent of brute-force grid search. Strikingly, our algorithms achieve essentially the same regret bound, but run in polynomial time. 2.3 The power of convex relaxations Much work in system identification, including the EM method, is concerned with explicitly finding the LDS parameters ? = (A, B, C, D, h0 ) which best explain the data. However, it is evident from Equation 3 that the CAi B terms cause the least-squares (or any other) loss to be non-convex in ?. Many methods used in practice, including EM and subspace identification, heuristically estimate each hidden state ht , after which estimating the parameters becomes a convex linear regression problem. However, this first step is far from guaranteed to work in theory or practice. Instead, we follow the paradigm of improper learning: in order to predict sequences as accurately as the best possible LDS ?? ? H, one need not predict strictly from an LDS. The central driver of our ? for which the best predictor algorithms is the construction of a slightly larger hypothesis class H, ? ? ? ? ? is nearly as good as ? . Furthermore, we construct H so that the loss functions are convex under this new parameterization. From this will follow our efficient online algorithm. As a warmup example, consider the following overparameterization: pick some time window ?  T , and let the predictions y?t be linear in the concatenation [xt , . . . , xt?? ] ? R? d . When kAk is bounded away from 1, this is a sound assumption.4 However, in general, this approximation is doomed to either truncate longer-term input-output dependences (short ? ), or suffer from overfitting (long ? ). Our main theorem uses an overparameterization whose approximation factor ? is ? independent of kAk, and whose sample complexity scales only as O(polylog(T, 1/?)). 2.4 Low approximate rank of Hankel matrices Our analysis relies crucially on the spectrum of a certain Hankel matrix, a square matrix whose anti-diagonal stripes have equal entries (i.e. Hij is a function of i + j). An important example is the 3 4 This is a standard construction; see, e.g. Theorem 3.2 in [Haz16]. This assumption is used in autoregressive models; see Section 6 of [HMR16] for a theoretical treatment. 4 1 Hilbert matrix Hn,? , the n-by-n matrix whose (i, j)-th entry is i+j+? . For example, " # 1 1/2 1/3 H3,?1 = 1/2 1/3 1/4 . 1/3 1/4 1/5 This and related matrices have been studied under various lenses for more than a century: see, e.g., [Hil94, Cho83]. A basic fact is that Hn,? is a positive definite matrix for every n ? 1, ? > ?2. The property we are most interested in is that the spectrum of a positive semidefinite Hankel matrix decays exponentially, a difficult result derived in [BT16] via Zolotarev rational approximations. We state these technical bounds in Appendix E. 3 The wave-filtering algorithm Our online algorithm (Algorithm 1) runs online projected gradient descent [Zin03] on the squared def loss ft (Mt ) = kyt ? y?t (Mt )k2 . Here, each Mt is a matrix specifying a linear map from fea? t to predictions y?t . Specifically, after choosing a certain bank of k filters {?j }, turized inputs X nk+2n+m ? Xt ? R consists of convolutions of the input time series with each ?j (scaled by certain constants), along with xt?1 , xt , and yt?1 . The number of filters k will turn out to be polylogarithmic in T . 1/4 The filters {?j } and scaling factors {?j } are given by the top eigenvectors and eigenvalues of the Hankel matrix ZT ? RT ?T , whose entries are given by 2 . Zij := 3 (i + j) ? (i + j) In the language of Section 2.3, one should think of each Mt as arising from an ? ? which replaces the original O((m + n + O(poly(m, n, d, log T ))-dimensional hypothesis class H, d)2 )-dimensional class H of LDS parameters (A, B, C, D, h0 ). Theorem 3 gives the key fact that ? approximately contains H. H Algorithm 1 Online wave-filtering algorithm for LDS sequence prediction 1: Input: time horizon T , filter parameter k, learning rate ?, radius parameter RM . 2: Compute {(?j , ?j )}kj=1 , the top k eigenpairs of ZT . 0 def 3: Initialize M1 ? Rm?k , where k 0 = nk + 2n + m. 4: for t = 1, . . . , T do ? ? Rk0 , with first nk entries X ? (i,j) := ? 1/4 PT ?1 ?j (u)xt?u (i), followed by 5: Compute X j u=1 the 2n + m entries of xt?1 , xt , and yt?1 . ? 6: Predict y?t := Mt X. 7: Observe yt . Suffer loss kyt ? y?t k2 . ? 8: Gradient update: Mt+1 ? Mt ? 2?(yt ? y?t ) ? X. 9: if kMt+1 kF ? RM then M 10: Perform Frobenius norm projection: Mt+1 ? kMRt+1 kF Mt+1 . 11: end if 12: end for In Section 4, we provide the precise statement and proof of Theorem 1, the main regret bound for Algorithm 1, with some technical details deferred to the appendix. We also obtain analogous sample complexity results for batch learning; however, on account of some definitional subtleties, we defer all discussion of the offline case, including the statement and proof of Theorem 2, to Appendix A. We make one final interesting note here, from which the name wave-filtering arises: when plotted coordinate-wise, our filters {?j } look like the vibrational modes of an inhomogeneous spring (see Figure 1). We provide some insight on this phenomenon (along with some other implementation concerns) in Appendix B. Succinctly: in the scaling limit, (ZT /kZT k2 )T ?? commutes with a certain second-order Sturm-Liouville differential operator D. This allows us to approximate filters with eigenfunctions of D, using efficient numerical ODE solvers. 5 0.20 ?1 ?3 ?5 ?10 ?15 ?20 ?27 ?ODE (500) ?ODE (97) ?ODE (5000) 0.10 0.00 ?0.10 ?0.20 0 200 400 600 800 1000 (a) 0 200 400 600 800 1000 0 (b) 200 400 600 800 1000 (c) Figure 1: (a) The entries of some typical eigenvectors of Z1000 , plotted coordinate-wise. (b) ?27 of Z1000 (?27 ? 10?16 ) computed with finite-precision arithmetic, along with a numerical solution to the ODE in Appendix B.1 with ? = 97. (c) Some very high-order filters, computed using the ODE, would be difficult to obtain by eigenvector computations. 4 Analysis We first state the full form of the regret bound achieved by Algorithm 1:5 Theorem 1 (Main). On any sequence
{(xt , yt )}Tt=1 , Algorithm 1, with ? 2 k), and ? choice of k = ? log2 T log(R? Rx Ly n) , RM = ?(R? ? ?((Rx2 Ly log(R? Rx Ly n) n T log4 T )?1 ), achieves regret   ? 4 Regret(T ) ? O R? Rx2 Ly log2 (R? Rx Ly n) ? n T log6 T , a = competing with LDS predictors (A, B, C, D, h0 ) with 0 4 A 4 I and kBkF , kCkF , kDkF , kh0 k ? R? . Note that the dimensions m, d do not appear explicitly in this bound, though they typically factor into R? . In Section 4.1, we state and prove Theorem 3, the convex relaxation guarantee for the filters, which may be of independent interest. This allows us to approximate the optimal LDS in ? 7? y?t . In Section 4.2, we hindsight (the regret comparator) by the loss-minimizing matrix Mt : X complete the regret analysis using Theorem 3, along with bounds on the diameter and gradient, to conclude Theorem 1. Since the batch analogue is less general (and uses the same ideas), we defer discussion of Algorithm 2 and Theorem 2 to Appendix A. 4.1 Approximate convex relaxation via wave filters Assume for now that h0 = 0; we will remove this at the end, and see that the regret bound is asymptotically the same. Recall (from Section 2.2) that we measure regret compared to predictions obtained by adding the derivative of the impulse response function of an LDS ? to yt?1 . Our ? which produces approximately approximation theorem states that for any ?, there is some M? ? H the same predictions. Formally: Theorem 3 (Spectral convex relaxation for symmetric LDSs). Let {? yt }Tt=1 be the online predictions made by an LDS ? = (A, B, C, D, h0 = 0). Let R? = max{kBkF , kCkF , kDkF }. Then, for any 0 ? > 0, with a choice of k = ? (log T log(R? Rx Ly nT /?)), there exists an M? ? Rm?k such that T T X X ? t ? yt k2 ? kM? X k? yt ? yt k2 + ?. t=1 t=1 ? t are defined as in Algorithm 1 (noting that X ? t includes the previous ground truth Here, k 0 and X yt?1 ). 5 Actually, for a slightly tighter proof, we analyze a restriction of the algorithm which does not learn the portion M (y) , instead always choosing the identity matrix for that block. 6 Proof. We construct this mapping ? 7? M? explicitly. Write M? as the block matrix   0 M (1) M (2) ? ? ? M (k) M (x ) M (x) M (y) , ? t , the concatenated vector where the blocks? dimensions are chosen to align with X h i 1/4 1/4 1/4 ?1 (X ? ?1 )t ?2 (X ? ?2 )t ? ? ? ?k (X ? ?k )t xt?1 xt yt?1 , so that the prediction is the block matrix-vector product ?t = M? X k X j=1 0 1/4 ?j M (j) (X ? ?j )t + M (x ) xt?1 + M (x) xt + M (y) yt?1 . Without loss of generality, assume that A is diagonal, with entries {?l }dl=1 .6 Let bl be the l-th row of B, and cl the l-th column of C. Also, we define a continuous family of vectors ? : [0, 1] ? RT , with entries ?(?)(i) = (?l ? 1)?li?1 . Then, our construction is as follows: ? M (j) = Pd l=1 0 ? M (x ) = ?D, ?1/4 ?j h?j , ?(?l )i (cl ? bl ), for each 1 ? j ? k. M (x) = CB + D, M (y) = Im?m . Below, we give the main ideas for why this M? works, leaving the full proof to Appendix C. Since M (y) is the identity, the online learner?s task is to predict the differences yt ? yt?1 as well as the derivative ?, which we write here: y?t ? yt?1 = (CB + D)xt ? Dxt?1 + = (CB + D)xt ? Dxt?1 + T ?1 X i=1 T ?1 X i=1 C(Ai ? Ai?1 )Bxt?i C d X l=1 !
i?1 ?li ? ?l el ? el d T ?1 X X = (CB + D)xt ? Dxt?1 + (cl ? bl ) ?(?l )(i) xt?i . l=1 Bxt?i (4) i=1 Notice that the inner sum is an inner product between each coordinate of the past inputs (xt , xt?1 , . . . , xt?T ) with ?(?l ) (or a convolution, viewed across the entire time horizon). The crux of our proof is that one can approximate ?(?) using a linear combination of the filters {?j }kj=1 . Writing Z := ZT for short, notice that Z 1 Z= ?(?) ? ?(?) d?, 0 since the (i, j) entry of the RHS is Z 1 (? ? 1)2 ?i+j?2 d? = 0 2 1 1 ? + = Zij . i+j?1 i+j i+j+1 What follows is a spectral bound for reconstruction error, relying on the low approximate rank of Z: Lemma 4.1. Choose any ? ? [0, 1]. Let ? ?(?) be the projection of ?(?) onto the k-dimensional subspace of RT spanned by {?j }kj=1 . Then, v u T   p u X ?k/ log T k?(?) ? ? ?(?)k2 ? t6 ?j ? O c0 log T , j=k+1 for an absolute constant c0 > 3.4. ? B, ? C, ? D, h0 ) := Write the eigendecomposition A = U ?U T . Then, the LDS with parameters (A, T ? (?, BU, U C, D, h0 ) makes the same predictions as the original, with A diagonal. 6 7 ? t replaces each ?(?l ) in Equation (4) with its approximation ? By construction of M (j) , M? X ?(?l ). Hence we conclude that ? t = yt?1 + (CB + D)xt ? Dxt?1 + M? X = yt?1 + (? yt ? yt?1 ) + ?t d X l=1 (cl ? bl ) T ?1 X ? ?(?l )(i) xt?i i=1 = y?t + ?t , letting {?t } denote some residual vectors arising from discarding the subspace of dimension T ? k. Theorem 3 follows by showing that these residuals are small, using Lemma 4.1: it turns out that k?t k is exponentially small in k/ log T , which implies the theorem. 4.2 From approximate relaxation to low regret ? be its image under the map Let ?? ? H denote the best LDS predictor, and let M?? ? H ? t is within ? from that of from Theorem 3, so that total squared error of predictions M?? X def ? 2 ? t k are quadratic in M , and thus con? . Notice that the loss functions ft (M ) = kyt ? M X vex. Algorithm 1 runs online gradient descent [Zin03] on these loss functions, with decision set 0  def M = {M ? Rm?k  kM kF ? RM }. Let Dmax := supM,M 0 ?M kM ? M 0 kF be the diameter of M, and Gmax := supM ?M,X? k?ft (M )kF be the largest norm of a gradient. We can invoke the classic regret bound: ? , has Lemma 4.2 (e.g. Thm. 3.1 in [Haz16]). Online gradient descent, using learning rate GDmax max T regret T T X ? def X ft (M ) ? 2Gmax Dmax T . RegretOGD (T ) = ft (Mt ) ? min t=1 M ?M t=1 To finish, it remains to show that Dmax and Gmax are small. In particular, since the gradients contain convolutions of the input by `2 (not `1 ) unit vectors, special care must be taken to ensure that these do not grow too quickly. These bounds are shown in Section D.2, giving the correct ? By Theorem 3, M ? competes regret of Algorithm 1 in comparison with the comparator M ? ? H. arbitrarily closely with the best LDS in hindsight, concluding the theorem. Finally, we discuss why it is possible to relax the earlier assumption h0 = 0 on the initial hidden state. Intuitively, as more of the ground truth responses {yt } are revealed, the largest possible effect of the initial state decays. Concretely, in Section D.4, we prove that a comparator who chooses a 2 ? nonzero h0 can only increase the regret by an additive O(log T ) in the online setting. 5 Experiments In this section, to highlight the appeal of our provable method, we exhibit two minimalistic cases where traditional methods for system identification fail, while ours successfully learns the system. Finally, we note empirically that our method seems not to degrade in practice on certain wellbehaved nonlinear systems. In each case, we use k = 25 filters, and a regularized follow-the-leader variant of Algorithm 1 (see Appendix B.2). 5.1 Synthetic systems: two hard cases for EM and SSID We construct two difficult systems, on which we run either EM or subspace identification7 (SSID), followed by Kalman filtering to obtain predictions. Note that our method runs significantly (>1000 times) faster than this traditional pipeline. In the first example (Figure 2(a), left), we have a SISO system (n = m = 1) and d = 2; all xt , ?t , and ?t are i.i.d. Gaussians, and B > = C = [1 1], D = 0. Most importantly, A = diag ([0.999, 0.5]) is ill-conditioned, so that there are long-term dependences between input and output. Observe that although EM and SSID both find reasonable guesses for the system?s dynamics, they turns out to be local optima. Our method learns to predict as well as the best possible LDS. 7 Specifically, we use ?Deterministic Algorithm 1? from page 52 of [VODM12]. 8 Time series (xt , yt ) System 1: ill-conditioned SISO System 2: 10-dimensional MIMO 100 1 0 0 ?100 ?200 Error ||? yt ? yt ||2 10 xt yt xt (1) ?1 yt (1) ?2 3 102 101 100 10?1 10?2 10?3 EM 0 SSID 100 ours 200 300 EM y?t = yt?1 400 500 0 SSID 200 ours 400 600 y?t = yt?1 10?4 1000 800 (a) Two synthetic systems. For clarity, error plots are smoothed by a median filter. Left: Noisy SISO system with a high condition number; EM and SSID finds a bad local optimum. Right: High-dimensional MIMO system; other methods fail to learn any reasonable model of the dynamics. (xt , yt , y?t ) 0.5 0.0 xt ?0.5 0 yt 200 y?t 400 600 800 1000 (b) Forced pendulum, a physical simulation our method learns in practice, despite a lack of theory. Figure 2: Visualizations of Algorithm 1. All plots: blue = ours, yellow = EM, red = SSID, black = true responses, green = inputs, dotted lines = ?guess the previous output? baseline. Horizontal axis is time. The second example (Figure 2(a), right) is a MIMO system (with n = m = d = 10), also with Gaussian noise. The transition matrix A = diag ([0, 0.1, 0.2, . . . , 0.9]) has a diverse spectrum, the observation matrix C has i.i.d. Gaussian entries, and B = In , D = 0. The inputs xt are random block impulses. This system identification problem is high-dimensional and non-convex; it is thus no surprise that EM and SSID consistently fail to converge. 5.2 The forced pendulum: a nonlinear, non-symmetric system We remark that although our algorithm has provable regret guarantees only for LDSs with symmetric transition matrices, it appears in experiments to succeed in learning some non-symmetric (even nonlinear) systems in practice, much like the unscented Kalman filter [WVDM00]. In Figure 2(b), we provide a typical learning trajectory for a forced pendulum, under Gaussian noise and random block impulses. Physical systems like this are widely considered in control and robotics, suggesting possible real-world applicability for our method. 6 Conclusion We have proposed a novel approach for provably and efficiently learning linear dynamical systems. Our online wave-filtering algorithm attains near-optimal regret in theory; and experimentally outperforms traditional system identification in both prediction quality and running time. Furthermore, we have introduced a ?spectral filtering? technique for convex relaxation, which uses convolutions by eigenvectors of a Hankel matrix. We hope that this theoretical tool will be useful in tackling more general cases, as well as other non-convex learning problems. Acknowledgments We thank Holden Lee and Yi Zhang for helpful discussions. We especially grateful to Holden for a thorough reading of our manuscript, and for pointing out a way to tighten the result in Lemma C.1. 9 References [Aud14] Koenraad MR Audenaert. A generalisation of mirsky?s singular value inequalities. arXiv preprint arXiv:1410.4941, 2014. [BM02] Peter L Bartlett and Shahar Mendelson. Rademacher and gaussian complexities: Risk bounds and structural results. Journal of Machine Learning Research, 3(Nov):463? 482, 2002. [BT16] Bernhard Beckermann and Alex Townsend. On the singular values of matrices with displacement structure. arXiv preprint arXiv:1609.09494, 2016. [Cho83] Man-Duen Choi. Tricks or treats with the hilbert matrix. The American Mathematical Monthly, 90(5):301?312, 1983. [DHS11] John Duchi, Elad Hazan, and Yoram Singer. Adaptive subgradient methods for online learning and stochastic optimization. The Journal of Machine Learning Research, 12:2121?2159, 2011. [GH96] Zoubin Ghahramani and Geoffrey E Hinton. Parameter estimation for linear dynamical systems. Technical report, Technical Report CRG-TR-96-2, University of Toronto, Deptartment of Computer Science, 1996. [Gr?u82] F Alberto Gr?unbaum. A remark on hilbert?s matrix. Linear Algebra and its Applications, 43:119?124, 1982. [Haz16] Elad Hazan. Introduction to online convex optimization. Foundations and Trends in Optimization, 2(3-4):157?325, 2016. [Hil94] David Hilbert. Ein beitrag zur theorie des legendre?schen polynoms. Acta mathematica, 18(1):155?159, 1894. [HMR16] Moritz Hardt, Tengyu Ma, and Benjamin Recht. Gradient descent learns linear dynamical systems. arXiv preprint arXiv:1609.05191, 2016. [HSC+ 16] Wenbing Huang, Fuchun Sun, Lele Cao, Deli Zhao, Huaping Liu, and Mehrtash Harandi. Sparse coding and dictionary learning with linear dynamical systems. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3938?3947, 2016. [Kal60] Rudolph Emil Kalman. A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82.1:35?45, 1960. [KV05] Adam Kalai and Santosh Vempala. Efficient algorithms for online decision problems. Journal of Computer and System Sciences, 71(3):291?307, 2005. [Lju98] Lennart Ljung. System identification: Theory for the User. Prentice Hall, Upper Saddle Riiver, NJ, 2 edition, 1998. [Lju02] Lennart Ljung. Prediction error estimation methods. Circuits, Systems and Signal Processing, 21(1):11?21, 2002. [LW94] Nick Littlestone and Manfred K. Warmuth. The weighted majority algorithm. Information and Computation, 108(2):212?261, 1994. [Mar10a] James Martens. Learning the linear dynamical system with asos. In Johannes Frnkranz and Thorsten Joachims, editors, Proceedings of the 27th International Conference on Machine Learning, pages 743?750. Omnipress, 2010. [Mar10b] James Martens. Learning the linear dynamical system with asos. 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Z-Forcing: Training Stochastic Recurrent Networks Anirudh Goyal MILA, Universit? de Montr?al Alessandro Sordoni Microsoft Maluuba Nan Rosemary Ke MILA, Polytechnique Montr?al Marc-Alexandre C?t? Microsoft Maluuba Yoshua Bengio MILA, Universit? de Montr?al Abstract Many efforts have been devoted to training generative latent variable models with autoregressive decoders, such as recurrent neural networks (RNN). Stochastic recurrent models have been successful in capturing the variability observed in natural sequential data such as speech. We unify successful ideas from recently proposed architectures into a stochastic recurrent model: each step in the sequence is associated with a latent variable that is used to condition the recurrent dynamics for future steps. Training is performed with amortised variational inference where the approximate posterior is augmented with a RNN that runs backward through the sequence. In addition to maximizing the variational lower bound, we ease training of the latent variables by adding an auxiliary cost which forces them to reconstruct the state of the backward recurrent network. This provides the latent variables with a task-independent objective that enhances the performance of the overall model. We found this strategy to perform better than alternative approaches such as KL annealing. Although being conceptually simple, our model achieves state-of-the-art results on standard speech benchmarks such as TIMIT and Blizzard and competitive performance on sequential MNIST. Finally, we apply our model to language modeling on the IMDB dataset where the auxiliary cost helps in learning interpretable latent variables. 1 Introduction Due to their ability to capture long-term dependencies, autoregressive models such as recurrent neural networks (RNN) have become generative models of choice for dealing with sequential data. By leveraging weight sharing across timesteps, they can model variable length sequences within a fixed parameter space. RNN dynamics involve a hidden state that is updated at each timestep to summarize all the information seen previously in the sequence. Given the hidden state at the current timestep, the network predicts the desired output, which in many cases corresponds to the next input in the sequence. Due to the deterministic evolution of the hidden state, RNNs capture the entropy in the observed sequences by shaping conditional output distributions for each step, which are usually of simple parametric form, i.e. unimodal or mixtures of unimodal. This may be insufficient for highly structured natural sequences, where there is correlation between output variables at the same step, i.e. simultaneities (Boulanger-Lewandowski et al., 2012), and complex dependencies between variables at different timesteps, i.e. long-term dependencies. For these reasons, recent efforts recur to highly multi-modal output distribution by augmenting the RNN with stochastic latent variables trained by amortised variational inference, or variational auto-encoding framework (VAE) (Kingma and Welling, 2014; Fraccaro et al., 2016; Kingma and Welling, 2014). The VAE framework allows efficient approximate inference by parametrizing the approximate posterior and generative model with neural networks trainable end-to-end by backpropagation. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Another motivation for including stochastic latent variables in autoregressive models is to infer, from the observed variables in the sequence (e.g. pixels or sound-waves), higher-level abstractions (e.g. objects or speakers). Disentangling in such way the factors of variations is appealing as it would increase high-level control during generation, ease semi-supervised and transfer learning, and enhance interpretability of the trained model (Kingma et al., 2014; Hu et al., 2017). Stochastic recurrent models proposed in the literature vary in the way they use the stochastic variables to perform output prediction and in how they parametrize the posterior approximation for variational inference. In this paper, we propose a stochastic recurrent generative model that incorporates into a single framework successful techniques from earlier models. We associate a latent variable with each timestep in the generation process. Similar to Fraccaro et al. (2016), we use a (deterministic) RNN that runs backwards through the sequence to form our approximate posterior, allowing it to capture the future of the sequence. However, akin to Chung et al. (2015); Bayer and Osendorfer (2014), the latent variables are used to condition the recurrent dynamics for future steps, thus injecting highlevel decisions about the upcoming elements of the output sequence. Our architectural choices are motivated by interpreting the latent variables as encoding a ?plan? for the future of the sequence. The latent plan is injected into the recurrent dynamics in order to shape the distribution of future hidden states. We show that mixing stochastic forward pass, conditional prior and backward recognition network helps building effective stochastic recurrent models. The recent surge in generative models suggests that extracting meaningful latent representations is difficult when using a powerful autoregressive decoder, i.e. the latter captures well enough most of the entropy in the data distribution (Bowman et al., 2015; Kingma et al., 2016; Chen et al., 2017; Gulrajani et al., 2017). We show that by using an auxiliary, task-agnostic loss, we ease the training of the latent variables which, in turn, helps achieving higher performance for the tasks at hand. The latent variables in our model are forced to contain useful information by predicting the state of the backward encoder, i.e. by predicting the future information in the sequence. Our work provides the following contributions: ? We unify several successful architectural choices into one generative stochastic model for sequences: backward posterior, conditional prior and latent variables that condition the hidden dynamics of the network. Our model achieves state-of-the-art in speech modeling. ? We propose a simple way of improving model performance by providing the latent variables with an auxiliary, task-agnostic objective. In the explored tasks, the auxiliary cost yielded better performance than other strategies such as KL annealing. Finally, we show that the auxiliary signal helps the model to learn interpretable representations in a language modeling task. 2 Background We operate in the well-known VAE framework (Kingma and Ba, 2014; Burda et al., 2015; Rezende and Mohamed, 2015), a neural network based approach for training generative latent variable models. Let x be an observation of a random variable, taking values in X . We assume that the generation of x involves a latent variable z, taking values in Z, by means of a joint density p? (x, z), parametrized by ?. Given a set of observed datapoints D = {x1 , . . . , xn }, the goal of maximum likelihood estimation (MLE) is to estimate the parameters ? that maximize the marginal log-likelihood L(?; D): Z n X ?? = arg max? L(?; D) = log p? (xi , z) dz . (1) i=1 z Optimizing the marginal log-likelihood is usually intractable, due to the integration over the latent variables. A common approach is to maximize a variational lower bound on the marginal loglikelihood. The evidence lower bound (ELBO) is obtained by introducing an approximate posterior q? (z|x) yielding:  
p? (x, z) log p? (x) ? E log = log p(x) ? DKL q? (z|x) k p(z|x) = F(x; ?, ?), (2) q? (z|x) q? (z|x) where KL denotes the Kullback-Leibler divergence. The ELBO is particularly appealing because the bound is tight when the approximate posterior matches the true posterior, i.e. it reduces to the 2 ht?1 ht ht?1 zt zt xt xt dt?1 ht?1 ht zt?1 dt (a) STORN ht ht zt zt xt xt bt?1 (b) VRNN ht?1 bt (c) SRNN bt?1 bt (d) Our model Figure 1: Computation graph for generative models of sequences that use latent variables: STORN (Bayer and Osendorfer, 2014), VRNN (Chung et al., 2015), SRNN (Fraccaro et al., 2016) and our model. In this picture, we consider that the task of the generative model consists in predicting the next observation in the sequence, given previous ones. Diamonds represent deterministic states, zt and xt are respectively the latent variables and the sequence input at step t. Dashed lines represent the computation that is part of the inference model. Double lines indicate auxiliary predictions implied by the proposed auxiliary cost. Differently from VRNN and SRNN, in STORN and our model the latent variable zt participates to the prediction of the next step xt+1 . marginal log-likelihood. The ELBO can also be rewritten as a minimum description length loss function (Honkela and Valpola, 2004): h i
F(x; ?, ?) = E log p? (x|z) ? DKL q? (z|x) k p? (z) , (3) q? (z|x) where the second term measures the degree of dependence between x and z, i.e. if
DKL q? (z|x) k p? (z) is zero then z is independent of x. Usually, the parameters of the generative model p? (x|z), the prior p? (z) and the inference model q? (z|x) are computed using neural networks. In this case, the ELBO can be maximized by gradient ascent on a Monte Carlo approximation of the expectation. For particularly simple parametric forms of q? (z|x), e.g. multivariate diagonal Gaussian or, more generally, for reparamatrizable distributions (Kingma and Welling, 2014), one can backpropagate through the sampling process z ? q? (z|x) by applying the reparametrization trick, which simulates sampling from q? (z|x) by first sampling from a fixed distribution u,  ? u(), and then by applying deterministic transformation z = f? (x, ). This makes the approach appealing in comparison to other approximate inference approaches. In order to have a better generative model overall, many efforts have been put in augmenting the capacity of the approximate posteriors (Rezende and Mohamed, 2015; Kingma et al., 2016; Louizos and Welling, 2017), the prior distribution (Chen et al., 2017; Serban et al., 2017a) and the decoder (Gulrajani et al., 2017; Oord et al., 2016). By having more powerful decoders p? (x|z), one could model more complex distributions over X . This idea has been explored while applying VAEs to sequences x = (x1 , . . . , Q xT ), where the decoding distribution p? (x|z) is modeled by an autoregressive model, p? (x|z) = t p? (xt |z, x1:t?1 ) (Bayer and Osendorfer, 2014; Chung et al., 2015; Fraccaro et al., 2016). In these models, Q z typically decomposes as a sequence of latent variables, z = (z1 , . . . , zT ), yielding p? (x|z) = t p? (xt |z1:t?1 , x1:t?1 ). We operate in this setting and, in the following section, we present our choices for parametrizing the generative model, the prior and the inference model. 3 Proposed Approach In Figure 1, we report the dependencies in the inference and the generative parts of our model, compared to existing models. From a broad perspective, we use a backward recurrent network for the approximate posterior (akin to SRNN (Fraccaro et al., 2016)), we condition the recurrent state of the forward auto-regressive model with the stochastic variables and use a conditional prior (akin to VRNN (Chung et al., 2015), STORN (Bayer and Osendorfer, 2014)). In order to make better use 3 of the latent variables, we use auxiliary costs (double arrows) to force the latent variables to encode information about the future. In the following, we describe each of these components. 3.1 Generative Model Decoder Given a sequence of observations x = (x1 , . . . , xT ), and desired set of labels or predictions y = (y1 , . . . , yT ), we assume that there exists a corresponding set of stochastic latent variables z = (z1 , . . . , zT ). In the following, without loss of generality, we suppose that the set of predictions corresponds to a shifted version of the input sequence, i.e. the model tries to predict the next observation given the previous ones, a common setting in language and speech modeling (Fraccaro et al., 2016; Chung et al., 2015). The generative model couples observations and latent variables by using an autoregressive model, i.e. by exploiting a LSTM architecture (Hochreiter and Schmidhuber, 1997), that runs through the sequence: ? ? ht = f (xt , ht?1 , zt ). (4) The parameters of the conditional probability distribution on the next observation p? (xt+1 |x1:t , z1:t ) are computed by a multi-layered feed-forward network that conditions on ht , f (o) (ht ). In the case of continuous-valued observations, f (o) may output the ?, log ? parameters of a Gaussian distribution, or the categorical proportions in the case of one-hot predictions. Note that, even if f (o) is a simple unimodal distribution, the marginal distribution p? (xt+1 |x1:t ) may be highly multimodal, due to the integration over the sequence of latent variables z. Note that f (o) does not condition on zt , i.e. zt is not directly used in the computation of the output conditional probabilities. We observed better performance by avoiding the latent variables from directly producing the next output. Prior The parameters of the prior distribution p? (zt |x1:t , z1:t?1 ) over each latent variable are obtained by using a non-linear transformation of the previous hidden state of the forward network. A common choice in the VAE framework is to use Gaussian latent variables. Therefore, f (p) produces the parameters of a diagonal multivariate Gaussian distribution: (p) (p) (p) (p) p? (zt |x1:t , z1:t?1 ) = N (zt ; ?t , ?t ) where [?t , log ?t ] = f (p) (ht?1 ). (5) This type of conditional prior has proven to be useful in previous work (Chung et al., 2015). 3.2 Inference Model The inference model is responsible for approximating the true posterior over the latent variables p(z1 , . . . , zT |x) in order to provide a tractable lower-bound on the log-likelihood. Our posterior approximation uses a LSTM processing the sequence x backwards: ? ? bt = f (xt+1 , bt+1 ). (6) Each state bt contains information about the future of the sequence and can be used to shape the approximate posterior for the latent zt . As the forward LSTM uses zt to condition future predictions, the latent variable can directly inform the recurrent dynamics about the future states, acting as a ?plan? of the future in the sequence. This information is channeled into the posterior distribution by a feed-forward neural network f (q) taking as input both the previous forward state ht?1 and the backward state bt : (q) (q) q? (zt |x) = N (zt ; ?t , ?t ) where (q) (q) [?t , log ?t ] = f (q) (ht?1 , bt ). (7) By injecting stochasticity in the hidden state of the forward recurrent model, the true posterior distribution for a given variable zt depends on all the variables zt+1:T after zt through dependence on ht+1:T . In order to formulate an efficient posterior approximation, we drop the dependence on zt+1:T . This is at the cost of introducing intrinsic bias in the posterior approximation, e.g. we may exclude the true posterior from the space of functions modelled by our function approximator. This is in contrast with SRNN (Fraccaro et al., 2016), in which the posterior distribution factorizes in a tractable manner at the cost of not including the latent variables in the forward autoregressive dynamics, i.e. the latent variables don?t condition the hidden state, but only help in shaping a multi-modal distribution for the current prediction. 4 3.3 Auxiliary Cost In various domains, such as text and images, it has been empirically observed that it is difficult to make use of latent variables when coupled with a strong autoregressive decoder (Bowman et al., 2015; Gulrajani et al., 2017; Chen et al., 2017). The difficulty in learning meaningful latent variables, in many cases of interest, is related to the fact that the abstractions underlying observed data may be encoded with a smaller number of bits than the observed variables. For example, there are multiple ways of picturing a particular ?cat? (e.g. different poses, colors or lightning) without varying the more abstract properties of the concept ?cat?. In these cases, the maximum-likelihood training objective may not be sensitive to how well abstractions are encoded, causing the latent variables to ?shut off?, i.e. the local correlations at the pixel level may be too strong and bias the learning process towards finding parameter solutions for which the latent variables are unused. In these cases, the posterior approximation tends to provide a too weak or noisy signal, due to the variance induced by the stochastic gradient approximation. As a result, the decoder may learn to ignore z and instead to rely solely on the autoregressive properties of x, causing x and z to be independent, i.e. the KL term in Eq. 2 vanishes. Recent solutions to this problem generally propose to reduce the capacity of the autoregressive decoder (Bowman et al., 2015; Bachman, 2016; Chen et al., 2017; Semeniuta et al., 2017). The constraints on the decoder capacity inherently bias the learning towards finding parameter solutions for which z and x are dependent. One of the shortcomings with this approach is that, in general, it may be hard to achieve the desired solutions by architecture search. Instead, we investigate whether it is useful to keep the expressiveness of the autoregressive decoder but force the latent variables to encode useful information by adding an auxiliary training signal for the latent variables alone. In practice, our results show that this auxiliary cost, albeit simple, helps achieving better performance on the objective of interest. Specifically, we consider training an additional conditional R generative model of the backward states b = {b1 , . . . , bT } given the forward states p? (b|h) = z p? (b, z|h)dz ? Eq? (z|b,h) [log p? (b|z) + log p? (z|h) ? log q? (z|b, h)]. This additional model is also trained through amortized variational inference. However, we share its prior p? (z|h) and approximate posterior q? (z|b, h) with those of the ?primary? model (b is a deterministic function of x per Eq. 6 and the approximate posterior is conditioned Q on b). In practice, we solely learn additional parameters ? for the decoding model p? (b|z) = t p? (bt |zt ). The auxiliary reconstruction model trains zt to contain relevant information about the future of the sequence contained in the hidden state of the backward network bt : (a) (a) (a) (a) p? (bt |zt ) = N (?t , ?t ) where [?t , log ?t ] = f (a) (zt ), (8) By means of the auxiliary reconstruction cost, the approximate posterior and prior of the primary model is trained with an additional signal that may help with escaping local minima due to short term reconstructions appearing in the lower bound, similarly to what has been recently noted in Karl et al. (2016). 3.4 Learning The training objective is a regularized version of the lower-bound on the data log-likelihood based on the variational free-energy, where the regularization is imposed by the auxiliary cost: h i X L(x; ?, ?, ?) = E log p? (xt+1 |x1:t , z1:t ) + ? log p? (bt |zt ) q? (zt |x) (9) t
?DKL q? (zt |x1:T ) k p? (zt |x1:t , z1:t?1 ) . We learn the parameters of our model by backpropagation through time (Rumelhart et al., 1988) and we approximate the expectation with one sample from the posterior q? (z|x) by using reparametrization. When optimizing Eq. 9, we disconnect the gradients of the auxiliary prediction from affecting the backward network, i.e. we don?t use the gradients ?? log p? (bt |zt ) to train the parameters ? of the approximate posterior: intuitively, the backward network should be agnostic about the auxiliary task assigned to the latent variables. It also performed better empirically. As the approximate posterior is trained only with the gradient flowing through the ELBO, the backward states b may be receiving a weak training signal early in training, which may hamper the usefulness of the auxiliary generative cost, i.e. all the backward states may be concentrated around the zero vector. Therefore, 5 we additionally train the backward network to predict the output variables in reverse (see Figure 1): h i X L(x; ?, ?, ?) = E log p? (xt+1 |x1:t , z1:t ) + ? log p? (bt |zt ) + ? log p? (xt |bt ) q? (zt |x) (10) t
?DKL q? (zt |x1:T ) k p? (zt |x1:t , z1:t?1 ) . 3.5 Connection to previous models Our model is similar to several previous stochastic recurrent models: similarly to STORN (Bayer and Osendorfer, 2014) and VRNN (Chung et al., 2015) the latent variables are provided as input to the autoregressive decoder. Differently from STORN, we use the conditional prior parametrization proposed in Chung et al. (2015). However, the generation process in the VRNN differs from our approach. In VRNN, zt are directly used, along with ht?1 , to produce the next output xt . We found that the model performed better if we relieved the latent variables from producing the next output. VRNN has a ?myopic? posterior in such that the latent variables are not informed about the whole future in the sequence. SRNN (Fraccaro et al., 2016) addresses the issue by running a posterior backward in the sequence and thus providing future context for the current prediction. However, the autoregressive decoder is not informed about the future of the sequence through the latent variables. Several efforts have been made in order to bias the learning process towards parameter solutions for which the latent variables are used (Bowman et al., 2015; Karl et al., 2016; Kingma et al., 2016; Chen et al., 2017; Zhao et al., 2017). Bowman et al. (2015) tackle the problem in a language modeling setting by dropping words from the input at random in order to weaken the autoregressive decoder and by annealing the KL divergence term during training. We achieve similar latent interpolations by using our auxiliary cost. Similarly, Chen et al. (2017) propose to restrict the receptive field of the pixel-level decoder for image generation tasks. Kingma et al. (2016) propose to reserve some free bits of KL divergence. In parallel to our work, the idea of using a task-agnostic loss for the latent variables alone has also been considered in (Zhao et al., 2017). The authors force the latent variables to predict a bag-of-words representation of a dialog utterance. Instead, we work in a sequential setting, in which we have a latent variable for each timestep in the sequence. 4 Experiments In this section, we evaluate our proposed model on diverse modeling tasks (speech, images and text). We show that our model can achieve state-of-the-art results on two speech modeling datasets: Blizzard (King and Karaiskos, 2013) and TIMIT raw audio datasets (also used in Chung et al. (2015)). Our approach also gives competitive results on sequential generation on MNIST (Salakhutdinov and Murray, 2008). For text, we show that the the auxiliary cost helps the latent variables to capture information about latent structure of language (e.g. sequence length, sentiment). In all experiments, we used the ADAM optimizer (Kingma and Ba, 2014). 4.1 Speech Modeling and Sequential MNIST Blizzard and TIMIT We test our model in two speech modeling datasets. Blizzard consists in 300 hours of English, spoken by a single female speaker. TIMIT has been widely used in speech recognition and consists in 6300 English sentences read by 630 speakers. We train the model directly on raw sequences represented as a sequence of 200 real-valued amplitudes normalized using the global mean and standard deviation of the training set. We adopt the same train, validation and test split as in Chung et al. (2015). For Blizzard, we report the average log-likelihood for half-second sequences (Fraccaro et al., 2016), while for TIMIT we report the average log-likelihood for the sequences in the test set. In this setting, our models use a fully factorized multivariate Gaussian distribution as the output distribution for each timestep. In order to keep our model comparable with the state-of-the-art, we keep the number of parameters comparable to those of SRNN (Fraccaro et al., 2016). Our forward/backward networks are LSTMs with 2048 recurrent units for Blizzard and 1024 recurrent units for TIMIT. The dimensionality of the Gaussian latent variables is 256. The prior f (p) , inference f (q) and auxiliary networks f (a) have a single hidden layer, with 1024 units for Blizzard and 512 units for TIMIT, and use leaky rectified nonlinearities with leakiness 13 and clipped at ?3 (Fraccaro et al., 2016). For Blizzard, we use a learning rate of 0.0003 and batch size of 128, for TIMIT they are 6 Model Blizzard TIMIT RNN-Gauss RNN-GMM VRNN-I-Gauss VRNN-Gauss VRNN-GMM SRNN (smooth+resq ) 3539 7413 ? 8933 ? 9223 ? 9392 ? 11991 -1900 26643 ? 28340 ? 28805 ? 28982 ? 60550 Ours Ours + kla ? 14435 ? 14226 ? 68132 ? 68903 Ours + aux Ours + kla, aux ? 15430 ? 15024 Models DBN 2hl (Germain et al., 2015) NADE (Uria et al., 2016) EoNADE-5 2hl (Raiko et al., 2014) DLGM 8 (Salimans et al., 2014) DARN 1hl (Gregor et al., 2015) DRAW (Gregor et al., 2015) PixelVAE (Gulrajani et al., 2016) P-Forcing(3-layer) (Goyal et al., 2016) PixelRNN(1-layer) (Oord et al., 2016) PixelRNN(7-layer) (Oord et al., 2016) MatNets (Bachman, 2016) ? 69530 ? 70469 Ours(1 layer) Ours + aux(1 layer) MNIST ? 84.55 88.33 84.68 ? 85.51 ? 84.13 ? 80.97 ? 79.02H 79.58H 80.75 79.20H 78.50H ? 80.60 ? 80.09 Table 1: On the left, we report the average log-likelihood per sequence on the test sets for Blizzard and TIMIT datasets. ?kla? and ?aux? denote respectively KL annealing and the use of the proposed auxiliary costs. On the right, we report the test set negative log-likelihood for sequential MNIST, where H denotes lower performance of our model with respect to the baselines. For MNIST, we observed that KL annealing hurts overall performance. 0.001 and 32 respectively. Previous work reliably anneal the KL term in the ELBO via a temperature weight during training (KL annealing) (Fraccaro et al., 2016; Chung et al., 2015). We report the results obtained by our model by training both with KL annealing and without. When KL annealing is used, the temperature was linearly annealed from 0.2 to 1 after each update with increments of 0.00005 (Fraccaro et al., 2016). We show our results in Table 1 (left), along with results that were obtained by models of comparable size to SRNN. Similar to (Fraccaro et al., 2016; Chung et al., 2015), we report the conservative evidence lower bound on the log-likelihood. In Blizzard, the KL annealing strategy (Ours + kla) is effective in the first training iterations, but eventually converges to a slightly lower log-likelihood than the model trained without KL annealing (Ours). We explored different annealing strategies but we didn?t observe any improvements in performance. Models trained with the proposed auxiliary cost outperform models trained with KL annealing strategy in both datasets. In TIMIT, it appears that there is a slightly synergistic effect between KL annealing and auxiliary cost. Even if not explicitly reported in the table, similar performance gains were observed on the training sets. Sequential MNIST The task consists in pixel-by-pixel generation of binarized MNIST digits. We use the standard binarized MNIST dataset used in Larochelle and Murray (2011). Both forward and backward networks are LSTMs with one layer of 1024 hidden units. We use a learning rate of 0.001 and batch size of 32. We report the results in Table 1 (right). In this setting, we observed that KL annealing hurt performance of the model. Although being architecturally flat, our model is competitive with respect to strong baselines, e.g. DRAW (Gregor et al., 2015), and is outperformed by deeper version of autoregressive models with latent variables, i.e. PixelVAE (gated) (Gulrajani et al., 2016), and deep autoregressive models such as PixelRNN (Oord et al., 2016) and MatNets (Bachman, 2016). 4.2 Language modeling A well-known result in language modeling tasks is that the generative model tends to fit the observed data without storing information in the latent variables, i.e. the KL divergence term in the ELBO becomes zero (Bowman et al., 2015; Zhao et al., 2017; Serban et al., 2017b). We test our proposed stochastic recurrent model trained with the auxiliary cost on a medium-sized IMDB text corpus containing 350K movie reviews (Diao et al., 2014). Following the setting described in Hu et al. (2017), we keep only sentences with less than 16 words and fixed the vocabulary size to 16K words. We split the dataset into train/valid/test sets following these ratios respectively: 85%, 5%, 10%. Special delimiter tokens were added at the beginning and end of each sentence but we only learned to 7 3000 3500 2500 3000 2000 1500 ours ours + kla ours + aux ours + kla, aux 1000 500 0 1 2 Updates TIMIT 4000 KL (nats) KL (nats) Blizzard 3 4 2500 2000 1500 ours ours + kla ours + aux ours + kla, aux 1000 500 0 5 1e4 0 1 2 Updates 3 4 5 1e4 Figure 2: Evolution of the KL divergence term (measured in nats) in the ELBO with and without auxiliary cost during training for Blizzard (left) and TIMIT (right). We plot curves for models that performed best after hyper-parameter (KL annealing and auxiliary cost weights) selection on the validation set. The auxiliary cost puts pressure on the latent variables resulting in higher KL divergence. Models trained with the auxiliary cost (Ours + aux) exhibit a more stable evolution of the KL divergence. Models trained with auxiliary cost alone achieve better performance than using KL annealing alone (Ours + kla) and similar, or better performance for Blizzard, compared to both using KL annealing and auxiliary cost (Ours + kla, aux). Model Ours Ours + aux Ours + aux ?, ? 0 0.0025 0.005 Valid KL 0.12 3.03 9.82 Test ELBO IWAE ELBO IWAE 53.93 55.71 65.03 52.40 52.54 58.13 54.67 56.57 65.84 53.11 53.37 58.83 Table 2: IMDB language modeling results for models trained by maximizing the standard evidence lower-bound. We report word perplexity as evaluated by both the ELBO and the IWAE bound and KL divergence between approximate posterior and prior distribution, for different values of auxiliary cost hyperparameters ?, ?. The gap in perplexity between the ELBO and IWAE (evaluated with 25 samples) increases with greater KL divergence values. generate the end of sentence token. We use a single layered LSTM with 500 hidden recurrent units, fix the dimensionality of word embeddings to 300 and use 64 dimensional latent variables. All the f (?) networks are single-layered with 500 hidden units and leaky relu activations. We used a learning rate of 0.001 and a batch size of 32. Results are shown in Table 2. As expected, it is hard to obtain better perplexity than a baseline model when latent variables are used in language models. We found that using the IWAE (Importance Weighted Autoencoder) (Burda et al., 2015) bound gave great improvements in perplexity. This observation highlights the fact that, in the text domain, the ELBO may be severely underestimating the likelihood of the model: the approximate posterior may loosely match the true posterior and the IWAE bound can correct for this mismatch by tightening the posterior approximation, i.e. the IWAE bound can be interpreted as the standard VAE lower bound with an implicit posterior distribution (Bachman and Precup, 2015). On the basis of this observation, we attempted training our models with the IWAE bound, but observed no noticeable improvement on validation perplexity. We analyze whether the latent variables capture characteristics of language by interpolating in the latent space (Bowman et al., 2015). Given a sentence, we first infer the latent variables at each step by running the approximate posterior and then concatenate them in order to form a contiguous latent encoding for the input sentence. Then, we perform linear interpolation in the latent space between the latent encodings of two sentences. At each step of the interpolation, the latent encoding is run through the decoder network to generate a sentence. We show the results in Table 3. 8 this movie is so terrible . never watch ever a 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Argmax Sampling it ?s a movie that does n?t work ! it ?s a movie that does n?t work ! it ?s a movie that does n?t work ! it ?s a very powerful piece of film ! it ?s a very powerful story about it ! it ?s a very powerful story about a movie about life it ?s a very dark part of the film , eh ? it ?s a very dark movie with a great ending ! ! it ?s a very dark movie with a great message here ! it ?s a very dark one , but a great one ! it ?s a very dark movie , but a great one ! this film is more of a ? classic ? i give it a 5 out of 10 i felt that the movie did n?t have any i do n?t know what the film was about the acting is good and the acting is very good the acting is great and the acting is good too i give it a 7 out of 10 , kids the acting is pretty good and the story is great the best thing i ?ve seen before is in the film funny movie , with some great performances but the acting is good and the story is really interesting this movie is great . i want to watch it again ! (1 / 10) violence : yes . a 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Argmax Sampling greetings again from the darkness . ? oh , and no . ? oh , and it is . well ... i do n?t know . so far , it ?s watchable . so many of the characters are likable . so many of the characters were likable . so many of the characters have been there . so many of them have fun with it . so many of the characters go to the house ! so many of the characters go to the house ! greetings again from the darkness . ? let ?s screenplay it . rating : **** out of 5 . i do n?t know what the film was about ( pg-13 ) violence , no . just give this movie a chance . so far , but not for children so many actors were excellent as well . there are a lot of things to describe . so where ?s the title about the movie ? as much though it ?s going to be funny ! there was a lot of fun in this movie ! Table 3: Results of linear interpolation in the latent space. The left column reports greedy argmax decoding obtained by selecting, at each step of the decoding, the word with maximum probability under the model distribution, while the right column reports random samples from the model. a is the interpolation parameter. In general, latent variables seem to capture the length of the sentences. 5 Conclusion In this paper, we proposed a recurrent stochastic generative model that builds upon recent architectures that use latent variables to condition the recurrent dynamics of the network. We augmented the inference network with a recurrent network that runs backward through the input sequence and added a new auxiliary cost that forces the latent variables to reconstruct the state of that backward network, thus explicitly encoding a summary of future observations. The model achieves state-of-the-art results on standard speech benchmarks such as TIMIT and Blizzard. The proposed auxiliary cost, albeit simple, appears to promote the use of latent variables more effectively compared to other similar strategies such as KL annealing. In future work, it would be interesting to use a multitask learning setting, e.g. sentiment analysis as in (Hu et al., 2017). Also, it would be interesting to incorporate the proposed approach with more powerful autogressive models, e.g. PixelRNN/PixelCNN (Oord et al., 2016). Acknowledgments The authors would like to thank Phil Bachman, Alex Lamb and Adam Trischler for the useful discussions. AG and YB would also like to thank NSERC, CIFAR, Google, Samsung, IBM and Canada Research Chairs for funding, and Compute Canada and NVIDIA for computing resources. The authors would also like to express debt of gratitude towards those who contributed to Theano over the years (as it is no longer maintained), making it such a great tool. 9 References Bachman, P. (2016). An architecture for deep, hierarchical generative models. 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Stochastic Gradient VB and the Variational Auto-Encoder. 2nd International Conference on Learning Representationsm (ICLR), pages 1?14. Larochelle, H. and Murray, I. (2011). The neural autoregressive distribution estimator. In Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, pages 29?37. Louizos, C. and Welling, M. (2017). Multiplicative normalizing flows for variational bayesian neural networks. arXiv preprint arXiv:1703.01961. Oord, A. v. d., Kalchbrenner, N., and Kavukcuoglu, K. (2016). Pixel recurrent neural networks. arXiv preprint arXiv:1601.06759. Raiko, T., Li, Y., Cho, K., and Bengio, Y. (2014). Iterative neural autoregressive distribution estimator nade-k. In Advances in neural information processing systems, pages 325?333. Rezende, D. J. and Mohamed, S. (2015). Variational inference with normalizing flows. arXiv preprint arXiv:1505.05770. Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1988). 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A hierarchical latent variable encoder-decoder model for generating dialogues. In In Proc. of AAAI. Uria, B., C?t?, M.-A., Gregor, K., Murray, I., and Larochelle, H. (2016). Neural autoregressive distribution estimation. Journal of Machine Learning Research, 17(205):1?37. Zhao, T., Zhao, R., and Eskenazi, M. (2017). Learning discourse-level diversity for neural dialog models using conditional variational autoencoders. arXiv preprint arXiv:1703.10960. 11 

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Learning Hierarchical Information Flow with Recurrent Neural Modules Danijar Hafner ? Google Brain mail@danijar.com Alex Irpan Google Brain alexirpan@google.com James Davidson Google Brain jcdavidson@google.com Nicolas Heess Google DeepMind heess@google.com Abstract We propose ThalNet, a deep learning model inspired by neocortical communication via the thalamus. Our model consists of recurrent neural modules that send features through a routing center, endowing the modules with the flexibility to share features over multiple time steps. We show that our model learns to route information hierarchically, processing input data by a chain of modules. We observe common architectures, such as feed forward neural networks and skip connections, emerging as special cases of our architecture, while novel connectivity patterns are learned for the text8 compression task. Our model outperforms standard recurrent neural networks on several sequential benchmarks. 1 Introduction Deep learning models make use of modular building blocks such as fully connected layers, convolutional layers, and recurrent layers. Researchers often combine them in strictly layered or task-specific ways. Instead of prescribing this connectivity a priori, our method learns how to route information as part of learning to solve the task. We achieve this using recurrent modules that communicate via a routing center that is inspired by the thalamus. Warren McCulloch and Walter Pitts invented the perceptron in 1943 as the first mathematical model of neural information processing [22], laying the groundwork for modern research on artificial neural networks. Since then, researchers have continued looking for inspiration from neuroscience to identify new deep learning architectures [11, 13, 16, 31]. While some of these efforts have been directed at learning biologically plausible mechanisms in an attempt to explain brain behavior, our interest is to achieve a flexible learning model. In the neocortex, communication between areas can be broadly classified into two pathways: Direct communication and communication via the thalamus [28]. In our model, we borrow this latter notion of a centralized routing system to connect specializing neural modules. In our experiments, the presented model learns to form connection patterns that process input hierarchically, including skip connections as known from ResNet [12], Highway networks [29], and DenseNet [14] and feedback connections, which are known to both play an important role in the neocortex and improve deep learning [7, 20]. The learned connectivity structure is adapted to the task, allowing the model to trade-off computational width and depth. In this paper, we study these properties with the goal of building an understanding of the interactions between recurrent neural modules. ? Work done during an internship with Google Brain. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. x1 x2 x3 f1 f1 f1 f2 f2 x1 f1 f4 ? f3 ?1 y4 y3 (a) Module f 1 receives the task input, f 2 can be used for side computation, f 3 is trained on an auxiliary task, and f 4 produces the output for the main task. f2 ?2 f2 f3 f3 f3 y1 y2 y3 (b) Computation of 3 modules unrolled in time. One possible path of hierarchical information flow is highlighted in green. We show that our model learns hierarchical information flow, skip connections and feedback connections in Section 4. Figure 1: Several modules share their learned features via a routing center. Dashed lines are used for dynamic reading only. We define both static and dynamic reading mechanisms in Section 2.2. Section 2 defines our computational model. We point out two critical design axes, which we explore experimentally in the supplementary material. In Section 3 we compare the performance of our model on three sequential tasks, and show that it consistently outperforms multi-layer recurrent networks. In Section 4, we apply the best performing design to a language modeling task, where we observe that the model automatically learns hierarchical connectivity patterns. 2 Thalamus Gated Recurrent Modules We find inspiration for our work in the neurological structure of the neocortex. Areas of the neocortex communicate via two principal pathways: The cortico-cortico-pathway comprises direct connections between nuclei, and the cortico-thalamo-cortico comprises connections relayed via the thalamus. Inspired by this second pathway, we develop a sequential deep learning model in which modules communicate via a routing center. We name the proposed model ThalNet. 2.1 Model Definition Our system comprises a tuple of computation modules F = (f 1 , ? ? ? , f I ) that route their respective features into a shared center vector ?. An example instance of our ThalNet model is shown in Figure 1a. At every time step t, each module f i reads from the center vector via a context input cit and an optional task input xit . The features ?it = f i (cit , xit ) that each module produces are directed into the center ?.2 Output modules additionally produce task output from their feature vector as a function oi (?i ) = y i . All modules send their features to the routing center, where they are merged to a single feature vector ?t = m(?1t , ? ? ? , ?It ). In our experiments, we simply implement m as the concatenation of all ?i . At the next time step, the center vector ?t is then read selectively by each module using a reading mechanism to obtain the context input cit+1 = ri (?t , ?it ).3 This reading mechanism allows modules to read individual features, allowing for complex and selective reuse of information between modules. The initial center vector ?0 is the zero vector. 2 In practice, we experiment with both feed forward and recurrent implementations of the modules f i . For simplicity, we omit the hidden state used in recurrent modules in our notation. 3 The reading mechanism is conditioned on both ?t and ?it separately as the merging does not preserve ?it in the general case. 2 y ?1 ?2 ?3 ?4 ? ? x f2 = c Figure 2: The ThalNet model from the perspective of a single module. In this example, the module receives input xi and produces features to the center ? and output y i . Its context input ci is determined as a linear mapping of the center features from the previous time step. In practice, we apply weight normalization to encourage interpretable weight matrices (analyzed in Section 4). In summary, ThalNet is governed by the following equations: Module features: ?it = f i (cit , xit ) (1) Module output: yti = (2) Center features: ?t i ct+1 = Read context input: = i o (?it ) m(?1t , ? ? ? , ?It ) ri (?t , ?it ) (3) (4) The choice of input and output modules depends on the task at hand. In a simple scenario (e.g., single task), there is exactly one input module receiving task input, some number of side modules, and exactly one output module producing predictions. The output modules get trained using appropriate loss functions, with their gradients flowing backwards through the fully differentiable routing center into all modules. Modules can operate in parallel as reads target the center vector from the previous time step. An unrolling of the multi-step process can be seen in Figure 1b. This figure illustrates the ability to arbitrarily route between modules between time steps This suggest a sequential nature of our model, even though application to static input is possible by allowing observing the input for multiple time steps. We hypothesize that modules will use the center to route information through a chain of modules before producing the final output (see Section 4). For tasks that require producing an output at every time step, we repeat input frames to allow the model to process through multiple modules first, before producing an output. This is because communication between modules always spans a time step.4 2.2 Reading Mechanisms We now discuss implementations of the reading mechanism ri (?, ?i ) and modules f i (ci , xi ), as defined in Section 2.1. We draw a distinction between static and dynamic reading mechanisms for ThalNet. For static reading, ri (?) is conditioned on independent parameters. For dynamic reading, ri (?, ?i ) is conditioned on the current corresponding module state, allowing the model to adapt its connectivity within a single sequence. We investigate the following reading mechanisms: ? Linear Mapping. In its simplest form, static reading consists of a fully connected layer r(?, ?) = W ? with weights W ? R|c|?|?| as illustrated in Figure 2. This approach performs reasonably well, but can exhibit unstable learning dynamics and learns noisy weight matrices that are hard to interpret. Regularizing weights using L1 or L2 penalties does not help here since it can cause side modules to not get read from anymore. ? Weight Normalization. We found linear mappings with weight normalization [26] paW rameterization to be effective. For this, the context input is computed as r(?, ?) = ? |W |? with scaling factor ? ? R, weights W ? R|c|?|?| , and the Euclidean matrix norm |W |. 4 Please refer to Graves [8] for a study of a similar approach. 3 Normalization results in interpretable weights since increasing one weight pushes other, less important, weights closer to zero, as demonstrated in Section 4. ? Fast Softmax. To achieve dynamic routing, we condition the reading weight matrix on the current module features ?i . This can be seen as a form of fast weights, providing a biologically plausible method for attention [2, 27]. We then apply softmax normalization to the computed weights so that each element of the context is computed as a weighted average over center elements, rather than just a weighted sum. Specifically,
P|?| r(?, ?)(j) = e(W ?+b)(j) / k=1 e(W ?+b)(jk) ? with weights W ? R|?|?|?|?|c| , and biases b ? R|?|?|c| . While this allows for a different connectivity pattern at each time step, it introduces |?i + 1| ? |?| ? |ci | learned parameters per module. ? Fast Gaussian. As a compact parameterization for dynamic routing, we consider choosing each context element as a Gaussian weighted average of ?, with only mean and variance vectors learned conditioned on ?i . The context input is computed as r(?, ?)(j) =
f (1, 2, ? ? ? , |?|)|(W ? + b)(j) , (U ? + d)(j) ? with weights W, U ? R|c|?|?| , biases b, d ? R|c| , and the Gaussian density function f (x|?, ? 2 ). The density is evaluated for each index in ? based on its distance from the mean. This reading mechanism only requires |?i + 1| ? 2 ? |ci | parameters per module and thus makes dynamic reading more practical. Reading mechanisms could also select between modules on a high level, instead of individual feature elements. We do not explore this direction since it seems less biologically plausible. Moreover, we demonstrate that such knowledge about feature boundaries is not necessary, and hierarchical information flow emerges when using fine-grained routing (see Figure 4). Theoretically, this also allows our model to perform a wider class of computations. 3 Performance Comparison We investigate the properties and performance of our model on several benchmark tasks. First, we compare reading mechanisms and module designs on a simple sequential task, to obtain a good configuration for the later experiments. Please refer to the supplementary material for the precise experiment description and results. We find that the weight normalized reading mechanism provides best performance and stability during training. We will use ThalNet models with four modules of configuration for all experiments in this section. To explore the performance of ThalNet, we now conduct experiments on three sequential tasks of increasing difficulty: ? Sequential Permuted MNIST. We use images from the MNIST [19] data set, the pixels of every image by a fixed random permutation, and show them to the model as a sequence of rows. The model outputs its prediction of the handwritten digit at the last time step, so that it must integrate and remember observed information from previous rows. This delayed prediction combined with the permutation of pixels makes the task harder than the static image classification task, with a multi-layer recurrent neural network achieving ~65 % test error. We use the standard split of 60,000 training images and 10,000 testing images. ? Sequential CIFAR-10. In a similar spirit, we use the CIFAR-10 [17] data set and feed images to the model row by row. We flatten the color channels of every row so that the model observes a vector of 96 elements at every time step. The classification is given after observing the last row of the image. This task is more difficult than the MNIST task, as the image show more complex and often ambiguous objects. The data set contains 50,000 training images and 10,000 testing images. ? Text8 Language Modeling. This text corpus consisting of the first 108 bytes of the English Wikipedia is commonly used as a language modeling benchmark for sequential models. At every time step, the model observes one byte, usually corresponding to 1 character, encoded as a one-hot vector of length 256. The task it to predict the distribution of the next character in the sequence. Performance is measured in bits per character (BPC) PN computed as ? N1 i=1 log2 p(xi ). Following Cooijmans et al. [4], we train on the first 90% and evaluate performance on the following 5% of the corpus. For the two image classification tasks, we compare variations of our model to a stacked Gated Recurrent Unit (GRU) [3] network of 4 layers as baseline. The variations we compare are different 4 Sequential CIFAR-10 Testing Sequential Permuted MNIST Testing 0.6 20 40 60 Epochs 80 0.45 ThalNet GRU-FF ThalNet FF-GRU ThalNet GRU GRU Baseline ThalNet FF-GRU-FF ThalNet FF 0.40 0.35 100 Sequential Permuted MNIST Training Bits per character (BPC) ThalNet FF-GRU-FF ThalNet FF-GRU ThalNet FF ThalNet GRU-FF ThalNet GRU GRU Baseline 20 40 60 Epochs 80 100 Sequential CIFAR-10 Training 1.0 0.8 0.7 ThalNet FF-GRU ThalNet FF-GRU-FF ThalNet GRU-FF ThalNet FF ThalNet GRU GRU Baseline 0.6 0.5 20 40 60 Epochs 80 100 Accuracy (%) 0.9 0.55 0.50 GRU Baseline ThalNet GRU ThalNet FF-GRU ThalNet GRU-FF ThalNet FF-GRU-FF ThalNet FF 0.45 0.40 20 40 60 Epochs 80 1.46 GRU (1 step) GRU (2 steps) ThalNet (2 steps) 1.44 1.42 1.40 10 20 30 Epochs 40 50 Text8 Language Modeling Training 0.60 100 Bits per character (BPC) 0.7 0.5 Accuracy (%) Text8 Language Modeling Evaluation 0.50 0.8 Accuracy (%) Accuracy (%) 0.9 1.45 1.40 1.35 1.30 1.25 1.20 ThalNet (2 steps) GRU (1 step) GRU (2 steps) 10 20 30 Epochs 40 50 Figure 3: Performance on the permuted sequential MNIST, sequential CIFAR, and text8 language modeling tasks. The stacked GRU baseline reaches higher training accuracy on CIFAR, but fails to generalize well. On both tasks, ThalNet clearly outperforms the baseline in testing accuracy. On CIFAR, we see how recurrency within the modules speeds up training. The same pattern is shows for the text8 experiment, where ThalNet using 12M parameters matches the performance of the baseline with 14M parameters. The step number 1 or 2 refers to repeated inputs as discussed in Section 2. We had to smooth the graphs using a running average since the models were evaluated on testing batches on a rolling basis. choices of feed-forward layers and GRU layers for implementing the modules f i (ci , xi ): We test with two fully connected layers (FF), a GRU layer (GRU), fully connected followed by GRU (FF-GRU), GRU followed by fully connected (GRU-FF), and a GRU sandwiched between fully connected layers (FF-GRU-FF).5 For all models, we pick the largest layer sizes such that the number of parameters does not exceed 50,000. Training is performed for 100 epochs on batches of size 50 using RMSProp [30] with a learning rate of 10?3 . For language modeling, we simulate ThalNet for 2 steps per token, as described in Section 2 to allow the output module to read information about the current input before making its prediction. Note that on this task, our model uses only half of its capacity directly, since its side modules can only integrate longer-term dependencies from previous time steps. We run the baseline once without extra steps and once with 2 steps per token, allowing it to apply its full capacity once and twice on each token, respectively. This makes the comparison a bit difficult, but only by favouring the baseline. This suggests that architectural modifications, such as explicit skip-connections between modules, could further improve performance. The Text8 task requires larger models. We train ThalNet with 4 modules of a size 400 feed forward layer and a size 600 GRU layer each, totaling in 12 million model parameters. We compare to a standard baseline in language modeling, a single GRU with 2000 units, totaling in 14 million parameters. We train on batches of 100 sequences, each containing 200 bytes, using the Adam optimizer [15] with a default learning rate of 10?3 . We scale down gradients exceeding a norm of 1. Results for 50 epochs of training are shown in Figure 3. The training took about 8 days for ThalNet with 2 steps per token, 6 days for the baseline with 2 steps per token, and 3 days for the baseline without extra steps. Figure 3 shows the training and testing and training curves for the three tasks described in this section. ThalNet outperforms standard GRU networks in all three tasks. Interestingly, ThalNet experiences a 5 Note that the modules require some amount of local structure to allow them to specialize. Implementing the modules as a single fully connected layer recovers a standard recurrent neural network with one large layer. 5 much smaller gap between training and testing performance than our baseline ? a trend we observed across all experimental results. On the Text8 task, ThalNet scores 1.39 BPC using 12M parameters, while our GRU baseline scores 1.41 BPC using 14M parameters (lower is better). Our model thus slightly improves on the baseline while using fewer parameters. This result places ThalNet in between the baseline and regularization methods designed for language modeling, which can also be applied to our model. The baseline performance is consistent with published results of LSTMs with similar number of parameters [18]. We hypothesize the information bottleneck at the reading mechanism acting as an implicit regularizer that encourages generalization. Compared to using one large RNN that has a lot of freedom of modeling the input-output mapping, ThalNet imposes local structure to how the input-output mapping can be implemented. In particular, it encourages the model to decompose into several modules that have stronger intra-connectivity than extra-connectivity. Thus, to some extend every module needs to learn a self-contained computation. 4 Hierarchical Connectivity Patterns Using its routing center, our model is able to learn its structure as part of learning to solve the task. In this section, we explore the emergent connectivity patterns. We show that our model learns to route features in hierarchical ways as hypothesized, including skip connections and feedback connections. For this purpose, we choose the text8 corpus, a medium-scale language modeling benchmark consisting of the first 108 bytes of Wikipedia, preprocessed for the Hutter Prize [21]. The model observes one one-hot encoded byte per time step, and is trained to predict its future input at the next time step. We use comparably small models to be able to run experiments quickly, comparing ThalNet models of 4 FF-GRU-FF modules with layer sizes 50, 100, 50 and 50, 200, 50. Both experiments use weight normalized reading. Our focus here is on exploring learned connectivity patterns. We show competitive results on the task using larger models in Section 3. We simulate two sub time steps to allow for the output module to receive information of the current input frame as discussed in Section 2. Models are trained for 50 epochs on batches of size 10 containing sequences of length 50 using RMSProp with a learning rate of 10?3 . In general, we observe different random seeds converging to similar connectivity patterns with recurring elements. 4.1 Trained Reading Weights Figure 4 shows trained reading weights for various reading mechanisms, along with their connectivity graphs that were manually deduced.6 Each image represents a reading weight matrix for the modules 1 to 4 (top to bottom). Each pixel row shows the weight factors that get multiplied with ? to produce a single element of the context vector of that module. The weight matrices thus has dimensions of |?| ? |ci |. White pixels represent large magnitudes, suggesting focus on features at those positions. The weight matrices of weight normalized reading clearly resemble the boundaries of the four concatenated module features ?1 , ? ? ? , ?4 in the center vector ?, even though the model has no notion of the origin and ordering of elements in the center vector. A similar structure emerges with fast softmax reading. These weight matrices are sparser than the weights from weight normalization. Over the course of a sequence, we observe some weights staying constant while others change their magnitudes at each time step. This suggests that optimal connectivity might include both static and dynamic elements. However, this reading mechanism leads to less stable training. This problem could potentially alleviated by normalizing the fast weight matrix. With fast Gaussian reading, we see that the distributions occasionally tighten on specific features in the first and last modules, the modules that receive input and emit output. The other modules learn large variance parameters, effectively spanning all center features. This could potentially be addressed by reading using mixtures of Gaussians for each context element instead. We generally find that weight normalized and fast softmax reading select features with in a more targeted way. 6 Developing formal measurements for this deduction process seems beneficial in the future. 6 skip connection skip connection feedback connection x 1 2 3 4 y x skip connection (a) Weight Normalization 1 2 3 4 y feedback connection (b) Fast Softmax (c) Fast Gaussian Figure 4: Reading weights learned by different reading mechanisms with 4 modules on the text8 language modeling task, alongside manually deducted connectivity graphs. We plot the weight matrices that produce the context inputs to the four modules, top to bottom. The top images show focus of the input modules, followed by side modules, and output modules at the bottom. Each pixel row gets multiplied with the center vector ? to produce one scalar element of the context input ci . We visualize the magnitude of weights between the 5 % to the 95 % percentile. We do not include the connectivity graph for Fast Gaussian reading as its reading weights are not clearly structured. 4.2 Commonly Learned Structures The top row in Figure 4 shows manually deducted connectivity graphs between modules. Arrows represent the main direction of information flow in the model. For example, the two incoming arrows to module 4 in Figure 4a indicate that module 4 mainly attends to features produced by modules 1 and 3. We infer the connections from the larger weight magnitudes in the first and third quarters of the reading weights for module 4 (bottom row). A typical pattern that emerges during the experiments can be seen in the connectivity graphs of both weight normalized and fast softmax reading (Figures 4a and 4b). Namely, the output module reads features directly from the input module. This direction connection is established early on during training, likely because this is the most direct gradient path from output to input. Later on, the side modules develop useful features to support the input and output modules. In another pattern, one module reads from all other modules and combines their information. In Figure 4b, module 2 takes this role, reading from modules 1, 3, 4, and distributing these features via the input module. In additional experiments with more than four modules, we observed this pattern to emerge predominantly. This connection pattern provides a more efficient way of information sharing than cross-connecting all modules. Both connectivity graphs in Figure 4 include hierarchical computation paths through the modules. They include learn skip connections, which are known to improve gradient flow from popular models such as ResNet [12], Highway networks [29], and DenseNet [14]. Furthermore, the connectivity graphs contain backward connections, creating feedback loops over two or more modules. Feedback connections are known to play a critical role in the neocortex, which inspired our work [7]. 5 Related Work We describe a recurrent mixture of experts model, that learns to dynamically pass information between the modules. Related approaches can be found in various recurrent and multi-task methods as outlined in this section. 7 Modular Neural Networks. ThalNet consists of several recurrent modules that interact and exploit each other. Modularity is a common property of existing neural models. [5] learn a matrix of tasks and robot bodies to improve both multitask and transfer learning. [1] learn modules modules specific to objects present in the scene, which are selected by an object classifier. These approaches specify modules corresponding to a specific task or variable manually. In contrast, our model automatically discovers and exploits the inherent modularity of the task and does not require a one-to-one correspondence of modules to task variables. The Column Bundle model [23] consists of a central column and several mini-columns around it. While not applied to temporal data, we observe a structural similarity between our modules and the mini-columns, in the case where weights are shared among layers of the mini-columns, which the authors mention as a possibility. Learned Computation Paths. We learn the connectivity between modules alongside the task. There are various methods in the multi-task context that also connectivity between modules. Fernando et al. [6] learn paths through multiple layers of experts using an evolutionary approach. Rusu et al. [25] learn adapter connections to connect to fixed previously trained experts and exploit their information. These approaches focus on feed-forward architectures. The recurrency in our approach allows for complex and flexible computational paths. Moreover, we learn interpretable weight matrices that can be examined directly without performing costly sensitivity analysis. The Neural Programmer Interpreted presented by Reed and De Freitas [24] is related to our dynamic gating mechanisms. In their work, a network recursively calls itself in a parameterized way to perform tree-shaped computations. In comparison, our model allows for parallel computation between modules and for unrestricted connectivity patterns between modules. Memory Augmented RNNs. The center vector in our model can be interpreted as an external memory, with multiple recurrent controllers operating on it. Preceding work proposes recurrent neural networks operating on external memory structures. The Neural Turing Machine proposed by Graves et al. [9], and follow-up work [10], investigate differentiable ways to address a memory for reading and writing. In the ThalNet model, we use multiple recurrent controllers accessing the center vector. Moreover, our center vector is recomputed at each time step, and thus should not be confused with a persistent memory as is typical for model with external memory. 6 Conclusion We presented ThalNet, a recurrent modular framework that learns to pass information between neural modules in a hierarchical way. Experiments on sequential and permuted variants of MNIST and CIFAR-10 are a promising sign of the viability of this approach. In these experiments, ThalNet learns novel connectivity patterns that include hierarchical paths, skip connections, and feedback connections. In our current implementation, we assume the center features to be a vector. Introducing a matrix shape for the center features would open up ways to integrate convolutional modules and similaritybased attention mechanisms for reading from the center. While matrix shaped features are easily interpretable for visual input, it is less clear how this structure will be leveraged for other modalities. A further direction of future work is to apply our paradigm to tasks with multiple modalities for inputs and outputs. It seems natural to either have a separate input module for each modality, or to have multiple output modules that can all share information through the center. We believe this could be used to hint specialization into specific patterns and create more controllable connectivity patterns between modules. Similarly, we an interesting direction is to explore the proposed model can be leveraged to learn and remember a sequence of tasks. We believe modular computation in neural networks will become more important as researchers approach more complex tasks and employ deep learning to rich, multi-modal domains. Our work provides a step in the direction of automatically organizing neural modules that leverage each other in order to solve a wide range of tasks in a complex world. 8 References [1] J. Andreas, M. Rohrbach, T. Darrell, and D. Klein. Neural module networks. In IEEE Conference on Computer Vision and Pattern Recognition, pages 39?48, 2016. [2] J. Ba, G. E. Hinton, V. Mnih, J. Z. Leibo, and C. Ionescu. Using fast weights to attend to the recent past. In Advances in Neural Information Processing Systems, pages 4331?4339, 2016. [3] K. Cho, B. van Merri?nboer, D. Bahdanau, and Y. Bengio. On the properties of neural machine translation: Encoder?decoder approaches. Syntax, Semantics and Structure in Statistical Translation, page 103, 2014. [4] T. Cooijmans, N. Ballas, C. Laurent, ?. G?l?ehre, and A. Courville. 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Implementing Intelligence on Silicon Using Neuron-Like Functional MOS Transistors Tadashi Shibata t Koji Kotani t Takeo Yamashita t Hiroshi Ishii Hideo Kosaka t and Tadahiro Ohmi Department of Electronic Engineering Tohoku University Aza-Aoba, Aramaki, Aobaku, Sendai 980 lAPAN Abstract We will present the implementation of intelligent electronic circuits realized for the first time using a new functional device called Neuron MOS Transistor (neuMOS or vMOS in short) simulating the behavior of biological neurons at a single transistor level. Search for the most resembling data in the memory cell array, for instance, can be automatically carried out on hardware without any software manipulation. Soft Hardware, which we named, can arbitrarily change its logic function in real time by external control signals without any hardware modification. Implementation of a neural network equipped with an on-chip self-learning capability is also described. Through the studies of vMOS intelligent circuit implementation, we noticed an interesting similarity in the architectures of vMOS logic circuitry and biological systems. 1 INTRODUCTION The motivation of this work has stemmed from the invention of a new functional transistor which simulates the behavior of biological neurons (Shibata and Ohmi, 1991; 1992a). The transistor can perfOlID weighted summation of multiple input signals and squashing on the sum all at a single transistor level. Due to its functional similarity, the transistor was named Neuron MOSFET (abbreviated as neuMOS or vMOS). What is of significance with this new device is that a number of new architecture electronic circuits can be build using vMOS' which are different from conventional ones both in operational principles and functional capabilities. They are charactetized by a high degree of parallelism in hardware computation, a large flexibility in hardware configuration and a dramatic reduction in the circuit complexity as compared to conventional integrated 919 920 Shibata, Kotani, Yamashita, Ishii, Kosaka, and Ohmi circuits. During the course of studies in exploring vMOS circuit applications an interesting similarity has been noticed between the basic vMOS logic circuit architecture and the common structure found in biological neuronal systems, i. e., the competitive processes of excitatory and inhibitory connections. The purpose of this article is to demonstrate how powerful the neuron-like functionality in an elemental device is in implementing intelligent functions in silicon integrated circuits. 2 NEURON MOSFET AND SOFT-HARDWARE LOGIC CIRCUITS The symbolic representation of a vMOS is given in Fig. 1. A vMOS is a regular MOS transistor except that its gate electrode is made electrically floating and multiple input terminals are capacitively coupled to the floating gate. The potential of the floating gate ~ is determined as a linear weighted sum of multiple input voltages where each weighting factor is given by the magnitude of a coupling capacitance. When <l>F' the weighted sum, exceeds the threshold voltage of the transistor, it turns on. Thus the function of a neuron model (McCulloch and Pitts, 1943) has been directly implemented in a simple transistor structure. vMOS transistors were fabricated using the doublepolysilicon gate technology and a CMOS process was employed for vMOS integrated circuits fabrication. It should be noted here that no floating-gate charging effect was employed in the operation of vMOS logic circuits. v2 V, 4>F " 1.1.---------J. _...-J,I ~ SOURCE vn '" FLOATING DRAIN Cl>F- c1 v.1 +C2 V.2 +?????+Cn Vn ) V ~ Cror GATE Transistor "Turns ON" Figure 1: Schematic of a neuron MOS transistor. Since the weighting factors in a vMOS are detennilled by the overlapping areas of the first poly (floating gate) and second poly (input gate) patterns, they are not alterable. For this reason, in vMOS applications to self-learning neural network synthesis, a synapse cell circuit was provided to each input temlinal of a vMOS to represent an alterable connection strength. Here the plasticity of a synaptic weight was created by charging/discharging of the floating-gate in a vMOS synapse circuitry as described in 4. TheI-Vcharacteristics ofa two-input-gate vMOS having identical coupling capacitances are shown in Fig. 2, where one of the input gates is used as a gate terminal and the other as a threshold-control terminal. The apparent threshold voltage as seen from the gate terminal is changed from a depletion-mode to an enhancement-mode threshold by the voltage given to the control terminal. This variable threshold nature of a vMOS, we believe, is most essential in creating flexibility in electronic hardware systems. Figure 3(a) shows a two-input-variable Soft Hardu:are Logic (SHL) circuit which can represent all possible sh.1een Boolean functions for two binary inputs Xl and X2 by adjusting the control signals VA' VB and Ve. The inputs, Xl and X2, are directly coupled to the floating gate of a complementary vMOS inverter in the output stage with a 1:2 coupling ratio. The vMOS inve11er, which we call the main inve11er, deternlines the logic. Implementing Intelligence on Silicon Using Neuron-Like Functional MOS Transistors -2.5 0.0 2.5 50 Figure 2: Measured characteristics of a variable threshold transistor. Voltage at the threshold-control tenninal was varied from +5V to -5V (from left to right). GATE VOLTAGE (V) x , o---.,----t X2 o--~----t Oms 2ms/div 20ms (a) (b) Figure 3: Two-input-variable soft hardware logic circuit(a) and measured characteristics(b). The slow operation is due to the loading effect. (The test circuit has no output buffers.) The inputs are also coupled to the main inve11er via three inter-stage vMOS inverters (pre-inverters). When the analog variable represented by the binary inputs Xl and X2 increases~ the inputs tend to turn on the main inverter via direct connection~ while the indirect connection via pre-inverters tend to turn off the main invelter because preinverter outputs change from VDO to 0 when they turn on. This competitive process creates logics. The turn-on thresholds of pre-inverters are made alterable by control signals utilizing the variable threshold characteristics of vMOS'. Thus the real-time alteration of logic functions has been achieved and are demonstrated by experiments in Fig. 3(b). With the basic circuit architecture of the two-staged vMOS inverter configuration shown in Fig. 3(a)~ any Boolean function can be generated. We found the inverting connections via preinverters are most essential in logic synthesis. The structure indicates an interesting similarity to neuronal functional modules in which intramodular inhibitory connections play essential roles. Fixed function logics can be generated much more simply using the basic two-staged structure~ resulting in a dramatic reduction in transistor counts and interconnections. It has been demonstrated that a full adder~ 3-b and 4-b NO conve11ers can be constructed only with 8~ 16 and 28 transistors~ respectively, which should be compared to 30~ 174 and 398 transistors by conventional CMOS design, respectively. The details on vMOS circuit design are desClibed in Refs. (Shibata and Ohmi, 1993a; 1993b) and experimental verification in Ref. (Kotani et al.~ 1992). 921 922 Shibata, Kotani, Yamashita, Ishii, Kosaka, and Ohmi Voo p '" y ( 60 VOUT \:c '" ~ :::. p X Analog Inverter ~ '" ~1 :::. INPUT '"1 '" -01 ~ N r ~ It) N OUTPUT +-+ 2msec/dlv +--+ Smsec/dlv (a) (b) (c) Figure 4: Real-time rule-variable data matching circuit (a) and measured wave forms (b & c). In (c), l) is changed as 0.5, 1, 1.5, and 2 [V] from top to bottom. A unique vMOS circuit based on the basic structure of Fig. 3(a) is the real-time rulevariable data matching circuitry shown in Fig. 4(a). The circuit output becomes high when X - y < l). X is the input data, Y the template data and l) the window width for data matching where X, Y and l) are all time variables. Measured data are shown in Figs. 4(b) and 4(c), where it is seen the triple peaks are merged into a single peak as l) increases (Shibata et al., 1993c). The circuit is composed of only 10 vMOS' and can be easily integrated with each pixel on a image sensor chip. If vMOS circuitry is combined with a bipolar image sensor cell having an amplification function (fanaka et al., 1989), for instance, in situ image processing such as edge detection and variable-template matching would become possible, leading to an intelligent image sensor chip. I I 3 BINARY-MULTIVALUED-ANALOG MERGED HARDWARE COMPUTATION A winner-take-all circuit (WTA) implemented by vMOS circuitry is given in Fig. 5. Each cell is composed of a vMOS variable threshold inverter in which the apparent threshold is modified by an analog input signals VA - Vc' When the common input signal VR is ramped up, the lowest threshold cell (a cell receiving the largest analog input) turns on firstly, at which instance a feedback loop is formed in each cell and the state of the cell is self-latched. As a result, only the winner cell yields an output of 1. The circuit has been applied to building an associative memory as demonstrated in Fig. 6. The binary data stored in a SRAM cell array are all simultaneously matched to the sample data by taking XNOR, and the number of matched bits are transfeITed to the floating gate of each WfA cell by capacitance coupling. The WI'A action finds the location of data having the largest number of matched bits. This principle has been also applied to an sorting circuitry (Yamashita et aI., 1993). In these circuits all computations are conducted by an algOlithm directly imbedded in the hardware. Such an analog-digital merged hardware computation algorithm is a key to implement intelligent data processing architecture on silicon. A multivalued DRAM cell equipped with the association function and a multivalued SRAM cell having self-quantizing and self-classification functions have been also developed based on the binary-multivalued-analog merged hardware algorithm (Rita et aI., 1994). Implementing Intelligence on Silicon Using Neuron-Like Functional MOS Transistors WINNER LATCH INITIAL ~1 ~o ~o V~o V'~ V~o VR VR V~o TIME CONTROL SIGNAL Figure 5: Operational principle of vMOS Winner-Take-All circuit. - 0 0 1 0 0 0 + o t t ! t 1 SAMPLE DATA fr ~ ~ g] a c: =ti c: ...., .... .... - WINNER- TAKE-ALL NETWORK (a) (b) Figure 6: vMOS associative memory: (a) circuit diagram; (b) photomicrograph of a test chip. 4 HARDWARE SELF-LEARNING NEURAL NETWORKS Since vMOS itself has the basic function of a neuron, a neuron cell is very easily implemented by a complementary vMOS inve11er. The learning capability of a neural network is due to the plasticity of synaptic connections. Therefore its circuit implementation is a key issue. A stand-by power dissipation free synapse circuit which has been developed using vMOS circuitry is shown in Fig. 7(a). The circuit is a differential pair of N-channel and P-channel vMOS source followers sharing the same floating gate, which are both merged into CMOS inverters to cut off dc cunent paths. When the pre-synaptic neuron fires, both source followers are activated. Then the analog weight value stored as charges in the common floating gate is read out and transferred to the floating gate (dendrite) of the post-synaptic neuron by capacitance coupling as shown in Figs. 7 (b) and (c). The outputs of N-vMOS (V+) and P-vMOS (V-) source followers are averaged at the dendrite level, yielding an effective synapse output equal to (V+ + V-)/2. The synapse can represent both positive (excitatory) and negative (inhibitory) weights depending on whether the effective output is larger or smaller than Vnrj2, respectively. The operation of the synapse cell is demonstrated in Fig. 8(a). 923 924 Shibata, Kotani, Yamashita, Ishii, Kosaka, and Ohmi Previous Neuron at Rest Previous Neuron Fired v"" v.. P-UMOS ? I v- tunneling electrode V- V- Voo V =." 2 7 T FLOATING GATE (DENDRITE) OF NEURON (a) (b) (c) Figure 7: Synapse cell circuit implemented by vMOS circuitry. 3~----------------~ Previous-Layer Neuron Fired 2 1 VDD VI ......... >,0 "';-1 :>-2 0 Voe v- f 2 Veo Voe 2 0 -3 -4 EXCITATORY Ylm. 0 CONVENTIONAL CELL V+ v- 0 V+ V+ + V- 2 t= 4 NEW CELL 3 ~2 INHIBITORY x -~ V" + V2 :> 1 0 -1 -2 _3L-~L-~--~---L--~ 400nsec o 5 10 15 20 25 Number of Programming Pulses (b) (a) Figure 8: (a) Measured synapse cell output characteristics; (b) weight updating characteristics as represented by N-vMOS threshold with (our new cell:bottom) or without (conventional EEPROM cell: top) feed back. The weight updating is conducted by giving high programming pulses to both Vx and Vy tenninals. (Their coupling capacitances are made much larger than others). Then the common floating gate is pulled up to the programming voltage~ allowing electrons to flow into the floating gate via Fowler-Nordheim tunneling. When either Vx or Vy is low, tunneling injection does not occur because the tunneling current is very sensitive to the electric field intensity, being exponentially dependent upon the tunnel oxide field (Hieda Implementing Intelligence on Silicon Using Neuron-Like Functional MOS Transistors et al.~ 1985). The data updating occurs only at the crossing point of Vx and Vy lines~ allowing Hebb-rule-like learning directly implemented on the hardware (Shibata and Ohmi~ 1992b). Hardware-Backpropagation (HEP) learning algorithm~ which is a simplified version of the original BP ~ has been also developed in order to facilitate its hardware implementation (Ishii et al.~ 1992) and has been applied to build self-learning vMOS neural networks (Ishii et al.~ 1993). One of the drawbacks of programming by tunneling is the non-linearity in the data updating characteristics under constant pulses as shown in Fig. 8(b) (top). This difficulty has been beautifully resolved in our cell. With Vs high~ the output of the N-vMOS source follower is fed back to the tunneling electrode and the floating-gate potential is set to the tunneling electrode. In this manner~ the voltage across the tunneling oxide is always preset to a constant voltage (equal to the N-vMOS threshold) before a programming pulse is applied~ thus allowing constant charge to be injected or extracted at each pulse (Kosaka et al~ 1993) as demonstrated in Fig. 8(b) (bottom). A test self-learning circuit that leamed XOR is shown in Fig. 9. INPUT1 "XOR" INPUT2 INPUT1 ! \l I I ---"~ JI-; ; [ INPUT2 ! 400nsecldiv Figure 9: Test circuit of vMOS neural network and its response when XOR is learnt. 5 SUMMARY Development of intelligent electronic circuit systems using a new functional device called Neuron MOS Transistor has been described. vMOS circuitry is charactedzed by its high parallelism in computation scheme and the large flexibility in altering hardware functions and also by its great simplicity in the circuit organization. The ideas of Soft Hardware and the vMOS associative memory were not directly inspired from biological systems. However~ an interesting similarity is found in their basic structures. It is also demonstrated that the vMOS circuitry is very powerful in building neural networks in which learning algorithms are imbedded in the hardware. We conclude that the neuron-like functionality at an elementary device level is essentially imp0l1ant in implementing sophisticated information processing algorithms directly in the hardware. 925 926 Shibata, Kotani, Yamashita, Ishii, Kosaka, and Ohmi ACKNOWLEDGMENT This work was paltially supported by the Grant-in-Aid for Scientific Research (04402029) and Grant-in-Aid for Developmental Scientitlc Research (05505003) from the Ministry of Education, Science and Culture, Japan. A palt of this work was carried out in the Super Clean Room of Laboratory for Microelectronics, Research Institute of Electrical Communication, Tohoku University. REFERENCES [1] T. Shibata and T. Ohmi, "An intelligent MOS transistor featuring gate-level weighted sum and threshold operations," in IEDM Tech. Dig., 1991, pp. 919-922. [2] T. Shibata and T. Ohmi, "A functional MOS transistor featuring gate-level weighted sum and threshold operations," IEEE Trans. Electron Devices, Vol. 39, No.6, pp.14441455 (1992a). [3] W. S. McCulloch and W. Pitts, "A logical calculus of the ideas immanent in nervous activity," Bull. Math. Biophys., Vol. 5, pp. 115-133, 1943. [4] T. Shibata and T. Onmi, "Neuron MOS binary-logic integrated circuits: Part I, Design fundamentals and soft-hardware-logic circuit implementation," IEEE Trans. Electron Devices, Vol. 40, No.3, pp. 570-576 (1993a). [5] T. Shibata and T. Ohmi, "Neuron MOS binary-logic integrated circuits: Palt II, Simplifying techniques of circuit configuration and their practical applications," IEEE Trans. Electron Devices, Vol. 40, No.5, 974-979 (1993b). [6] K. Kotani, T. Shibata, and T. Ohmi, "Neuron-MOS binary-logic circuits featuring dramatic reduction in transistor count and interconnections," in IEDM Tech. Dig., 1992, pp. 431-434. [7] T. Shibata, K. Kotani, and T. Ohmi, "Real-time reconfigurable logic circuits using neuron MOS transistors," in ISSCC Dig. Technical papers, 1993c, FA 15.3, pp. 238-239. [8] N. Tanaka, T. Ohmi, and Y. Nakamura, "A novel bipolar imaging device with selfnoise reduction capability," IEEE Trans. Electron Devices, VoL 36, No.1, pp. 31-38 (1989). [9] T. Yamashita, T. Shibata, and T. Ohmi, "Neuron MOS winner-take-all circuit and its application to associative memory," in ISSCC Dig. Technical papers, 1993, FA 15.2, pp. 236-237. [10] R. Au, T. Yamashita, T. Shibata, and T. Ohmi, "Neuron-MOS multiple-valued memory technology for intelligent data processing," in ISSCC Dig. Technical papers, 1994, FA 16.3. [11] K. Hieda, M. Wada, T. Shibata, and H. Iizuka, "Optimum design of dual-control gate cell for high-density EEPROM's," IEEE Trans. Electron Devices, vol. ED-32, no. 9, pp. 1776-1780, 1985. [12] T. Shibata and T. Ohmi, itA self-leaming neural-network LSI using neuron MOSFET's," in Dig. Tech. Papers, 1992 Symposium on VLSI Technology, Seattle, June, 1992, pp. 84-85. [13] H. Ishii, T. Shibata, H. Kosaka, and T. Ohmi, "Hardware-Backpropagation learning of neuron MOS neural networks," in IEDM Tech. Dig., 1992, pp. 435-438. [14] H. Ishii, T. Shibata, H. Kosaka, and T. Ohmi, "Hardware-learning neural network LSI using a highly functional transistor simulating neuron actions," in Proc. Intemational Joint Conference on Neural Networks '93, Nagoya, Oct. 25-29, 1993, pp. 907-910. [15] H. Kosaka, T. Shibata, H. Ishii, and T. Ohmi, "An excellent weight-updatinglinearity synapse memory cell for self-Ieaming neuron MOS neural networks," in IEDM Tech. Dig., 1993, pp. 626-626. 

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Neural Variational Inference and Learning in Undirected Graphical Models Volodymyr Kuleshov Stanford University Stanford, CA 94305 kuleshov@cs.stanford.edu Stefano Ermon Stanford University Stanford, CA 94305 ermon@cs.stanford.edu Abstract Many problems in machine learning are naturally expressed in the language of undirected graphical models. Here, we propose black-box learning and inference algorithms for undirected models that optimize a variational approximation to the log-likelihood of the model. Central to our approach is an upper bound on the logpartition function parametrized by a function q that we express as a flexible neural network. Our bound makes it possible to track the partition function during learning, to speed-up sampling, and to train a broad class of hybrid directed/undirected models via a unified variational inference framework. We empirically demonstrate the effectiveness of our method on several popular generative modeling datasets. 1 Introduction Many problems in machine learning are naturally expressed in the language of undirected graphical models. Undirected models are used in computer vision [1], speech recognition [2], social science [3], deep learning [4], and other fields. Many fundamental machine learning problems center on undirected models [5]; however, inference and learning in this class of distributions give rise to significant computational challenges. Here, we attempt to tackle these challenges via new variational inference and learning techniques aimed at undirected probabilistic graphical models p. Central to our approach is an upper bound on the log-partition function of p parametrized by a an approximating distribution q that we express as a flexible neural network [6]. Our bound is tight when q = p and is convex in the parameters of q for interesting classes of q. Most interestingly, it leads to a lower bound on the log-likelihood function log p, which enables us to fit undirected models in a variational framework similar to black-box variational inference [7]. Our approach offers a number of advantages over previous methods. First, it enables training undirected models in a black-box manner, i.e. we do not need to know the structure of the model to compute gradient estimators (e.g., as in Gibbs sampling); rather, our estimators only require evaluating a model?s unnormalized probability. When optimized jointly over q and p, our bound also offers a way to track the partition function during learning [8]. At inference-time, the learned approximating distribution q may be used to speed-up sampling from the undirected model by initializing an MCMC chain (or it may itself provide samples). Furthermore, our approach naturally integrates with recent variational inference methods [6, 9] for directed graphical models. We anticipate that our approach will be most useful in automated probabilistic inference systems [10]. As a practical example for how our methods can be used, we study a broad class of hybrid directed/undirected models and show how they can be trained in a unified black-box neural variational inference framework. Hybrid models like the ones we consider have been popular in the early deep learning literature [4, 11] and take inspiration from the principles of neuroscience [12]. They also 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. possess a higher modeling capacity for the same number of variables; quite interestingly, we identify settings in which such models are also easier to train. 2 Background Undirected graphical models. Undirected models form one of the two main classes of probabilistic graphical models [13]. Unlike directed Bayesian networks, they may express more compactly relationships between variables when the directionality of a relationship cannot be clearly defined (e.g., as in between neighboring image pixels). In this paper, we mainly focus on Markov random fields (MRFs), a type of undirected model corresponding to a probability distribution of the form p? (x) = p?? (x)/Z(?), where p?? (x) = exp(? ? x) R is an unnormalized probability (also known as energy function) with parameters ?, and Z(?) = p?? (x)dx is the partition function, which is essentially a normalizing constant. Our approach also admits natural extensions to conditional random field (CRF) undirected models. Importance sampling. In general, the partition function of an MRF is often an intractable integral over p?(x). We may, however, rewrite it as Z Z Z p?? (x) q(x)dx = w(x)q(x)dx, (1) I := p?? (x)dx = x x x q(x) where q is a proposal distribution. Integral I can in turn be approximated by a Monte-Carlo estimate Pn I? := n1 i=1 w(xi ), where xi ? q. This approach, called importance sampling [14], may reduce the variance of an estimator and help compute intractable integrals. The variance of an importance
sampling estimate I? has a closed-form expression: n1 Eq(x) [w(x)2 ] ? I 2 . By Jensen?s inequality, it equals 0 when p = q. Variational inference. Inference in undirected models is often intractable. Variational approaches approximate this process by optimizing the evidence lower bound log Z(?) ? max Eq(x) [log p?? (x) ? log q(x)] q over a distribution q(x); this amounts to finding a q that approximates p in terms of KL(q||p). Ideal q?s should be expressive, easy to optimize over, and admit tractable inference procedures. Recent work has shown that neural network-based models possess many of these qualities [15, 16, 17]. Auxiliary-variable deep generative models. Several families of q have been proposed to ensure that the approximating distribution is sufficiently flexible to fit p. This work makes use of a class of distributions q(x, a) = q(x|a)q(a) that contain auxiliary variables a [18, 19]; these are latent variables that make the marginal q(x) multimodal, which in turn enables it to approximate more closely a multimodal target distribution p(x). 3 Variational Bounds on the Partition Function This section introduces a variational upper bound on the partition function of an undirected graphical model. We analyze its properties and discuss optimization strategies. In the next section, we use this bound as an objective for learning undirected models. 3.1 A Variational Upper Bound on Z(?) We start with the simple observation that the variance of an importance sampling estimator (1) of the partition function naturally yields an upper bound on Z(?):   p?(x)2 Eq(x) ? Z(?)2 . (2) q(x)2 As mentioned above, this bound is tight when q = p. Hence, it implies a natural algorithm for computing Z(?): minimize (2) over q in some family Q. 2 We immediately want to emphasize that this algorithm will not be directly applicable to highly peaked and multimodal distributions p? (such as an Ising model near its critical point). If q is initially very far from p?, Monte Carlo estimates will tend to under-estimate the partition function. However, in the context of learning p, we may expect a random initialization of p? to be approximately uniform; we may thus fit an initial q to this well-behaved distribution, and as we gradually learn or anneal p, q should be able to track p and produce useful estimates of the gradients of p? and of Z(?). Most importantly, these estimates are black-box and do not require knowing the structure of p? to compute. We will later confirm that our intuition is correct via experiments. 3.2 Properties of the Bound Convexity properties. A notable feature of our objective is that if q is an exponential family with parameters ?, the bound is jointly log-convex in ? and ?. This lends additional credibility to the bound as an optimization objective. If we choose to further parametrize ? by a neural net, the resulting non-convexity will originate solely from the network, and not from our choice of loss function. To establish log-convexity, it suffices to look at p?? (x)2 /q(x) for one x, since the sum of log-convex 2 ? (x) functions is log-convex. Note that log p?q(x) = 2?T x ? log q? (x). One can easily check that a non-negative concave function is also log-concave; since q is in the exponential family, the second term is convex, and our claim follows. Importance sampling. Minimizing the bound on Z(?) may be seen as a form of adaptive importance sampling, where the proposal distribution q is gradually adjusted as more samples are taken [14, 20]. This provides another explanation for why we need q ? p; note that when q = p, the variance is zero, and a single sample computes the partition function, demonstrating that the bound is Pn ? i ) indeed tight. This also suggests the possibility of taking n1 i=1 p(x q(xi ) as an estimate of the partition function, with the xi being all the samples that have been collected during the optimization of q. 2 ? ?2 -divergence minimization. Observe that optimizing (2) is equivalent to minimizing Eq (p?q) q2 , 2 which is the ? -divergence, a type of ?-divergence with ? = 2 [21, 22]. This connections highlights the variational nature of our approach and potentially suggests generalizations to other divergences. Moreover, many interesting properties of the bound can be easily established from this interpretation, such as convexity in terms of q, p? (in functional space). 3.3 Auxiliary-Variable Approximating Distributions A key part of our approach is the choice of approximating family Q: it needs to be expressive, easy to optimize over, and admit tractable inference procedures. In particular, since p?(x) may be highly multi-modal and peaked, q(x) should ideally be equally complex. Note that unlike earlier methods that parametrized conditional distributions q(z|x) over hidden variables z (e.g. variational autoencoders [15]), our setting does not admit a natural conditioning variable, making the task considerably more challenging. Here, we propose to address these challenges via an approach based on auxiliary-variable approximations [18]: we introduce a set of latent variables a into q(x, a) = q(x|a)q(a) making the marginal q(x) multi-modal. Computing the marginal q(x) may no longer be tractable; we therefore apply the variational principle one more time and introduce an additional relaxation of the form     p(a|x)2 p?(x)2 p?(x)2 Eq(a,x) ? Eq(x) ? Z(?)2 , (3) q(x|a)2 q(a)2 q(x)2 where, p(a|x) is a probability distribution over a that lifts p? to the joint space of (x, a). To establish the first inequality, observe that        p(a|x)2 p?(x)2 p(a|x)2 p?(x)2 p?(x)2 p(a|x)2 Eq(a,x) = E = E ? E . q(x)q(a|x) q(x) q(a|x) q(x|a)2 q(a)2 q(a|x)2 q(x)2 q(x)2 q(a|x)2   2 The factor Eq(a|x) p(a|x) is an instantiation of bound (2) for the distribution p(a|x), and is 2 q(a|x) therefore lower-bounded by 1. 3 This derivation also sheds light on the role of p(a|x): it is an approximating distribution for the intractable posterior q(a|x). When p(a|x) = q(a|x), the first inequality in (3) is tight, and we are optimizing our initial bound. 3.3.1 Instantiations of the Auxiliary-Variable Framework The above formulation is sufficiently general to encompass several different variational inference approaches. Either could be used to optimize our objective, although we focus on the latter, as it admits the most flexible approximators for q(x). Non-parametric variational inference. First, as suggested by Gershman et al. [23], we may take PK 1 q to be a uniform mixture of K exponential families: q(x) = k=1 K qk (x; ?k ). This is equivalent to letting a be a categorical random variable with a fixed, uniform prior. The qk may be either Gaussians or Bernoulli, depending on whether x is discrete or continuous. This choice of q lets us potentially model arbitrarily complex p given enough components. Note that for distributions of this form it is easy to compute the marginal q(x) (for small K), and the bound in (3) may not be needed. MCMC-based variational inference. Alternatively, we may set q(a|x) to be an MCMC transition operator T (x0 |x) (or a sequence of operators) as in Salimans et al. [24]. The prior q(a) may be set to a flexible distribution, such as normalizing flows [25] or another mixture distribution. This gives a distribution of the form q(x, a) = T (x|a)q(a). (4) For example, if T (x|a) is a Restricted Boltzmann Machine (RBM; Smolensky [26]), the Gibbs sampling operator T (x0 |x) has a closed form that can be used to compute importance samples. This is in contrast to vanilla Gibbs sampling, where there is no closed form density for weighting samples. The above approach also has similarities to persistent contrastive divergence (PCD; Tieleman and Hinton [27]), a popular approach for training RBM models, in which samples are taken from a Gibbs chain that is not reset during learning. The distribution q(a) may be thought of as a parametric way of representing a persistent distribution from which samples are taken throughout learning; like the PCD Gibbs chain, it too tracks the target probability p during learning. Auxiliary-variable neural networks. Lastly, we may also parametrize q(a|x) by an flexible function approximator such as a neural network [18]. More concretely, we set q(a) to a simple continuous prior (e.g. normal or uniform) and set q? (x|a) to an exponential family distribution whose natural parameters are parametrized by a neural net. For example, if x is continuous, we may set q(x|a) = N (?(a), ?(a)I), as in a variational auto-encoder. Since the marginal q(x) is intractable, we use the variational bound (3) and parametrize the approximate posterior p(a|x) with a neural network. For example, if a ? N (0, 1), we may again set p(a|x) = N (?(x), ?(x)I). 3.4 Optimization In the rest of the paper, we focus on the auxiliary-variable neural network approach for optimizing bound (3). This approach affords us the greatest modeling flexibility and allows us to build on previous neural variational inference approaches. The key challenge with this choice of representation is optimizing (3) with respect to the parameters ?, ? of p, q. Here, we follow previous work on black-box variational inference [6, 7] and compute Monte Carlo estimates of the gradient of our neural network architecture. p(x,a) ? The gradient with respect to p has the form 2Eq q(x,a) ?(x, a) and can be estimated directly via 2 ?? p Monte Carlo. We use the score function estimator to compute the gradient of q, which can be written 2 ? as ?Eq(x,a) p(x,a) q(x,a)2 ?? log q(x, a) and estimated again using Monte Carlo samples. In the case of a PK 1 non-parametric variational approximation k=1 K qk (x; ?k ), the gradient has a simple expression h i p(x) ? 2 p(x) ? 2 ??k Eq q(x)2 = ?Eqk q(x)2 dk (x) , where dk (x) is the difference of x and its expectation under qk . 4 Note also that if our goal is to compute the partition function, we may collect all intermediary samples for computing the gradient and use them as regular importance samples. This may be interpreted as a form of adaptive sampling [20]. Variance reduction. A well-known shortcoming of the score function gradient estimator is its high variance, which significantly slows down optimization. We follow previous work [6] and introduce two variance reduction techniques to mitigate this problem. We first use a moving average ?b of p?(x)2 /q(x)2 to center the learning signal. This leads to a gradient ? 2 ? estimate of the form Eq(x) ( p(x) 2 ? b)?? log q(x); this yields the correct gradient by well known q(x) properties of the score function [7]. Furthermore, we use variance normalization, a form of adaptive step size. More specifically, we keep a running average ? ? 2 of the variance of the p?(x)2 /q(x)2 and 0 2 use a normalized form g = g/ max(1, ? ? ) of the original gradient g. Note that unlike the standard evidence lower bound, we cannot define a sample-dependent baseline, as we are not conditioning on any sample. Likewise, many advanced gradient estimators [9] do not apply in our setting. Developing better variance reduction techniques for this setting is likely to help scale the method to larger datasets. 4 Neural Variational Learning of Undirected Models Next, we turn our attention to the problem of learning the parameters of an MRF. Given data D = {x(i) }ni=1 , our training objective is the log-likelihood n n 1 X T (i) 1X log p? (x(i) ) = ? x ? log Z(?). (5) n i=1 n i=1   2 ? (x) We can use our earlier bound to upper bound the log-partition function by log Ex?q p?q(x) . By 2 our previous discussion, this expression is convex in ?, ? if q is an exponential family distribution. The resulting lower bound on the log-likelihood may be optimized jointly over ?, ?; as discussed earlier, by training p and q jointly, the two distributions may help each other. In particular, we may start learning at an easy ? (where p is not too peaked) and use q to slowly track p, thus controlling the variance in the gradient. log p(D|?) := Linearizing the logarithm. Since the log-likelihood contains the logarithm of the bound (2), our Monte Carlo samples will produce biased estimates of the gradient. We did not find this to pose problems in practice; however, to ensure unbiased gradients one may further linearize the log using the identity log(x) ? ax ? log(a) ? 1, which is tight for a = 1/x. Together with our bound on the log-partition function, this yields   n 1 X T (i) 1 p?? (x)2 log p(D|?) ? max ? x ? aEx?q ? log(a) ? 1 . (6) ?,q n 2 q? (x)2 i=1 This expression is convex in each of (?, ?) and a, but is not jointly convex. However, it is straightforward to show that equation (6) and its unlinearized version have a unique point satisfying first-order stationarity conditions. This may be done by writing out the KKT conditions of both problems and 2 ? (x) ?1 using the fact that a? = (Ex?q p?q(x) at the optimum. See Gopal and Yang [28] for more details. 2 ) 4.1 Variational Inference and Learning in Hybrid Directed/Undirected Models We apply our framework to a broad class of hybrid directed/undirected models and show how they can be trained in a unified variational inference framework. The models we consider are best described as variational autoencoders with a Restricted Boltzmann Machine (RBM; Smolensky [26]) prior. More formally, they are latent-variable distributions of the form p(x, z) = p(x|z)p(z), where p(x|z) is an exponential family whose natural parameters are parametrized by a neural network as a function of z, and p(z) is an RBM. The latter is an undirected latent variable model with hidden variables h and unnormalized log-probability log p?(z, h) = z T W h + bT z + cT h, where W, b, c are parameters. 5 We train the model using two applications of the variational principle: first, we apply the standard evidence lower bound with an approximate posterior r(z|x); then, we apply our lower bound on the RBM log-likelihood log p(z), which yields the objective log p(x) ? Er(z|x) [log p(x|z) + log p?(z) + log B(? p, q) ? log r(z|x)] . (7) Here, B denotes our bound (3) on the partition function of p(z) parametrized with q. Equation (7) may be optimized using standard variational inference techniques; the terms r(z|x) and p(x|z) do not appear in B and their gradients may be estimated using REINFORCE and standard Monte Carlo, respectively. The gradients of p?(z) and q(z) are obtained using methods described above. Note also that our approach naturally extends to models with multiple layers of latent directed variables. Such hybrid models are similar in spirit to deep belief networks [11]. From a statistical point of view, a latent variable prior makes the model more flexible and allows it to better fit the data distribution. Such models may also learn structured feature representations: previous work has shown that undirected modules may learn classes of digits, while lower, directed layers may learn to represent finer variation [29]. Finally, undirected models like the ones we study are loosely inspired by the brain and have been studied from that perspective [12]. In particular, the undirected prior has been previously interpreted as an associative memory module [11]. 5 5.1 Experiments Tracking the Partition Function We start with an experiment aimed at visualizing the importance of tracking the target distribution p using q during learning. logZ 36 We use Equation 6 to optimize the likeliTrue value 34 Our method hood of a 5 ? 5 Ising MRF with coupling 32 Loopy BP 30 factor J and unaries chosen randomly Gibbs sampling 28 ?2 ?2 in {10 , ?10 }. We set J = ?0.6, 26 sampled 1000 examples from the model, 24 22 and fit another Ising model to this data. 20 0 5 10 15 20 25 30 35 40 We followed a non-parametric inference Iteration approach with a mixture of K = 8 Bernoullis. We optimized (6) using SGD and alternated between ten steps over the ?k and one step over ?, a. We drew 100 Monte Carlo samples per qk . Our method converged in about 25 steps over ?. At each iteration we computed log Z via importance sampling. The adjacent figure shows the evolution of log Z during learning. It also plots log Z computed by exact inference, loopy BP, and Gibbs sampling (using the same number of samples). Our method accurately tracks the partition function after about 10 iterations. In particular, our method fares better than the others when J ? ?0.6, which is when the Ising model is entering its phase transition. 5.2 Learning Restricted Boltzmann Machines Next, we use our method to train Restricted Boltzmann Machines (RBMs) on the UCI digits dataset [30], which contains 10,992 8 ? 8 images of handwritten digits; we augment this data by moving each image 1px to the left, right, up, and down. We train an RBM with 100 hidden units using ADAM [31] with batch size 100, a learning rate of 3 ? 10?4 , ?1 = 0.9, and ?2 = 0.999; we choose q to be a uniform mixture of K = 10 Bernoulli distributions. We alternate between training p and q, performing either 2 or 10 gradient steps on q for each step on p and taking 30 samples from q per step; the gradients of p are estimated via adaptive importance sampling. We compare our method against persistent contrastive divergence (PCD; Tieleman and Hinton [27]), a standard method for training RBMs. The same ADAM settings were used to optimize the model with the PCD gradient. We used k = 3 Gibbs steps and 100 persistent chains. Both PCD and our method were implemented in Theano [32]. In Figure 1, we plot the true log-likelihood of the model (computed with annealed importance sampling with step size 10?3 ) as a function of the epoch; we use 10 gradient steps on q for each step on p. Both PCD and our method achieve comparable performance. Interestingly, we may use our 6 Figure 1: Learning curves for an RBM trained with PCD-3 and with neural variational inference on the UCI digits dataset. Log-likelihood was computed using annealed importance sampling. Neural variational inference 26 Log-likelihood Log-likelihood Persistent contrastive divergence 28 30 0 50 100 150 Epochs 200 26 28 true predicted 30 250 0 20 40 60 80 Epochs 100 120 Table 1: Test set negative log likelihood on binarized MNIST and Omniglot for VAE and ADGM models with Bernoulli (200 vars) and RBM priors with 64 visible and either 8 or 64 hidden variables. Binarized MNIST Model VAE ADGM Omniglot Ber(200) RBM(64,8) RBM(64,64) Ber(200) RBM(64,8) RBM(64,64) 111.9 107.9 105.4 104.3 102.3 100.7 135.1 136.8 130.2 134.4 128.5 131.1 approximating distribution q to estimate the log-likelihood via importance sampling. Figure 1 (right) shows that this estimate closely tracks the true log-likelihood; thus, users may periodically query the model for reasonably accurate estimates of the log-likelihood. In our implementation, neural variational inference was approximately eight times slower than PCD; when performing two gradient steps on q, our method was only 50% slower with similar samples and pseudo-likelihood; however log-likelihood estimates were noisier. Annealed importance sampling was always more than order of magnitude slower than neural variational inference. Visualizing the approximating distribution. Next, we trained another RBM model performing two gradient steps for q for each step of p. The adjacent figure shows the mean distribution of each component of the mixture of Bernoullis q; one may distinguish in them the shapes of various digits. This confirms that q indeed approximates p. Speeding up sampling from undirected models. After the model has finished training, we can use the approximating q to initialize an MCMC sampling chain. Since q is a rough approximation of p, the resulting chain should mix faster. To confirm this intuition, we plot in the adjacent figure samples from a Gibbs sampling chain that has been initialized randomly (top), as well as from a chain that was initialized with a sample from q (bottom). The latter method reaches a plausible-looking digit in a few steps, while the former produces blurry samples. 5.3 Learning Hybrid Directed/Undirected Models Next, we use the variational objective (7) to learn two types of hybrid directed/undirected models: a variational autoencoder (VAE) and an auxiliary variable deep generative model (ADGM) [18]. We consider three types of priors: a standard set of 200 uniform Bernoulli variables, an RBM with 64 visible and 8 hidden units, and an RBM with 64 visible and 64 hidden units. In the ADGM, the approximate posterior r(z, u|x) = r(z|u, x)r(u|x) includes auxiliary variables u ? R10 . All the conditional probabilities r(z|u, x), r(u|x), r(z|x), p(x|z) are parametrized with dense neural networks with one hidden layer of size 500. 7 Figure 2: Samples from a deep generative model using different priors over the discrete latent variables z. On the left, the prior p(z) is a Bernoulli distribution (200 vars); on the right, p(z) is an RBM (64 visible and 8 hidden vars). All other parts of the model are held fixed. We train all neural networks for 200 epochs with ADAM (same parameters as above) and neural variational inference (NVIL) with control variates as described in Mnih and Rezende [9]. We parametrize q with a neural network mapping 10-dimensional auxiliary variables a ? N (0, I) to x via one hidden layer of size 32. We show in Table 1 the test set negative log-likelihoods on the binarized MNIST [33] and 28 ? 28 Omniglot [17] datasets; we compute these using 103 Monte Carlo samples and using annealed importance sampling for the 64 ? 64 RBM. Overall, adding an RBM prior with as little as 8 latent variables results in significant log-likelihood improvements. Most interestingly, this prior greatly improves sample quality over the discrete latent variable VAE (Figure 2). Whereas the VAE failed to generate correct digits, replacing the prior with a small RBM resulted in smooth MNIST images. We note that both methods were trained with exactly the same gradient estimator (NVIL). We observed similar behavior for the ADGM model. This suggests that introducing the undirected component made the models more expressive and easier to train. 6 Related Work and Discussion Our work is inspired by black-box variational inference [7] for variational autoencoders and related models [15], which involve fitting approximate posteriors parametrized by neural networks. Our work presents analogous methods for undirected models. Popular classes of undirected models include Restricted and Deep Boltzmann Machines [4, 26] as well as Deep Belief Networks [11]. Closest to our work is the discrete VAE model; however, Rolfe [29] seeks to efficiently optimize p(x|z), while the RBM prior p(z) is optimized using PCD; our work optimizes p(x|z) using standard techniques and focuses on p(z). Our bound has also been independently studied in directed models [22]. More generally, our work proposes an alternative to sampling-based learning methods; most variational methods for undirected models center on inference. Our approach scales to small and medium-sized datasets, and is most useful within hybrid directed-undirected generative models. It approaches the speed of the PCD method and offers additional benefits, such as partition function tracking and accelerated sampling. Most importantly, our algorithms are black-box, and do not require knowing the structure of the model to derive gradient or partition function estimators. We anticipate that our methods will be most useful in automated inference systems such as Edward [10]. The scalability of our approach is primarily limited by the high variance of the Monte Carlo estimates of the gradients and the partition function when q does not fit p sufficiently well. In practice, we found that simple metrics such as pseudo-likelihood were effective at diagnosing this problem. When training deep generative models with RBM priors, we noticed that weak q?s introduced mode collapse (but training would still converge). Increasing the complexity of q and using more samples resolved these problems. Finally, we also found that the score function estimator of the gradient of q does not scale well to higher dimensions. Better gradient estimators are likely to further improve our method. 7 Conclusion In summary, we have proposed new variational learning and inference algorithms for undirected models that optimize an upper-bound on the partition function derived from the perspective of 8 importance sampling and ?2 divergence minimization. Our methods allow training undirected models in a black-box manner and will be useful in automated inference systems [10]. Our framework is competitive with sampling methods in terms of speed and offers additional benefits such as partition function tracking and accelerated sampling. Our approach can also be used to train hybrid directed/undirected models using a unified variational framework. Most interestingly, it makes generative models with discrete latent variables both more expressive and easier to train. Acknowledgements. This work is supported by the Intel Corporation, Toyota, NSF (grants 1651565, 1649208, 1522054) and by the Future of Life Institute (grant 2016-158687). References [1] Yongyue Zhang, Michael Brady, and Stephen Smith. Segmentation of brain mr images through a hidden markov random field model and the expectation-maximization algorithm. IEEE transactions on medical imaging, 20(1):45?57, 2001. [2] Jeffrey A Bilmes. Graphical models and automatic speech recognition. In Mathematical foundations of speech and language processing, pages 191?245. Springer, 2004. [3] John Scott. Social network analysis. Sage, 2012. [4] Ruslan Salakhutdinov and Geoffrey Hinton. Deep boltzmann machines. In Artificial Intelligence and Statistics, pages 448?455, 2009. [5] Martin J Wainwright, Michael I Jordan, et al. Graphical models, exponential families, and R in Machine Learning, 1(1?2):1?305, 2008. variational inference. Foundations and Trends [6] Andriy Mnih and Karol Gregor. Neural variational inference and learning in belief networks. arXiv preprint arXiv:1402.0030, 2014. [7] Rajesh Ranganath, Sean Gerrish, and David Blei. Black box variational inference. In Artificial Intelligence and Statistics, pages 814?822, 2014. [8] Guillaume Desjardins, Yoshua Bengio, and Aaron C Courville. On tracking the partition function. In Advances in neural information processing systems, pages 2501?2509, 2011. [9] Andriy Mnih and Danilo J Rezende. Variational inference for monte carlo objectives. arXiv preprint arXiv:1602.06725, 2016. [10] Dustin Tran, Alp Kucukelbir, Adji B Dieng, Maja Rudolph, Dawen Liang, and David M Blei. Edward: A library for probabilistic modeling, inference, and criticism. arXiv preprint arXiv:1610.09787, 2016. [11] Geoffrey E. Hinton, Simon Osindero, and Yee Whye Teh. A fast learning algorithm for deep belief nets. Neural Computation, 18:1527?1554, 2006. [12] Geoffrey E Hinton, Peter Dayan, Brendan J Frey, and Radford M Neal. The" wake-sleep" algorithm for unsupervised neural networks. Science, 268(5214):1158, 1995. [13] Daphne Koller and Nir Friedman. Probabilistic graphical models: principles and techniques. MIT press, 2009. [14] Rajan Srinivasan. Importance sampling: Applications in communications and detection. Springer Science & Business Media, 2013. [15] Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. CoRR, abs/1312.6114, 2013. URL http://arxiv.org/abs/1312.6114. [16] Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of the 31th International Conference on Machine Learning, ICML 2014, Beijing, China, 21-26 June 2014, pages 1278? 1286, 2014. URL http://jmlr.org/proceedings/papers/v32/rezende14.html. [17] Yuri Burda, Roger B. Grosse, and Ruslan Salakhutdinov. Importance weighted autoencoders. CoRR, abs/1509.00519, 2015. URL http://arxiv.org/abs/1509.00519. [18] Lars Maal?e, Casper Kaae S?nderby, S?ren Kaae S?nderby, and Ole Winther. Auxiliary deep generative models. arXiv preprint arXiv:1602.05473, 2016. 9 [19] Rajesh Ranganath, Dustin Tran, and David Blei. Hierarchical variational models. In International Conference on Machine Learning, pages 324?333, 2016. [20] Ernest K. Ryu and Stephen P. Boyd. Adaptive importance sampling via stochastic convex programming. Unpublished manuscript, November 2014. [21] Tom Minka et al. Divergence measures and message passing. Technical report, Technical report, Microsoft Research, 2005. [22] Adji B Dieng, Dustin Tran, Rajesh Ranganath, John Paisley, and David M Blei. Variational inference via chi-upper bound minimization. Advances in Neural Information Processing Systems, 2017. [23] Samuel Gershman, Matthew D. Hoffman, and David M. Blei. Nonparametric variational inference. In Proceedings of the 29th International Conference on Machine Learning, ICML 2012, Edinburgh, Scotland, UK, 2012. [24] Tim Salimans, Diederik Kingma, and Max Welling. Markov chain monte carlo and variational inference: Bridging the gap. In Proceedings of the 32nd International Conference on Machine Learning (ICML-15), pages 1218?1226, 2015. [25] Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. arXiv preprint arXiv:1505.05770, 2015. [26] Paul Smolensky. Information processing in dynamical systems: Foundations of harmony theory. Technical report, DTIC Document, 1986. [27] Tijmen Tieleman and Geoffrey Hinton. Using fast weights to improve persistent contrastive divergence. In Proceedings of the 26th Annual International Conference on Machine Learning, pages 1033?1040. ACM, 2009. [28] Siddharth Gopal and Yiming Yang. Distributed training of large-scale logistic models. In Proceedings of the 30th International Conference on Machine Learning, ICML 2013, Atlanta, GA, USA, pages 289?297, 2013. [29] Jason Tyler Rolfe. Discrete variational autoencoders. arXiv preprint arXiv:1609.02200, 2016. [30] Fevzi Alimoglu, Ethem Alpaydin, and Yagmur Denizhan. Combining multiple classifiers for pen-based handwritten digit recognition. 1996. [31] Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. [32] Fr?d?ric Bastien, Pascal Lamblin, Razvan Pascanu, James Bergstra, Ian Goodfellow, Arnaud Bergeron, Nicolas Bouchard, David Warde-Farley, and Yoshua Bengio. Theano: new features and speed improvements. arXiv preprint arXiv:1211.5590, 2012. [33] Hugo Larochelle and Iain Murray. The neural autoregressive distribution estimator. In AISTATS, volume 1, page 2, 2011. 10 

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Subspace Clustering via Tangent Cones Amin Jalali Wisconsin Institute for Discovery University of Wisconsin Madison, WI 53715 amin.jalali@wisc.edu Rebecca Willett Department of Electrical and Computer Engineering University of Wisconsin Madison, WI 53706 willett@discovery.wisc.edu Abstract Given samples lying on any of a number of fixed subspaces, subspace clustering is the task of grouping the samples based on the their corresponding subspaces. Many subspace clustering methods operate by assigning a measure of affinity to each pair of points and feeding these affinities into a graph clustering algorithm. This paper proposes a new paradigm for subspace clustering that computes affinities based on the corresponding conic geometry. The proposed conic subspace clustering (CSC) approach considers the convex hull of a collection of normalized data points and the corresponding tangent cones. The union of subspaces underlying the data imposes a strong association between the tangent cone at a sample x and the original subspace containing x. In addition to describing this novel geometric perspective, this paper provides a practical algorithm for subspace clustering that leverages this perspective, where a tangent cone membership test is used to estimate the affinities. This algorithm is accompanied with deterministic and stochastic guarantees on the properties of the learned affinity matrix, on the true and false positive rates and spread, which directly translate into the overall clustering accuracy. 1 Introduction Finding a low-dimensional representation of high-dimensional data is central to many tasks in science and engineering. Union-of-subspaces have been a popular data representation tool for the past decade. These models, while still parsimonious, offer more flexibility and better approximations to non-linear data manifolds than single-subspace models. To fully leverage union-of-subspaces models, we must be able to determine which data point lies in which subspace. This subproblem is referred to as subspace clustering [16]. Formally, given a set of points x1 , . . . , xN ? Rn lying on k linear subspaces S1 , . . . , Sk ? Rn , subspace clustering is the pursuit of partitioning those points into k clusters so that all points in each cluster lie within the same subspace among S1 , . . . , Sk . Once the points have been clustered into subspaces, standard dimensionality reduction methods such as principal component analysis can be used to identify the underlying subspaces. A generic approach in the literature is to construct a graph with each vertex corresponding to one of the given samples and each edge indicating whether (or the degree to which) a pair of points could have come from the same subspace. We refer to the (weighted) adjacency matrix of this graph as the affinity matrix. An ideal affinity matrix A would have A(i, j) = 1 if and only if xi and xj are in the same subspace, and otherwise A(i, j) = 0. Given an estimated affinity matrix, a variety of graph clustering methods, such as spectral clustering [17], can be used to cluster the samples, so forming the affinity matrix is a critical step. Many existing methods for subspace clustering with provable guarantees leverage the self-expressive property of the data. Such approaches pursue a representation of each data point in terms of the other data points, and then the representation coefficients are used to construct an affinity matrix. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. For example, the celebrated sparse subspace clustering (SSC) approach of [3] seeks a representation of each sample as a weighted combination of the other points, with minimal ?1 norm. However, such sparse self-expression can lead to graph connectivity issues, e.g., see [10, 8, 20, 5, 19, 18], where clusters can be arbitrarily broken into separate components. This paper proposes a new paradigm for devising subspace clustering algorithms: Conic Subspace Clustering (CSC): exploiting the association of the tangent cones to the convex hull of normalized samples with the original subspaces for computing affinities and subsequent clustering. CSC leverages new insights into the geometry of subspace clustering. One of the key effects of this approach is that the learned affinity matrix is generally denser among samples from the same subspace, which in turn can mitigate graph connectivity issues. In Proposition 1 below, we hint on what we mean by the strong association of the tangent cones with the underlying subspaces for an ideal dataset. In Section 2, we show how a similar idea can be implemented with finite number of samples. Given a set of nonzero samples from a union of linear subspaces, we normalize them to fall on the unit sphere and henceforth assume X = {x1 , . . . , xN } ? S n?1 is the set of samples. We further overload the notation to define X = [x1 , x2 , ? ? ? , xN ] ? Rn?N . Data hull refers to the convex hull of samples. The tangent cone at x ? conv(X) with respect to conv(X) is defined as !" # ? ? ? ? T (x) := cl conv cone(X ? {x}) = cl x? ?X ?x (x ? x) : ?x ? 0, x ? X where the Minkowski sum of two sets A and B is denoted by A + B, while A + {x} may be simplified to A + x. The linear space of a cone C is defined as lin C := C ? (?C). We term the intersection of a subspace S with the unit sphere as a ring R = S ? S n?1 . Proposition 1. For a union of rings, namely X = (S1 ? . . . ? Sk ) ? S n?1 , and for every x ? X, S(x) = span{x} + lin T (x), where S(x) is the convex hull of the union of all subspaces Si , i = 1, . . . , k, to which x belongs. 1.1 Our contributions We introduce a new paradigm for subspace clustering, conic subspace clustering (CSC), inspired by ideas from convex geometry. More specifically, we propose to consider the convex hull of normalized samples, and exploit the structure of the tangent cone to this convex body at each sample to estimate the relationships for pairs of samples (to construct an affinity matrix for clustering). We provide an algorithm which implements CSC (Section 2) along with deterministic guarantees on how to choose the single parameter in this algorithm, ?, guaranteeing no false positives (Section 5) and any desired true positive rate (Section 4), in the range allowed by the provided samples. We specialize our results to random models, to showcase our guarantees in terms of the few parameters defining said random generative models and to compare with existing methods. Aside from statistical guarantees, we also provide different optimization problems for implementing our algorithm that can be used for faster computation and increased robustness (Section 7). In Section 6, we elaborate on the true positive rate and spread for CSC and compare it to what is known about a sparsity-based subspace clustering approach, namely sparse subspace clustering, SSC [3]. This comparison provides us with insight on situations where methods such as SSC would face the so called graph connectivity issue, demonstrating the advantage of CSC in such situations. 2 Conic Subspace Clustering (CSC) via Rays: Intuition and Algorithm In this section, we discuss an intuitive algorithm for subspace clustering under the proposed conic subspace clustering paradigm. We present the underlying idea without worrying about the computational aspects, and relegate such discussions to Section 7. All proofs are presented in the Appendix. Henceforth, lower case letters represent vectors, while specific letters such as x and x? are reserved to represent columns of X, and x is commonly used as the reference point. Start by considering Figure 1(a) and the point x ? R := (S1 ? ? ? ? ? Sk ) ? S n?1 from which all the rays are emanating. Moreover, define Rt := St ? S n?1 for t = 1, . . . , k, which gives 2 (a) x + cl cone(R ? x) (b) x? ? S(x) (c) x? ? / S(x) Figure 1: Illustration of the idea behind our implementation of Conic Subspace Clustering (CSC) via rays. The union of the red and blue rings is R, and x is the point from which all the rays are emanating. 1(a) The union of the red and blue surfaces is x+cl cone(R?x). 1(b) When x? and x are from the same subspace, the points d? (x, x? ) for different values of ? ? 0 lie within cl cone(R ? x) ? specifically, in the blue shaded cone associated with the blue ring. 1(c) When x? and x are from different subspaces, the points d? (x, x? ) lie outside cl cone(R ? x) for large enough values of ?. R = R1 ? . . . ? Rk . Only two subspaces are shown and the reference point x is in R1 . The red and blue rays correspond to elements of x + cone(R ? x) = x + cone({x? ? x : x? ? R}), where cone(A) := {?y : y ? A, ? ? 0}.1 We leverage the geometry of this cone to determine subspace membership. Specifically, Figure 1(b) considers a point x? ? R1 different from x. The dashed line segment represents points ?sign(?x, x? ?)?x? for different values of ? ? 0; where sign(0) can be arbitrarily chosen as ?1. The vectors emanating from x and reaching these points represent d? (x, x? ) := ?sign(?x, x? ?)?x? ? x. (1) For x, x ? R1 , this illustration shows that d? (x, x ) ? cl cone(R ? x) for any ? ? 0. In contrast, Figure 1(c) considers x? ? R2 , while x ? R1 . In this case, there exist ? > 0 such that d? (x, x? ) ? / cl cone(R ? x), indicating that x? ?? S(x). Formally, Proposition 2. For any x, x? ? R and any scalar value ? ? 0, ! # x? ? S(x) ?? d? (x, x? ) : ? ? 0 ? cl cone(R ? x) (2) ! # Equivalently, x? ? S(x) if and only if ? ? R : ?x? ? x ? cl cone(R ? x) is unbounded. ? ? In other words, we can test whether or not x? ? S(x) by testing the cone membership for d? (x, x? ). Of course, such a test would not be practical: we cannot compute d? (x, x? ) for an infinite set of ? values, the set cl cone(R ? x) is generally non-convex (in Figure 1(a), the cone is the union of the red and blue surfaces), and cl cone(R ? x) is not known exactly because we only observe a finite collection of points from R instead of all of R. We now develop an alternative test to (2) that addresses these challenges and can be computed within a convex optimization framework. We first address the convexity issue: Proposition 3. For the closed convex cone W(x) := conv cl cone(R ? x), and for any x, x? ? R, ! # x? ? S(x) =? d? (x, x? ) : ? ? 0 ? W(x). (3) ! # In other words, x? ? S(x) implies that ? ? R : d? (x, x? ) ? cl cone(R ? x) is unbounded. Next, we formulate the test as a convex optimization program, when a finite number of samples are given. Specifically, using the samples in X ? R instead of all the points in R, we can define an approximation of W(x) as ! # WN (x) := (X ? x1TN )? : ? ? RN (4) + which is the tangent cone (also known as the descent cone) at x with respect to the data hull conv(X). The implementation of CSC via rays, as sketched above and detailed below, is based on testing the membership of d? (x, x? ) in the tangent cone WN (x) for all pairs of samples x, x? to determine their affinity. More specifically, the cone membership test can be stated as a feasibility program, tagged as the Cone Representability (CR) program: min??RN 0 subject to d? (x, x? ) = (X ? x1TN )? , ? ? 0N . 1 Note that this is not the same as a conic hull. 3 (CR) If there exists a ? ? 0 for which (CR) is infeasible, then we conclude x? ?? S(x). Later, in our theoretical results in Sections 4 and 5 we characterize a range (dependent on a target error rate) of possible values of ? such that for any single ? from this range, checking the feasibility of (CR) for all x, x? reveals the true relationships within a target error rate. In Section 7, we discuss a number of variations for the above optimization program. While our upcoming guarantees are all concerned with the cone membership test itself and not the specific implementation, these variations provide better algorithmic options and are more robust to noise. Specifically, we choose to use a variation that is a bounded feasible linear program for our implementation of the cone membership test. We refer to solving any of the variations of the cone membership test for an ordered pair of samples (x, x? ) and a fixed value of ? as CSC1 (?, x, x? ): Compute ? $(x, x? ) = min {? : (1 ? ?)(?x? ? x) = (X ? x1TN )? , ? ? 0 , ? ? 0}. Set A(x, x? ) ? {0, 1} by rounding 1 ? ? $(x, x? ) to either 0 or 1, whichever is closest. We refer to the optimization program used in the above as the Robust Cone Membership (RCM) program. Similarly, solving a collection of these tests for all samples x? and for a fixed x, or for all pairs x, x? , are referred to as CSC1 (?, x) and CSC1 (?), respectively. When CSC1 (?) is followed by spectral clustering for the constructed affinity matrix, we refer to the whole process as CSC(?). It is worth mentioning that the linear program used in CSC1 (?, x, x? ) is equivalent to (CR) in a sense made clear in Section 7, and provides the same affinity matrix with a variety of algorithmic advantages, as discussed in Section 7. 3 Theoretical Guarantees In this section, we discuss our approach to providing theoretical guarantees for the aforementioned implementation of CSC via rays. Let us first set some conventions. We refer to a declaration x? ? S(x) (or x? ?? S(x)) as positive (or negative), regardless of the ground truth. Hence, a true positive is an affinity of 1 when the samples are from the same subspace, and a false positive is an affinity of 1 when the samples are from different subspaces. We provide guarantees for CSC1 (?) to give no false positives. This makes the affinity matrix a permuted block diagonal matrix. In this case, if there are enough well-spread ones in each row of the affinity matrix, spectral clustering or any other reasonable clustering algorithm will be able to perfectly recover the underlying grouping; see graph connectivity in spectral clustering literature [17]. These two phenomena, no false positives and enough well-spread true positives per sample, are the focus of our theoretical results in Sections 4 and 5. In a nutshell, the guarantees boil down to characterizing a range of ??s for which CSC has controlled degrees of errors: no false positives and a certain true positive rate per row. We also examine the distribution of true positives recovered by our method and illustrate a favorable spread. Through the intuition behind the cone membership test, namely (CR), it is easy to observe that the number of true positives and the number of false positives are monotonically non-increasing in ? (which can be observed in Figure 2 as well). This is because we only have a finite number of samples, and using a larger value of ? would throw points out of WN (x), which is not an ideal wedge and rather is a pointed cone. Hence, to have a high number of true positives we need to use an upper bounded ?, and to have a few number of false positives we need to use a lower bounded value of ?. To assess the strength of our deterministic results, we assume probabilistic models on the subspaces and/or samples and study the ranges of ? for which CSC1 (?) has controlled errors of both types, with high probability. For the random models, we take the number of subspaces to be fixed, namely k. However, CSC1 (?) need not know the number of subspaces and spectral clustering can use the gap in the eigenvalues of the Laplacian matrix to determine the number of clusters; e.g., see [15]. In the random-sample model, we assume k subspaces are given and samples from each subspace are drawn uniformly at random from the unit sphere on that subspace. In the random-subspace model, each subspace is chosen independently and uniformly at random with respect to the Haar measure. 3.1 Examples In this section, we illustrate the performance of the CSC method on some small examples. First, we examine the role of the parameter ? in CSC1 (?, x, x? ) and its effect on the false positive and true positive rates in practice. In the first experiment, we have k = 5 subspaces, each with dimension 4 d = 5, in an n = 10 dimensional space, and draw 30 samples from each of the k 5-dimensional subspaces. We then run CSC1 (?, x, x? ) for a variety of values of ? between one and six over 15 random trials; In Figures 2(b), 2(c), and 2(d), we show the results of each trial in thin lines and the means across trials in thick lines (Figure 2(c) shows the median). Figure 2(a) shows, for each value of ?, the histogram of true positive rates across rows. Superimposed on this histogram plot is the empirical mean of the histogram (green solid line) and our theoretical bound from Theorem 6 (purple dashed curve corresponding to the purple dashed curve in Figure 2(b)): for each value of ?, the true positive rate will be above this curve, with high probability. Our theoretical bounds correspond to sufficient but not necessary conditions, and empirically we see good error rates outside the range of ??s for which we have guarantees. Nonetheless, this small example helps illustrating the tightness of the theory for true positive rate in relation to ?. 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 1 600 0.8 500 0.6 400 300 0.4 200 0.2 100 0 0 1 2 3 4 1 5 (a) 2 3 4 0 1 5 2 (b) 3 4 5 1 (c) 2 3 4 5 (d) Figure 2: Illustration of the role of ? (horizontal axis) in determining (a) the histogram of true positive rates across rows, with the empirical mean (solid line) and the theoretical bound (dashed curve, corresponding to the dashed curve in (b)), (b) maximum, mean, and minimum (across rows of the affinity matrix) true positive rates along with the theoretical bound (dashed curve), (c) the clustering mismatch rate after performing spectral clustering, and, (d) the false positive rate. This experiment is in a 10 dimensional space, with 30 random samples from each of 5 random 5-dimensional subspaces, over 15 random trials. Bold curves correspond to averages across trials in (a), (b), and (d), but to the median in (c). Next, we look at learned affinity matrices output by the proposed CSC method and SSC [3], which is a widely-used benchmark and the foundation of much current subspace clustering research. As described at length in Section D, the true positive rate of SSC is necessarily bounded because of the ?1 regularization used to learn the affinity matrix. This is not true of CSC ? in fact, ? can be used to control the true positive rate (in an admissible range) as long as it exceeds some lower bound (? ? ?L ). The difference between the true positive rates of SSC and CSC are illustrated in Figure 3. In this experiment, CSC naturally outputs a 0/1 affinity matrix, while the affinity matrix of SSC has a broader diversity of values. We show this matrix and a thresholded version for comparison purposes, where the threshold is set to correspond to a 5% false positive rate. (a) (b) (c) (d) (e) (f) Figure 3: Affinity matrices for two toy models in an ambient dimension n = 12. (a-c) k = 3 subspaces, each of rank d = 4 and each with 3d = 12 samples. (a) Result of CSC1 (?). (b) Result of SSC. (c) Thresholded version of (b). (d-e) k = 3 subspaces, each of rank d = 6 and each with 3d = 18 samples. (d) Result of CSC1 (?). (e) Result of SSC. (f) Thresholded version of (e), with threshold set so that the false positive rate is 5%. As predicted by the theory, CSC achieves higher true positive rates than SSC can. 5 4 Guarantees on True Positive Rates We study conditions under which a fraction ? ? (0, 1) of samples x? ? S(x) are declared as such. As discussed before, the number of true positives is non-increasing in ?. Therefore, we are interested in an upper bound ?U,? on ? so that CSC1 (?, x) for ? ? ?U,? returns at least ?Nt true positives (Nt is the number of samples from St ) for any x ? X t := X ? St and t = 1, . . . , k. Consider {x}? := {y : ?x, y? = 0}. For a close convex set A containing the origin, denote by r(A) the radius of the largest Euclidean sphere in span(A) that is centered at the origin and is a subset of A. Theorem 4 (Deterministic condition for any true positive rate). The conic subspace clustering algorithm at x with parameter ?, namely CSC1 (?, x), returns a ratio ? ? (0, 1) of relationships between x x ? X t and other samples as true positives, provided that ? ? ?U,? where x := ?U,? sin2 (?tx ) cos(?tx ) ? cos(??tx ) (5) in which, for m := ??(N ? 1)?, cos(??tx ) is the (m + 1)-st largest value among |?x, x? ?| for t x? ? X t , and for r(?) denoting the inner radius, ?tx := arctan(r((x + WN (x)) ? {x}? )). Then, CSC1 (?) is guaranteed to return a fraction ? of true positives per sample provided that x ? ? ?U,? := minx?X ?U,? . x As it can be seen from the above characterization, ?tx and ?U,? can vary from sample to sample even within the same subspace. When samples are drawn uniformly at random from a given subspace (the random-sample model), the next theorem provides a uniform lower bound on the inner radius x and ?tx for all such samples. Note that ?U,? is non-decreasing in ?tx and non-increasing in ??tx . Theorem 5. Under a random-sample model, and for a choice pt ? (0, 1), with probability at least 1 ? pt , a solution ? to (cos ?)dt ?1 log(Nt /pt ) ? = Nt 6 dt sin ? is a lower bound on ?tx , which is defined in Theorem 4 and is a function of the inradius of a base of t the t-th cone WN (x). Theorem 5 is proved in the Appendix using ideas from inversive geometry [1]. In a random-sample model, we can quantify the aforementioned m-th order statistic. Therefore, we can explicitly compute the upper bound ?U,? (with high probability) in terms of quantities dt and Nt . The final result is given in Theorem 6. Note that both the inradius and the m-th order statistic are random variables defined through the samples, hence are dependent. Therefore, a union bound is used. Theorem 6. Under a random-sample model, CSC1 (?, x) for any x ? X t yields a fraction ? of x x true positives with high probability, provided that ? ? ?u,? , where ?u,? is computed similar to (5), ? m x x using the lower bound on ?t from Theorem 5 and ??t = 2 ( N + ?). The probability is at least I( m N + ?; m, N ? m) ? pt , where I(?; ?, ?) denotes the incomplete Beta function. 5 Guarantees for Zero False Positives In this section, we provide guarantees for CSC1 (?, x) to return no false positives, in terms of the value of ?. Specifically, we guarantee this by examining a lower bound ?L for ? in CSC1 (?, x). For a fixed column x of the data matrix X, we will use x? as a pointer to any other column of X. Recall d? (x, x? ) from (1) and consider ?L (x) := inf {? ? 0 : d? (x, x? ) ?? WN (x) ?x? ?? S(x)} (6) = sup {? ? 0 : d? (x, x? ) ? WN (x) for some x? ?? S(x)} . If the above value is finite, then using any value even slightly larger than this would declare any x? ?? S(x) correctly as such, hence no false positives. However, the above infimum may not exist for a general configuration. In other words, there might be a sample x? ?? S(x) for which d? (x, x? ) ? WN (x) for all values of ? ? 0. The following condition prohibits such a situation. 6 Theorem 7 (Deterministic condition for zero false positives). For x ? X, without loss of generality, suppose S(x) = S1 ? . . . ? Sj for some j < k. Provided that all of the columns of X that are not in S(x) are also not in WN (x), then ?L (x) in (6) is finite. This condition is equivalent to St ? WN (x) = {0} for all t = j + 1, . . . , k and all x ? X \ St . In case this condition holds for all x ? X, we define ?L := max ?L (x). x?X (7) If this condition is met and ? ? ?L is used, then CSC1 (?) will return no false positives. We note that the condition of Theorem 7 becomes harder to satisfy as the number of samples grow (which makes WN (x) larger). While this is certainly not desired, such an artifact is present in other subspace clustering algorithms. See the discussion after Theorem 1 in [11] for examples. Next, we specialize Theorem 7 to a random-subspace model. Under such model, for t = j+1, . . . , k, St and WN (x) are two random objects and are dependent (all samples, including those from St , take part in forming WN (x), hence the orientation and the dimension of St affect the definition of WN (x)), which makes the analysis harder. However, these two can be decoupled by massaging the ?t condition of Theorem 7 from St ? WN (x) = {0} into an equivalent condition St ? WN (x) = {0} ?t where WN (x) = convcone{x? ? x : x? ?? St }; see Lemma 10 in the Appendix. Next, the event of a random subspace and a cone having trivial intersection can be studied using the notion of the statistical dimension of the cone and the brilliant Gordon?s Lemma [6]. The statistical dimension of a closed convex cone C ? Rn is defined as ?(C) := E supy?C?S n?1 ?y, g?2 ? n where g ? N (0, In ). Now, we can state the following lemma based on Gordon?s Lemma. Lemma 8. With the notation in Theorem 7, and under the random-subspace model, ?L is finite ?t provided that ?(WN (x)) + dim(St ) ? n for t = 1, . . . , k. "k Furthermore, for the above to hold, it is sufficient to have t=1 dim(St ) ? n (Lemma 11 in the Appendix). Under the above conditions, we are guaranteed that a finite ? exists such that with high probability, CSC1 (?) results in zero false positives. It is easy to compute ?L for certain configurations of subspaces. For example, when the subspaces are independent (the dimension of their Minkowski sum is equal to sum of their dimensions) we have ?L = 1 (Lemma 13 in the Appendix). Independent subspaces have been assumed before in the subspace clustering literature for providing guarantees; e.g., [2, 3]. Also see [21] and references therein. However, it remains as an open question how one should compute this value for more general configurations. We provide some theoretical tools for such computation in Appendix B.5. Finally, if ?L does not exceed ?U,? from above, then CSC1 (?) successfully returns a (permuted) block diagonal matrix with a density of ones (per row) of at least ?. This allows us to have a good idea about the performance of the post-processing step (e.g., spectral clustering) and hence CSC(?). 6 True Positives? Rate and Distribution Because sparse subspace clustering (SSC) relies upon sparse representations, the number of true positives is inherently limited. In fact, it can be shown that SSC will find a representation of each column x as a weighted sum of columns that correspond to the extreme rays of WN ; as shown in Lemma 17 in the Appendix. This phenomenon is closely linked to the graph connectivity issues associated with SSC, mentioned before. In particular, under a random-sample model, the true positive rate for SSC will go to zero as Nt /dt grows, where Nt is the number of samples from St with dim(St ) = dt . In contrast, the true positive behavior for CSC has several favorable characteristics. First, if the subspaces are all independent, then the true positive rate ? can approach one. Second, in unfavorable settings in which the true positive rate is low, it can be shown that the true positives are distributed in such a way that precludes graph connectivity issues (see Section D.3 for more details). Specifically, for each subspace St , there is a matrix Asub defined below, whose support is contained within the true positive support of the output of CSC(?) for ? ? (?L , ?U,? ). Let Xt have i.i.d. standard normal entries, and ? be the m-th largest element of |XtT Xt |. Then, Asub is defined by (Asub )i,j = |XtT Xt | when |XtT Xt | > ?, and zero otherwise. The distribution of Asub when columns of Xt are drawn uniformly at random from the unit sphere ensures that graph connectivity issues are avoided with high probability as soon as the true positive rate ? exceeds O(log Nt /Nt ). As a 7 result, even if the values of ? which provide ?U,? > ?L are small, there is still the potential of perfect clustering. These distributional arguments cannot be made for sparsity-based methods like SSC. We refer to Appendix D for more details. 7 CSC Optimization and Variations In Table 1, we provide a number of optimization programs that implement the cone membership test. These formulations possess different computational and robustness properties. Let us introduce a notation of equivalence. We say an optimization program P , implementing the cone membership test, is in CR-class if the possible set of its optimal values can be divided into two disjoint sets Oin and Oout corresponding to whether d? (x, x? ) ? WN (x) or d? (x, x? ) ?? WN (x), respectively. Then we write [[P : Oin , Oout ]]; e.g., [[(CR) : 0, infeasible]]. All of the problems in Table 1 are in CR-class. Table 1: Different formulations for the cone membership (second column) with their set of outputs when d? (x, x? ) ? WN (x) (third column) and when d? (x, x? ) ?? WN (x). In all of the variations, A = X ? x1TN , y and b = d? (x, x? ) live in Rn , and ? lives in RN . Tag Formulation Oin Oout P1 P2 P3 P4 min? 0 s.t. b = A? , ? ? 0 miny ?y, b? s.t. y T A ? 0 miny ?y, b? s.t. y T A ? 0 , ?y, b? ? ?? min?,? ? s.t. (1 ? ?)b = A? , ? ? 0, ? ? 0 {0} {0} {0} {0} infeasible unbounded {??} {1} The first optimization problem (P1) is merely the statement of the cone membership test as a linear ? feasibility program and (P2) is its Lagrangian dual. (P2) looks for a certificate y ? WN (x) (in the dual cone) that rejects the membership of d? (x, x? ) in WN (x). However, neither (P1) nor (P2) are robust or computationally appealing. Next, observe that restricting y to any set with the origin in its relative interior yields a program that is in CR-class. (P3) is defined by augmenting (P2) with a linear constraint, which not only makes the problem bounded and feasible, but the freedom in choosing ? allows for controlling the bit-length of the optimal solution and hence allows for optimizing the computational complexity of solving (P3) via interior point methods. Furthermore, this program can be solved approximately, up to a precision ?? ? (0, ?), and provides the same desired set of results: an ?? -inexact solution for (P3) has a nonnegative objective value if and only if d? (x, x? ) ? WN (x). If we dualize (P3) and divide the objective by ?? we get (P4) which can also be derived by handtweaking (P1). However, the duality relationship with (P3) is helpful in understanding the dual space and devising efficient optimization algorithms. Notice that (?, ?) = (1, 0N ) is always feasible, and the optimal solution is in [0, 1]. Moreover, [[(P 4) : 0, 1]], which makes it a desirable candidate as a proxy for x? ?? S(x). We use this program in our experiments reported in Section 3.1. 8 Discussions and Future Directions This paper describes a new paradigm for understanding subspace clustering in relation to the underlying conic geometry. With this new perspective, we design an algorithm, CSC via rays, with guarantees on false and true positive rates and spreads, that sidesteps graph connectivity issues that arise with methods like sparse subspace clustering. This paper should be seen as the first introduction to the idea of, and tools for, conic subspace clustering, rather than establishing CSC as the new state-of-the-art, and as a means to ignite future work on several directions in subspace clustering. We focus on our novel geometric perspective and its potential to lead to new algorithms by providing a rigorous theoretical understanding (statistical and computational) and hope that this publication will spur discussions and insights that can inform the suggested future work. A cone membership test is just one approach to exploits this geometry. Remaining Questions. While more extensive comparisons with existing methods are necessary, many comparisons are non-trivial because CSC reveals important properties of subspace clustering methods (e.g. spread of true positives) that are not understood for other methods. The limited small-scale experiments were simply intended to illustrate these properties. 8 Our study of the parameter choice is theoretical in nature and beyond heuristics for implementation. But some questions are still open. Firstly, while we have a clear deterministic characterization for ?U , tighter characterizations would lead to a larger range for ?. In Figure 2(b), such pursuit would result in a new theoretical curve (instead of the current dashed purple curve) that stays closer to the minimum true positive rate across rows (the lowest thick solid curve). On the other hand, outside of the case of independent subspaces, where ?L = 1, we only have a deterministic guarantee on the finiteness of ?L and computing it for the semi-random model is a topic of current research. Therefore, we do not have a guarantee on the non-triviality of the resulting range (?L , ?U ). However, as observed in the small numerical examples in Section 3.1, as well as in our more extensive experiments that are not reported here, there often exists a big range of ? with which we can get perfect clustering. Extensions. While the presented algorithm assumes noiseless data points from the underlying subspaces, our intuition and simulations indicate stability towards stochastic noise. Moreover, the current analysis is suggestive of algorithmic variants that exhibit robust empirical performance in the presence of stochastic noise. This is why, similar to advances in other subspace clustering methods, we hope that the analysis for the noiseless setup provides essential insights to provably generalize the method to noisy settings. Furthermore, there remain several other open avenues for exploration, particularly with respect to theoretical and large-scale empirical comparisons with other methods, and extensions to measurements corrupted by adversarial perturbations, with outliers among the data points, as well as with missing entries in the data points. By design, SSC and other similar methods require a full knowledge of data points. CSC imposes the same requirement and an open question is how to extend the CSC framework when some entries are missing from the data points. References [1] David E. Blair. Inversion theory and conformal mapping, volume 9 of Student Mathematical Library. American Mathematical Society, Providence, RI, 2000. [2] Jo?o Paulo Costeira and Takeo Kanade. A multibody factorization method for independently moving objects. Int. J. Comput. Vis., 29(3):159?179, 1998. [3] Ehsan Elhamifar and Ren? Vidal. Sparse subspace clustering. In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pages 2790?2797. IEEE, 2009. [4] Ehsan Elhamifar and Ren? Vidal. Clustering disjoint subspaces via sparse representation. In 2010 IEEE International Conference on Acoustics, Speech and Signal Processing, pages 1926? 1929. IEEE, 2010. [5] Ehsan Elhamifar and Rene Vidal. Sparse subspace clustering: Algorithm, theory, and applications. IEEE Trans. Pattern Anal. Mach. Intell, 35(11):2765?2781, 2013. [6] Yehoram Gordon. On Milman?s inequality and random subspaces which escape through a mesh in Rn . In Geometric aspects of functional analysis, volume 1317 of Lecture Notes in Math., pages 84?106. Springer, 1988. [7] Reinhard Heckel and Helmut B?lcskei. Robust subspace clustering via thresholding. IEEE Trans. Inf. Theory, 61(11):6320?6342, 2015. 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Confidence bounds and intervals for parameters relating to the binomial, negative binomial, poisson and hypergeometric distributions with applications to rare events. 2008. [14] Mahdi Soltanolkotabi and Emmanuel J. Cand?s. A geometric analysis of subspace clustering with outliers. Ann. Statist., 40(4):2195?2238, 2012. [15] Mahdi Soltanolkotabi, Ehsan Elhamifar, and Emmanuel J. Cand?s. Robust subspace clustering. Ann. Statist., 42(2):669?699, 2014. [16] Ren? Vidal. Subspace clustering. IEEE Signal Process. Mag., 28(2):52?68, 2011. [17] Ulrike von Luxburg. A tutorial on spectral clustering. Statistics and Computing, 17(4):395? 416, 2007. [18] Yining Wang, Yu-Xiang Wang, and Aarti Singh. Graph connectivity in noisy sparse subspace clustering. In Artificial Intelligence and Statistics, pages 538?546, 2016. [19] Yu-Xiang Wang and Huan Xu. Noisy sparse subspace clustering. J. Mach. Learn. Res., 17:Paper No. 12, 41, 2016. [20] Yu-Xiang Wang, Huan Xu, and Chenlei Leng. 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The Neural Hawkes Process: A Neurally Self-Modulating Multivariate Point Process Hongyuan Mei Jason Eisner Department of Computer Science, Johns Hopkins University 3400 N. Charles Street, Baltimore, MD 21218 U.S.A {hmei,jason}@cs.jhu.edu Abstract Many events occur in the world. Some event types are stochastically excited or inhibited?in the sense of having their probabilities elevated or decreased?by patterns in the sequence of previous events. Discovering such patterns can help us predict which type of event will happen next and when. We model streams of discrete events in continuous time, by constructing a neurally self-modulating multivariate point process in which the intensities of multiple event types evolve according to a novel continuous-time LSTM. This generative model allows past events to influence the future in complex and realistic ways, by conditioning future event intensities on the hidden state of a recurrent neural network that has consumed the stream of past events. Our model has desirable qualitative properties. It achieves competitive likelihood and predictive accuracy on real and synthetic datasets, including under missing-data conditions. 1 Introduction Some events in the world are correlated. A single event, or a pattern of events, may help to cause or prevent future events. We are interested in learning the distribution of sequences of events (and in future work, the causal structure of these sequences). The ability to discover correlations among events is crucial to accurately predict the future of a sequence given its past, i.e., which events are likely to happen next and when they will happen. We specifically focus on sequences of discrete events in continuous time (?event streams?). Modeling such sequences seems natural and useful in many applied domains: ? Medical events. Each patient has a sequence of acute incidents, doctor?s visits, tests, diagnoses, and medications. By learning from previous patients what sequences tend to look like, we could predict a new patient?s future from their past. ? Consumer behavior. Each online consumer has a sequence of online interactions. By modeling the distribution of sequences, we can learn purchasing patterns. Buying cookies may temporarily depress purchases of all desserts, yet increase the probability of buying milk. ? ?Quantified self? data. Some individuals use cellphone apps to record their behaviors? eating, traveling, working, sleeping, waking. By anticipating behaviors, an app could perform helpful supportive actions, including issuing reminders and placing advance orders. ? Social media actions. Previous posts, shares, comments, messages, and likes by a set of users are predictive of their future actions. ? Other event streams arise in news, animal behavior, dialogue, music, etc. A basic model for event streams is the Poisson process (Palm, 1943), which assumes that events occur independently of one another. In a non-homogenous Poisson process, the (infinitesimal) probability of an event happening at time t may vary with t, but it is still independent of other events. A Hawkes process (Hawkes, 1971; Liniger, 2009) supposes that past events can temporarily raise the probability of future events, assuming that such excitation is ? positive, ? additive over the past events, and ? exponentially decaying with time. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Intensity-1 Intensity-2 BaseRate-1 BaseRate-2 LSTM-Unit Type-1 Type-2 Figure 1: Drawing an event stream from a neural Hawkes process. An LSTM reads the sequence of past events (polygons) to arrive at a hidden state (orange). That state determines the future ?intensities? of the two types of events?that is, their time-varying instantaneous probabilities. The intensity functions are continuous parametric curves (solid lines) determined by the most recent LSTM state, with dashed lines showing the steady-state asymptotes that they would eventually approach. In this example, events of type 1 excite type 1 but inhibit type 2. Type 2 excites itself, and excites or inhibits type 1 according to whether the count of type 2 events so far is odd or even. Those are immediate effects, shown by the sudden jumps in intensity. The events also have longer-timescale effects, shown by the shifts in the asymptotic dashed lines. However, real-world patterns often seem to violate these assumptions. For example, ? is violated if one event inhibits another rather than exciting it: cookie consumption inhibits cake consumption. ? is violated when the combined effect of past events is not additive. Examples abound: The 20th advertisement does not increase purchase rate as much as the first advertisement did, and may even drive customers away. Market players may act based on their own complex analysis of market history. Musical note sequences follow some intricate language model that considers melodic trajectory, rhythm, chord progressions, repetition, etc. ? is violated when, for example, a past event has a delayed effect, so that the effect starts at 0 and increases sharply before decaying. We generalize the Hawkes process by determining the event intensities (instantaneous probabilities) from the hidden state of a recurrent neural network. This state is a deterministic function of the past history. It plays the same role as the state of a deterministic finite-state automaton. However, the recurrent network enjoys a continuous and infinite state space (a high-dimensional Euclidean space), as well as a learned transition function. In our network design, the state is updated discontinuously with each successive event occurrence and also evolves continuously as time elapses between events. Our main motivation is that our model can capture effects that the Hawkes process misses. The combined effect of past events on future events can now be superadditive, subadditive, or even subtractive, and can depend on the sequential ordering of the past events. Recurrent neural networks already capture other kinds of complex sequential dependencies when applied to language modeling?that is, generative modeling of linguistic word sequences, which are governed by syntax, semantics, and habitual usage (Mikolov et al., 2010; Sundermeyer et al., 2012; Karpathy et al., 2015). We wish to extend their success (Chelba et al., 2013) to sequences of events in continuous time. Another motivation for a more expressive model than the Hawkes process is to cope with missing data. Even in a domain where Hawkes might be appropriate, it is hard to apply Hawkes when sequences are only partially observed. Real datasets may systematically omit some types of events (e.g., illegal drug use, or offline purchases) which, in the true generative model, would have a strong influence on the future. They may also have stochastically missing data, where the missingness mechanism?the probability that an event is not recorded?can be complex and data-dependent (MNAR). In this setting, we can fit our model directly to the observation sequences, and use it to predict observation sequences that were generated in the same way (using the same complete-data distribution and the same missingness mechanism). Note that if one knew the true complete-data distribution?perhaps Hawkes?and the true missingness mechanism, one would optimally predict the incomplete future from the incomplete past in Bayesian fashion, by integrating over possible completions (imputing the missing events and considering their influence on the future). Our hope is that the neural model is expressive enough that it can learn to approximate this true predictive distribution. Its hidden state after observing the past should implicitly encode the Bayesian posterior, and its update rule for this hidden state should emulate the ?observable operator? that updates the posterior upon each new observation. See Appendix A.4 for further discussion. 2 A final motivation is that one might wish to intervene in a medical, economic, or social event stream so as to improve the future course of events. Appendix D discusses our plans to deploy our model family as an environment model within reinforcement learning, where an agent controls some events. 2 Notation We are interested in constructing distributions over event streams (k1 , t1 ), (k2 , t2 ), . . ., where each ki ? {1, 2, . . . , K} is an event type and 0 < t1 < t2 < ? ? ? are times of occurrence.1 That is, there are K types of events, tokens of which are observed to occur in continuous time. For any distribution P in our proposed family, an event stream is almost surely infinite. However, when we observe the process only during a time interval [0, T ], the number I of observed events is almost surely finite. The log-likelihood ` of the model P given these I observations is I X  log P ((ki , ti ) | Hi , ?ti ) + log P (tI+1 > T | HI ) (1) i=1 def where the history Hi is the prefix sequence (k1 , t1 ), (k2 , t2 ), . . . , (ki?1 , ti?1 ), and ?ti = ti ? ti?1 , and P ((ki , ti ) | Hi , ?ti ) dt is the probability that the next event occurs at time ti and has type ki . Throughout the paper, the subscript i usually denotes quantities that affect the distribution of the next event (ki , ti ). These quantities depend only on the history Hi . We use (lowercase) Greek letters for parameters related to the classical Hawkes process, and Roman letters for other quantities, including hidden states and affine transformation parameters. We denote vectors by bold lowercase letters such as s and ?, and matrices by bold capital Roman letters such as U. Subscripted bold letters denote distinct vectors or matrices (e.g., wk ). Scalar quantities, including vector and matrix elements such as sk and ?j,k , are written without bold. Capitalized scalars represent upper limits on lowercase scalars, e.g., 1 ? k ? K. Function symbols are notated like their return type. All R ? R functions are extended to apply elementwise to vectors and matrices. 3 The Model In this section, we first review Hawkes processes, and then introduce our model one step at a time. Formally, generative models of event streams are multivariate point processes. A (temporal) point process is a probability distribution over {0, 1}-valued functions on a given time interval (for us, [0, ?)). A multivariate point process is formally a distribution over K-tuples of such functions. The k th function indicates the times at which events of type k occurred, by taking value 1 at those times. 3.1 Hawkes Process: A Self-Exciting Multivariate Point Process (SE-MPP) A basic model of event streams is the non-homogeneous multivariate Poisson process. It assumes that an event of type k occurs at time t?more precisely, in the infinitesimally wide interval [t, t + dt)?with probability ?k (t)dt. The value ?k (t) ? 0 can be regarded as a rate per unit time, just like the parameter ? of an ordinary Poisson process. ?k is known as the intensity function, and the total PK intensity of all event types is given by ?(t) = k=1 ?k (t). A well-known generalization that captures interactions is the self-exciting multivariate point process (SE-MPP), or Hawkes process (Hawkes, 1971; Liniger, 2009), in which past events h from the history conspire to raise the intensity of each type of event. Such excitation is positive, additive over the past events, and exponentially decaying with time: X ?k (t) = ?k + ?kh ,k exp(??kh ,k (t ? th )) (2) h:th <t where ?k ? 0 is the base intensity of event type k, ?j,k ? 0 is the degree to which an event of type j initially excites type k, and ?j,k > 0 is the decay rate of that excitation. When an event occurs, all intensities are elevated to various degrees, but then will decay toward their base rates ?. 1 More generally, one could allow 0 ? t1 ? t2 ? ? ? ? , where ti is a immediate event if ti?1 = ti and a delayed event if ti?1 < ti . It is not too difficult to extend our model to assign positive probability to immediate events, but we will disallow them here for simplicity. 3 3.2 Self-Modulating Multivariate Point Processes The positivity constraints in the Hawkes process limit its expressivity. First, the positive interaction parameters ?j,k fail to capture inhibition effects, in which past events reduce the intensity of future events. Second, the positive base rates ? fail to capture the inherent inertia of some events, which are unlikely until their cumulative excitation by past events crosses some threshold. To remove such limitations, we introduce two self-modulating models. Here the intensities of future events are stochastically modulated by the past history, where the term ?modulation? is meant to encompass both excitation and inhibition. The intensity ?k (t) can even fluctuate non-monotonically between successive events, because the competing excitatory and inhibitory influences may decay at different rates. 3.2.1 Hawkes Process with Inhibition: A Decomposable Self-Modulating MPP (D-SM-MPP) Our first move is to enrich the Hawkes model?s expressiveness while still maintaining its decomposable structure. We relax the positivity constraints on ?j,k and ?k , allowing them to range over R, which allows inhibition (?j,k < 0) and inertia (?k < 0). However, the resulting total activation could now be negative. We therefore pass it through a non-linear transfer function fk : R ? R+ to obtain a positive intensity function as required: X ? k (t)) (3a) ? k (t) = ?k + ?k (t) = fk (? ? ?k ,k exp(??k ,k (t ? th )) (3b) h h h:th <t As t increases between events, the intensity ?k (t) may both rise and fall, but eventually approaches the base rate f (?k +0), as the influence of each previous event still decays toward 0 at a rate ?j,k > 0. What non-linear function fk should we use? The ReLU function f (x) = max(x, 0) is not strictly positive as required. A better choice is the scaled ?softplus? function f (x) = s log(1 + exp(x/s)), which approaches ReLU as s ? 0. We learn a separate scale parameter sk for each event type k, ? k (t)) = sk log(1 + which adapts to the rate of that type. So we instantiate (3a) as ?k (t) = fk (? ? k (t)/sk )). Appendix A.1 graphs this and motivates the ?softness? and the scale parameter. exp(? 3.2.2 Neural Hawkes Process: A Neurally Self-Modulating MPP (N-SM-MPP) Our second move removes the restriction that the past events have independent, additive influence ? k (t). Rather than predict ? ? k (t) as a simple summation (3b), we now use a recurrent neural on ? network. This allows learning a complex dependence of the intensities on the number, order, and timing of past events. We refer to our model as a neural Hawkes process. Just as before, each event type k has an time-varying intensity ?k (t), which jumps discontinuously at each new event, and then drifts continuously toward a baseline intensity. In the new process, however, these dynamics are controlled by a hidden state vector h(t) ? (?1, 1)D , which in turn depends on a vector c(t) ? RD of memory cells in a continuous-time LSTM.2 This novel recurrent neural network architecture is inspired by the familiar discrete-time LSTM (Hochreiter and Schmidhuber, 1997; Graves, 2012). The difference is that in the continuous interval following an event, each memory cell c exponentially decays at some rate ? toward some steady-state value c?. At each time t > 0, we obtain the intensity ?k (t) by (4a), where (4b) shows how the hidden states h(t) are continually obtained from the memory cells c(t) as the cells decay: ?k (t) = fk (w> k h(t)) (4a) h(t) = oi  (2?(2c(t)) ? 1) for t ? (ti?1 , ti ] (4b) This says that on the interval (ti?1 , ti ]?in other words, after event i?1 up until event i occurs at some time ti ?the h(t) defined by equation (4b) determines the intensity functions via equation (4a). So for t in this interval, according to the model, h(t) is a sufficient statistic of the history (Hi , t ? ti?1 ) with respect to future events (see equation (1)). h(t) is analogous to hi in an LSTM language model (Mikolov et al., 2010), which summarizes the past event sequence k1 , . . . , ki?1 . But in our decay architecture, it will also reflect the interarrival times t1 ? 0, t2 ? t1 , . . . , ti?1 ? ti?2 , t ? ti?1 . This interval (ti?1 , ti ] ends when the next event ki stochastically occurs at some time ti . At this point, the continuous-time LSTM reads (ki , ti ) and updates the current (decayed) hidden cells c(t) to new initial values ci+1 , based on the current (decayed) hidden state h(ti ). 2 We use one-layer LSTMs with D hidden units in our present experiments, but a natural extension is to use multi-layer (?deep?) LSTMs (Graves et al., 2013), in which case h(t) is the hidden state of the top layer. 4 How does the continuous-time LSTM make those updates? Other than depending on decayed values, the update formulas resemble the discrete-time case:3 ii+1 f i+1 zi+1 oi+1 ? ? (Wi ki + Ui h(ti ) + di ) ? ? (Wf ki + Uf h(ti ) + df ) ? 2? (Wz ki + Uz h(ti ) + dz ) ? 1 ? ? (Wo ki + Uo h(ti ) + do ) ci+1 ? f i+1  c(ti ) + ii+1  zi+1 (6a) ?i+1 ? ?f i+1  c ?i + ??i+1  zi+1 c (6b) ? i+1 ? f (Wd ki + Ud h(ti ) + dd ) (6c) (5a) (5b) (5c) (5d) The vector ki ? {0, 1}K is the ith input: a one-hot encoding of the new event ki , with non-zero value only at the entry indexed by ki . The above formulas will make a discrete update to the LSTM state. They resemble the discrete-time LSTM, but there are two differences. First, the updates do not depend on the ?previous? hidden state from just after time ti?1 , but rather its value h(ti ) at time ti , after it has decayed for a period of ti ? ti?1 . Second, equations (6b)?(6c) are new. They define how in future, as t > ti increases, the elements of c(t) will continue to deterministically decay (at ?i+1 . Specifically, c(t) is given by (7), which continues to different rates) from ci+1 toward targets c control h(t) and thus ?k (t) (via (4), except that i has now increased by 1). def ?i+1 + (ci+1 ? c ?i+1 ) exp (?? i+1 (t ? ti )) for t ? (ti , ti+1 ] c(t) = c (7) In short, not only does (6a) define the usual cell values ci+1 , but equation (7) defines c(t) on R>0 . On the interval (ti , ti+1 ], c(t) follows an exponential curve that begins at ci+1 (in the sense that ?i+1 (which it would approach as t ? ?, if extrapolated). limt?t+ c(t) = ci+1 ) and decays toward c i A schematic example is shown in Figure 1. As in the previous models, ?k (t) drifts deterministically between events toward some base rate. But the neural version is different in three ways: ? The base rate is not a constant ?k , but shifts upon each event.4 ? The drift can be non-monotonic, because the excitatory and inhibitory influences on ?k (t) from different elements of h(t) may decay at different rates. ? The sigmoidal transfer function means that the behavior of h(t) itself is a little more interesting than exponential decay. Suppose that ci is very negative but increases toward a ?i > 0. Then h(t) will stay close to ?1 for a while and then will rapidly rise past 0. This target c usefully lets us model a delayed response (e.g. the last green segment in Figure 1). We point out two behaviors that are naturally captured by our LSTM?s ?forget? and ?input? gates: ? if f i+1 ? 1 and ii+1 ? 0, then ci+1 ? c(ti ). So c(t) and h(t) will be continuous at ti . There is no jump due to event i, though the steady-state target may change. ?i+1 ? c ?i . So although there may be a jump in activation, ? if ?f i+1 ? 1 and ??i+1 ? 0, then c it is temporary. The memory cells will decay toward the same steady states as before. Among other benefits, this lets us fit datasets in which (as is common) some pairs of event types do not influence one another. Appendix A.3 explains why all the models in this paper have this ability. The drift of c(t) between events controls how the system?s expectations about future events change as more time elapses with no event having yet occured. Equation (7) chooses a moderately flexible parametric form for this drift function (see Appendix D for some alternatives). Equation (6a) was designed so that c in an LSTM could learn to count past events with discrete-time exponential discounting; and (7) can be viewed as extending that to continuous-time exponential discounting. Our memory cell vector c(t) is a deterministic function of the past history (Hi , t ? ti ).5 Thus, the event intensities at any time are also deterministic via equation (4). The stochastic part of the model is the random choice?based on these intensities?of which event happens next and when it happens. The events are in competition: an event with high intensity is likely to happen sooner than an event with low intensity, and whichever one happens first is fed back into the LSTM. If no event type has high intensity, it may take a long time for the next event to occur. Training the model means learning the LSTM parameters in equations (5) and (6c) along with the other parameters mentioned in this section, namely sk ? R and wk ? RD for k ? {1, 2, . . . , K}. 3 The upright-font subscripts i, f, z and o are not variables, but constant labels that distinguish different W, U and d tensors. The ?f and ?? in equation (6) are defined analogously to f and i but with different weights. 4 Equations (4b) and (7) imply that after event i ? 1, the base rate jumps to fk (w> (oi  (2?(2? ci ) ? 1))). 5 Appendix A.2 explains how our LSTM handles the start and end of the sequence. 5 4 Algorithms For the proposed models, the log-likelihood (1) of the parameters turns out to be given by a simple formula?the sum of the log-intensities of the events that happened, at the times they happened, minus an integral of the total intensities over the observation interval [0, T ]: Z T X log ?ki (ti ) ? ?(t)dt `= (8) i:ti ?T | t=0{z } call this ? The full derivation is given in Appendix B.1. Intuitively, the ?? term (which is ? 0) sums the log-probabilities of infinitely many non-events. Why? The probability that there was not an event of any type in the infinitesimally wide interval [t, t + dt) is 1 ? ?(t)dt, whose log is ??(t)dt. We can locally maximize ` using any stochastic gradient method. A detailed recipe is given in Appendix B.2, including the Monte Carlo trick we use to handle the integral in equation (8). If we wish to draw random sequences from the model, we can adopt the thinning algorithm (Lewis and Shedler, 1979; Liniger, 2009) that is commonly used for the Hawkes process. See Appendix B.3. Given an event stream prefix (k1 , t1 ), (k2 , t2 ), . . . , (ki?1 , ti?1 ), we may wish to predict the time and type of the single next  event. The next event?s time ti has density pi (t) = P (ti = t | Hi ) = Rt ?(t) exp ? ti?1 ?(s)ds . To predict a single time whose expected L2 loss is as low as possible, R? we should choose t?i = E[ti | Hi ] = tpi (t)dt. Given the next event time ti , the most likely ti?1 type would be argmaxk ?k (ti )/?(ti ), but the most likely next event type without knowledge of ti R ? ?k (t) pi (t)dt. The integrals in the preceding equations can be estimated by is k?i = argmax k ti?1 ?(t) Monte Carlo samlping much as before (Appendix B.2). For event type prediction, we recommend a paired comparison that uses the same t values for each k in the argmax; this also lets us share the ?(t) and pi (t) computations across all k. 5 Related Work The Hawkes process has been widely used to model event streams, including for topic modeling and clustering of text document streams (He et al., 2015; Du et al., 2015a), constructing and inferring network structure (Yang and Zha, 2013; Choi et al., 2015; Etesami et al., 2016), personalized recommendations based on users? temporal behavior (Du et al., 2015b), discovering patterns in social interaction (Guo et al., 2015; Lukasik et al., 2016), learning causality (Xu et al., 2016), and so on. Recent interest has focused on expanding the expressivity of Hawkes processes. Zhou et al. (2013) describe a self-exciting process that removes the assumption of exponentially decaying influence (as we do). They replace the scaled-exponential summands in equation (2) with learned positive functions of time (the choice of function again depends on ki , k). Lee et al. (2016) generalize the constant excitation parameters ?j,k to be stochastic, which increases expressivity. Our model also allows non-constant interactions between event types, but arranges these via deterministic, instead of stochastic, functions of continuous-time LSTM hidden states. Wang et al. (2016) consider non-linear effects of past history on the future, by passing the intensity functions of the Hawkes process through a non-parametric isotonic link function g, which is in the same place as our non-linear function fk . In contrast, our fk has a fixed parametric form (learning only the scale parameter), and is approximately linear when x is large. This is because we model non-linearity (and other complications) with a continuous-time LSTM, and use fk only to ensure positivity of the intensity functions. Du et al. (2016) independently combined Hawkes processes with recurrent neural networks (and Xiao et al. (2017a) propose an advanced way of estimating the parameters of that model). However, Du et al.?s architecture is different in several respects. They use standard discrete-time LSTMs without our decay innovation, so they must encode the intervals between past events as explicit numerical inputs to the LSTM. They have only a single intensity function ?(t), and it simply decays exponentially toward 0 between events, whereas our more modular model creates separate (potentially transferrable) functions ?k (t), each of which allows complex and non-monotonic dynamics en route to a non-zero steady state intensity. Some structural limitations of their design are that ti and ki are conditionally independent given h (they are determined by separate distributions), and that their model cannot avoid a positive probability of extinction at all times. Finally, since they take 6 f = exp, the effect of their hidden units on intensity is effectively multiplicative, whereas we take f = softplus to get an approximately additive effect inspired by the classical Hawkes process. Our rationale is that additivity is useful to capture independent (disjunctive) causes; at the same time, the hidden units that our model adds up can each capture a complex joint (conjunctive) cause. Experiments6 6 We fit our various models on several simulated and real-world datasets, and evaluated them in each case by the log-probability that they assigned to held-out data. We also compared our approach with that of Du et al. (2016) on their prediction task. The datasets that we use in this paper range from one extreme with only K = 2 event types but mean sequence length > 2000, to the other extreme with K = 5000 event types but mean sequence length 3. Dataset details can be found in Table 1 in Appendix C.1. Training details (e.g., hyperparameter selection) can be found in Appendix C.2. 6.1 Synthetic Datasets In a pilot experiment with synthetic data (Appendix C.4), we confirmed that the neural Hawkes process generates data that is not well modeled by training an ordinary Hawkes process, but that ordinary Hawkes data can be successfully modeled by training an neural Hawkes process. In this experiment, we were not limited to measuring the likelihood of the models on the stochastic event sequences. We also knew the true latent intensities of the generating process, so we were able to directly measure whether the trained models predicted these intensities accurately. The pattern of results was similar. 6.2 Real-World Media Datasets Retweets Dataset (Zhao et al., 2015). On Twitter, novel tweets are generated from some distribution, which we do not model here. Each novel tweet serves as the beginning-of-stream event (see Appendix A.2) for a subsequent stream of retweet events. We model the dynamics of these streams: how retweets by various types of users (K = 3) predict later retweets by various types of users. Details of the dataset and its preparation are given in Appendix C.5. The dataset is interesting for its temporal pattern. People like to retweet an interesting post soon after it is created and retweeted by others, but may gradually lose interest, so the intervals between retweets become longer over time. In other words, the stream begins in a self-exciting state, in which previous retweets increase the intensities of future retweets, but eventually interest dies down and events are less able to excite one another. The decomposable models are essentially incapable of modeling such a phase transition, but our neural model should have the capacity to do so. We generated learning curves (Figure 2) by training our models on increasingly long prefixes of the training set. As we can see, our self-modulating processes significantly outperform the Hawkes process at all training sizes. There is no obvious a priori reason to expect inhibition or even inertia in this application domain, which explains why the D-SM-MPP makes only a small improvement over the Hawkes process when the latter is well-trained. But D-SM-MPP requires much less data, and also has more stable behavior (smaller error bars) on small datasets. Our neural model is even better. Not only does it do better on the average stream, but its consistent superiority over the other two models is shown by the per-stream scatterplots in Figure 3, demonstrating the importance of our model?s neural component even with large datasets. MemeTrack Dataset (Leskovec and Krevl, 2014). This dataset is similar in conception to Retweets, but with many more event types (K = 5000). It considers the reuse of fixed phrases, or ?memes,? in online media. It contains time-stamped instances of meme use in articles and posts from 1.5 million different blogs and news sites. We model how the future occurrence of a meme is affected by its past trajectory across different websites?that is, given one meme?s past trajectory across websites, when and where it will be mentioned again. On this dataset,7 the advantage of our full neural models was dramatic, yielding cross-entropy per event of around ?8 relative to the ?15 of D-SM-MPP?which in turn is far above the ?800 of the 6 7 Our code and data are available at https://github.com/HMEIatJHU/neurawkes. Data preparation details are given in Appendix C.6. 7 20 30 40 125 250 7 8 9 500 1000 2000 4000 8000 16000 number of training sequences 500 0 10 N-SM-MPP D-SM-MPP SE-MPP 500 1000 1500 250 500 1000 2000 4000 8000 16000 number of training sequences 10 20 30 40 50 2000 125 N-SM-MPP D-SM-MPP 0 log-likelihood per event 10 N-SM-MPP D-SM-MPP SE-MPP 6 log-likelihood per event log-likelihood per event log-likelihood per event 5 N-SM-MPP D-SM-MPP SE-MPP 0 1000 2000 4000 8000 16000 number of training sequences 60 32000 4000 8000 16000 32000 number of training sequences 2 0 0 1.0 2 2 1.2 4 1.4 4 6 8 SE-MPP 2 D-SM-MPP SE-MPP Figure 2: Learning curve (with 95% error bars) of all three models on the Retweets (left two) and MemeTrack (right two) datasets. Our neural model significantly outperforms our decomposable model (right graph of each pair), and both significantly outperform the Hawkes process (left of each pair?same graph zoomed out). 6 1.8 8 10 2.0 10 10 8 6 4 N-SM-MPP 2 0 2 1.6 10 8 6 4 N-SM-MPP 2 0 2 Figure 3: Scatterplots of N-SM-MPP vs. SE-MPP (left) and N-SM-MPP vs. D-SM-MPP (right), comparing the held-out log-likelihood of the two models (when trained on our full Retweets training set) with respect to each of the 2000 test sequences. Nearly all points fall to the right of y = x, since N-SM-MPP (the neural Hawkes process) is consistently more predictive than our non-neural model and the Hawkes process. 2.0 1.8 1.6 1.4 N-SM-MPP 1.2 1.0 Figure 4: Scatterplot of N-SMMPP vs. SE-MPP, comparing their log-likelihoods with respect to each of the 31 incomplete sequences? test sets. All 31 points fall to the right of y = x. Hawkes process. Figure 2 illustrates the persistent gaps among the models. A scatterplot similar to Figure 3 is given in Figure 13 of Appendix C.6. We attribute the poor performance of the Hawkes process to its failure to capture the latent properties of memes, such as their topic, political stance, or interestingness. This is a form of missing data (section 1), as we now discuss. As the table in Appendix C.1 indicates, most memes in MemeTrack are uninteresting and give rise to only a short sequence of mentions. Thus the base mention probability is low. An ideal analysis would recognize that if a specific meme has been mentioned several times already, it is a posteriori interesting and will probably be mentioned in future as well. The Hawkes process cannot distinguish the interesting memes from the others, except insofar as they appear on more influential websites. By contrast, our D-SM-MPP can partly capture this inferential pattern by using negative base rates ? to create ?inertia? (section 3.2.1). Indeed, all 5000 of its learned ?k parameters were negative, with values ranging from ?10 to ?30, which numerically yields 0 intensity and is hard to excite. An ideal analysis would also recognize that if a specific meme has appeared mainly on conservative websites, it is a posteriori conservative and unlikely to appear on liberal websites in the future. The D-SM-MPP, unlike the Hawkes process, can again partly capture this, by having conservative websites inhibit liberal ones. Indeed, 24% of its learned ? parameters were negative. (We re-emphasize that this inhibition is merely a predictive effect?probably not a direct causal mechanism.) And our N-SM-MPP process is even more powerful. The LSTM state aims to learn sufficient statistics for predicting the future, so it can learn hidden dimensions (which fall in (?1, 1)) that encode useful posterior beliefs in boolean properties of the meme such as interestingness, conservativeness, timeliness, etc. The LSTM?s ?long short-term memory? architecture explicitly allows these beliefs to persist indefinitely through time in the absence of new evidence, without having to be refreshed by redundant new events as in the decomposable models. Also, the LSTM?s hidden dimensions are computed by sigmoidal activation rather than softplus activation, and so can be used implicitly to perform logistic regression. The flat left side of the sigmoid resembles softplus and can model inertia as we saw above: it takes several mentions to establish interestingness. Symmetrically, the flat right side can model saturation: once the posterior probability of interestingness is at 80%, it cannot climb much farther no matter how many more mentions are observed. A final potential advantage of the LSTM is that in this large-K setting, it has fewer parameters than the other models (Appendix C.3), sharing statistical strength across event types (websites) to generalize better. The learning curves in Figure 2 suggest that on small data, the decomposable 8 (non-neural) models may overfit their O(K 2 ) interaction parameters ?j,k . Our neural model only has to learn O(D2 ) pairwise interactions among its D hidden nodes (where D  K), as well as O(KD) interactions between the hidden nodes and the K event types. In this case, K = 5000 but D = 64. This reduction by using latent hidden nodes is analogous to nonlinear latent factor analysis. 6.3 Modeling Streams With Missing Data We set up an artificial experiment to more directly investigate the missing-data setting of section 1, where we do not observe all events during [0, T ], but train and test our model just as if we had. We sampled synthetic event sequences from a standard Hawkes process (just as in our pilot experiment from 6.1), removed all the events of selected types, and then compared the neural Hawkes process (N-SM-MPP) with the Hawkes process (SE-MPP) as models of these censored sequences. Since we took K = 5, there were 25 ? 1 = 31 ways to construct a dataset of censored sequences. As shown in Figure 4, for each of the 31 resulting datasets, training a neural Hawkes model achieves better generalization. Appendix A.4 discusses why this kind of behavior is to be expected. 6.4 Prediction Tasks?Medical, Social and Financial To compare with Du et al. (2016), we evaluate our model on the prediction tasks and datasets that they proposed. The Financial Transaction dataset contains long streams of high frequency stock transactions for a single stock, with the two event types ?buy? and ?sell.? The electrical medical records (MIMIC-II) dataset is a collection of de-identified clinical visit records of Intensive Care Unit patients for 7 years. Each patient has a sequence of hospital visit events, and each event records its time stamp and disease diagnosis. The Stack Overflow dataset represents two years of user awards on a question-answering website: each user received a sequence of badges (of 22 different types). We follow Du et al. (2016) and attempt to predict every held-out event (ki , ti ) from its history Hi , evaluating the prediction k?i with 0-1 loss (yielding an error rate, or ER) and evaluating the prediction t?i with L2 loss (yielding a root-mean-squared error, or RMSE). We make minimum Bayes risk predictions as explained in section 4. Figure 8 in Appendix C.7 shows that our model consistently outperforms that of Du et al. (2016) on event type prediction on all the datasets, although for time prediction neither model is consistently better. 6.5 Sensitivity to Number of Parameters Does our method do well because of its flexible nonlinearities or just because it has more parameters? The answer is both. We experimented on the Retweets data with reducing the number of hidden units D. Our N-SM-MPP substantially outperformed SE-MPP (the Hawkes process) on held-out data even with very few parameters, although more parameters does even better: number of hidden units number of parameters log-likelihood Hawkes 21 -7.19 1 31 -6.51 2 87 -6.41 4 283 -6.36 8 1011 -6.24 16 3811 -6.18 32 14787 -6.16 256 921091 -6.10 We also tried halving D across several datasets, which had negligible effect, always decreasing held-out log-likelihood by < 0.2% relative. More information about model sizes is given in Appendix C.3. Note that the neural Hawkes process does not always have more parameters. When K is large, we can greatly reduce the number of params below that of a Hawkes process, by choosing D  K, as for MemeTrack in section 6.2. 7 Conclusion We presented two extensions to the multivariate Hawkes process, a popular generative model of streams of typed, timestamped events. Past events may now either excite or inhibit future events. They do so by sequentially updating the state of a novel continuous-time recurrent neural network (LSTM). Whereas Hawkes sums the time-decaying influences of past events, we instead sum the time-decaying influences of the LSTM nodes. Our extensions to Hawkes aim to address real-world phenomena, missing data, and causal modeling. Empirically, we have shown that both extensions yield a significantly improved ability to predict the course of future events. There are several exciting avenues for further improvements (discussed in Appendix D), including embedding our model within a reinforcement learner to discover causal structure and learn an intervention policy. 9 Acknowledgments We are grateful to Facebook for enabling this work through a gift to the second author. Nan Du kindly helped us by making his code public and answering questions, and the NVIDIA Corporation kindly donated two Titan X Pascal GPUs. We also thank our lab group at Johns Hopkins University?s Center for Language and Speech Processing for helpful comments. The first version of this work appeared on arXiv in December 2016. References Ciprian Chelba, Tomas Mikolov, Mike Schuster, Qi Ge, Thorsten Brants, Phillipp Koehn, and Tony Robinson. One billion word benchmark for measuring progress in statistical language modeling. Computing Research Repository, arXiv:1312.3005, 2013. URL http://arxiv.org/abs/ 1312.3005. Edward Choi, Nan Du, Robert Chen, Le Song, and Jimeng Sun. 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Inverse Reward Design Dylan Hadfield-Menell Smitha Milli Pieter Abbeel? Stuart Russell Anca Dragan Department of Electrical Engineering and Computer Science University of California, Berkeley Berkeley, CA 94709 {dhm, smilli, pabbeel, russell, anca}@cs.berkeley.edu Abstract Autonomous agents optimize the reward function we give them. What they don?t know is how hard it is for us to design a reward function that actually captures what we want. When designing the reward, we might think of some specific training scenarios, and make sure that the reward will lead to the right behavior in those scenarios. Inevitably, agents encounter new scenarios (e.g., new types of terrain) where optimizing that same reward may lead to undesired behavior. Our insight is that reward functions are merely observations about what the designer actually wants, and that they should be interpreted in the context in which they were designed. We introduce inverse reward design (IRD) as the problem of inferring the true objective based on the designed reward and the training MDP. We introduce approximate methods for solving IRD problems, and use their solution to plan risk-averse behavior in test MDPs. Empirical results suggest that this approach can help alleviate negative side effects of misspecified reward functions and mitigate reward hacking. 1 Introduction Robots2 are becoming more capable of optimizing their reward functions. But along with that comes the burden of making sure we specify these reward functions correctly. Unfortunately, this is a notoriously difficult task. Consider the example from Figure 1. Alice, an AI engineer, wants to build a robot, we?ll call it Rob, for mobile navigation. She wants it to reliably navigate to a target location and expects it to primarily encounter grass lawns and dirt pathways. She trains a perception system to identify each of these terrain types and then uses this to define a reward function that incentivizes moving towards the target quickly, avoiding grass where possible. When Rob is deployed into the world, it encounters a novel terrain type; for dramatic effect, we?ll suppose that it is lava. The terrain prediction goes haywire on this out-of-distribution input and generates a meaningless classification which, in turn, produces an arbitrary reward evaluation. As a result, Rob might then drive to its demise. This failure occurs because the reward function Alice specified implicitly through the terrain predictors, which ends up outputting arbitrary values for lava, is different from the one Alice intended, which would actually penalize traversing lava. In the terminology from Amodei et al. (2016), this is a negative side effect of a misspecified reward ? a failure mode of reward design where leaving out important aspects leads to poor behavior. Examples date back to King Midas, who wished that everything he touched turn to gold, leaving out that he didn?t mean his food or family. Another failure mode is reward hacking, which happens when, e.g., a vacuum cleaner ejects collected dust so that it can collect even more (Russell & Norvig, 2010), or a racing boat in a game loops in place to collect points instead of actually winning the race (Amodei & Clark, 2016). Short of requiring that the reward designer anticipate and penalize all possible misbehavior in advance, how can we alleviate the impact of such reward misspecification? ? 2 OpenAI, International Computer Science Institute (ICSI) Throughout this paper, we will use robot to refer generically to any artificial agent. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. true reward function 1 2 10 .. . 100 .. . intended environment + actual environment proxy reward function 1 2 10 .. . .. . 0 0 0 Figure 1: An illustration of a negative side effect. Alice designs a reward function so that her robot navigates to the pot of gold and prefers dirt paths. She does not consider that her robot might encounter lava in the real world and leaves that out of her reward specification. The robot maximizing this proxy reward function drives through the lava to its demise. In this work, we formalize the (Bayesian) inverse reward design (IRD) problem as the problem of inferring (a distribution on) the true reward function from the proxy. We show that IRD can help mitigate unintended consequences from misspecified reward functions like negative side effects and reward hacking. We leverage a key insight: that the designed reward function should merely be an observation about the intended reward, rather than the definition; and should be interpreted in the context in which it was designed. First, a robot should have uncertainty about its reward function, instead of treating it as fixed. This enables it to, e.g., be risk-averse when planning in scenarios where it is not clear what the right answer is, or to ask for help. Being uncertain about the true reward, however, is only half the battle. To be effective, a robot must acquire the right kind of uncertainty, i.e. know what it knows and what it doesn?t. We propose that the ?correct? shape of this uncertainty depends on the environment for which the reward was designed. In Alice?s case, the situations where she tested Rob?s learning behavior did not contain lava. Thus, the lava-avoiding reward would have produced the same behavior as Alice?s designed reward function in the (lava-free) environments that Alice considered. A robot that knows the settings it was evaluated in should also know that, even though the designer specified a lava-agnostic reward, they might have actually meant the lava-avoiding reward. Two reward functions that would produce similar behavior in the training environment should be treated as equally likely, regardless of which one the designer actually specified. We formalize this in a probabilistic model that relates the proxy (designed) reward to the true reward via the following assumption: Assumption 1. Proxy reward functions are likely to the extent that they lead to high true utility behavior in the training environment. Formally, we assume that the observed proxy reward function is the approximate solution to a reward design problem (Singh et al., 2010). Extracting the true reward is the inverse reward design problem. The idea of using human behavior as observations about the reward function is far from new. Inverse reinforcement learning uses human demonstrations (Ng & Russell, 2000; Ziebart et al., 2008), shared autonomy uses human operator control signals (Javdani et al., 2015), preference-based reward learning uses answers to comparison queries (Jain et al., 2015), and even what the human wants (HadfieldMenell et al., 2017). We observe that, even when the human behavior is to actually write down a reward function, this should still be treated as an observation, demanding its own observation model. Our paper makes three contributions. First, we define the inverse reward design (IRD) problem as the problem of inferring the true reward function given a proxy reward function, an intended decision problem (e.g., an MDP), and a set of possible reward functions. Second, we propose a solution to IRD and justify how an intuitive algorithm which treats the proxy reward as a set of expert demonstrations can serve as an effective approximation. Third, we show to that this inference approach, combined with risk-averse planning, leads to algorithms that are robust to misspecified rewards, alleviating both negative side effects as well as reward hacking. We build a system that ?knows-what-it-knows? about reward evaluations that automatically detects and avoids distributional shift in situations with high-dimensional features. Our approach substantially outperforms the baseline of literal reward interpretation. 2 2 Inverse Reward Design Definition 1. (Markov Decision Process Puterman (2009)) A (finite-horizon) Markov decision process (MDP), M , is a tuple M = hS, A, T, r, Hi. S is a set of states. A is a set of actions. T is a probability distribution over the next state, given the previous state and action. We write this as T (st+1 |st , a). r is a reward function that maps states to rewards r : S 7? R. H ? Z+ is the finite planning horizon for the agent. A solution to M is a policy: a mapping from the current timestep and state to a distribution over actions. The optimal policy maximizes the expected sum of rewards. We will use ? to represent trajectories. In this work, we consider reward functions that are linear combinations of feature vectors ?(?). Thus, the reward for a trajectory, given weights w, is r(?; w) = w> ?(?). The MDP formalism defines optimal behavior, given a reward function. However, it provides no information about where this reward function comes from (Singh et al., 2010). We refer to an MDP without rewards as a world model. In practice, a system designer needs to select a reward function that encapsulates the intended behavior. This process is reward engineering or reward design: Definition 2. (Reward Design Problem (Singh et al., 2010)) A reward design problem (RDP) is ? ? ? ? ? defined as a tuple P = hr? , M , R, ?(?| r , M )i. r? is the true reward function. M is a world model. ? ? ? R is a set of proxy reward functions. ?(?| r , M ) is an agent model, that defines a distribution on trajectories given a (proxy) reward function and a world model. ? ? In an RDP, the designer believes that an agent, represented by the policy ?(?| r , M ), will be deployed ? ? ? ? in M . She must specify a proxy reward function r ? R for the agent. Her goal is to specify r so that ? ? ? ?(?| r , M ) obtains high reward according to the true reward function r? . We let w represent weights for the proxy reward function and w? represent weights for the true reward function. In this work, our motivation is that system designers are fallible, so we should not expect that they perfectly solve the reward design problem. Instead we consider the case where the system designer is approximately optimal at solving a known RDP, which is distinct from the MDP that the robot currently finds itself in. By inverting the reward design process to infer (a distribution on) the true reward function r? , the robot can understand where its reward evaluations have high variance and plan to avoid those states. We refer to this inference problem as the inverse reward design problem: Definition 3. (Inverse Reward Design) The inverse reward design (IRD) problem is defined by a ? ? ? ? ? ? tuple hR, M , R, ?(?| r , M ), r i. R is a space of possible reward functions. M is a world model. ? ? ? ? h?, M , R, ?(?| r , M )i partially specifies an RDP P , with an unobserved reward function r? ? R. ? ? r ? R is the observed proxy reward that is an (approximate) solution to P . In solving an IRD problem, the goal is to recover r? . We will explore Bayesian approaches to IRD, so ? ? ? we will assume a prior distribution on r? and infer a posterior distribution on r? given r P (r? | r , M ). 3 Related Work Optimal reward design. Singh et al. (2010) formalize and study the problem of designing optimal rewards. They consider a designer faced with a distribution of environments, a class of reward functions to give to an agent, and a fitness function. They observe that, in the case of bounded agents, it may be optimal to select a proxy reward that is distinct from the fitness function. Sorg et al. (2010) and subsequent work has studied the computational problem of selecting an optimal proxy reward. In our work, we consider an alternative situation where the system designer is the bounded agent. In this case, the proxy reward function is distinct from the fitness function ? the true utility function in our terminology ? because system designers can make mistakes. IRD formalizes the problem of determining a true utility function given an observed proxy reward function. This enables us to design agents that are robust to misspecifications in their reward function. Inverse reinforcement learning. In inverse reinforcement learning (IRL) (Ng & Russell, 2000; Ziebart et al., 2008; Evans et al., 2016; Syed & Schapire, 2007) the agent observes demonstrations of (approximately) optimal behavior and infers the reward function being optimized. IRD is a similar 3 problem, as both approaches infer an unobserved reward function. The difference is in the observation: IRL observes behavior, while IRD directly observes a reward function. Key to IRD is assuming that this observed reward incentivizes behavior that is approximately optimal with respect to the true reward. In Section 4.2, we show how ideas from IRL can be used to approximate IRD. Ultimately, we consider both IRD and IRL to be complementary strategies for value alignment (Hadfield-Menell et al., 2016): approaches that allow designers or users to communicate preferences or goals. Pragmatics. The pragmatic interpretation of language is the interpretation of a phrase or utterance in the context of alternatives (Grice, 1975). For example, the utterance ?some of the apples are red? is often interpreted to mean that ?not all of the apples are red? although this is not literally implied. This is because, in context, we typically assume that a speaker who meant to say ?all the apples are red? would simply say so. Recent models of pragmatic language interpretation use two levels of Bayesian reasoning (Frank et al., 2009; Goodman & Lassiter, 2014). At the lowest level, there is a literal listener that interprets language according to a shared literal definition of words or utterances. Then, a speaker selects words in order to convey a particular meaning to the literal listener. To model pragmatic inference, we consider the probable meaning of a given utterance from this speaker. We can think of IRD as a model of pragmatic reward interpretation: the speaker in pragmatic interpretation of language is directly analogous to the reward designer in IRD. 4 Approximating the Inference over True Rewards We solve IRD problems by formalizing Assumption 1: the idea that proxy reward functions are likely to the extent that they incentivize high utility behavior in the training MDP. This will give us a ? ? probabilistic model for how w is generated from the true w? and the training MDP M . We will invert ? ? this probability model to compute a distribution P (w = w? |w, M ) on the true utility function. 4.1 Observation Model ? ? Recall that ?(?|w, M ) is the designer?s model of the probability that the robot will select trajectory ? ? ? ?, given proxy reward w. We will assume that ?(?|w, M ) is the maximum entropy trajectory distribution from Ziebart et al. (2008), i.e. the designer models the robot as approximately optimal: ? ? ? ?(?|w, M ) ? exp(w> ?(?)). An optimal designer chooses w to maximize expected true value, i.e. ? ? E[w? > ?(?)|? ? ?(?|w, M )] is high. We model an approximately optimal designer:    ? ? ? ? P (w|w? , M ) ? exp ? E w? > ?(?)|? ? ?(?|w, M ) (1) with ? controlling how close to optimal we assume the person to be. This is now a formal statement of ? ? ? Assumption 1. w? can be pulled out of the expectation, so we let ? = E[?(?)|? ? ?(?|w, M )]. Our ? ? ? ? goal is to invert (1) and sample from (or otherwise estimate) P (w? |w, M ) ? P (w|w? , M )P (w? ). ? ? The primary difficulty this entails is that we need to know the normalized probability P (w|w? , M ). ? This depends on its normalizing constant, Z(w), which integrates over possible proxy rewards.   ? >   Z exp ?w ? ? ? ? ? ? ? > P (w = w |w, M ) ? P (w), Z(w) = ? exp ?w ? dw. (2) ? w Z(w) 4.2 Efficient approximations to the IRD posterior ? ? ? ? To compute P (w = w? |w, M ), we must compute Z, which is intractable if w lies in an infinite or ? large finite set. Notice that computing the value of the integrand for Z is highly non-trivial as it involves solving a planning problem. This is an example of what is referred to as a doubly-intractable likelihood (Murray et al., 2006). We consider two methods to approximate this normalizing constant. 4 Figure 2: An example from the Lavaland domain. Left: The training MDP where the designer specifies a proxy reward function. This incentivizes movement toward targets (yellow) while preferring dirt (brown) to grass (green), and generates the gray trajectory. Middle: The testing MDP has lava (red). The proxy does not penalize lava, so optimizing it makes the agent go straight through (gray). This is a negative side effect, which the IRD agent avoids (blue): it treats the proxy as an observation in the context of the training MDP, which makes it realize that it cannot trust the (implicit) weight on lava. Right: The testing MDP has cells in which two sensor indicators no longer correlate: they look like grass to one sensor but target to the other. The proxy puts weight on the first, so the literal agent goes to these cells (gray). The IRD agent knows that it can?t trust the distinction and goes to the target on which both sensors agree (blue). Sample to approximate the normalizing constant. This approach, inspired by methods in approximate Bayesian computation (Sunn?ker et al., 2013), samples a finite set of weights {wi } to approximate the integral in Equation 2. We found empirically that it helped to include the candidate sample w in the sum. This leads to the normalizing constant ? Z(w) = w > ?w + N ?1 X
exp ?w> ?i . (3) i=0 Where ?i and ?w are the vector of feature counts realized optimizing wi and w respectively. Bayesian inverse reinforcement learning. During inference, the normalizing constant serves a calibration purpose: it computes how good the behavior produced by all proxy rewards in that MDP would be with respect to the true reward. Reward functions which increase the reward for all trajectories are not preferred in the inference. This creates an invariance to linear shifts in the feature encoding. If we were to change the MDP by shifting features by some vector ?0 , ? ? ? + ?0 , the posterior over w would remain the same. We can achieve a similar calibration and maintain the same property by directly integrating over the possible trajectories in the MDP:   ? Z ? exp ?w> ? ? Z(w) = exp(w> ?(?))d? ; P? (w|w) ? (4) Z(w) ? Proposition 1. The posterior distribution that the IRD model induces on w? (i.e., Equation 2) and the posterior distribution induced by IRL (i.e., Equation 4) are invariant to linear translations of the features in the training MDP. Proof. See supplementary material. This choice of normalizing constant approximates the posterior to an IRD problem with the posterior from maximum entropy IRL (Ziebart et al., 2008). The result has an intuitive interpretation. The ? proxy w determines the average feature counts for a hypothetical dataset of expert demonstrations ? ? and ? determines the effective size of that dataset. The agent solves M with reward w and computes ? the corresponding feature expectations ?. The agent then pretends like it got ? demonstrations with ? features counts ?, and runs IRL. The more the robot believes the human is good at reward design, the more demonstrations it pretends to have gotten from the person. The fact that reducing the proxy to ? behavior in M approximates IRD is not surprising: the main point of IRD is that the proxy reward is merely a statement about what behavior is good in the training environment. 5 ?k Is ?k s Is 2 {grass, dirt, target, unk} s ? N (?Is , ?Is ) Figure 3: Our challenge domain with latent rewards. Each terrain type (grass, dirt, target, lava) induces a different distribution over high-dimensional features: ?s ? N (?Is , ?Is ). The designer never builds an indicator for lava, and yet the agent still needs to avoid it in the test MDPs. 5 Evaluation 5.1 Experimental Testbed We evaluated our approaches in a model of the scenario from Figure 1 that we call Lavaland. Our system designer, Alice, is programming a mobile robot, Rob. We model this as a gridworld with movement in the four cardinal direction and four terrain types: target, grass, dirt, and lava. The true objective for Rob, w? , encodes that it should get to the target quickly, stay off the grass, and avoid lava. Alice designs a proxy that performs well in a training MDP that does not contain lava. Then, we measure Rob?s performance in a test MDP that does contain lava. Our results show that combining IRD and risk-averse planning creates incentives for Rob to avoid unforeseen scenarios. We experiment with four variations of this environment: two proof-of-concept conditions in which the reward is misspecified, but the agent has direct access to feature indicators for the different categories (i.e. conveniently having a feature for lava); and two challenge conditions, in which the right features are latent; the reward designer does not build an indicator for lava, but by reasoning in the raw observation space and then using risk-averse planning, the IRD agent still avoids lava. 5.1.1 Proof-of-Concept Domains These domains contain feature indicators for the four categories: grass, dirt, target, and lava. Side effects in Lavaland. Alice expects Rob to encounter 3 types of terrain: grass, dirt, and target, ? and so she only considers the training MDP from Figure 2 (left). She provides a w to encode a trade-off between path length and time spent on grass. The training MDP contains no lava, but it is introduced when Rob is deployed. An agent that treats the proxy reward literally might go on the lava in the test MDP. However, an agent that runs IRD will know that it can?t trust the weight on the lava indicator, since all such weights would produce the same behavior in the training MDP (Figure 2, middle). Reward Hacking in Lavaland. Reward hacking refers generally to reward functions that can be gamed or tricked. To model this within Lavaland, we use features that are correlated in the training domain but are uncorrelated in the testing environment. There are 6 features: three from one sensor and three from another sensor. In the training environment the features from both sensors are correct indicators of the state?s terrain category (grass, dirt, target). At test time, this correlation gets broken: lava looks like the target category to the second sensor, but the grass category to the first sensor. This is akin to how in a racing game (Amodei & Clark, 2016), winning and game points can be correlated at reward design time, but test environments might contain loopholes for maximizing points without winning. We want agents to hedge their bets between winning and points, or, in Lavaland, between the two sensors. An agent that treats the proxy reward function literally might go to the these new cells if they are closer. In contrast, an agent that runs IRD will know that a reward function with the same weights put on the first sensor is just as likely as the proxy. Risk averse planning makes it go to the target for which both sensors agree (Figure 2, right). 6 MaxEnt Z Sample Z Proof-of-Concept Proxy Latent Rewards Fraction of ? with Lava 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.0 0.2 Negative Side Effects 0.0 Reward Hacking Raw Observations Classifier Features Figure 4: The results of our experiment comparing our proposed method to a baseline that directly plans with the proxy reward function. By solving an inverse reward design problem, we are able to create generic incentives to avoid unseen or novel states. 5.1.2 Challenge Domain: Latent Rewards, No More Feature Indicators The previous examples allow us to explore reward hacking and negative side effects in an isolated experiment, but are unrealistic as they assume the existence of a feature indicator for unknown, unplanned-for terrain. To investigate misspecified objectives in a more realistic setting, we shift to the terrain type being latent, and inducing raw observations: we use a model where the terrain category determines the mean and variance of a multivariate Gaussian distribution over observed features. Figure 3 shows a depiction of this scenario. The designer has in mind a proxy reward on dirt, target, and grass, but forgets that lava might exist. We consider two realistic ways through which a designer might actually specify the proxy reward function, which is based on the terrain types that the robot does not have access to: 1) directly on the raw observations ? collect samples of the training terrain types (dirt, grass, target) and train a (linear) reward predictor; or 2) classifier features ? build a classifier to classify terrain as dirt, grass, or target, and define a proxy on its output. Note that this domain allows for both negative side effects and reward hacking. Negative side effects can occur because the feature distribution for lava is different from the feature distribution for the three safe categories, and the proxy reward is trained only on the three safe categories. Thus in the testing MDP, the evaluation of the lava cells will be arbitrary so maximizing the proxy reward will likely lead the agent into lava. Reward hacking occurs when features that are correlated for the safe categories are uncorrelated for the lava category. 5.2 Experiment Lavaland Parameters. We defined a distribution on map layouts with a log likelihood function that prefers maps where neighboring grid cells are the same. We mixed this log likelihood with a quadratic cost for deviating from a target ratio of grid cells to ensure similar levels of the lava feature in the testing MDPs. Our training MDP is 70% dirt and 30% grass. Our testing MDP is 5% lava, 66.5% dirt, and 28.5% grass. In the proof-of-concept experiments, we selected the proxy reward function uniformly at random. For latent rewards, we picked a proxy reward function that evaluated to +1 for target, +.1 for dirt, and ?.2 for grass. To define a proxy on raw observations, we sampled 1000 examples of grass, dirt, and target and did a linear regression. With classifier features, we simply used the target rewards as the weights on the classified features. We used 50 dimensions for our feature vectors. We selected trajectories via risk-averse trajectory optimization. Details of our planning method, and our approach and rationale in selecting it can be found in the supplementary material. IVs and DVs. We measured the fraction of runs that encountered a lava cell on the test MDP as our dependent measure. This tells us the proportion of trajectories where the robot gets ?tricked? by the misspecified reward function; if a grid cell has never been seen then a conservative robot should plan to avoid it. We manipulate two factors: literal-optimizer and Z-approx. literal-optimizer is true if the robot interprets the proxy reward literally and false otherwise. Z-approx varies the approximation technique used to compute the IRD posterior. It varies across the two levels described in Section 4.2: sample to approximate the normalizing constant (Sample-Z) or use the normalizing constant from maximum entropy IRL (MaxEnt-Z) (Ziebart et al., 2008). 7 Results. Figure 4 compares the approaches. On the left, we see that IRD alleviates negative side effects (avoids the lava) and reward hacking (does not go as much on cells that look deceptively like the target to one of the sensors). This is important, in that the same inference method generalizes across different consequences of misspecified rewards. Figure 2 shows example behaviors. In the more realistic latent reward setting, the IRD agent avoids the lava cells despite the designer forgetting to penalize it, and despite not even having an indicator for it: because lava is latent in the space, and so reward functions that would implicitly penalize lava are as likely as the one actually specified, risk-averse planning avoids it. We also see a distinction between raw observations and classifier features. The first essentially matches the proof-of-concept results (note the different axes scales), while the latter is much more difficult across all methods. The proxy performs worse because each grid cell is classified before being evaluated, so there is a relatively good chance that at least one of the lava cells is misclassified as target. IRD performs worse because the behaviors considered in inference plan in the already classified terrain: a non-linear transformation of the features. The inference must both determine a good linear reward function to match the behavior and discover the corresponding uncertainty about it. When the proxy is a linear function of raw observations, the first job is considerably easier. 6 Discussion Summary. In this work, we motivated and introduced the Inverse Reward Design problem as an approach to mitigate the risk from misspecified objectives. We introduced an observation model, identified the challenging inference problem this entails, and gave several simple approximation schemes. Finally, we showed how to use the solution to an inverse reward design problem to avoid side effects and reward hacking in a 2D navigation problem. We showed that we are able to avoid these issues reliably in simple problems where features are binary indicators of terrain type. Although this result is encouraging, in real problems we won?t have convenient access to binary indicators for what matters. Thus, our challenge evaluation domain gave the robot access to only a high-dimensional observation space. The reward designer specified a reward based on this observation space which forgets to penalize a rare but catastrophic terrain. IRD inference still enabled the robot to understand that rewards which would implicitly penalize the catastrophic terrain are also likely. Limitations and future work. IRD gives the robot a posterior distribution over reward functions, but much work remains in understanding how to best leverage this posterior. Risk-averse planning can work sometimes, but it has the limitation that the robot does not just avoid bad things like lava, it also avoids potentially good things, like a giant pot of gold. We anticipate that leveraging the IRD posterior for follow-up queries to the reward designer will be key to addressing misspecified objectives. Another limitation stems from the complexity of the environments and reward functions considered here. The approaches we used in this work rely on explicitly solving a planning problem, and this is a bottleneck during inference. In future work, we plan to explore the use of different agent models that plan approximately or leverage, e.g., meta-learning (Duan et al., 2016) to scale IRD up to complex environments. Another key limitation is the use of linear reward functions. We cannot expect IRD to perform well unless the prior places weights on (a reasonable approximation to) the true reward function. If, e.g., we encoded terrain types as RGB values in Lavaland, there is unlikely to be a reward function in our hypothesis space that represents the true reward well. Finally, this work considers one relatively simple error model for the designer. This encodes some implicit assumptions about the nature and likelihood of errors (e.g., IID errors). In future work, we plan to investigate more sophisticated error models that allow for systematic biased errors from the designer and perform human subject studies to empirically evaluate these models. Overall, we are excited about the implications IRD has not only in the short term, but also about its contribution to the general study of the value alignment problem. Acknowledgements This work was supported by the Center for Human Compatible AI and the Open Philanthropy Project, the Future of Life Institute, AFOSR, and NSF Graduate Research Fellowship Grant No. DGE 1106400. 8 References Amodei, Dario and Clark, Jack. Faulty Reward Functions in the Wild. https://blog.openai. com/faulty-reward-functions/, 2016. Amodei, Dario, Olah, Chris, Steinhardt, Jacob, Christiano, Paul, Schulman, John, and Man?, Dan. Concrete Problems in AI Safety. CoRR, abs/1606.06565, 2016. URL http://arxiv.org/abs/ 1606.06565. Duan, Yan, Schulman, John, Chen, Xi, Bartlett, Peter L., Sutskever, Ilya, and Abbeel, Pieter. RL2 : Fast Reinforcement Learning via Slow Reinforcement Learning. CoRR, abs/1611.02779, 2016. URL http://arxiv.org/abs/1611.02779. Evans, Owain, Stuhlm?ller, Andreas, and Goodman, Noah D. Learning the Preferences of Ignorant, Inconsistent Agents. In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, pp. 323?329. AAAI Press, 2016. Frank, Michael C, Goodman, Noah D, Lai, Peter, and Tenenbaum, Joshua B. Informative Communication in Word Production and Word Learning. In Proceedings of the 31st Annual Conference of the Cognitive Science Society, pp. 1228?1233. Cognitive Science Society Austin, TX, 2009. Goodman, Noah D and Lassiter, Daniel. Probabilistic Semantics and Pragmatics: Uncertainty in Language and Thought. Handbook of Contemporary Semantic Theory. Wiley-Blackwell, 2, 2014. Grice, H. Paul. Logic and Conversation, pp. 43?58. Academic Press, 1975. Hadfield-Menell, Dylan, Dragan, Anca, Abbeel, Pieter, and Russell, Stuart. Cooperative Inverse Reinforcement Learning. In Proceedings of the Thirtieth Annual Conference on Neural Information Processing Systems, 2016. Hadfield-Menell, Dylan, Dragan, Anca D., Abbeel, Pieter, and Russell, Stuart J. The Off-Switch Game. In Proceedings of the International Joint Conference on Artificial Intelligence, 2017. 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Structured Bayesian Pruning via Log-Normal Multiplicative Noise Kirill Neklyudov 1,2 k.necludov@gmail.com 1 Dmitry Molchanov 1,3 dmolchanov@hse.ru Arsenii Ashukha 1,2 Dmitry Vetrov 1,2 aashukha@hse.ru National Research University Higher School of Economics 3 Skolkovo Institute of Science and Technology dvetrov@hse.ru 2 Yandex Abstract Dropout-based regularization methods can be regarded as injecting random noise with pre-defined magnitude to different parts of the neural network during training. It was recently shown that Bayesian dropout procedure not only improves generalization but also leads to extremely sparse neural architectures by automatically setting the individual noise magnitude per weight. However, this sparsity can hardly be used for acceleration since it is unstructured. In the paper, we propose a new Bayesian model that takes into account the computational structure of neural networks and provides structured sparsity, e.g. removes neurons and/or convolutional channels in CNNs. To do this we inject noise to the neurons outputs while keeping the weights unregularized. We establish the probabilistic model with a proper truncated log-uniform prior over the noise and truncated log-normal variational approximation that ensures that the KL-term in the evidence lower bound is computed in closed-form. The model leads to structured sparsity by removing elements with a low SNR from the computation graph and provides significant acceleration on a number of deep neural architectures. The model is easy to implement as it can be formulated as a separate dropout-like layer. 1 Introduction Deep neural networks are a flexible family of models which provides state-of-the-art results in many machine learning problems [14, 20]. However, this flexibility often results in overfitting. A common solution for this problem is regularization. One of the most popular ways of regularization is Binary Dropout [19] that prevents co-adaptation of neurons by randomly dropping them during training. An equally effective alternative is Gaussian Dropout [19] that multiplies the outputs of the neurons by Gaussian random noise. In recent years several Bayesian generalizations of these techniques have been developed, e.g. Variational Dropout [8] and Variational Spike-and-Slab Neural Networks [13]. These techniques provide theoretical justification of different kinds of Dropout and also allow for automatic tuning of dropout rates, which is an important practical result. Besides overfitting, compression and acceleration of neural networks are other important challenges, especially when memory or computational resources are restricted. Further studies of Variational Dropout show that individual dropout rates for each weight allow to shrink the original network architecture and result in a highly sparse model [16]. General sparsity provides a way of neural network compression, while the time of network evaluation may remain the same, as most modern DNN-oriented software can?t work with sparse matrices efficiently. At the same time, it is possible to achieve acceleration by enforcing structured sparsity in convolutional filters or data tensors. In the simplest case it means removing redundant neurons or convolutional filters instead of separate weights; but more complex patterns can also be considered. This way Group-wise Brain Damage 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. [10] employs group-wise sparsity in convolutional filters, Perforated CNNs [3] drop redundant rows from the intermediate dataframe matrices that are used to compute convolutions, and Structured Sparsity Learning [24] provides a way to remove entire convolutional filters or even layers in residual networks. These methods allow to obtain practical acceleration with little to no modifications of the existing software. In this paper, we propose a tool that is able to induce an arbitrary pattern of structured sparsity on neural network parameters or intermediate data tensors. We propose a dropout-like layer with a parametric multiplicative noise and use stochastic variational inference to tune its parameters in a Bayesian way. We introduce a proper analog of sparsity-inducing log-uniform prior distribution [8, 16] that allows us to formulate a correct probabilistic model and avoid the problems that come from using an improper prior. This way we obtain a novel Bayesian method of regularization of neural networks that results in structured sparsity. Our model can be represented as a separate dropout-like layer that allows for a simple and flexible implementation with almost no computational overhead, and can be incorporated into existing neural networks. Our experiments show that our model leads to high group sparsity level and significant acceleration of convolutional neural networks with negligible accuracy drop. We demonstrate the performance of our method on LeNet and VGG-like architectures using MNIST and CIFAR-10 datasets. 2 Related Work Deep neural networks are extremely prone to overfitting, and extensive regularization is crucial. The most popular regularization methods are based on injection of multiplicative noise over layer inputs, parameters or activations [8, 19, 22]. Different kinds of multiplicative noise have been used in practice; the most popular choices are Bernoulli and Gaussian distributions. Another type of regularization of deep neural networks is based on reducing the number of parameters. One approach is to use low-rank approximations, e.g. tensor decompositions [4, 17], and the other approach is to induce sparsity, e.g. by pruning [5] or L1 regularization [24]. Sparsity can also be induced by using the Sparse Bayesian Learning framework with empirical Bayes [21] or with sparsity-inducing priors [12, 15, 16]. High sparsity is one of the key factors for the compression of DNNs [5, 21]. However, in addition to compression it is beneficial to obtain acceleration. Recent papers propose different approaches to acceleration of DNNs, e.g. Spatial Skipped Convolutions [3] and Spatially Adaptive Computation Time [2] that propose different ways to reduce the number of computed convolutions, Binary Networks [18] that achieve speedup by using only 1 bit to store a single weight of a DNN, Low-Rank Expansions [6] that use low-rank filter approximations, and Structured Sparsity Learning [24] that allows to remove separate neurons or filters. As reported in [24] it is possible to obtain acceleration of DNNs by introducing structured sparsity, e.g. by removing whole neurons, filters or layers. However, non-adaptive regularization techniques require tuning of a huge number of hyperparameters that makes it difficult to apply in practice. In this paper we apply the Bayesian learning framework to obtain structured sparsity and focus on acceleration of neural networks. 3 Stochastic Variational Inference Given a probabilistic model p(y | x, ?) we want to tune parameters ? of the model using training dataset D = {(xi , yi )}N i=1 . The prior knowledge about parameters ? is defined by prior distribution p(?). Using the Bayes rule we obtain the posterior distribution p(? | D) = p(D | ?)p(?)/p(D). However, computing posterior distribution using the Bayes rule usually involves computation of intractable integrals, so we need to use approximation techniques. One of the most widely used approximation techniques is Variational Inference. In this approach the unknown distribution p(? | D) is approximated by a parametric distribution q (?) by minimization of the Kullback-Leibler divergence KL(q (?) k p(? | D)). Minimization of the KL divergence is equivalent to maximization of the variational lower bound L( ). L( ) = LD ( ) where LD ( ) = N X Eq i=1 2 KL(q (?) k p(?)), (1) log p(yi | xi , ?) (2) (?) LD ( ) is a so-called expected log-likelihood function which is intractable in case of complex probabilistic model p(y | x, ?). Following [8] we use the Reparametrization trick to obtain an unbiased differentiable minibatch-based Monte Carlo estimator of the expected log-likelihood. Here N is the total number of objects, M is the minibatch size, and f ( , ") provides samples from the approximate posterior q (?) as a deterministic function of a non-parametric noise " ? p("). B LD ( ) ' LSGV ( )= D M N X log p(yik | xik , wik = f ( , "ik )) M L( ) ' LSGV B ( ) = k=1 B LSGV ( D r LD ( ) ' r ) KL(q (w) k p(w)) B LSGV ( D ) (3) (4) (5) This way we obtain a procedure of approximate Bayesian inference where we solve optimization problem (4) by stochastic gradient ascent w.r.t. variational parameters . This procedure can be efficiently applied to Deep Neural Networks and usually the computational overhead is very small, as compared to ordinary DNNs. If the model p(y | x, ?, w) has another set of parameters w that we do not want to be Bayesian about, we can still use the same variational lower bound objective: L( , w) = LD ( , w) where LD ( , w) = N X KL(q (?) k p(?)) ! max, (6) Eq (7) ,w i=1 (?) log p(yi | xi , ?, w) This objective corresponds the maximum likelihood estimation wM L of parameters w, while finding the approximate posterior distribution q (?) ? p(? | D, wM L ). In this paper we denote the weights of the neural networks, the biases, etc. as w and find their maximum likelihood estimation as described above. The parameters ? that undergo the Bayesian treatment are the noisy masks in the proposed dropout-like layer (SBP layer). They are described in the following section. 4 Group Sparsity with Log-normal Multiplicative Noise Variational Inference with a sparsity-inducing log-uniform prior over the weights of a neural network is an efficient way to enforce general sparsity on weight matrices [16]. However, it is difficult to apply this approach to explicitly enforce structured sparsity. We introduce a dropout-like layer with a certain kind of multiplicative noise. We also make use of the sparsity-inducing log-uniform prior, but put it over the noise variables rather than weights. By sharing those noise variables we can enforce group-wise sparsity with any form of groups. 4.1 Variational Inference for Group Sparsity Model We consider a single dropout-like layer with an input vector x 2 RI that represents one object with I features, and an output vector y 2 RI of the same size. The input vector x is usually supposed to come from the activations of the preceding layer. The output vector y would then serve as an input vector for the following layer. We follow the general way to build dropoutlike layers (8). Each input feature xi is multiplied by a noise variable ?i that comes from some distribution pnoise (?). For example, for Binary Dropout pnoise (?) would be a fully factorized Bernoulli distribution with pnoise (?i ) = Bernoulli(p), and for Gaussian dropout it would be a fully-factorized Gaussian distribution with pnoise (?i ) = N (1, ?). yi = x i ? ? i ? ? pnoise (?) (8) Note that if we have a minibatch X M ?I of M objects, we would independently sample a separate noise vector ?m for each object xm . This would be the case throughout the paper, but for the sake of simplicity we would consider a single object x in all following formulas. Also note that the noise ? is usually only sampled during the training phase. A common approximation during the testing phase is to use the expected value E? instead of sampling ?. All implementation details are provided and discussed in Section 4.5. 3 We follow a Bayesian treatment of the variable ?, as described in Section 3. In order to obtain a sparse solution, we choose the prior distribution p(?) to be a fully-factorized improper log-uniform distribution. We denote this distribution as LogU1 (?) to stress that it has infinite domain. This distribution is known for its sparsification properties and works well in practice for deep neural networks [16]. p(?) = I Y p(?i ) = LogU1 (?i ) / p(?i ) i=1 1 ?i ?i > 0 (9) In order to train the model, i.e. perform variational inference, we need to choose an approximation family q for the posterior distribution p(? | D) ? q (?). q (?) = I Y i=1 q(?i | ?i , ?i ? LogN(?i | ?i , 2 i) i) = I Y i=1 () LogN(?i | ?i , 2 i) (10) log ?i ? N (log ?i | ?i , 2 i) (11) A common choice of variational distribution q(?) is a fully-factorized Gaussian distribution. However, for this particular model we choose q(?) to be a fully-factorized log-normal distribution (10?11). To make this choice, we were guided by the following reasons: ? The log-uniform distribution is a specific case of the log-normal distribution when the parameter goes to infinity and ? remains fixed. Thus we can guarantee that in the case of no data our variational approximation can be made exact. Hence this variational family has no "prior gap". ? We consider a model with multiplicative noise. The scale of this noise corresponds to its shift in the logarithmic space. By establishing the log-uniform prior we set no preferences on different scales of this multiplicative noise. The usual use of a Gaussian as a posterior immediately implies very asymmetric skewed distribution in the logarithmic space. Moreover log-uniform and Gaussian distributions have different supports and that will require establishing two log-uniform distributions for positive and negative noises. In this case Gaussian variational approximation would have quite exotic bi-modal form (one mode in the log-space of positive noises and another one in the log-space of negative noises). On the other hand, the log-normal posterior for the multiplicative noise corresponds to a Gaussian posterior for the additive noise in the logarithmic scale, which is much easier to interpret. ? Log-normal noise is always non-negative both during training and testing phase, therefore it does not change the sign of its input. This is in contrast to Gaussian multiplicative noise N (?i | 1, ?) that is a standard choice for Gaussian dropout and its modifications [8, 19, 23]. During the training phase Gaussian noise can take negative values, so the input to the following layer can be of arbitrary sign. However, during the testing phase noise ? is equal to 1, so the input to the following layer is non-negative with many popular non-linearities (e.g. ReLU, sigmoid, softplus). Although Gaussian dropout works well in practice, it is difficult to justify notoriously different input distributions during training and testing phases. ? The log-normal approximate posterior is tractable. Specifically, the KL divergence term KL(LogN(? | ?, 2 ) k LogU1 (?)) can be computed analytically. The final loss function is presented in equation (12) and is essentially the original variational lower bound (4). B LSGV B ( ) = LSGV (?, , W ) D KL(q(? | ?, ) k p(?)) ! max , ?, ,W (12) where ? and are the variatianal parameters, and W denotes all other trainable parameters of the neural network, e.g. the weight matrices, the biases, batch normalization parameters, etc. Note that we can optimize the variational lower bound w.r.t. the parameters ? and of the log-normal noise ?. We do not fix the mean of the noise thus making our variational approximation more tight. 4.2 Problems of Variational Inference with Improper Log-Uniform Prior The log-normal posterior in combination with a log-uniform prior has a number of attractive features. However, the maximization of the variational lower bound with a log-uniform prior and a log-normal 4 posterior is an ill-posed optimization problem. As the log-uniform distribution is an improper prior, the KL-divergence between a log-normal distribution LogN(?, 2 ) and a log-uniform distribution LogU1 is infinite for any finite value of parameters ? and . KL LogN(x | ?, 2 ) k LogU1 (x) = C log (13) C = +1 A common way to tackle this problem is to consider the density of the log-uniform distribution to be equal to C? and to treat C as some finite constant. This trick works well for the case of a Gaussian posterior distribution [8, 16]. The KL divergence between a Gaussian posterior and a log-uniform prior has an infinite gap, but can be calculated up to this infinite constant in a meaningful way [16]. However, for the case of the log-normal posterior the KL divergence is infinite for any finite values of variational parameters, and is equal to zero for a fixed finite ? and infinite . As the data-term (3) is bounded for any value of variational parameters, the only global optimum of the variational lower bound is achieved when ? is finite and fixed, and goes to infinity. In this case the posterior distribution collapses into the prior distribution and the model fails to extract any information about the data. This effect is wholly caused by the fact that the log-uniform prior is an improper (non-normalizable) distribution, which makes the whole probabilistic model flawed. 4.3 Variational Inference with Truncated Approximation Family Due to the improper prior the optimization problem becomes ill-posed. But do we really need to use an improper prior distribution? The most common number format that is used to represent the parameters of a neural network is the floating-point format. The floating-point format is only able to represent numbers from a limited range. For example, a single-point precision variable can only represent numbers from the range 3.4 ? 1038 to +3.4 ? 1038 , and the smallest possible positive number is equal to 1.2 ? 10 38 . All of probability mass of the improper log-uniform prior is concentrated beyond the single-point precision (and essentially any practical floating point precision), not to mention that the actual relevant range of values of neural network parameters is much smaller. It means that in practice this prior is not a good choice for software implementation of neural networks. We propose to use a truncated log-uniform distribution (14) as a proper analog of the log-uniform distribution. Here I[a,b] (x) denotes the indicator function for the interval x 2 [a, b]. The posterior distribution should be defined on the same support as the prior distribution, so we also need to use a truncated log-normal distribution (14). LogU[a,b] (?i ) / LogU1 (?i ) ? I[a,b] (log ?i ) LogN[a,b] (?i ) / LogN(?i | ?i , 2 i) ? I[a,b] (log ?i ) (14) Our final model then can be formulated as follows. y i = x i ? ?i q(?i | ?i , p(?i ) = LogU[a,b] (?i ) i) = LogN[a,b] (?i | ?i , 2 i) (15) Note that all the nice facts about the log-normal posterior distribution from the Section 4.1 are also true for the truncated log-normal posterior. However, now we have a proper probabilistic model and the Stochastic Variational Inference can be preformed correctly. Unlike (13), now the KL divergence term (16?17) can be calculated correctly for all valid values of variational parameters (see Appendix A for details). KL(q(? | ?, ) k p(?)) = KL(q(?i | ?i , where ?i = distribution. a ?i i i ) k p(?i )) , i = b a = log p 2?e i2 b ?i i , (?) and I X i=1 KL(q(?i | ?i , log( ( i ) (16) i ) k p(?i )) (?i )) ?i (?i ) 2( ( i ) i ( i) , (?i )) (17) (?) are the density and the CDF of the standard normal The reparameterization trick also can still be performed (18) using the inverse CDF of the truncated normal distribution (see Appendix B). ?i = exp ?i + i 1 ( (?i ) + ( ( i ) (?i )) yi ) , where yi ? U (y | 0, 1) (18) The final loss and the set of parameters is the same as described in Section 4.1, and the training procedure remains the same. 5 4.4 Sparsity Log-uniform prior is known to lead to a sparse solution [16]. In the variational dropout paper authors interpret the parameter ? of the multiplicative noise N (1, ?) as a Gaussian dropout rate and use it as a thresholding criterion for weight pruning. Unlike the binary or Gaussian dropout, in the truncated log-normal model there is no "dropout rate" variable. However, we can use the signal-to-noise ratio p E?/ Var(?) (SNR) for thresholding. SNR(?i ) = p ( ( exp( 2 i )( (2 i i ?i ) ?i ) ( i i ))/ (2 i i )) p ( i) ( ( i (?i ) ?i ) ( i i )) 2 (19) The SNR can be computed analytically, the derivation can be found in the appendix. It has a simple interpretation. If the SNR is low, the corresponding neuron becomes very noisy and its output no longer contains any useful information. If the SNR is high, it means that the neuron output contains little noise and is important for prediction. Therefore we can remove all neurons or filters with a low SNR and set their output to constant zero. 4.5 Implementation details We perform a minibatch-based stochastic variational inference for training. The training procedure looks as follows. On each training step we take a minibatch of M objects and feed it into the neural network. Consider a single SBP layer with input X M ?I and output Y M ?I . We independently sample a separate noise vector ?m ? q(?) for each object xm and obtain a noise matrix ?M ?I . The output matrix Y M ?I is then obtained by component-wise multiplication of the input matrix and the noise matrix: ymi = xmi ? ?im . To be fully Bayesian, one would also sample and average over different dropout masks ? during testing, i.e. perform Bayesian ensembling. Although this procedure can be used to slightly improve the final accuracy, it is usually avoided. Bayesian ensembling essentially requires sampling of different copies of neural networks, which makes the evaluation K times slower for averaging over K samples. Instead, during the testing phase in most dropout-based techniques the noise variable ? is replaced with its expected value. In this paper we follow the same approach and replace all non-pruned ?i with their expectations (20) during testing. The derivation of the expectation of the truncated log-normal distribution is presented in Appendix C. E?i = exp(?i + i2 /2) ( i) (?i ) ? ? 2 i + ?i i a ? ? 2 i + ?i i b ? (20) We tried to use Bayesian ensembling with this model, and experienced almost no gain of accuracy. It means that the variance of the learned approximate posterior distribution is low and does not provide a rich ensemble. Throughout the paper we introduced the SBP dropout layer for the case when input objects are represented as one-dimensional vectors x. When defined like that, it would induce general sparsity on the input vector x. It works as intended for fully-connected layers, as a single input feature corresponds to a single output neuron of a preceding fully-connected layer and a single output neuron of the following layer. However, it is possible to apply the SBP layer in a more generic setting. Firstly, if the input object is represented as a multidimensional tensor X with shape I1 ? I2 ? ? ? ? ? Id , the noise vector ? of length I = I1 ? I2 ? ? ? ? ? Id can be reshaped into a tensor with the same shape. Then the output tensor Y can be obtained as a component-wise product of the input tensor X and the noise tensor ?. Secondly, the SBP layer can induce any form of structured sparsity on this input tensor X. To do it, one would simply need to use a single random variable ?i for the group of input features that should be removed simultaneously. For example, consider an input tensor X H?W ?C that comes from a convolutional layer, H and W being the size of the image, and C being the number of channels. Then, in order to remove redundant filters from the preceding layer (and at the same time redundant channels from the following layer), one need to share the random variables ? in the following way: yhwc = xhwc ? ?c ?c ? LogN[a,b] (?c | ?c , 2 c) (21) Note that now there is one sample ? 2 RC for one object X H?W ?C on each training step. If the signal-to-noise ratio becomes lower than 1 for a component ?c , that would mean that we can 6 Figure 1: The value of the SGVB for the case of fixed variational parameter ? = 0 (blue line) and for the case when both variational parameters ? and are trained (green line) Figure 2: The learned signal-to-noise ratio for image features on the MNIST dataset. permanently remove the c-th channel of the input tensor, and therefore delete the c-th filter from the preceding layer and the c-th channel from the following layer. All the experiments with convolutional architectures used this formulation of SBP. This is a general approach that is not limited to reducing the shape of the input tensor. It is possible to obtain any fixed pattern of group-wise sparsity using this technique. Similarly, the SBP layer can be applied in a DropConnect fashion. One would just need to multiply the weight tensor W by a noise tensor ? of similar shape. The training procedure remains the same. It is still possible to enforce any structured sparsity pattern for the weight tensor W by sharing the random variables as described above. 5 Experiments We perform an evaluation on different supervised classification tasks and with different architectures of neural networks including deep VGG-like architectures with batch normalization layers. For each architecture, we report the number of retained neurons and filters, and obtained acceleration. Our experiments show that Structured Bayesian Pruning leads to a high level of structured sparsity in convolutional filters and neurons of DNNs without significant accuracy drop. We also demonstrate that optimization w.r.t. the full set of variational parameters (?, ) leads to improving model quality and allows us to perform sparsification in a more efficient way, as compared to tuning of only one free parameter that corresponds to the noise variance. As a nice bonus, we show that Structured Bayesian Pruning network does not overfit on randomly labeled data, that is a common weakness of non-bayesian dropout networks. The source code is available in Theano [7] and Lasagne, and also in TensorFlow [1] (https://github.com/necludov/group-sparsity-sbp). 5.1 Experiment Setup The truncation parameters a and b are the hyperparameters of our model. As our layer is meant for regularization of the model, we would like our layer not to amplify the input signal and restrict the noise ? to an interval [0, 1]. This choice corresponds to the right truncation threshold b set to 0. We find empirically that the left truncation parameter a does not influence the final result much. We use values a = 20 and b = 0 in all experiments. We define redundant neurons by the signal-to-noise ratio of the corresponding multiplicative noise ?. See Section 4.4 for more details. By removing all neurons and filters with the SNR < 1 we experience no accuracy drop in all our experiments. SBP dropout layers were put after each convolutional layer to remove its filters, and before each fully-connected layer to remove its input neurons. As one filter of the last convolutional layer usually corresponds to a group of neurons in the following dense layer, it means that we can remove more input neurons in the first dense layer. Note that it means that we have two consecutive dropout layers between the last convolutional layer and the first fully-connected layer in CNNs, and a dropout layer before the first fully-connected layer in FC networks (see Fig. 2). 7 Table 1: Comparison of different structured sparsity inducing techniques on LeNet-5-Caffe and LeNet-500-300 architectures. SSL [24] is based on group lasso regularization, SparseVD [16]) is a Bayesian model with a log-uniform prior that induces weight-wise sparsity. For SparseVD a neuron/filter is considered pruned, if all its weights are set to 0. Our method provides the highest speed-up with a similar accuracy. We report acceleration that was computed on CPU (Intel Xeon E5-2630), GPU (Tesla K40) and in terms of Floating Point Operations (FLOPs). Network Method Error % Neurons per Layer Original SparseVD LeNet-500-300 SSL (ours) StructuredBP 1.54 1.57 1.49 1.55 784 537 434 245 500 217 174 160 Original SparseVD LeNet5-Caffe SSL (ours) StructuredBP 0.80 0.75 1.00 0.86 20 17 3 3 50 32 12 18 CPU GPU FLOPs 10 10 10 10 1.00? 1.19? 2.21? 2.33? 1.00? 1.03? 1.04? 1.08? 1.00? 3.73? 6.06? 11.23? 500 75 500 283 1.00? 1.48? 5.17? 5.41? 1.00? 1.41? 1.80? 1.91? 1.00? 2.19? 3.90? 10.49? 300 130 78 55 800 329 800 284 Table 2: Comparison of different structured sparsity inducing techniques (SparseVD [16]) on VGGlike architectures on CIFAR-10 dataset. StructuredBP stands for the original SBP model, and StructuredBPa stands for the SBP model with KL scaling. k is a width scale factor that determines the number of neurons or filters on each layer of the network (width(k) = k ? original width) k 1.0 (ours) (ours) 1.5 (ours) (ours) 5.2 Method Original SparseVD StructuredBP StructuredBPa Error % 7.2 7.2 7.5 9.0 64 64 64 44 64 62 62 54 128 128 128 92 128 126 126 115 256 234 234 234 256 155 155 155 Original SparseVD StructuredBP StructuredBPa 6.8 7.0 7.2 7.8 96 96 96 77 96 78 77 74 192 191 190 161 192 146 146 146 384 254 254 254 384 126 126 125 Units per Layer 256 512 512 31 81 76 31 79 73 31 76 55 384 27 26 26 768 79 79 78 768 74 70 66 512 9 9 9 512 138 59 34 512 101 73 35 512 413 56 21 512 373 27 280 CPU 1.00? 2.50? 2.71? 3.68? GPU 1.00? 1.69? 1.74? 2.06? FLOPs 1.00? 2.27? 2.30? 3.16? 768 9 9 9 768 137 71 47 768 100 82 55 768 416 79 54 768 479 49 237 1.00? 3.35? 3.63? 4.47? 1.00? 2.16? 2.17? 2.47? 1.00? 3.27? 3.32? 3.93? More Flexible Variational Approximation Usually during automatic training of dropout rates the mean of the noise distribution remains fixed. In the case of our model it is possible to train both mean and variance of the multiplicative noise. By using a more flexible distribution we obtain a tighter variational lower bound and a higher sparsity level. In order to demonstrate this effect, we performed an experiment on MNIST dataset with a fully connected neural network that contains two hidden layers with 1000 neurons each. The results are presented in Fig. 1. 5.3 LeNet5 and Fully-Connected Net on MNIST We compare our method with other sparsity inducing methods on the MNIST dataset using a fully connected architecture LeNet-500-300 and a convolutional architecture LeNet-5-Caffe. These networks were trained with Adam without any data augmentation. The LeNet-500-300 network was trained from scratch, and the LeNet-5-Caffe1 network was pretrained with weight decay. An illustration of trained SNR for the image features for the LeNet-500-3002 network is shown in Fig. 2. The final accuracy, group-wise sparsity levels and speedup for these architectures for different methods are shown in Table 1. 5.4 VGG-like on CIFAR-10 To prove that SBP scales to deep architectures, we apply it to a VGG-like network [25] that was adapted for the CIFAR-10 [9] dataset. The network consists of 13 convolutional and two fullyconnected layers, trained with pre-activation batch normalization and Binary Dropout. At the start of the training procedure, we use pre-trained weights for initialization. Results with different scaling of the number of units are presented in Table 2. We present results for two architectures with different scaling coefficient k 2 {1.0, 1.5} . For smaller values of scaling coefficient k 2 {0.25, 0.5} we obtain less sparse architecture since these networks have small learning capacities. Besides the results for the standard StructuredBP procedure, we also provide the results for SBP with KL scaling (StructuredBPa). Scaling the KL term of the variational lower bound proportional to the computational complexity of the layer leads to a higher sparsity level for the first layers, providing 1 2 A modified version of LeNet5 from [11]. Caffe Model specification: https://goo.gl/4yI3dL Fully Connected Neural Net with 2 hidden layers that contains 500 and 300 neurons respectively. 8 more acceleration. Despite the higher error values, we obtain the higher value of true variational lower bound during KL scaling, hence, we find its another local maximum. 5.5 Random Labels A recent work shows that Deep Neural Networks have so much capacity that they can easily memorize the data even with random labeling [26]. Binary dropout as well as other standard regularization techniques do not prevent the networks from overfitting in this scenario. However, recently it was shown that Bayesian regularization may help [16]. Following these works, we conducted similar experiments. We used a Lenet5 network on the MNIST dataset and a VGG-like network on CIFAR-10. Although Binary Dropout does not prevent these networks from overfitting, SBP decides to remove all neurons of the neural network and provides a constant prediction. In other words, in this case SBP chooses the simplest model that achieves the same testing error rate. This is another confirmation that Bayesian regularization is more powerful than other popular regularization techniques. 6 Conclusion We propose Structured Bayesian Pruning, or SBP, a dropout-like layer that induces multiplicative random noise over the output of the preceding layer. We put a sparsity-inducing prior over the noise variables and tune the noise distribution using stochastic variational inference. SBP layer can induce an arbitrary structured sparsity pattern over its input and provides adaptive regularization. We apply SBP to cut down the number of neurons and filters in convolutional neural networks and report significant practical acceleration with no modification of the existing software implementation of these architectures. Acknowledgments We would like to thank Christos Louizos and Max Welling for valuable discussions. Kirill Neklyudov and Arsenii Ashukha were supported by HSE International lab of Deep Learning and Bayesian Methods which is funded by the Russian Academic Excellence Project ?5-100?. Dmitry Molchanov was supported by the Ministry of Education and Science of the Russian Federation (grant 14.756.31.0001). Dmitry Vetrov was supported by the Russian Science Foundation grant 17-11-01027. References [1] Mart?n Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467, 2016. [2] Michael Figurnov, Maxwell D Collins, Yukun Zhu, Li Zhang, Jonathan Huang, Dmitry Vetrov, and Ruslan Salakhutdinov. Spatially adaptive computation time for residual networks. arXiv preprint arXiv:1612.02297, 2016. [3] Mikhail Figurnov, Aizhan Ibraimova, Dmitry P Vetrov, and Pushmeet Kohli. Perforatedcnns: Acceleration through elimination of redundant convolutions. In Advances in Neural Information Processing Systems, pages 947?955, 2016. [4] Timur Garipov, Dmitry Podoprikhin, Alexander Novikov, and Dmitry Vetrov. Ultimate tensorization: compressing convolutional and fc layers alike. arXiv preprint arXiv:1611.03214, 2016. [5] Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149, 2015. [6] Max Jaderberg, Andrea Vedaldi, and Andrew Zisserman. 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[22] Li Wan, Matthew Zeiler, Sixin Zhang, Yann L Cun, and Rob Fergus. Regularization of neural networks using dropconnect. In Proceedings of the 30th International Conference on Machine Learning (ICML-13), pages 1058?1066, 2013. [23] Sida I Wang and Christopher D Manning. Fast dropout training. In ICML (2), pages 118?126, 2013. [24] Wei Wen, Chunpeng Wu, Yandan Wang, Yiran Chen, and Hai Li. Learning structured sparsity in deep neural networks. In Advances in Neural Information Processing Systems, pages 2074?2082, 2016. [25] Sergey Zagoruyko. 92.45 on cifar-10 in torch, 2015. [26] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016. 10 

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Attend and Predict: Understanding Gene Regulation by Selective Attention on Chromatin Ritambhara Singh, Jack Lanchantin, Arshdeep Sekhon, Yanjun Qi Department of Computer Science University of Virginia yanjun@virginia.edu Abstract The past decade has seen a revolution in genomic technologies that enabled a flood of genome-wide profiling of chromatin marks. Recent literature tried to understand gene regulation by predicting gene expression from large-scale chromatin measurements. Two fundamental challenges exist for such learning tasks: (1) genome-wide chromatin signals are spatially structured, high-dimensional and highly modular; and (2) the core aim is to understand what the relevant factors are and how they work together. Previous studies either failed to model complex dependencies among input signals or relied on separate feature analysis to explain the decisions. This paper presents an attention-based deep learning approach, AttentiveChrome, that uses a unified architecture to model and to interpret dependencies among chromatin factors for controlling gene regulation. AttentiveChrome uses a hierarchy of multiple Long Short-Term Memory (LSTM) modules to encode the input signals and to model how various chromatin marks cooperate automatically. AttentiveChrome trains two levels of attention jointly with the target prediction, enabling it to attend differentially to relevant marks and to locate important positions per mark. We evaluate the model across 56 different cell types (tasks) in humans. Not only is the proposed architecture more accurate, but its attention scores provide a better interpretation than state-of-the-art feature visualization methods such as saliency maps.1 1 Introduction Gene regulation is the process of how the cell controls which genes are turned ?on? (expressed) or ?off? (not-expressed) in its genome. The human body contains hundreds of different cell types, from liver cells to blood cells to neurons. Although these cells include the same set of DNA information, their functions are different 2 . The regulation of different genes controls the destiny and function of each cell. In addition to DNA sequence information, many factors, especially those in its environment (i.e., chromatin), can affect which genes the cell expresses. This paper proposes an attention-based deep learning architecture to learn from data how different chromatin factors influence gene expression in a cell. Such understanding of gene regulation can enable new insights into principles of life, the study of diseases, and drug development. ?Chromatin? denotes DNA and its organizing proteins 3 . A cell uses specialized proteins to organize DNA in a condensed structure. These proteins include histones, which form ?bead?-like structures that DNA wraps around, in turn organizing and compressing the DNA. An important aspect of histone proteins is that they are prone to chemical modifications that can change the spatial arrangement of 1 Code shared at www.deepchrome.org. 2 DNA is a long string of paired chemical molecules or nucleotides that fall into four different types and are denoted as A, T, C, and G. DNA carries information organized into units such as genes. The set of genetic material of DNA in a cell is called its genome. 3 The complex of DNA, histones, and other structural proteins is called chromatin. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. DNA. These spatial re-arrangements result in certain DNA regions becoming accessible or restricted and therefore affecting expressions of genes in the neighborhood region. Researchers have established the ?Histone Code Hypothesis? that explores the role of histone modifications in controlling gene regulation. Unlike genetic mutations, chromatin changes such as histone modifications are potentially reversible ([5]). This crucial difference makes the understanding of how chromatin factors determine gene regulation even more impactful because this knowledge can help developing drugs targeting genetic diseases. At the whole genome level, researchers are trying to chart the locations and intensities of all the chemical modifications, referred to as marks, over the chromatin 4 . Recent advances in nextgeneration sequencing have allowed biologists to profile a significant amount of gene expression and chromatin patterns as signals (or read counts) across many cell types covering the full human genome. These datasets have been made available through large-scale repositories, the latest being the Roadmap Epigenome Project (REMC, publicly available) ([18]). REMC recently released 2,804 genome-wide datasets, among which 166 datasets are gene expression reads (RNA-Seq datasets) and the rest are signal reads of various chromatin marks across 100 different ?normal? human cells/tissues [18]. The fundamental aim of processing and understanding this repository of ?big? data is to understand gene regulation. For each cell type, we want to know which chromatin marks are the most important and how they work together in controlling gene expression. However, previous machine learning studies on this task either failed to model spatial dependencies among marks or required additional feature analysis to explain the predictions (Section 4). Computational tools should consider two important properties when modeling such data. ? First, signal reads for each mark are spatially structured and high-dimensional. For instance, to quantify the influence of a histone modification mark, learning methods typically need to use as input features all of the signals covering a DNA region of length 10, 000 base pair (bp) 5 centered at the transcription start site (TSS) of each gene. These signals are sequentially ordered along the genome direction. To develop ?epigenetic? drugs, it is important to recognize how a chromatin mark?s effect on regulation varies over different genomic locations. ? Second, various types of marks exist in human chromatin that can influence gene regulation. For example, each of the five standard histone proteins can be simultaneously modified at multiple different sites with various kinds of chemical modifications, resulting in a large number of different histone modification marks. For each mark, we build a feature vector representing its signals surrounding a gene?s TSS position. When modeling genome-wide signal reads from multiple marks, learning algorithms should take into account the modular nature of such feature inputs, where each mark functions as a module. We want to understand how the interactions among these modules influence the prediction (gene expression). In this paper we propose an attention-based deep learning model, AttentiveChrome, that learns to predict the expression of a gene from an input of histone modification signals covering the gene?s neighboring DNA region. By using a hierarchy of multiple LSTM modules, AttentiveChrome can discover interactions among signals of each chromatin mark, and simultaneously learn complex dependencies among different marks. Two levels of ?soft? attention mechanisms are trained, (1) to attend to the most relevant regions of a chromatin mark, and (2) to recognize and attend to the important marks. Through predicting and attending in one unified architecture, AttentiveChrome allows users to understand how chromatin marks control gene regulation in a cell. In summary, this work makes the following contributions: ? AttentiveChrome provides more accurate predictions than state-of-the-art baselines. Using datasets from REMC, we evaluate AttentiveChrome on 56 different cell types (tasks). ? We validate and compare interpretation scores using correlation to a new mark signal from REMC (not used in modeling). AttentiveChrome?s attention scores provide a better interpretation than state-of-the-art methods for visualizing deep learning models. ? AttentiveChrome can model highly modular inputs where each module is highly structured. AttentiveChrome can explain its decisions by providing ?what? and ?where? the model has focused 4 In biology this field is called epigenetics. ?Epi? in Greek means over. The epigenome in a cell is the set of chemical modifications over the chromatin that alter gene expression. 5 A base pair refers to one of the double-stranded DNA sequence characters (ACGT) 2 HM1 (Xj=1) Transcription Start Site (TSS) HM2 (Xj=2) Gene A HM3 (Xj=3) Gene A HM4 (Xj=4) Gene A HM5 (Xj=5) Gene A Bin # t=1 2 3 4 ...... j t Gene A GeneA = ON/OFF (y=+1/-1) j t=100 HM-level Attention Bin-level Attention Classification Figure 1: Overview of the proposed AttentiveChrome framework. It includes 5 important parts: (1) Bin-level LSTM encoder for each HM mark; (2) Bin-level ?-Attention across all bin positions of each HM mark; (3) HM-level LSTM encoder encoding all HM marks; (4) HM-level ?-Attention among all HM marks; (5) the final classification. on. This flexibility and interpretability make this model an ideal approach for many real-world applications. ? To the authors? best knowledge, AttentiveChrome is the first attention-based deep learning method for modeling data from molecular biology. In the following sections, we denote vectors with bold font and matrices using capital letters. To simplify notation, we use ?HM? as a short form for the term ?histone modification?. 2 Background: Long Short-Term Memory (LSTM) Networks Recurrent neural networks (RNNs) have been widely used in applications such as natural language processing due to their abilities to model sequential dependencies. Given an input matrix X of size nin ? T , an RNN produces a matrix H of size d ? T , where nin is the input feature size, T is the length of input feature , and d is the RNN embedding size. At each timestep t ? {1, ? ? ? , T }, an RNN takes an input column vector xt ? Rnin and the previous hidden state vector ht?1 ? Rd and produces the next hidden state ht by applying the following recursive operation: ????? ht = ?(Wxt + Uht?1 + b) = LST M (xt ), (1) where W, U, b are the trainable parameters of the model, and ? is an element-wise nonlinearity function. Due to the recursive nature, RNNs can capture the complete set of dependencies among all timesteps being modeled, like all spatial positions in a sequential sample. To handle the ?vanishing gradient? issue of training basic RNNs, Hochreiter et al. [13] proposed an RNN variant called the Long Short-term Memory (LSTM) network. An LSTM layer has an input-to-state component and a recurrent state-to-state component like that in Eq. (1). Additionally, it has gating functions that control when information is written to, read from, and forgotten. Though the LSTM formulation results in a complex form of Eq. (1) (see Supplementary), when given input vector xt and the state ht?1 from previous time step t ? 1, an LSTM module also produces a new state ht . The embedding vector ht encodes the learned representation summarizing feature dependencies from the time step 0 to the time step t. For our task, we call each bin position on the genome coordinate a ?time step?. 3 AttentiveChrome: A Deep Model for Joint Classification and Visualization Input and output formulation for the task: We use the same feature inputs and outputs as done previously in DeepChrome ([29]). Following Cheng et al. [7], the gene expression prediction is formulated as a binary classification task whose output represents if the gene expression of a gene is high(+1) or low(-1). As shown in Figure 1, the input feature of a sample (a particular gene) is denoted as a matrix X of size M ? T . Here M denotes the number of HM marks we consider in the input. T is the total number of bin positions we take into account from the neighboring region of a gene?s TSS site on the genome. We refer to this region as the ?gene region? in the rest of the paper. xj denotes the j-th row vector of X whose elements are sequentially structured (signals from the j-th HM mark) j ? {1, ..., M }. xjt in matrix X represents the signal from the t-th bin of the j-th HM 3 mark. t ? {1, ..., T }. We assume our training set D contains Ntr labeled pairs. We denote the n-th pair as (X(n) , y (n) ), X(n) is a matrix of size M ? T and y (n) ? {?1, +1}, where n ? {1, ..., Ntr }. An end-to-end deep architecture for predicting and attending jointly: AttentiveChrome learns to predict the expression of a gene from an input of HM signals covering its gene region. First, the signals of each HM mark are fed into a separate LSTM network to encode the spatial dependencies among its bin signals, and then another LSTM is used to model how multiple factors work together for predicting gene expression. Two levels of "soft" attention mechanisms are trained and dynamically predicted for each gene: (1) to attend to the most relevant positions of an HM mark, and (2) then to recognize and attend to the relevant marks. In summary, AttentiveChrome consists of five main modules (see Supplementary Figure S:2): (1) Bin-level LSTM encoder for each HM mark; (2) Bin-level Attention on each HM mark; (3) HM-level LSTM encoder encoding all HM marks; (4) HM-level Attention over all the HM marks; (5) the final classification module. We describe the details of each component as follows: Bin-Level Encoder Using LSTMs: For a gene of interest, the j-th row vector, xj , from X includes a total of T elements that are sequentially ordered along the genome coordinate. Considering the sequential nature of such signal reads, we treat each element (essentially a bin position) as a ?time step? and use a bidirectional LSTM to model the complete dependencies among elements in xj . A bidirectional LSTM contains two LSTMs, one in each direction (see Supplementary Figure S:2 ????? ????? (c)). It includes a forward LST M j that models xj from xj1 to xjT and a backward LST M j that ? ? ? ? models from xjT to xj1 . For each position t, the two LSTMs output hjt and hjt , each of size d. ? ? ????? ? ? ????? hjt = LST M j (xjt ) and hjt = LST M j (xjt ). The final embedding vector at the t-th position is the ? ? ? ? concatenation hjt = [hjt , hjt ]. By coupling these LSTM-based HM encoders with the final classification, they can learn to embed each HM mark by extracting the dependencies among bins that are essential for the prediction task. Bin-Level Attention, ?-attention: Although the LSTM can encode dependencies among the bins, it is difficult to determine which bins are most important for prediction from the LSTM. To automatically and adaptively highlight the most relevant bins for each sample, we use "soft" attention to learn the importance weights of bins. This means when representing j-th HM mark, AttentiveChrome follows a basic concept that not all bins contribute equally to the encoding of the entire j-th HM mark. The attention mechanism can help locate and recognize those bins that are important for the current gene sample of interest from j-th HM mark and can aggregate those important bins to form an embedding vector. This extraction is implemented through learning a weight vector ?j of size T for the j-th HM mark. For t ? {1, ..., T }, ?tj represents the importance of the t-th bin in the j-th HM. It is computed as: ?jt = exp(Wb hjt ) PT j . i=1 exp(Wb hi ) ?tj is a scalar and is computed by considering all bins? embedding vectors {hj1 , ? ? ? , hjT }. The context parameter Wb is randomly initialized and jointly learned with the other model parameters during training. Our intuition is that through Wb the model will automatically learn the context of the task (e.g., type of a cell) as well as the positional relevance to the context simultaneously. Once we have the importance weight of each bin position, we can represent the entire j-th HM mark as a PT weighted sum of all its bin embeddings: mj = t=1 ?tj ? hjt . Essentially the attention weights ?tj tell us the relative importance of the t-th bin in the representation mj for the current input X (both hjt and ?tj depend on X). HM-Level Encoder Using Another LSTM: We aim to capture the dependencies among HMs as some HMs are known to work together to repress or activate gene expression [6]. Therefore, next we model the joint dependencies among multiple HM marks (essentially, learn to represent a set). Even though there exists no clear order among HMs, we assume an imagined sequence as {HM1 , HM2 , HM3 , ..., HMM } 6 . We implement another bi-directional LSTM encoder, this time on the imagined sequence of HMs using the representations mj of the j-th HMs as LSTM inputs (Supplementary Figure S:2 (e)). Setting the embedding size as d0 , this set-based encoder, we denote ?????? ?????? as LST M s , encodes the j-th HM as: sj = [LST M s (mj ), LST M s (mj )]. Differently from mj , sj encodes the dependencies between the j-th HM and other HM marks. 6 We tried several different architectures to model the dependencies among HMs, and found no clear ordering. 4 Table 1: Comparison of previous studies for the task of quantifying gene expression using histone modification marks (adapted from [29]). AttentiveChrome is the only model that exhibits all 8 desirable properties. Computational Study Linear Regression ([14]) Support Vector Machine ([7]) Random Forest ([10]) Rule Learning ([12]) DeepChrome-CNN [29] AttentiveChrome Unified ? ? ? ? X X Nonlinear ? X X X X X Bin-Info Representation Learning ? Bin-specific Best-bin ? Automatic Automatic Neighbor Bins Whole Region ? ? ? ? X X X X X X X X Prediction Feature Inter. Interpretable X X X ? X X ? X ? X X X X ? ? X ? X HM-Level Attention, ?-attention: Now we want to focus on the important HM markers for classifying a gene?s expression as high or low. We do this by learning a second level of attention among HMs. Similar to learning ?tj , we learn another set of weights ? j for j ? {1, ? ? ? , M } representing j ss ) the importance of HMj . ? i is calculated as: ? j = PMexp(W i . The HM-level context parameter i=1 exp(Ws s ) Ws learns the context of the task and learns how HMs are relevant to that context. Ws is randomly initialized and jointly trained. We encode the entire "gene region" into a hidden representation v as PM a weighted sum of embeddings from all HM marks: v = j=1 ? j sj . We can interpret the learned attention weight ? i as the relative importance of HMi when blending all HM marks to represent the entire gene region for the current gene sample X. Training AttentiveChrome End-to-End: The vector v summarizes the information of all HMs for a gene sample. We feed it to a simple classification module f (Supplementary Figure S:2(f)) that computes the probability of the current gene being expressed high or low: f (v) = softmax(Wc v + bc ). Wc and bc are learnable parameters. Since the entire model, including the attention mechanisms, is differentiable, learning end-to-end is trivial by using backpropagation [21]. All parameters are learned together to minimize a negative log-likelihood loss function that captures the difference between true labels y and predicted scores from f (.). 4 Connecting to Previous Studies In recent years, there has been an explosion of deep learning models that have led to groundbreaking performance in many fields such as computer vision [17], natural language processing [30], and computational biology [1, 27, 38, 16, 19, 29]. Attention-based deep models: The idea of attention in deep learning arises from the properties of the human visual system. When perceiving a scene, the human vision gives more importance to some areas over others [9]. This adaptation of ?attention? allows deep learning models to focus selectively on only the important features. Deep neural networks augmented with attention mechanisms have obtained great success on multiple research topics such as machine translation [4], object recognition [2, 26], image caption generation [33], question answering [30], text document classification [34], video description generation[35], visual question answering -[32], or solving discrete optimization [31]. Attention brings in two benefits: (1) By selectively focusing on parts of the input during prediction the attention mechanisms can reduce the amount of computation and the number of parameters associated with deep learning model [2, 26]. (2) Attention-based modeling allows for learning salient features dynamically as needed [34], which can help improve accuracy. Different attention mechanisms have been proposed in the literature, including ?soft? attention [4], ?hard? attention [33, 24], or ?location-aware? [8]. Soft attention [4] calculates a ?soft? weighting scheme over all the component feature vectors of input. These weights are then used to compute a weighted combination of the candidate feature vectors. The magnitude of an attention weight correlates highly with the degree of significance of the corresponding component feature vector to the prediction. Inspired by [34], AttentiveChrome uses two levels of soft attention for predicting gene expression from HM marks. Visualizing and understanding deep models: Although deep learning models have proven to be very accurate, they have widely been viewed as ?black boxes?. Researchers have attempted to develop separate visualization techniques that explain a deep classifier?s decisions. Most prior studies have focused on understanding convolutional neural networks (CNN) for image classifications, including techniques such as ?deconvolution? [36], ?saliency maps? [3, 28] and ?class optimization? based 5 visualisation [28]. The ?deconvolution? approach [36] maps hidden layer representations back to the input space for a specific example, showing those features of an image that are important for classification. ?Saliency maps" [28] use a first-order Taylor expansion to linearly approximate the deep network and seek most relevant input features. The ?class optimization? based visualization [28] tries to find the best example (through optimization) that maximizes the probability of the class of interest. Recent studies [15, 22] explored the interpretability of recurrent neural networks (RNN) for text-based tasks. Moreover, since attention in models allows for automatically extracting salient features, attention-coupled neural networks impart a degree of interpretability. By visualizing what the model attends to in [34], attention can help gauge the predictive importance of a feature and hence interpret the output of a deep neural network. Deep learning in bioinformatics: Deep learning is steadily gaining popularity in the bioinformatics community. This trend is credited to its ability to extract meaningful representations from large datasets. For instance, multiple recent studies have successfully used deep learning for modeling protein sequences [23, 37] and DNA sequences [1, 20], predicting gene expressions [29], as well as understanding the effects of non-coding variants [38, 27]. Previous machine learning models for predicting gene expression from histone modification marks: Multiple machine learning methods have been proposed to predict gene expression from histone modification data (surveyed by Dong et al. [11]) including linear regression [14], support vector machines [7], random forests [10], rule-based learning [12] and CNNs [29]. These studies designed different feature selection strategies to accommodate a large amount of histone modification signals as input. The strategies vary from using signal averaging across all relevant positions, to a ?best position? strategy that selected the input signals at the position with the highest correlation to the target gene expression and automatically learning combinatorial interactions among histone modification marks using CNN (called DeepChrome [29]). DeepChrome outperformed all previous methods (see Supplementary) on this task and used a class optimization-based technique for visualizing the learned model. However, this class-level visualization lacks the necessary granularity to understand the signals from multiple chromatin marks at the individual gene level. Table 1 compares previous learning studies on the same task with AttentiveChrome across seven desirable model properties. The columns indicate properties (1) whether the study has a unified end-to-end architecture or not, (2) if it captures non-linearity among features, (3) how has the bin information been incorporated, (4) if representation of features is modeled on local and (5) global scales, (6) whether gene expression prediction is provided, (7) if combinatorial interactions among histone modifications are modeled, and finally (8) if the model is interpretable. AttentiveChrome is the only model that exhibits all seven properties. Additionally, Section 5 compares the attention weights from AttentiveChrome with the visualization from "saliency map" and "class optimization." Using the correlation to one additional HM mark from REMC, we show that AttentiveChrome provides better interpretation and validation. 5 Experiments and Results Dataset: Following DeepChrome [29], we downloaded gene expression levels and signal data of five core HM marks for 56 different cell types archived by the REMC database [18]. Each dataset contains information about both the location and the signal intensity for a mark measured across the whole genome. The selected five core HM marks have been uniformly profiled across all 56 cell types in the REMC study [18]. These five HM marks include (we rename these HMs in our analysis for readability): H3K27me3 as HreprA , H3K36me3 as Hstruct , H3K4me1 as Henhc , H3K4me3 as Hprom , and H3K9me3 as HreprB . HMs HreprA and HreprB are known to repress the gene expression, Hprom activates gene expression, Hstruct is found over the gene body, and Henhc sometimes helps in activating gene expression. Details of the Dataset: We divided the 10, 000 base pair DNA region (+/ ? 5000 bp) around the transcription start site (TSS) of each gene into bins, with each bin containing 100 continuous bp). For each gene in a specific celltype, the feature generation process generated a 5 ? 100 matrix, X, where columns represent T (= 100) different bins and rows represent M (= 5) HMs. For each cell type, the gene expression has been quantified for all annotated genes in the human genome and has been normalized. As previously mentioned, we formulated the task of gene expression prediction as a binary classification task. Following [7], we used the median gene expression across all genes for a particular cell type as the threshold to discretize expression values. For each cell type, we divided 6 Table 2: AUC score performances for different variations of AttentiveChrome and baselines Baselines AttentiveChrome Variations Model DeepChrome (CNN) [29] LSTM CNNAttn CNN?, ? LSTM- LSTM- LSTMAttn ? ?, ? Mean Median Max Min 0.8052 0.8036 0.9185 0.7073 0.7622 0.7617 0.8707 0.6469 0.7936 0.7914 0.9059 0.7001 0.8100 0.8118 0.9155 0.7237 0.8133 0.8143 0.9218 0.7250 0.8115 0.8123 0.9177 0.7215 36 0 16 49 50 49 0.8008 0.8009 0.9225 0.6854 Improvement over DeepChrome [29] (out of 56 cell types) our set of 19,802 gene samples into three separate, but equal-size folds for training (6601 genes), validation (6601 genes), and testing (6600 genes) respectively. Model Variations and Two Baselines: In Section 3, we introduced three main components of AttentiveChrome to handle the task of predicting gene expression from HM marks: LSTMs, attention mechanisms, and hierarchical attention. To investigate the performance of these components, our experiments compare multiple AttentiveChrome model variations plus two standard baselines. ? DeepChrome [29]: The temporal (1-D) CNN model used by Singh et al. [29] for the same classification task. This study did not consider the modular property of HM marks. ? LSTM: We directly apply an LSTM on the input matrix X without adding any attention. This setup does not consider the modular property of each HM mark, that is, we treat the signals of all HMs at t-th bin position as the t-th input to LSTM. ? LSTM-Attn: We add one attention layer on the baseline LSTM model over input X. This setup does not consider the modular property of HM marks. ? CNN-Attn: We apply one attention layer over the CNN model from DeepChrome [29], after removing the max-pooling layer to allow bin-level attention for each bin. This setup does not consider the modular property of HM marks. ? LSTM-?, ?: As introduced in Section 3, this model uses one LSTM per HM mark and add one ?-attention per mark. Then it uses another level of LSTM and ?-attention to combine HMs. ? CNN-?, ?: This considers the modular property among HM marks. We apply one CNN per HM mark and add one ?-attention per mark. Then it uses another level of CNN and ?-attention to combine HMs. ? LSTM-?: This considers the modular property of HM marks. We apply one LSTM per HM mark and add one ?-attention per mark. Then, the embedding of HM marks is concatenated as one long vector and then fed to a 2-layer fully connected MLP. We use datasets across 56 cell types, comparing the above methods over each of the 56 different tasks. Model Hyperparameters: For AttentiveChrome variations, we set the bin-level LSTM embedding size d to 32 and the HM-level LSTM embedding size as 16. Since we implement a bi-directional LSTM, this results in each embedding vector ht as size 64 and embedding vector mj as size 32. Therefore, we set the context vectors, Wb and Ws , to size 64 and 32 respectively.7 Performance Evaluation: Table 2 compares different variations of AttentiveChrome using summarized AUC scores across all 56 cell types on the test set. We find that overall the LSTM-attention based models perform better than CNN-based and LSTM baselines. CNN-attention model gives worst performance. To add the bin-level attention layer to the CNN model, we removed the max-pooling layer. We hypothesize that the absence of max-pooling is the cause behind its low performance. LSTM-? has better empirical performance than the LSTM-?, ? model. We recommend the use of the proposed AttentiveChrome LSTM-?, ? (from here on referred to as AttentiveChrome) for hypothesis generation because it provides a good trade-off between AUC and interpretability. Also, while the performance improvement over DeepChrome [29] is not large, AttentiveChrome is better as it allows interpretability to the "black box" neural networks. 7 We can view Wb as 1 ? 64 matrix. 7 Table 3: Pearson Correlation values between weights assigned for Hprom (active HM) by different visualization techniques and Hactive read coverage (indicating actual activity near "ON" genes) for predicted "ON" genes across three major cell types. Viz. Methods H1-hESC GM12878 K562 ? Map (LSTM-?) ? Map (LSTM-?, ?) Class-based Optimization (CNN) Saliency Map (CNN) 0.8523 0.8995 0.0562 0.1822 0.8827 0.8456 0.1741 -0.1421 0.9147 0.9027 0.1116 0.2238 Using Attention Scores for Interpretation: Unlike images and text, the results for biology are hard to interpret by just looking at them. Therefore, we use additional evidence from REMC as well as introducing a new strategy to qualitatively and quantitatively evaluate the bin-level attention weights or ?-map LSTM-? model and AttentiveChrome. To specifically validate that the model is focusing its attention at the right bins, we use the read counts of a new HM signal - H3K27ac from REMC database. We represent this HM as Hactive because this HM marks the region that is active when the gene is ?ON". H3K27ac is an important indication of activity in the DNA regions and is a good source to validate the results. We did not include H3K27ac Mark as input because it has not been profiled for all 56 cell types we used for prediction. However, the genome-wide reads of this HM mark are available for three important cell types in the blood lineage: H1-hESC (stem cell), GM12878 (blood cell), and K562 (leukemia cell). We, therefore, chose to compare and validate interpretation in these three cell types. This HM signal has not been used at any stage of the model training or testing. We use it solely to analyze the visualization results. We use the average read counts of Hactive across all 100 bins and for all the active genes (gene=ON) in the three selected cell types to compare different visualization methods. We compare the attention ?-maps of the best performing LSTM-? and AttentiveChrome models with the other two popular visualization techniques: (1) the Class-based optimization method and (2) the Saliency map applied on the baseline DeepChrome-CNN model. We take the importance weights calculated by all visualization methods for our active input mark, Hprom , across 100 bins and then calculate their Pearson correlation to Hactive counts across the same 100 bins. Hactive counts indicate the actual active regions. Table 3 reports the correlation coefficients between Hprom weights and read coverage of Hactive . We observe that attention weights from our models consistently achieve the highest correlation with the actual active regions near the gene, indicating that this method can capture the important signals for predicting gene activity. Interestingly, we observe that the saliency map on the DeepChrome achieves a higher correlation with Hactive than the Class-based optimization method for two cell types: H1-hESC (stem cell) and K562 (leukemia cell). Next, we obtain the attention weights learned by AttentionChrome, representing the important bins and HMs for each prediction of a particular gene as ON or OFF. For a specific gene sample, we can visualize and inspect the bin-level and HM-level attention vectors ?tj and ? j generated by AttentionChrome. In Figure 2(a), we plot the average bin-level attention weights for each HM for cell type GM12878 (blood cell) by averaging ?-maps of all predicted ?ON" genes (top) and ?OFF" genes (bottom). We see that on average for ?ON" genes, the attention profiles near the TSS region are well defined for Hprom , Henhc , and Hstruct . On the contrary, the weights are low and close to uniform for HreprA and HreprB . This average trend reverses for ?OFF" genes in which HreprA and HreprB seem to gain more importance over Hprom , Henhc , and Hstruct . These observations make sense biologically as Hprom , Henhc , and Hstruct are known to encourage gene activation while HreprA and HreprB are known to repress the genes 8 . On average, while Hprom is concentrated near the TSS region, other HMs like Hstruct show a broader distribution away from the TSS. In summary, the importance of each HM and its position varies across different genes. E.g., Henhc can affect a gene from a distant position. In Figure 2(b), we plot the average read coverage of Hactive (top) for the same 100 bins, that we used for input signals, across all the active genes (gene=ON) for GM12878 cell type. We also plot the bin-level attention weights ?tj for AttentiveChrome (bottom) averaged over all genes predicted as ON for GM12878. Visually, we can tell that the average Hprom profile is similar to Hactive . This 8 The small dips at the TSS in both subfigures of Figure 2(a) are caused by missing signals at the TSS due to the inherent nature of the sequencing experiments. 8 0 Hstruct 20 TSS 40 60 Bins (t) Henhc 80 100 Hprom HreprB (c) Genes=ON 0 20 40 60 Bins (t) 80 100 Gene: PAX5 0 1 HreprA Bins (t) 0 20 40 TSS 60 80 Hstruct 100 Henhc Hprom HreprB 0.014 0.008 Bins (t) Henhc Hprom HreprB 0.012 0.010 Hstruct H1-hESC GM12878 K562 Attention Weights TSS Hactive Gene = OFF ON OFF HreprA Genes=OFF Avg. Attention Weights 0.0 0.2 0.4 0.6 0.8 1.0 HreprA (b) Cell Type: GM12878 Read Coverage of Hactive Genes=ON Avg. Attention Weights 0.0 0.2 0.4 0.6 0.8 1.0 (a) 0 0 20 40 TSS Map 60 80 100 Maps Figure 2: (Best viewed in color) (a) Bin-level attention weights (?tj ) from AttentiveChrome averaged for all genes when predicting gene=ON and gene=OFF in GM12878 cell type. (b) Top: Cumulative Hactive signal across all active genes. Bottom: Plot of the bin-level attention weights (?tj ). These weights are averaged for gene=ON predictions. Hprom weights are concentrated near the TSS and corresponds well with the Hactive indicating actual activity near the gene. This indicates that AttentiveChrome is focusing on the correct bin positions for this case (c) Heatmaps visualizing the HM-level weights (? j ), with j ? {1, ..., 5} for an important differentially regulated gene (PAX5) across three blood lineage cell types: H1-hESC (stem cell), GM12878 (blood cell), and K562 (leukemia cell). The trend of HM-level ? j weights for PAX5 have been verified through biological literature. observation makes sense because Hprom is related to active regions for ?ON" genes. Thus, validating our results from Table 3. Finally in Figure 2(c) we demonstrate the advantage of AttentiveChrome over LSTM-? model by printing out the ? j weights for genes with differential expressions across the three cell types. That is, we select genes with varying ON(+1)/OFF(?1) states across the three chosen cell types using a heatmap. Figure 2(c) visualizes the ? j weights for a certain differentially regulated gene, PAX5. PAX5 is critical for the gene regulation when stem cells convert to blood cells ([25]). This gene is OFF in the H1-hESC cell stage (left column) but turns ON when the cell develops into GM12878 cell (middle column). The ? j weight of repressor mark HreprA is high when gene=OFF in H1-hESC (left column). This same weight decreases when gene=ON in GM12878 (middle column). In contrast, the ? j weight of the promoter mark Hprom increases from H1-hESC (left column) to GM12878 (middle column). These trends have been observed in [25] showing that PAX5 relates to the conversion of chromatin states: from a repressive state (Hprom (H3K4me3):?, HreprA (H3K27me3):+) to an active state (Hprom (H3K4me3):+, HreprA (H3K27me3):?). This example shows that our ? j weights visualize how different HMs work together to influence a gene?s state (ON/OFF). We would like to emphasize that the attention weights on both bin-level (?-map) and HM-level (?-map) are gene (i.e. sample) specific. The proposed AttentiveChrome model provides an opportunity for a plethora of downstream analyses that can help us understand the epigenomic mechanisms better. Besides, relevant datasets are big and noisy. A predictive model that automatically selects and visualizes essential features can significantly reduce the potential manual costs. 6 Conclusion We have presented AttentiveChrome, an attention-based deep-learning approach that handles prediction and understanding in one architecture. The advantages of this work include: ? AttentiveChrome provides more accurate predictions than state-of-the-art baselines (Table 2). ? The attention scores of AttentiveChrome provide a better interpretation than saliency map and class optimization (Table 3). This allows us to view what the model ?sees? when making its prediction. ? AttentiveChrome can model highly modular feature inputs in which each is sequentially structured. ? To the authors? best knowledge, AttentiveChrome is the first implementation of deep attention mechanism for understanding data about gene regulation. We can gain insights and understand the predictions by locating ?what? and ?where? AttentiveChrome has focused (Figure 2). Many real-world applications are seeking such knowledge from data. 9 References [1] Babak Alipanahi, Andrew Delong, Matthew T Weirauch, and Brendan J Frey. Predicting the sequence specificities of dna-and rna-binding proteins by deep learning. Nature Publishing Group, 2015. [2] Jimmy Ba, Volodymyr Mnih, and Koray Kavukcuoglu. Multiple object recognition with visual attention. 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Acceleration and Averaging In Stochastic Descent Dynamics Walid Krichene Google, Inc. walidk@google.com Peter Bartlett U.C. Berkeley bartlett@cs.berkeley.edu Abstract We formulate and study a general family of (continuous-time) stochastic dynamics for accelerated first-order minimization of smooth convex functions. Building on an averaging formulation of accelerated mirror descent, we propose a stochastic variant in which the gradient is contaminated by noise, and study the resulting stochastic differential equation. We prove a bound on the rate of change of an energy function associated with the problem, then use it to derive estimates of convergence rates of the function values (almost surely and in expectation), both for persistent and asymptotically vanishing noise. We discuss the interaction between the parameters of the dynamics (learning rate and averaging rates) and the covariation of the noise process. In particular, we show how the asymptotic rate of covariation affects the choice of parameters and, ultimately, the convergence rate. 1 Introduction We consider the constrained convex minimization problem min f (x), x?X where X is a closed, convex, compact subset of Rn , and f is a proper closed convex function, assumed to be differentiable with Lipschitz gradient, and we denote X ? ? X the set of its minimizers. First-order methods play an important role in minimizing such functions, in particular in large-scale machine learning applications, in which the dimensionality (number of features) and size (number of samples) in typical datasets makes higher-order methods intractable. Many such algorithms can be viewed as a discretization of continuous-time dynamics. The simplest example is gradient descent, which can be viewed as the discretization of the gradient flow dynamics x(t) ? = ??f (x(t)), where x(t) ? denotes the time derivative of a C 1 trajectory x(t). An important generalization of gradient descent was developed by Nemirovsky and Yudin [1983], and termed mirror descent: it couples a dual variable z(t) and its ?mirror? primal variable x(t). More specifically, the dynamics are given by  z(t) ? = ??f (x(t)) MD (1) x(t) = ?? ? (z(t)), where ?? ? : Rn ? X is a Lipschitz function defined on the entire dual space Rn , with values in the feasible set X ; it is often referred to as a mirror map, and we will recall its definition and properties in Section 2. Mirror descent can be viewed as a generalization of projected gradient descent, where the Euclidean projection is replaced by the mirror map ?? ? [Beck and Teboulle, 2003]. This makes it possible to adapt the choice of the mirror map to the geometry of the problem, leading to better dependence on the dimension n, see [Ben-Tal and Nemirovski, 2001], [Ben-Tal et al., 2001]. Continuous-time dynamics Although optimization methods are inherently discrete, the continuous-time point of view can help in their design and analysis, since it can leverage the 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. rich literature on dynamical systems, control theory, and mechanics, see [Helmke and Moore, 1994], [Bloch, 1994], and the references therein. Continuous-time models are also commonly used in financial applications, such as option pricing [Black and Scholes, 1973], even though the actions are taken in discrete time. In convex optimization, beyond simplifying the analysis, continuous-time models have also motivated new algorithms: mirror descent is one such example, since it was originally motivated in continuous time (Chapter 3 in [Nemirovsky and Yudin, 1983]). In a more recent line of work ([Su et al., 2014], [Krichene et al., 2015], [Wibisono et al., 2016]), Nesterov?s accelerated method [Nesterov, 1983] was shown to be the discretization of a second-order ordinary differential equation (ODE), which, in the unconstrained case, can be interpreted as a damped non-linear oscillator [Cabot et al., 2009, Attouch et al., 2015]. This motivated a restarting heuristic [O?Donoghue and Cand?s, 2015], which aims at further dissipating the energy. Krichene et al. [2015] generalized this ODE to mirror descent, and gave an averaging interpretation of accelerated dynamics by writing it as two coupled first-order ODEs. This is the starting point of this paper, in which we introduce and study a stochastic variant of accelerated mirror descent. Stochastic dynamics and related work The dynamics that we have discussed so far are deterministic first-order dynamics, since they use the exact gradient ?f . However, in many machine learning applications, evaluating the exact gradient ?f can be prohibitively expensive, e.g. in regularized empirical risk minimization problems, where the P objective function f involves the sum 1 of loss functions over a training set, of the form f (x) = |I| i?I fi (x) + g(x), where I indexes 1 the training samples, and g is a regularization function . Instead of computing the exact gradient P 1 ?f (x) = |I| i?I ?fi (x) + ?g(x), a common approach is to compute an unbiased, stochastic P ? estimate of the gradient, given by 1 ? ?fi (x) + ?g(x), where I is a uniformly random subset ? |I| i?I of I, indexing a random batch of samples from the training set. This approach motivates the study of stochastic dynamics for convex optimization. But despite an extensive literature on stochastic gradient and mirror descent in discrete time, e.g. [Nemirovski et al., 2009], [Duchi et al., 2010], [Lan, 2012], [Johnson and Zhang, 2013], [Xiao and Zhang, 2014], and many others, few results are known for stochastic mirror descent in continuous-time. To the best of our knowledge, the only published results are by Raginsky and Bouvrie [2012] and Mertikopoulos and Staudigl [2016]. In its simplest form, the stochastic gradient dynamics can be described by the (underdamped) Langevin equation dX(t) = ??f (X(t)) + ?dB(t), where B(t) denotes a standard Wiener process (Brownian motion). It has a long history in optimization [Chiang et al., 1987], dating back to simulated annealing, and it is known to have a unique 2f (x) invariant measure with density proportional to the Gibbs distribution e? ? (see, e.g., [Pavliotis, 2014]). Langevin dynamics have recently played an important role in the analysis of sampling methods [Dalalyan, 2017, Bubeck et al., 2015, Durmus and Moulines, 2016, Cheng and Bartlett, 2017, Eberle et al., 2017, Cheng et al., 2017], where f is taken to be proportional to the logarithm of a target density. It has also been used to derive convergence rates for smooth, non-convex optimization where the objective is dissipative [Raginsky et al., 2017]. For mirror descent dynamics, Raginsky and Bouvrie [2012] were the first to propose a stochastic variant of the mirror descent ODE (1), given by the SDE:  dZ(t) = ??f (X(t)) + ?dB(t) SMD (2) X(t) = ?? ? (Z(t)), where ? is a constant volatility. They argued that the function values f (X(t)) along sample trajectories do not converge to the minimum value of f due to the persistent noise, but the optimality gap is bounded by a quantity proportional to ? 2 . They also proposed a method to reduce the variance by simultaneously sampling multiple trajectories and linearly coupling them. Mertikopoulos and Staudigl [2016] extended the analysis in some important directions: they replaced the constant ? with a general volatility matrix ?(x, t) which can be space ? and time dependent, and studied two regimes: the small noise limit (?(x, t) vanishes at a O(1/ log t) rate), in which case they prove almost sure convergence; and the persistent noise regime (?(x, t) is uniformly bounded), in which case they define 1 In statistical learning, one seeks to minimize the expected risk (with respect to the true, unknown data distribution). A common approach is to minimize the empirical risk (observed on a given training set) then bound the distance between empirical and expected risk. Here we only focus on the optimization part. 2 a rectified variant of SMD, obtained by replacing the second equation by X(t) = ?? ? (Z(t)/s(t)), where 1/s(t) is a sensitivity parameter (intuitively, decreasing?the sensitivity reduces the impact of accumulated noise). ? In particular, they prove that with s(t) = t, the expected function values converge at a O(1/ t) rate. While these recent results paint a broad picture of mirror descent dynamics, they leave many questions open: in particular, they do not provide estimates for convergence rates in the vanishing noise limit, which is an important regime in machine learning applications, since one can often control the variance of the gradient estimate, for example by gradually increasing the batch size, as done by Xiao and Zhang [2014]. Besides, they do not study accelerated dynamics, and the interaction between acceleration and noise remains unexplored in continuous time. Our contributions In this paper, we answer many of the questions left open in previous works. We formulate and study a family of stochastic accelerated mirror descent dynamics, and we characterize the interaction between its different parameters: the volatility of the noise, the (primal and dual) learning rates, and the sensitivity of the mirror map. More specifically: ? In Theorem 1, we give sufficient conditions for almost sure convergence of solution trajectories to the set of minimizers X ? . In particular, we show that it is possible to guarantee almost sure convergence even when the volatility grows unbounded asymptotically. ? In Theorem 2, we derive a bound on the expected function values. In particular, we can prove that in the vanishing noise regime, acceleration (with appropriate averaging) achieves a faster rate, see Corollary 2 and the discussion in Remark 3. ? In Theorem 3, we provide estimates of sample trajectory convergence rates. The rest of the paper is organized as follows: We review the building blocks of our construction in Section 2, then formulate the stochastic dynamics in Section 3, and prove two instrumental lemmas. Section 4 is dedicated to the convergence results. We conclude with a brief discussion in Section 5. 2 2.1 Accelerated Mirror Descent Dynamics Smooth mirror map We start by reviewing some definitions and preliminaries. Let (E, k ? k), be a normed vector space, and (E ? , k ? k? ) be its dual space equipped with the dual norm, and denote by hx, zi the pairing between x ? E, z ? E ? . To simplify, both E and E ? can be identified with Rn , but we make the distinction for clarity. We say that a map F : E ? E ? is Lipschitz continuous on X ? E with constant L if for all x, x0 ? X , kF (x) ? F (x0 )k? ? Lkx ? x0 k. Let ? : E ? R ? {+?} be a convex function with effective domain X (i.e. X = {x ? E : ?(x) < ?}). Its convex conjugate ? ? is defined on E ? by ? ? (z) = supx?X hz, xi ? ?(x). One can show that if ? is strongly convex, then ? ? is differentiable on all of E ? , and its gradient ?? ? is a Lipschitz function that maps E ? to X (see the supplementary material). This map is often called a mirror map [Nemirovsky and Yudin, 1983]. To give a concrete example, take ? to be the squared Euclidean norm, ?(x) = 12 kxk22 . Then one can show ? ? (z) = arg minx?X kz ? xk22 , and the mirror map reduces to the Euclidean projection on X . For additional examples, see e.g. Banerjee et al. [2005]. We make the following assumptions throughout the paper: Assumption 1. X is closed, convex and compact, ? is non-negative (without loss of generality), ? ? is twice differentiable with a Lipschitz gradient, and f is differentiable with Lipschitz gradient. We denote by L?? the Lipschitz constant of ?? ? , and by Lf the Lipschitz constant of ?f . 2.2 Averaging formulation of accelerated mirror descent We start from the averaging formulation of Krichene et al. [2015], and include a sensitivity parameter similar to Mertikopoulos and Staudigl [2016]. This results in the following ODE:  z(t) ? = ??(t)?f (x(t)) AMD?,a,s (3) x(t) ? = a(t)(?? ? (z(t)/s(t)) ? x(t)), 3 with initial conditions2 (x(t0 ), z(t0 )) = (x0 , z0 ). The ODE system is parameterized by the following functions, all assumed to be positive and continuous on [t0 , ?) (see Figure 1 for an illustration): ? s(t) is a non-decreasing, inverse sensitivity parameter. As we will see, s(t) will be helpful in the stochastic case in scaling the noise term, in order to reduce its impact. ? ?(t) is a learning rate in the dual space. ? a(t) is an averaging rate in the primal space. Indeed, the second ODE in (3) can be written inR integral form as a weighted average of the mirror trajectory as follows: let t a(? )d? ? w(t) = e t0 (equivalently, a(t) = w(t) ? + w(t) ), then the ODE is equivalent to w(t)x(t) ? w(t)x(t) ? = w(t)?? ? (z(t)/s(t)), and integrating and rearranging, Rt x(t0 )w(t0 ) + t0 w(? ? )?? ? (Z(? )/s(? ))d? x(t) = . w(t) There are other, different ways of formulating the accelerated dynamics: instead of two first-order ODEs, one can write one second-order ODE (such as in Su et al. [2014], Wibisono et al. [2016]), which has interesting interpretations related to Lagrangian dynamics. The averaging formulation given in Equation (3) is better suited to our analysis. 2.3 Energy function The analysis of continuous-time dynamics often relies on a Lyapunov argument (in reference to Lyapunov [1892]): one starts by defining a non-negative energy function, then bounding its rate of change along solution trajectories. This bound can then be used to prove convergence to the set of minimizers X ? . We will consider a modified version of the energy function used by Krichene et al. [2016]: given a positive, C 1 function r(t), and a pair of optimal primal-dual points (x? , z ? ) such that x? ? X ? and ?? ? (z ? ) = x? , let L(x, z, t) = r(t)(f (x) ? f (x? )) + s(t)D?? (z(t)/s(t), z ? ). (4) Here, D?? is the Bregman divergence associated with ? ? , defined by D?? (z 0 , z) = ? ? (z 0 ) ? ? ? (z) ? h?? ? (z), z 0 ? zi , for all z, z 0 ? E ? . Then we can prove a bound on the time derivative of L along solution trajectories of AMD?,a,s , given in the following proposition. To keep the equations compact, we will occasionally omit explicit dependence on time, and write, e.g. ?/r instead of ?(t)/r(t). Lemma 1. Suppose that a = ?/r. Then under AMD?,?/r,s , for all t ? t0 , d L(x(t), z(t), t) ? (f (x(t)) ? f (x? ))(r(t) ? ? ?(t)) + ?(x? )s(t). ? (5) dt Proof. We start by bounding the rate of change of the Bregman divergence term: d 2 s(t)D?? (z(t)/s(t), z ? ) = sD ? ?? (z/s, z ? ) + s ?? ? (z/s) ? ?? ? (z ? ), z/s ? ? sz/s ? dt = h?? ? (z/s) ? x? , zi ? + s(D ? ?? (z/s, z ? ) ? h?? ? (z/s) ? ?? ? (z ? ), z/si) ? = h?? ? (z/s) ? x? , zi ? + s(?(x ? ) ? ?(?? ? (z/s))) ? ? h?? ? (z/s) ? x? , zi ? + s?(x ? ), where the third equality can be proved using the fact that ?(x) + ? ? (z) = hx, zi ? x ? ?? ? (z) ? z ? ??(x) (Theorem 23.5 in Rockafellar [1970]), and the last inequality follows from the assumption that s is non-decreasing, and that ? is non-negative. Using this expression, we can then compute d L(x(t), z(t), t) ? r(f ? (x) ? f (x? )) + r h?f (x), xi ? + h?? ? (z/s) ? x? , zi ? + ?(x? )s? dt = r(f ? (x) ? f (x? )) + r h?f (x), xi ? + hx/a ? + x ? x? , ???f (x)i + ?(x? )s? ? (f (x) ? f (x? ))(r? ? ?) + h?f (x), xi ? (r ? ?/a) + ?(x? )s, ? where we plugged in the expression of z? and ?? ? (z/s) from AMD?,a,s in the second equality, and used convexity of f in the last inequality. The assumption a = ?/r ensures that the middle term vanishes3 , which concludes the proof. The initial conditions typically satisfy ?? ? (z0 ) = x0 which ensures that the trajectory starts with zero velocity, but this is not necessary in general. 3 Note that this assumption can be replaced by an adaptive rate a(t) similar to [Krichene et al., 2016]. 2 4 As a consequence of the previous proposition, we can prove the following convergence rate: Corollary 1. Suppose that a = ?/r and that ? ? r. ? Then under AMD?,?/r,s , for all t ? t0 ? ?(x )(s(t) ? s(t0 )) + L(x0 , z0 , t0 ) f (x(t)) ? f (x? ) ? . r(t) ? Proof. Starting from the bound (5), the first term is non-positive by the assumption that ? ? r. Integrating, we have L(x(t), z(t), t) ? L(x0 , z0 , t0 ) ? ?(x? )(s(t) ? s(t0 )), thus, L(x(t), z(t), t) ?(x? )(s(t) ? s(t0 )) + L(x0 , z0 , t0 ) f (x(t) ? f (x? )) ? ? . r(t) r(t) Remark 1. Corollary 1 can be interpreted as follows: given a desired convergence rate r(t), one can choose parameters a, ?, s that satisfy the conditions of the corollary (e.g. by first setting ? = r, ? then choosing a = ?/r). This defines an ODE, the solutions of which are guaranteed to converge at the rate r(t). While the convergence rate can seemingly be arbitrary for continuous time dynamics, discretizing the ODE does not always preserve the convergence rate. Wibisono et al. [2016], Wilson et al. [2016] give sufficient conditions on the discretization scheme to preserve polynomial rates, for example, a first-order discretization can preserve quadratic rates, and a higher-order discretization (using cubic-regularized Newton updates) can preserve cubic rates. Discretization is an important topic, it is however outside the scope of this paper. Remark 2. As a special case, one can recover Nesterov?s ODE by taking r(t) = t2 , ?(t) = ?t, a(t) = ?/t (i.e. w(t) = w(t0 )(t/t0 )? ), and s(t) = 1 (see the supplement for additional details). It is worth observing that in this case, both the primal and dual rates ?(t) and w(t) are increasing. A different choice of parameters leads to dynamics similar to Nesterov?s but with different weights. 3 Stochastic dynamics We now formulate the stochastic variant of accelerated mirror descent dynamics (SAMD). Intuitively, we would like to replace the gradient term ?f (x) in AMD?,a,s by a noisy gradient. Writing the noisy dynamics as an It? SDE [?ksendal, 2003], we consider the system  dZ(t) = ??(t)[?f (X(t))dt + ?(X(t), t)dB(t)] (6) SAMD?,a,s dX(t) = a(t)[?? ? (Z(t)/s(t)) ? X(t)]dt, with initial condition (X(t0 ), Z(t0 )) = (x0 , z0 ) (we assume deterministic initial conditions for simplicity). Here, B(t) ? Rn is a standard Wiener process with respect to a given filtered probability space (?, F, {Ft }t?t0 , P), and ? : (x, t) 7? ?(x, t) ? Rn?n is a volatility matrix assumed measurable and Lipschitz in x (uniformly in t), and continuous in t for all x. The drift term in SAMD?,a,s is identical to the deterministic case, and the volatility term ??(t)?(X(t), t)dB(t) represents the noise in the gradient. In particular, we note that the learning rate ?(t) multiplies ?(X(t), t)dB(t), to capture the fact that the gradient noise is scaled by the learning rate ?. This formulation is fairly general, and does not assume, in particular, that the different components of the noise are independent, as we can see in the quadratic covariation of the dual process Z(t): d[Zi (t), Zj (t)] = ?(t)2 (?(X(t), t)?(X(t), t)T )i,j dt = ?(t)2 ?ij (X(t), t)dt, T where we defined the infinitesimal covariance matrix ?(x, t) = ?(x, t)?(x, t) ? R analysis, we will focus on different noise regimes, which can be characterized using4 ??2 (t) = sup k?(x, t)ki , n?n (7) . In our (8) x?X where k?ki = supkzk? ?1 k?zk is the induced matrix norm. Since ?(x, t) is Lipschitz in x and continuous in t, and X is compact, ?? (t) is finite for all t, and continuous. Contrary to the work of [Raginsky and Bouvrie, 2012, Mertikopoulos and Staudigl, 2016], we do not assume, a priori, that ?? (t) is uniformly bounded in t. We give an illustration of the stochastic dynamics in Figure 1 (see the supplement for details). 4 In our model, we focus on the time-dependence of the volatility. Note that in some problems, the variance of the gradient estimates typically scales with the squared norm of the gradient, thus one can potentially consider a model where the volatility ?(x, t) scales with k?f (x)k? , which could lead to different convergence results. See e.g. [Bottou et al., 2016] in the discrete case. 5 ?? ? E? X Z(t) s(t) ?? ?  Z(t) s(t)  X(t) Figure 1: Illustration of the SAMD dynamics. The dual variable Z(t) cumulates gradients. It is scaled by the sensitivity 1/s(t) then mapped to the primal space via the mirror map, resulting in ?? ? (Z/s) (dotted line). The primal variable is then a weighted average of the mirror trajectory. Existence and uniqueness First, we give the following existence and uniqueness result: Proposition 1. For all T > t0 , SAMD?,a,s has a unique (up to redefinition on a P-null set) solution (X(t), Z(t)) continuous on [0, T ], with the property that (X(t), Z(t)) is adapted to the filtration RT RT {Ft }, and t0 kX(t)k2 dt, t0 kZ(t)k2? dt have finite expectations. Proof. By assumption, ?? ? and ?f are Lipschitz continuous, thus the function (x, z) 7? (??(t)?f (x), a(t)[?? ? (z/s(t)) ? x]) is Lipschitz on [t0 , T ] (since a, ?, s are positive continuous). Additionally, the function x 7? ?(x, t) is also Lipschitz. Therefore, we can invoke the existence and uniqueness theroem for stochastic differential equations [?ksendal, 2003, Theorem 5.2.1]. Since T is arbitrary, we can conclude that there exists a unique continuous solution on [t0 , ?). Energy decay Next, in order to analyze the convergence properties of the solution trajectories (X(t), Z(t)), we will need to bound the time-derivative of the energy function L. Lemma 2. Suppose that the primal rate a = ?/r, and let (X(t), Z(t)) be the unique solution to SAMD?,?/r,s . Then for all t ? t0 ,   nL?? ? 2 (t)??2 (t) dL(X(t), Z(t), t) ? (f (X(t)) ? f (x? ))(r(t) ? ? ?(t)) + ?(x? )s(t) ? + dt+hV (t), dB(t)i , 2 s(t) where V (t) is the continuous process given by V (t) = ??(t)?(X(t), t)T (?? ? (Z(t)/s(t)) ? ?? ? (z ? )). (9) Proof. By definition of the energy function L, ?x L(x, z, t) = r(t)?f (x) and ?z L(x, z, t) = ?? ? (z/s(t)) ? ?? ? (z ? ), which are Lipschitz continuous in (x, z) (uniformly in t on any bounded interval, since s, r are continuous positive functions of t). Thus by the It? formula for functions with Lipschitz continuous gradients [Errami et al., 2002], we have dL = ?t Ldt + h?x L, dXi + h?z L, dZi +  1  T 2 tr ?? ?zz L?? dt 2 = ?t Ldt + h?x L, dXi + h?z L, ???f (X)i dt + h?z L, ???dBi +
?2 tr ??2zz L dt. 2 The first three terms correspond exactly to the deterministic case, and we can bound them by (5) from Lemma 1. The last two terms are due to the stochastic noise, and consist of a volatility term ?? h?z L(X, Z, t), ?dBi = ?? h?? ? (Z/s) ? ?? ? (z ? ), ?dBi = hV, dBi , and the It? correction term

?2 ?2 tr ?(X, t)?2zz L(X, Z, t) dt = tr ?(X, t)?2 ? ? (Z/s) dt. 2 2s 6 We can bound the last term using the fact that ?? ? is, by assumption, L?? -Lipschitz, and the definition (8) of ? ? : for all x ? E, z ? E ? , and t ? t0 , tr(?(x, t)?2 ? ? (z)) ? nL?? ??2 (t). Combining the previous inequalities, we obtain the desired bound. Integrating the bound ofR Lemma 2 will allow us to bound changes in energy. This bound will involve t the It? martingale term t0 hV (? ), dB(? )i, and in order to control this term, we give, in the following lemma, an asymptotic envelope (a consequence of the law of the iterated logarithm). Rt Lemma 3. Let b(t) = t0 ? 2 (? )??2 (? )d? . Then Z t p hV (? ), dB(? )i = O( b(t) log log b(t)) a.s. as t ? ?. (10) t0 Rt Pn R t Proof. Let us denote the It? martingale by V(t) = t0 hV (? ), dB(? )i = i=1 t0 Vi (? )dBi (? ), and its quadratic variation by ?(t) = [V(t), V(t)]. By definition of V, we have d? = n X n X Vi Vj d[Bi , Bj ] = i=1 j=1 n X i=1 Vi2 dt = hV, V i dt. By the Dambis-Dubins-Schwartz time change theorem (e.g. Corollary 8.5.4 in [?ksendal, 2003]), ? such that there exists a Wiener process B ? V(t) = B(?(t)). (11) ?(t) ? D2 b(t) a.s. (12) We now proceed to bound ?(t). Using the expression (9) of V , we have hV, V i = ? 2 (t)?T (t)?(X, t)?(t), where ?(t) = ?? ? (Z(t)/s(t)) ? ?? ? (z ? ). Since the mirror map has values in X and X is assumed compact, the diameter D = supx,x0 ?X kx ? x0 k is finite, and ?(t) ? D for all t. Thus, d?(t) ? D2 ?(t)2 ??2 (t)dt, and integrating, Since ?(t) is a non-decreasing process, two cases are possible: if limt?? ?(t) is finite, then lim supt?? |V(t)| is a.s. finite and the result follows immediately. If limt?? ?(t) = ?, then ? B(?(t)) V(t) lim sup p ? lim sup q ?(t) t?? t?? b(t) log log b(t) log log D2 ? =D 2 a.s. ?(t) D2 where the inequality combines (11) and (12), and the equality is by the law of the iterated logarithm. 4 Convergence results Equipped with Lemma 2 and Lemma 3, which bound, respectively, the rate of change of the energy and the asymptotic growth of the martingale term, we are now ready to prove our convergence results. Rt ? Theorem 1. Suppose that ?(t)?? (t) = o(1/ log t), and that t0 ?(? )d? dominates b(t) and p Rt b(t) log log b(t) (where b(t) = t0 ? 2 (? )??2 (? )d? as defined in Lemma 3). Consider SAMD dynamics with r = s = 1. Let (X(t), Z(t)) be the unique continuous solution of SAMD?,?,1 . Then lim f (X(t)) ? f (x? ) = 0 t?? a.s. Proof sketch. We give a sketch of the proof here (the full argument is deferred to the supplement). (i) The first step is to prove that under the conditions of the theorem, the continuous solution of SAMD?,?,1 , (X(t), Z(t)), is an asymptotic pseudo trajectory (a notion defined and studied by Bena?m and Hirsch [1996] and Bena?m [1999]) of the deterministic flow AMD?,?,1 . The rigorous definition is given in the supplementary material, but intuitively, this means that for large enough times, the sample paths of the process (X(t), Z(t)) get arbitrarily close to (x(t), z(t)), the solution trajectories of the deterministic dynamics. 7 (ii) The second step is to show that under the deterministic flow, the energy L decreases enough for large enough times. (iii) The third step is to prove that under the stochastic process, f (X(t)) cannot stay bounded away from f (x? ) for all t. Note that under the conditions of the theorem, integrating the bound of Lemma 2, and using the asymptotic envelope of Lemma 3, gives Z t L(X(t), Z(t), t)?L(x0 , z0 , t0 ) ? ? (f (X(? ))?f (x? ))?(? )d? +O(b(t))+O( p b(t) log log b(t)), t0 and if say f (X(t)) ? f (x? ) ? c > 0 for all t, then the first term dominates the bound, and the energy would decrease to ??, a contradiction. Combining these steps, we argue that f (X(t)) eventually becomes close to f (x? ) by (iii), then stays close by virtue of (i) and (ii). The result of Theorem 1 makes it possible to guarantee almost sure convergence (albeit without guaranteeing a convergence rate) when the noise is persistent (?? (t) is constant, or even increasing). To give a concrete example, suppose ?? (t) = O(t? ) (with ? < 21 but can be positive), and Rt 1 1 1 let ?(t) = t??? 2 . Then ?(t)?? (t) = O(t? 2 ), t0 ?(? )d? = ?(t??+ 2 ), b(t) = O(log t), and p ? b(t) log log b(t) = O( log t log log log t), and the conditions of the theorem are satisfied. Therefore, with the appropriate choice of learning rate ?(t) (and the corresponding averaging in the primal space given by a(t) = ?(t)), one can guarantee almost sure convergence. Next, we derive explicit bounds on convergence rates. We start by bounding expected function values. Theorem 2. Suppose that a = ?/r and ? ? r. ? Let (X(t), Z(t)) be the unique continuous solution to SAMD?,?/r,s . Then for all t ? t0 , nL ? R t ? 2 (? )??2 (? ) L(x0 , z0 , t0 ) + ?(x? )(s(t) ? s(t0 )) + 2? t0 d? s(? ) ? . E[f (X(t))] ? f (x ) ? r(t) Proof. Integrating the bound of Lemma 2, and using the fact that (f (X(t)) ? f (x? ))(r? ? ?) ? 0 by assumption on ?, we have L(X(t), Z(t), t) ? L(x0 , z0 , t0 ) ? ?(x? )(s(t) ? s(t0 )) + nL?? 2 Z t t0 ? 2 (? )??2 (? ) d? + s(? ) Z t hV (? ), dB(? )i , t0 (13) Taking expectations, the last term vanishes since it is an It? martingale, and we conclude by observing that E[f (X(t))] ? f (x? ) ? E[L(X(t), Z(t), t)]/r(t). To give a concrete example, suppose that ?? (t) = O(t? ) is given, and let r(t) = t? and s(t) = t? , ?, ? > 0. To simplify, we will take ?(t) = r(t) ? = ?t??1 . Then the bound of Theorem 2 shows that ? ??? ?+2????1 E[f (X(t))] ? f (x ) = O(t +t ). To minimize the asymptotic rate, we can choose 1 ? ? ? = ? + 2? ? ? ? 1, i.e. ? + ? ? ? ? 21 = 0, and the resulting rate is O(t?? 2 ). In particular, we have: Corollary 2. Suppose that ?? (t) = O(t? ), ? < 21 . Then with ?(t) = (1 ? ?)t?? , a(t) = 1?? t , and 1 1 ? ?? s(t) = t 2 , we have E[f (X(t))] ? f (x ) = O(t 2 ). Remark 3. Corollary 2 can be interpreted as follows: Given a polynomial bound ?? (t) = O(t? ) on the volatility of the noise process, one can adapt the choice of primal and dual averaging rates (a(t) 1 and ?(t)), which leads to an O(t?? 2 ) convergence rate. ? In the persistent noise regime (? = 0), the dynamics use a constant ?, and result in a O(1/ t) rate, similar to the rectified dynamics proposed by Mertikopoulos and Staudigl [2016]. In the vanishing noise regime (? < 0), we can take advantage of the decreasing volatility by making ?(t) increasing. With the appropriate averaging rate a(t), this leads to the improved rate 1 O(t?? 2 ). It is worth observing here that when ? ? ? 21 , the same rate can be obtained without 1 acceleration. It is not hard to show that the rectified SMD dynamics with s(t) = tmax(0,?+ 2 ) achieves 1 1 a O(tmax(?? 2 ,?1) ). However for ? < ? 12 , acceleration improves the rate from O(t?1 ) to O(t?? 2 ). 8 In the increasing noise regime (? > 0), as long as the volatility does not increase too fast (? < 12 ), one can still guarantee convergence by decreasing ?(t) with the appropriate rate. Finally, we give an estimate of the asymptotic convergence rate along solution trajectories. Theorem 3. Suppose that a = ?/r and ? ? r. ? Let (X(t), Z(t)) be the unique continuous solution to SAMD?,?/r,s . Then ? ? R t ?2 (? )??2 (? ) p + b(t) log log b(t) s(t) + n s(? ) t0 ? a.s. as t ? ?, f (X(t)) ? f (x? ) = O ? r(t) where b(t) = Rt t0 ? 2 (? )??2 (? )d? . Proof. Integrating the bound of Lemma 2 once again, we get inequality (13), where we can bound Rt the It? martingale term t0 hV (? ), dB(? )i using Lemma 3. This concludes the proof. p Comparing the last bound to that of Theorem 2, we have the additional b(t) log log b(t)/r(t) term due to the envelope of the martingale term. This results in a slower a.s. convergence rate. Suppose again that ?? (t) = O(t? ), and that r(t) = t? and ?(t) = r(t) ? = ?t??1 to simRt 2 2 2?+2??1 plify. Then b(t) = t0 ? (? )?? (? )d? = O(t ), and the martingale term becomes p ? ?? 21 O( b(t) log log b(t)/r(t)) = O(t log log t). Remarkably, the asymptotic rate of sample ? trajectories is, up to a log log t factor, the same as the asymptotic rate in expectation; one should observe, however, that the constant in the O notation is trajectory-dependent. Corollary 3. Suppose that ?? (t) = O(t? ), ? < 21 . Then with ?(t) = (1 ? ?)t?? , a(t) = 1?? t , and ? 1 ? ?? 12 2 s(t) = t , we have f (X(t)) ? f (x ) = O(t log log t) a.s. 5 Conclusion Starting from the averaging formulation of accelerated mirror descent in continuous-time, and motivated by stochastic optimization, we formulated a stochastic variant and studied the resulting SDE. We discussed the role played by each parameter: the dual learning rate ?(t), the inverse sensitivity parameter s(t), and the noise covariation bound ?? (t). Our results show that in the persistent noise regime, thanks to averaging, it is possible ? to guarantee a.s. convergence, remarkably even when ?? (t) is increasing (as long as ?? (t) = o( t)). In the vanishing noise regime, adapting the choice of ?(t) to the rate of ?? (t) (with the appropriate averaging) leads to improved convergence 1 1? rates, e.g. to O(t?? 2 ) in expectation and O(t?? 2 log log t) almost surely, when ?? (t) = O(t? ). These asymptotic bounds in continuous-time can provide guidelines in setting the different parameters of accelerated stochastic mirror descent. It is also worth observing that in the deterministic case, one can theoretically obtain arbitrarily fast convergence, through a time change as observed by Wibisono et al. [2016] ? a time-change would simply result in using different weights ?(t) and a(t); this can also be seen in Corollary 1, where the rate r(t) can be arbitrarily fast. In the stochastic dynamics, such a time-change would also lead to re-scaling the noise co-variation, and does not lead to a faster rate. To some extent, adding the noise prevents us from ?artificially? accelerating convergence using a simple time-change. Finally, we believe this continuous-time analysis can be extended in several directions. For instance, it will be interesting to carry out a similar analysis for strongly convex functions, for which we expect faster optimal rates. Acknowledgments We gratefully acknowledge the support of the NSF through grant IIS-1619362 and of the Australian Research Council through an Australian Laureate Fellowship (FL110100281) and through the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS). We thank the anonymous reviewers for their insightful comments and suggestions. 9 References H. Attouch, J. Peypouquet, and P. Redont. Fast convergence of an inertial gradient-like system with vanishing viscosity. CoRR, abs/1507.04782, 2015. A. Banerjee, S. Merugu, I. S. Dhillon, and J. Ghosh. Clustering with Bregman divergences. J. Mach. Learn. Res., 6:1705?1749, Dec. 2005. A. Beck and M. Teboulle. Mirror descent and nonlinear projected subgradient methods for convex optimization. Oper. Res. Lett., 31(3):167?175, May 2003. A. Ben-Tal and A. Nemirovski. Lectures on Modern Convex Optimization. SIAM, 2001. A. Ben-Tal, T. Margalit, and A. Nemirovski. 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Kernel functions based on triplet comparisons Matth?us Kleindessner? Department of Computer Science Rutgers University Piscataway, NJ 08854 mk1572@cs.rutgers.edu Ulrike von Luxburg Department of Computer Science University of T?bingen Max Planck Institute for Intelligent Systems, T?bingen luxburg@informatik.uni-tuebingen.de Abstract Given only information in the form of similarity triplets ?Object A is more similar to object B than to object C? about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a lowdimensional Euclidean embedding of the data set that reflects the given similarity triplets, we aim at defining kernel functions that correspond to high-dimensional embeddings. These kernel functions can subsequently be used to apply any kernel method to the data set. 1 Introduction Assessing similarity between objects is an inherent part of many machine learning problems, be it in an unsupervised task like clustering, in which similar objects should be grouped together, or in classification, where many algorithms are based on the assumption that similar inputs should produce similar outputs. In a typical machine learning setting one assumes to be given a data set D of objects together with a dissimilarity function d (or, equivalently, a similarity function s) quantifying how ?close? objects are to each other. In recent years, however, a new branch of the machine learning literature has emerged that relaxes this scenario (see the next paragraph and Section 3 for references). Instead of being able to evaluate d itself, we only get to see a collection of similarity triplets of the form ?Object A is more similar to object B than to object C?, which claims that d(A, B) < d(A, C). The main motivation for this relaxation comes from human-based computation: It is widely accepted that humans are better and more reliable at providing similarity triplets, which means assessing similarity on a relative scale, than at providing similarity estimates on an absolute scale (?The similarity between objects A and B is 0.8?). This can be seen as a special case of the general observation that humans are better at comparing two stimuli than at identifying a single one (Stewart et al., 2005). For this reason, whenever one is lacking a meaningful dissimilarity function that can be evaluated automatically and has to incorporate human expertise into the machine learning process, collecting similarity triplets (e.g., via crowdsourcing) may be an appropriate means. Given a data set D and similarity triplets for its objects, it is not immediately clear how to solve machine learning problems on D. A general approach is to construct an ordinal embedding of D, that is to map objects to a Euclidean space of a small dimension such that the given triplets are preserved as well as possible (Agarwal et al., 2007; Tamuz et al., 2011; van der Maaten and Weinberger, 2012; Terada and von Luxburg, 2014; Amid and Ukkonen, 2015; Heim et al., 2015; Amid et al., 2016; Jain et al., 2016). Once such an ordinal embedding has been constructed, one can solve a problem on D by solving it on the embedding. Only recently, algorithms have been proposed for solving various specific problems directly without constructing an ordinal embedding as an intermediate step (Heikinheimo and Ukkonen, 2013; Kleindessner and von Luxburg, 2017). With this paper we provide another generic means for solving machine learning problems based on similarity triplets that is different from ? Work done while being a PhD student at the University of T?bingen. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. the ordinal embedding approach. We define two data-dependent kernel functions on D, corresponding to high-dimensional embeddings of D, that can subsequently be used by any kernel method. Our proposed kernel functions measure similarity between two objects in D by comparing to which extent the two objects give rise to resembling similarity triplets. The intuition is that this quantifies the relative difference in the locations of the two objects in D. Experiments on both artificial and real data show that this is indeed the case and that the similarity scores defined by our kernel functions are meaningful. Our approach is appealingly simple, and other than ordinal embedding algorithms our kernel functions are deterministic and parameter-free. We observe them to run significantly faster than well-known embedding algorithms and to be ideally suited for a landmark design. Setup Let X be an arbitrary set and d : X ? X ! R+ 0 be a symmetric dissimilarity function on X : a higher value of d means that two elements of X are more dissimilar to each other. The terms dissimilarity and distance are used synonymously. To simplify presentation, we assume that for all triples of distinct objects A, B, C 2 X either d(A, B) < d(A, C) or d(A, B) > d(A, C) is true. Note that we do not require d to be a metric. We formally define a similarity triplet as binary answer to a dissimilarity comparison ? (1) d(A, B) < d(A, C). We refer to A as the anchor object. A similarity triplet can be incorrect, meaning that it claims a positive answer to the comparison (1) although in fact the negative answer is true. In the following, we deal with a finite data set D = {x1 , . . . , xn } ? X and collections of similarity triplets that are encoded as follows: an ordered triple of distinct objects (xi , xj , xk ) means d(xi , xj ) < d(xi , xk ). A collection of similarity triplets is the only information that we are given about D. Note that such a collection does not necessarily provide an answer to every possible dissimilarity comparison (1). 2 Our kernel functions Assume we are given a collection S of similarity triplets for the objects of D. Similarity triplets in S can be incorrect, but for the moment assume that contradicting triples (xi , xj , xk ) and (xi , xk , xj ) cannot be present in S at the same time. We will discuss how to deal with the general case below. Kernel function k1 Our first kernel function is based on the following idea: We fix two objects xa and xb . In order to compute a similarity score between xa and xb we would like to rank all objects in D with respect to their distance from xa and also rank them with respect to their distance from xb , and take a similarity score between these two rankings as similarity score between xa and xb . One possibility to measure similarity between rankings is given by the famous Kendall tau correlation coefficient (Kendall, 1938), which is also known as Kendall?s ? : for two rankings of n items, Kendall?s ? between the two rankings is the fraction of concordant pairs of items minus the fraction of discordant pairs of items. Here, a pair of two items i1 and i2 is concordant if i1 i2 or i1 i2 according to both rankings, and discordant if it satisfies i1 i2 according to one and i1 i2 according to the other ranking. Formally, a ranking is represented by a permutation : {1, . . . , n} ! {1, . . . , n} such that (i) 6= (j), i 6= j, and (i) = m means that item i is ranked at the m-th position. Given two rankings 1 and 2 , the number of concordant pairs equals X fc ( 1 , 2 ) = [ { 1 (i) < 1 (j)} { 2 (i) < 2 (j)} + { 1 (i) > 1 (j)} { 2 (i) > 2 (j)}], i<j the number of discordant pairs equals X fd ( 1 , 2 ) = [ { 1 (i) < 1 (j)} { 2 (i) > 2 (j)} i<j and Kendall?s ? between 1 and 2 is given by ? ( 1, + { 2) 1 (i) = [fc ( 1, > 2) 1 (j)} fd ( { 1, 2 (i) < n 2 )] / 2 2 (j)}], . By measuring similarity between the two rankings of objects (one with respect to their distance from xa and one with respect to their distance from xb ) with Kendall?s ? we would compute a similarity score between xa and xb . This idea is illustrated with an example in Figure 1 (left). It has been established recently that Kendall?s ? is actually a kernel function on the set of total rankings (Jiao and Vert, 2015). Hence, by measuring similarity on D in the described way we would even end up with a 2 x4 d(x1 , x4 ) x3 x1 d(x1 , x3 ) x4 x7 x5 d(x1 , x3 ) x6 x1 x2 x2 x7 x5 x3 d(x3 , x5 ) x6 d(x2 , x3 ) Figure 1: Illustrations of the ideas behind k1 (left) and k2 (right). For k1 : In order to compute a similarity score between x1 (in red) and x2 (in blue) we would like to rank all objects with respect to their distance from x1 and also with respect to their distance from x2 and compute Kendall?s ? between the two rankings. In this example, the objects would rank as x1 x3 x2 x4 x5 x6 x7 and x2 x3 x6 x1 x5 x4 x7 , respectively. Kendall?s ? between these two rankings is 1/3, and this would be the similarity score between x1 and x2 . For comparison, the score between x1 and x7 (in green) would be 5/7, and between x2 and x7 it would be 3/7. For k2 : In order to compute a similarity score between x1 and x2 we would like to check for every pair ? ? of objects (xi , xj ) whether the distance comparisons d(xi , x1 ) < d(xi , xj ) and d(xi , x2 ) < d(xi , xj ) yield the same result or not. Here, we have 32 pairs for which they yield the same result and 17 pairs for which they do not. We would assign 7 2 ? (32 17) = 15/49 as similarity score between x1 and x2 . The score between x1 and x7 would be 3/49, and between x2 and x7 it would be 1/49. kernel function on D since the following holds: for any mapping h : D ! Z and kernel function k : Z ? Z ! R, k (h, h) : D ? D ! R is a kernel function. In our situation, the problem is that in most cases S will contain only a small fraction of all possible similarity triplets and also that some of the triplets in S might be incorrect, so that there is no way of ranking all objects with respect to their distance from any fixed object based on the similarity triplets in S. To adapt the procedure, we consider a feature map that corresponds to the kernel function just described. By a feature map corresponding to a kernel function k : D ? D ! R we mean a mapping : D ! Rm for some m 2 N such that k(xi , xj ) = h (xi ), (xj )i = (xi )T ? (xj ). It is easy to see from the above formulas (also compare with Jiao and Vert, 2015) that a feature map n corresponding to the described kernel function is given by k? : D ! R( 2 ) with ? ? 1 q (x ) = ? {d(x , x ) < d(x , x )} {d(x , x ) > d(x , x )} . k? a a i a j a i a j n 2 1?i<j?n In our situation, where we are only given S and cannot evaluate k? in most cases, we have to replace k? by an approximation: up to a normalizing factor, we replace an entry in k? (xa ) by zero if we cannot evaluate it based on the triplets in S. More precisely, we consider the feature map ( n) k1 : D ! R 2 given by k1 (xa ) = ([ k1 (xa )]i,j )1?i<j?n with ? ? 1 [ k1 (xa )]i,j = p ? {(xa , xi , xj ) 2 S} {(xa , xj , xi ) 2 S} |{(xi , xj , xk ) 2 S : xi = xa }| (2) and define our first proposed kernel function k1 : D ? D ! R by k1 (xi , xj ) = k1 (xi )T ? k1 (xj ). (3) Note that the scaling factor in the definition of k1 , ensuring that the feature embedding lies on the unit sphere, is crucial whenever the number of similarity triplets in which an object appears as anchor object is not approximately constant over the different objects. For ease of exposition we have assumed that every object in D appears at least once as an anchor object in a similarity triplet in S. In the unlikely case that xa does not appear at least once as an anchor object, meaning that we do not have any information for ranking the objects in D with respect to their distance from xa at all, we simply set k1 (xa ) to zero (which is consistent with (2) under the convention ?0/0=0?). Kernel function k2 Our second kernel function is based on a similar idea. Now we do not consider xa and xb as anchor objects when measuring their similarity, but compare whether they rank similarly with respect to their distances from the various other objects. Concretely, we would like to count the number of pairs of objects (xi , xj ) for which the comparisons ? d(xi , xa ) < d(xi , xj ) and 3 ? d(xi , xb ) < d(xi , xj ) (4) {y : d(y, xn ) < d(y, x2 )} {y : d(y, x5 ) < d(y, x4 )} x1 xn .. . x5 x4 x3 x2 Figure 2: k1 measures similarity between two objects by counting in how many of the halfspaces that are obtained from distance comparisons the two objects reside at the same time. The outcome does not only depend on the distance between the two objects, but also on their location within the data set: although x1 and x2 are located far apart, k1 considers them to be very similar. See the running text for details. yield the same result and subtract the number of pairs for which these comparisons yield different results. See the right-hand side of Figure 1 for an illustration of this idea. Adapted to our situation of 2 being only given S it corresponds to considering the feature map k2 : D ! Rn given by k2 (xa ) 1 =p ? |{(xi , xj , xk ) 2 S : xj = xa _ xk = xa }| ? ? {(xi , xa , xj ) 2 S} {(xi , xj , xa ) 2 S} and defining our second proposed kernel function k2 : D ? D ! R by k2 (xi , xj ) = k2 (xi ) T ? k2 (xj ). 1?i,j?n (5) Again, the scaling factor in the definition of k2 is crucial whenever there are objects appearing in more similarity triplets than others and we apply the convention ?0/0=0?. Contradicting similarity triplets If S contains contradicting triples (xi , xj , xk ) and (xi , xk , xj ) and there might be triples being present repeatedly, one can alter the definition of k1 or k2 as follows: if #{(xa , xi , xj ) 2 S} denotes the number of how often the triple (xa , xi , xj ) appears in S, set k1 (xa ) = e k1 (xa )/ e k1 (xa ) where e k1 (xa ) equals ? ? #{(xa , xi , xj ) 2 S} #{(xa , xj , xi ) 2 S} . #{(xa , xi , xj ) 2 S} + #{(xa , xj , xi ) 2 S} 1?i<j?n The definition of k2 can be revised in an analogous way. In doing so, we incorporate a simple estimate of the likelihood of a triple being correct. 2.1 Reducing diagonal dominance If the number |S| of given similarity triplets is small, our kernel functions suffer from a problem that is shared by many other kernel functions defined on complex data: k1 and k2 map the objects in D to sparse vectors, that is almost all of their entries are zero. As a consequence, two different feature vectors ki (xa ) and ki (xb ) appear to be almost orthogonal and the similarity score ki (xa , xb ) is much smaller than the self-similarity scores ki (xa , xa ) or ki (xb , xb ). This phenomenon, usually referred to as diagonal dominance of the kernel function, has been observed to pose difficulties for the kernel methods using the kernel function, and several ways have been proposed for dealing with it (Sch?lkopf et al., 2002; Greene and Cunningham, 2006). In all our experiments we deal with diagonal dominance in the following simple way: Let k denote a kernel function and K the kernel matrix on D, that is K = (k(xi , xj ))ni,j=1 , which would be the input to a kernel method. Then we replace K n?n by K denotes the identity matrix and min is the smallest eigenvalue of K. min I where I 2 R 2.2 Geometric intuition Intuitively, our kernel functions measure similarity between xa and xb by quantifying to which extent xa and xb can be expected to be located in the same region of D: Think of D as a subset of Rm and d being the Euclidean metric. A similarity triplet d(xa , xi ) < d(xa , xj ) then tells us that xa resides in the halfspace defined by the hyperplane that is perpendicular to the line segment connecting xi and xj and goes through the segment?s midpoint. If there is also a similarity triplet d(xb , xi ) < d(xb , xj ), xa and xb thus are located in the same halfspace (assuming the correctness of the similarity triplets) and this is reflected by a higher value of k1 (xa , xb ). Similarly, a similarity triplet d(xi , xa ) < d(xi , xj ) 4 400 points Distance matrix K1 Similarity scores K2 8 8 0.2 6 6 0.15 4 4 0.1 2 2 0.05 0 0 0 -2 -2 -6 -4 -2 0 2 4 -0.05 -5 6 0 5 10 0.2 6 4 0.15 5 2 0.1 0 0 -2 0.05 -4 0 -5 -6 -0.05 -8 -5 0 5 10 -10 -5 15 0 5 10 15 Figure 3: Kernel matrices for two data sets, each consisting of 400 points, based on 10% of all similarity triplets. 1st plot of a row: Data points. 2nd plot: Distance matrix. 3rd / 4th plot: Kernel matrix for k1 / k2 . 6th plot: Similarity scores between a fixed point and the other points (for k1 ). tells us that xa is located in a ball with radius d(xi , xj ) centered at xi , and the value of k2 (xa , xb ) is higher if there is a similarity triplet d(xi , xb ) < d(xi , xj ) telling us that xb is located in this ball too and it is smaller if there is a triplet d(xi , xj ) < d(xi , xb ) telling us that xb is not located in this ball. Note that the similarity scores between xa and xb defined by k1 or k2 do not only depend on d(xa , xb ), but rather on the locations of xa and xb within D and on how the points in D are spread in the space since this affects how the various hyperplanes or balls are related to each other. Consider the example illustrated in Figure 2: Let d(x3 , xn ) = 1 implying that d(xi , xi+1 ) = ?(1/n), 3 ? i < n, and d(x1 , x2 ) > d(x2 , xn ) > d(x1 , xn ) > d(x2 , x3 ) > d(x1 , x3 ) > 1 be arbitrarily large. Although x1 and x2 are located at the maximum distance to each other, they satisfy d(x1 , xi ) < d(x1 , xj ) and d(x2 , xi ) < d(x2 , xj ) for all 3 ? i < j ? n, and hence both x1 and x2 are jointly located in all the halfspaces obtained from these distance comparisons. We end up with k1 (x1 , x2 ) ! 1, n ! 1, assuming k1 is computed based on all possible similarity triplets, all of which are correct. The distance between x3 and xn is much smaller, but there are many points in between them and the hyperplanes obtained from the distance comparisons with these points separate x3 and xn . We end up with k1 (x3 , xn ) ! 1, n ! 1. Depending on the task at hand, this may be desirable or not. Let us examine the meaningfulness of our kernel functions by calculating them on five visualizable data sets. Each of the first four data sets consists of 400 points in R2 and d equals the Euclidean metric. The fifth data set consists of 400 vertices of an undirected graph from a stochastic block model and d equals the shortest path distance. We computed k1 and k2 based on 10% of all possible similarity triplets (chosen uniformly at random from all triplets). The results for the first two data sets are shown in Figure 3. The results for the remaining data sets are shown in Figure 6 in Section A.1 in the supplementary material. The first plot of a row shows the data set. The second plot shows the distance matrix on the data set. Next, we can see the kernel matrices. The last plot of a row shows the similarity scores (encoded by color) based on k1 between one fixed point (shown as a black cross) and the other points in the data set. Clearly, the kernel matrices reflect the block structures of the distance matrices, and the similarity scores between a fixed point and the other points tend to decrease as the distances to the fixed point increase. A situation like in the example of Figure 2 does not occur. 2.3 Landmark design Our kernel functions are designed as to extract information from an arbitrary collection S of similarity triplets. However, by construction, a single triplet is useless, and what matters is the concurrent pres? ence of two triplets: k1 (xa , xb ) is only affected by pairs of triplets answering d(xa , xi ) < d(xa , xj ) ? and d(xb , xi ) < d(xb , xj ), while k2 (xa , xb ) is only affected by pairs of triplets answering (4). Hence, when we can choose which dissimilarity comparisons of the form (1) are evaluated for creating S (e.g., in crowdsourcing), we should aim at maximizing the number of appropriate pairs of triplets. This can easily be achieved by means of a landmark design inspired from landmark multidimensional scaling (de Silva and Tenenbaum, 2004): We choose a small subset of landmark objects L ? D. Then, ? for k1 , only comparisons of the form d(xi , xj ) < d(xi , xk ) with xi 2 D and xj , xk 2 L are evalu5 ? ated. For k2 , only comparisons of the form d(xj , xi ) < d(xj , xk ) with xi 2 D and xj , xk 2 L are evaluated. The landmark objects can be chosen either randomly or, if available, based on additional knowledge about D and the task at hand. 2.4 Computational complexity General S A naive implementation of our kernel functions explicitly computes the feature vectors k1 (xi ) or k2 (xi ), i = 1, . . . , n, and subsequently calculates the kernel matrix K by means of (3) or (5). In doing so, we store the feature vectors in the feature matrix k1 (D) = ( k1 (xi ))ni=1 2 n 2 R( 2 )?n or k2 (D) = ( k2 (xi ))ni=1 2 Rn ?n . Proceeding this way is straightforward and simple, requiring to go through S only once, but comes with a computational cost of O(|S| + n4 ) operations. Note that the number of different distance comparisons of the form (1) is O(n3 ) and hence one might expect that |S| 2 O(n3 ) and O(|S| + n4 ) = O(n4 ). By performing (3) or (5) in terms of matrix multiplication k1 (D)T ? k1 (D) or k2 (D)T ? k2 (D) and applying Strassen?s algorithm (Higham, 1990) one can reduce the number of operations to O(|S| + n3.81 ), but still this is infeasible for many data sets. Infeasibility for large data sets, however, is even more the case for ordinal embedding algorithms, which are the current state-of-the-art method for solving machine learning problems based on similarity triplets. All existing ordinal embedding algorithms iteratively solve an optimization problem. For none of these algorithms theoretical bounds for their complexity are available in the literature, but it is widely known that their running times are prohibitively high (Heim et al., 2015; Kleindessner and von Luxburg, 2017). Landmark design If we know that S contains only dissimilarity comparisons involving landmark |L| 2 objects, we can adapt the feature matrices such that k1 (D) 2 R( 2 )?n or k2 (D) 2 R|L| ?n and reduce the number of operations to O(|S| + min{|L|2 , n}log2 (7/8) |L|2 n2 ), which is O(|S| + |L|1.62 n2 ) if |L|2 ? n. Note that in this case we might expect that |S| 2 O(|L|2 n). In both cases, whenever the number of given similarity triplets |S| is small compared to the number of all different distance comparisons under consideration, the feature matrix k1 (D) or k2 (D) is sparse with only O(|S|) non-zero entries and methods for sparse matrix multiplication decrease computational complexity (Gustavson, 1978; Kaplan et al., 2006). 3 Related work ? Similarity triplets are a special case of answers to the general dissimilarity comparisons d(A, B) < d(C, D), A, B, C, D 2 X . We refer to any collection of answers to these general comparisons as ordinal data. In recent years, ordinal data has become popular in machine learning. Among the work on ordinal data in general (see Kleindessner and von Luxburg, 2014, 2017, for references), similarity triplets have been paid particular attention: Jamieson and Nowak (2011) deal with the question of how many similarity triplets are required for uniquely determining an ordinal embedding of Euclidean data. This work has been carried on and generalized by Jain et al. (2016). Algorithms for constructing an ordinal embedding based on similarity triplets (but not on general ordinal data) are proposed in Tamuz et al. (2011), van der Maaten and Weinberger (2012), Amid et al. (2016), and Jain et al. (2016). Heikinheimo and Ukkonen (2013) present a method for medoid estimation based on statements ?Object A is the outlier within the triple of objects (A, B, C)?, which correspond to the two similarity triplets d(B, C) < d(B, A) and d(C, B) < d(C, A). Ukkonen et al. (2015) use the same kind of statements for density estimation and Ukkonen (2017) uses them for clustering. Wilber et al. (2014) examine how to minimize time and costs when collecting similarity triplets via crowdsourcing. Producing a number of ordinal embeddings at the same time, each corresponding to a different dissimilarity function based on which a comparison (1) might have been evaluated, is studied in Amid and Ukkonen (2015). In Heim et al. (2015), one of the algorithms by van der Maaten and Weinberger (2012) is adapted from the batch setting to an online setting, in which similarity triplets are observed in a sequential way, using stochastic gradient descent. In Kleindessner and von Luxburg (2017), we propose algorithms for medoid estimation, outlier detection, classification, and clustering based on statements ?Object A is the most central object within (A, B, C)?, which comprise the two similarity triplets d(B, A) < d(B, C) and d(C, A) < d(C, B). Finally, Haghiri et al. (2017) study the problem of efficient nearest neighbor search based on similarity triplets. There 6 Figure 4: Best viewed magnified on screen. Left: Clustering of the food data set. Part of the dendrogram obtained from complete-linkage clustering using k1 . Right: Kernel PCA on the car data set based on the kernel function k2 . is also a number of papers that consider similarity triplets as side information to vector data (e.g., Schultz and Joachims, 2003; McFee and Lanckriet, 2011; Wilber et al., 2015). 4 Experiments We performed experiments that demonstrate the usefulness of our kernel functions. We first apply them to three small image data sets for which similarity triplets have been gathered via crowdsourcing. We then study them more systematically and compare them to an ordinal embedding approach in clustering tasks on subsets of USPS and MNIST digits using synthetically generated triplets. 4.1 Crowdsourced similarity triplets In this section we present experiments on real crowdsourcing data that show that our kernel functions can capture the structure of a data set. Note that for the following data sets there is no ground truth available and hence there is no way other than visual inspection for evaluating our results. Food data set We applied the kernelized version of complete-linkage clustering based on our kernel function k1 to the food data set introduced in Wilber et al. (2014). This data set consists of 100 images2 of a wide range of foods and comes with 190376 (unique) similarity triplets, which contain 9349 pairs of contradicting triplets. Figure 4 (left) shows a part of the dendrogram that we obtained. Each of the ten clusters depicted there contains pretty homogeneous images. For example, the fourth row only shows vegetables and salads whereas the ninth row only shows fruits and the last row only shows desserts. To give an impression of accelerated running time of our approach compared to an ordinal embedding approach: computation of k1 or k2 on this data set took about 0.1 seconds while computing an ordinal embedding using the GNMDS algorithm (Agarwal et al., 2007) took 18 seconds (embedding dimension equaling two; all computations performed in Matlab?see Section 4.2 for details; the embedding is shown in Figure 9 in Section A.1 in the supplementary material). Car data set We applied kernel PCA (Sch?lkopf et al., 1999) based on our kernel function k2 to the car data set, which we have introduced in Kleindessner and von Luxburg (2017). It consists of 60 images of cars. For this data set we have collected statements of the kind ?Object A is the most central object within (A, B, C)?, meaning that d(B, A) < d(B, C) and d(C, A) < d(C, B), via crowdsourcing. We ended up with 13514 similarity triplets, of which 12502 were unique. The projection of the car data set onto the first two kernel principal components can be seen in Figure 4 (right). The result looks reasonable, with the cars arranged in groups of sports cars (top left), ordinary cars (middle right) and off-road/sport utility vehicles (bottom left). Also within these groups there is some reasonable structure. For example, the race-like sports cars are located near to each other and close to the Formula One car, and the sport utility vehicles from German manufacturers are placed next to each other. Nature data set We performed similar experiments on the nature data set introduced in Heikinheimo and Ukkonen (2013). The results are presented in Section A.2 in the supplementary material. 2 According to Wilber et al., the data set contains copyrighted material under the educational fair use exemption to the U.S. copyright law. 7 We would like to discuss a question raised by one of the reviewers: in our setup (see Section 1), we assume that similarity triplets are noisy evaluations of dissimilarity comparisons (1), where d is some fixed dissimilarity function. This leads to our (natural) way of dealing with contradicting similarity triplets as described in Section 2. In a different setup one could drop the dissimilarity function d and consider similarity triplets as elements of some binary relation on D ? D that is not necessarily transitive or antisymmetric. In the latter setup it is not clear whether our way of dealing with contradicting triplets is the right thing to do. However, we believe that the experiments of this section show that our setup is valid in a wide range of scenarios and our approach works in practice. 4.2 Synthetically generated triplets We studied our kernel functions with respect to the number of input similarity triplets that they require in order to produce a valuable solution in clustering tasks. We found that in the scenario of a general collection S of triplets our approach is highly superior compared to an ordinal embedding approach in terms of running time, but on most data sets it is inferior regarding the required number of triplets. The full benefit of our kernel functions emerges in a landmark design. There our approach can compete with an embedding approach in terms of the required number of triplets and is so much faster as to being easily applicable to large data sets to which ordinal embedding algorithms are not. In this section we want to demonstrate this claim. We studied k1 and k2 in a landmark design by applying kernel k-means clustering (Dhillon et al., 2001) to subsets of USPS and MNIST digits, respectively. Collections S of similarity triplets were generated as follows: We chose a certain number of landmark objects uniformly at random from all objects of the data set under consideration. Choosing d as the Euclidean metric, we created answers to all possible distance comparisons with the landmark objects as explained in Section 2.3. Answers were incorrect with some probability 0 ? ep ? 1 independently of each other. From the set of all answers we chose triplets in S uniformly at random without replacement. We compared our approach to an ordinal embedding approach with ordinary k-means clustering. We tried the GNMDS (Agarwal et al., 2007), the CKL (Tamuz et al., 2011), and the t-STE (van der Maaten and Weinberger, 2012) embedding algorithms in the Matlab implementation made available by van der Maaten and Weinberger (2012). In doing so, we set all parameters except the embedding dimension to the provided default parameters. The parameter ? of the CKL algorithm was set to 0.1 since we observed good results with this value. Note that in these unsupervised clustering tasks there is no immediate way of performing cross-validation for choosing parameters. We compared to the embedding algorithms in two scenarios: in one case they were provided the same triplets as input as our kernel functions, in the other case (denoted by the additional ?rand? in the plots) they were provided a same number of triplets chosen uniformly at random with replacement from all possible triplets (no landmark design) and incorrect with the same probability ep. For further comparison, we considered ordinary k-means applied to the original point set and a random clustering. We always provided the correct number of clusters as input, and set the number of replicates in k-means and kernel k-means to five and the maximum number of iterations to 100. For assessing the quality of a clustering we computed its purity (e.g., Manning et al., 2008), which measures the accordance with the known ground truth partitioning according to the digits? values. A high purity value indicates a good clustering. Note that the limitation for the scale of our experiments only comes from the running time of the embedding algorithms and not from our kernel functions. Still, in terms of the number of data points our experiments are comparable or actually even superior to all the papers on ordinal embedding cited in Section 3. In terms of the number of similarity triplets per data point, we used comparable numbers of triplets. USPS digits We chose 1000 points uniformly at random from the subset of USPS digits 1, 2, and 3. Using 15 landmark objects, we studied the performance of our approach and the ordinal embedding approach as a function of the number of input triplets. The first and the second row of Figure 5 show the results (average over 10 runs of an experiment) for k1 . The results for k2 are shown in Figure 7 in Section A.1 in the supplementary material. The first two plots of a row show the purity values of the various clusterings for ep = 0 and ep = 0.3, respectively. The third and the fourth plot show the corresponding time (in sec) that it took to compute our kernel function or an ordinal embedding. We set the embedding dimension to 2 (1st row) or 10 (2nd row). Based on the achieved purity values no method can be considered superior. Our kernel function k2 performs slightly worse than k1 and the ordinal embedding algorithms. The GNMDS algorithm apparently cannot deal with the landmark triplets at all and yields the same purity values as a random clustering when provided with the landmark triplets. Our approach is highly superior regarding running time. The running times of the 8 0.6 0.6 0 1 2 # input triplets 3 0.4 4 0 2 104 k 1 on USPS (ep=0, embedding dim=10) 4 6 # input triplets 104 0.6 0 1 2 3 4 0.4 k 1 on MNIST (ep=0.15, embedding dim=5) 2 Purity 0.2 6 104 2000 4000 6000 8000 # points 10000 2000 3 4000 6000 8000 10000 # points 30 20 10 0 4 0 2 104 4 6 # input triplets 104 k 1 on USPS (ep=0.3, embedding dim=10) 150 60 40 20 0 1 2 3 2000 1000 0 2000 4000 6000 # points 50 0 4 k 1 on MNIST (ep=0.15, embedding dim=5) 0 100 0 2 104 8000 10000 4 6 # input triplets 104 k 2 on MNIST (ep=0.15, embedding dim=5) 3000 0.4 0 2 # input triplets 0.2 0 1 80 0 k2 0.6 0.4 4 # input triplets k 2 on MNIST (ep=0.15, embedding dim=5) 0.6 Purity 0 104 # input triplets Running time [s] 0.4 0 100 Running time [s] Purity Purity 0.8 0.6 10 k 1 on USPS (ep=0, embedding dim=10) 1 0.8 20 # input triplets k 1 on USPS (ep=0.3, embedding dim=10) 1 30 0 40 Running time [s] Purity Purity 0.8 Running time [s] GNMDS t-STE CKL GNMDS rand t-STE rand CKL rand Coordinates Random k 1 on USPS (ep=0.3, embedding dim=2) 40 k1 Running time [s] 1 0.8 0.4 k 1 on USPS (ep=0, embedding dim=2) k 1 on USPS (ep=0.3, embedding dim=2) 4000 Running time [s] k 1 on USPS (ep=0, embedding dim=2) 1 3000 2000 1000 0 0 2000 4000 6000 8000 10000 # points Figure 5: 1st & 2nd row (USPS digits for k1 ): Clustering 1000 points from USPS digits 1, 2, and 3. Purity and running time as a function of the number of input triplets. 3rd row (MNIST digits): Clustering subsets of MNIST digits. Purity and running time as a function of the number of points. ordinal embedding algorithms depend on the embedding dimension and ep and in these experiments the dependence is monotonic. All computations were performed in Matlab R2016a on a MacBook Pro with 2.9 GHz Intel Core i7 and 8 GB 1600 MHz DDR3. In order to make a fair comparison we did not use MEX files or sparse matrix operations in the implementation of our kernel functions. MNIST digits We studied the performance of the various methods as a function of the size n of the data set with the number of input triplets growing linearly with n. For i = 1, . . . , 10, we chose n = i ? 103 points uniformly at random from MNIST digits. We used 30 landmark objects and provided 150n input similarity triplets. The third row of Figure 5 shows the purity values of the various methods for k1 / k2 (1st / 2nd plot) and the corresponding running times (3rd / 4th plot) when ep = 0.15. The embedding dimension was set to 5. A spot check suggested that setting it to 2 would have given worse results, while setting it to 10 would have given similar results, but would have led to a higher running time. We computed the t-STE embedding only for n ? 6000 due to its high running time. It seems that GNMDS with random input triplets performs best, but for large values of n our kernel function k1 can compete with it. For 10000 points, computing k1 or k2 took 100 or 180 seconds, while even the fastest embedding algorithm ran for 2000 seconds. For further comparison, Figure 8 in Section A.1 in the supplementary material shows a kernel PCA embedding based on k1 (150n landmark triplets) and a 2-dim GNMDS embedding (150n random triplets) of n = 20000 digits. Here, computation of k1 took 900 seconds, while GNMDS ran for more than 6000 seconds. 5 Conclusion We proposed two data-dependent kernel functions that can be evaluated when given only an arbitrary collection of similarity triplets for a data set D. Our kernel functions can be used to apply any kernel method to D. Hence they provide a generic alternative to the standard ordinal embedding approach based on numerical optimization for machine learning with similarity triplets. In a number of experiments we demonstrated the meaningfulness of our kernel functions. A big advantage of our kernel functions compared to the ordinal embedding approach is that our kernel functions run significantly faster. A drawback is that, in general, they seem to require a higher number of similarity triplets for capturing the structure of a data set. However, in a landmark design our kernel functions can compete with the ordinal embedding approach in terms of the required number of triplets. 9 Acknowledgements This work has been supported by the Institutional Strategy of the University of T?bingen (DFG, ZUK 63). References S. Agarwal, J. Wills, L. Cayton, G. Lanckriet, D. Kriegman, and S. Belongie. Generalized non-metric multidimensional scaling. In International Conference on Artificial Intelligence and Statistics (AISTATS), 2007. E. Amid and A. Ukkonen. Multiview triplet embedding: Learning attributes in multiple maps. In International Conference on Machine Learning (ICML), 2015. E. Amid, N. Vlassis, and M. Warmuth. t-exponential triplet embedding. arXiv:1611.09957v1 [cs.AI], 2016. V. de Silva and J. Tenenbaum. Sparse multidimensional scaling using landmark points. Technical report, Stanford University, 2004. I. Dhillon, Y. Guan, and B. Kulis. Kernel k-means, spectral clustering and normalized cuts. In International Conference on Knowledge Discovery and Data Mining (KDD), 2001. D. Greene and P. Cunningham. Practical solutions to the problem of diagonal dominance in kernel document clustering. In International Conference on Machine Learning (ICML), 2006. F. G. Gustavson. Two fast algorithms for sparse matrices: Multiplication and permuted transposition. ACM Transactions on Mathematical Software, 4(3):250?269, 1978. S. Haghiri, U. von Luxburg, and D. Ghoshdastidar. Comparison based nearest neighbor search. In International Conference on Artificial Intelligence and Statistics (AISTATS), 2017. H. Heikinheimo and A. Ukkonen. The crowd-median algorithm. In Conference on Human Computation and Crowdsourcing (HCOMP), 2013. Data available on http://www.anttiukkonen.com/. E. Heim, M. Berger, L. M. Seversky, and M. Hauskrecht. Efficient online relative comparison kernel learning. In SIAM International Conference on Data Mining (SDM), 2015. N. Higham. Exploiting fast matrix multiplication within the level 3 BLAS. ACM Transactions on Mathematical Software, 16(4):352?368, 1990. L. Jain, K. Jamieson, and R. Nowak. Finite sample prediction and recovery bounds for ordinal embedding. In Neural Information Processing Systems (NIPS), 2016. K. Jamieson and R. Nowak. Low-dimensional embedding using adaptively selected ordinal data. In Allerton Conference on Communication, Control, and Computing, 2011. Y. Jiao and J.-P. Vert. The Kendall and Mallows kernels for permutations. In International Conference on Machine Learning (ICML), 2015. H. Kaplan, M. Sharir, and E. Verbin. Colored intersection searching via sparse rectangular matrix multiplication. In Symposium on Computational Geometry (SoCG), 2006. M. Kendall. A new measure of rank correlation. Biometrika, 30(1?2):81?93, 1938. M. Kleindessner and U. von Luxburg. Uniqueness of ordinal embedding. In Conference on Learning Theory (COLT), 2014. M. Kleindessner and U. von Luxburg. Lens depth function and k-relative neighborhood graph: Versatile tools for ordinal data analysis. JMLR, 18(58):1?52, 2017. Data available on http://www.tml.cs.uni-tuebingen.de/team/luxburg/code_and_data/. C. D. Manning, P. Raghavan, and H. Sch?tze. Introduction to Information Retrieval. Cambridge University Press, 2008. 10 B. McFee and G. Lanckriet. Learning multi-modal similarity. JMLR, 12:491?523, 2011. B. Sch?lkopf, A. Smola, and K.-R. M?ller. Kernel principal component analysis. In B. Sch?lkopf, C. Burges, and A. Smola, editors, Advances in Kernel Methods: Support Vector Learning, pages 327?352. MIT Press, 1999. B. Sch?lkopf, J. Weston, E. Eskin, C. Leslie, and W. Noble. A kernel approach for learning from almost orthogonal patterns. In European Conference on Machine Learning (ECML), 2002. M. Schultz and T. Joachims. Learning a distance metric from relative comparisons. In Neural Information Processing Systems (NIPS), 2003. N. Stewart, G. D. A. Brown, and N. Chater. Absolute identification by relative judgment. Psychological Review, 112(4):881?911, 2005. O. Tamuz, C. Liu, S. Belongie, O. Shamir, and A. Kalai. Adaptively learning the crowd kernel. In International Conference on Machine Learning (ICML), 2011. Y. Terada and U. von Luxburg. Local ordinal embedding. In International Conference on Machine Learning (ICML), 2014. A. Ukkonen. Crowdsourced correlation clustering with relative distance comparisons. In International Conference on Data Mining series (ICDM), 2017. A. Ukkonen, B. Derakhshan, and H. Heikinheimo. Crowdsourced nonparametric density estimation using relative distances. In Conference on Human Computation and Crowdsourcing (HCOMP), 2015. L. van der Maaten and K. Weinberger. Stochastic triplet embedding. In IEEE International Workshop on Machine Learning for Signal Processing (MLSP), 2012. Code available on http://homepage.tudelft.nl/19j49/ste/. M. Wilber, I. Kwak, and S. Belongie. Cost-effective hits for relative similarity comparisons. In Conference on Human Computation and Crowdsourcing (HCOMP), 2014. Data available on http://vision.cornell.edu/se3/projects/cost-effective-hits/. M. Wilber, I. Kwak, D. Kriegman, and S. Belongie. Learning concept embeddings with combined human-machine expertise. In International Conference on Computer Vision (ICCV), 2015. 11 

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An Error Detection and Correction Framework for Connectomics Jonathan Zung Princeton University jzung@princeton.edu Ignacio Tartavull Princeton University tartavull@princeton.edu Kisuk Lee Princeton University and MIT kisuklee@mit.edu H. Sebastian Seung Princeton University sseung@princeton.edu Abstract We define and study error detection and correction tasks that are useful for 3D reconstruction of neurons from electron microscopic imagery, and for image segmentation more generally. Both tasks take as input the raw image and a binary mask representing a candidate object. For the error detection task, the desired output is a map of split and merge errors in the object. For the error correction task, the desired output is the true object. We call this object mask pruning, because the candidate object mask is assumed to be a superset of the true object. We train multiscale 3D convolutional networks to perform both tasks. We find that the error-detecting net can achieve high accuracy. The accuracy of the error-correcting net is enhanced if its input object mask is ?advice? (union of erroneous objects) from the error-detecting net. 1 Introduction While neuronal circuits can be reconstructed from volumetric electron microscopic imagery, the process has historically [30] and even recently [28] been highly laborious. One of the most timeconsuming reconstruction tasks is the tracing of the brain?s ?wires,? or neuronal branches. This task is an example of instance segmentation, and can be automated through computer detection of the boundaries between neurons. Convolutional nets were first applied to neuronal boundary detection a decade ago [10, 29]. Since then convolutional nets have become the standard approach, and the accuracy of boundary detection has become impressively high [31, 1, 15, 6]. Given the low error rates, it becomes helpful to think of subsequent processing steps in terms of modules that detect and correct errors. In the error detection task (Figure 1a), the input is the raw image and a binary mask that represents a candidate object. The desired output is a map containing the locations of split and merge errors in the candidate object. Related work on this problem has been restricted to detection of merge errors only by either hand-designed [18] or learned [24] computations. However, a typical segmentation contains both split and merge errors, so it would be desirable to include both in the error detection task. In the error correction task (Figure 1b), the input is again the raw image and a binary mask that represents a candidate object. The candidate object mask is assumed to be a superset of a true object, which is the desired output. With this assumption, error correction is formulated as object mask pruning. Object mask pruning can be regarded as the splitting of undersegmented objects to create true objects. In this sense, it is the opposite of agglomeration, which merges oversegmented objects to create true objects [11, 21]. Object mask pruning can also be viewed as the subtraction of voxels 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. (a) Error detection task for split (top) and merge (bottom) errors. The desired output is an error map (red). A voxel in the error map is red if and only if a window centered on it contains a split or merge error. We also consider a variant of the task in which the object mask is the sole input; the grayscale image is not used. (b) The object mask pruning task. The input mask is assumed to be a superset of a true object. The desired output (right) is the true object containing the central voxel (black dot). In the first case there is nothing to prune, while in the second case the object not overlapping the central voxel is erased. Figure 1: Error detection and correction tasks. For both tasks, the inputs are a candidate object mask (blue) and the original image (grayscale). Note that diagrams are 2D for illustrative purposes, but in reality the inputs and outputs are 3D. from an object to create a true object. In this sense, it is the opposite of a flood-filling net [13, 12] or MaskExtend [18], each iteration of which is the addition of voxels to an object to create a true object. Iterative mask extension has been studied in other work on instance segmentation in computer vision [25, 23]. The task of generating an object mask de novo from an image has also been studied in computer vision [22]. We implement both error detection and error correction using 3D multiscale convolutional networks. One can imagine multiple uses for these nets in a connectomics pipeline. For example, the errordetecting net could be used to reduce the amount of labor required for proofreading by directing human attention to locations in the image where errors are likely. This labor reduction could be substantial because the declining error rate of automated segmentation has made it more time-consuming for a human to find an error. We show that the error-detecting net can provide ?advice? to the error-correcting net in the following way. To create the candidate object mask for the error-correcting net from a baseline segmentation, one can simply take the union of all erroneous segments as found by the error-detecting net. Since the error rate in the baseline segmentation is already low, this union is small and it is easy to select out a single object. The idea of using the error detector to choose locations for the error corrector was proposed previously though not actually implemented [18]. Furthermore, the idea of using the error detector to not only choose locations but provide ?advice? is novel as far as we know. We contend that our approach decomposes the neuron segmentation problem into two strictly easier pieces. First, we hypothesize that recognizing an error is much easier than producing the correct answer. Indeed, humans are often able to detect errors using only morphological cues such as abrupt terminations of axons, but may have difficulty actually finding the correct extension. On the other hand, if the error-detection network has high accuracy and the initial set of errors is sparse, then the error correction module only needs to prune away a small number of irrelevant parts from the candidate mask described above. This contrasts with the flood-filling task which involves an unconstrained search for new parts to add. Given that most voxels are not a part of the object to be reconstructed, an upper bound on the object is usually more informative than a lower bound. As an added benefit, selective application of the error correction module near likely errors makes efficient use of our computational budget [18]. 2 In this paper, we support the intuition above by demonstrating high accuracy detection of both split and merge errors. We also demonstrate a complete implementation of the stated error detection-correction framework, and report significant improvements upon our baseline segmentation. Some of the design choices we made in our neural networks may be of interest to other researchers. Our error-correcting net is trained to produce a vector field via metric learning instead of directly producing an object mask. The vector field resembles a semantic labeling of the image, so this approach blurs the distinction between instance and semantic segmentation. This idea is relatively new in computer vision [7, 4, 3]. Our multiscale convolutional net architecture, while similar in spirit to the popular U-Net [26], has some novelty. With proper weight sharing, our model can be viewed as a feedback recurrent convolutional net unrolled in time (see the appendix for details). Although our model architecture is closely related to the independent works of [27, 9, 5], we contribute a feedback recurrent convolutional net interpretation. 2 2.1 Error detection Task specification: detecting split and merge errors Given a single segment in a proposed segmentation presented as an object mask Obj, the error detection task is to produce a binary image called the error map, denoted Errpx ?py ?pz (Obj). The definition of the error map depends on a choice of a window size px ? py ? pz . A voxel i in the error map is 0 if and only if the restriction of the input mask to a window centred at i of size px ? py ? pz is voxel-wise equal to the restriction of some object in the ground truth. Observe that the error map is sensitive to both split and merge errors. A smaller window size allows us to localize errors more precisely. On the other hand, if the window radius is less than the width of a typical boundary between objects, it is possible that two objects participating in a merge error never appear in the same window. These merge errors would not be classified as an error in any window. We could use a less stringent measure than voxel-wise equality that disregards small perturbations of the boundaries of objects. However, our proposed segmentations are all composed of the same building blocks (supervoxels) as the ground truth segmentation, so this is not an issue for us. P We define the combined error map as Obj Err(Obj) ? Obj where ? represents pointwise multiplication. In other words, we restrict the error map for each object to the object itself, and then sum the results. The figures in this paper show the combined error map. 2.2 Architecture of the error-detecting net We take a fully supervised approach to error detection. We implement error detection using a multiscale 3D convolutional network. The architecture is detailed in Figure 2. Its design is informed by experience with convolutional networks for neuronal boundary detection (see [15]) and reflects recent trends in neural network design [26, 8]. Its field of view is Px ? Py ? Pz = 318 ? 318 ? 33 (which is roughly cubic in physical size given the anisotropic resolution of our dataset). The network computes (a downsampling of) Err46?46?7 . At test time, we perform inference in overlapping windows and conservatively blend the output from overlapping windows using a maximum operation. We trained two variants, one of which takes as input only Obj, and another which additionally receives as input the raw image. 3 3.1 Error correction Task specification: object mask pruning Given an image patch of size Px ? Py ? Pz and a candidate object mask of the same dimensions, the object mask pruning task is to erase all voxels which do not belong to the true object overlapping the central voxel. The candidate object mask is assumed to be a superset of the true object. 3 64 64 64 4x 2x 48 48 48 48 4x 2x 4x4 2x 2 32 4x 2x 4x4 2x 2 2 32 28 28 24 2x 2x 4x 4x 4 Skip connection 2x Strided transposed convolution 2 4x 4 Strided convolution 24 32 32 28 28 24 28 Output 18x18x7 24 4x 2x 4x1 2x 1 18 2x 4x 4x 2x 1 1 24 32 18 18 2 Summation joining Input (2 channels) 318x318x33 Figure 2: Architectures for the error-detecting and error-correcting nets respectively. Each node represents a layer and the number inside represents the number of feature maps. The layers closer to the top of the diagram have lower resolution than the layers near the bottom. We make savings in computation by minimizing the number of high resolution feature maps. The diagonal arrows represent strided convolutions, while the horizontal arrows represent skip connections. Associated with the diagonal arrows, black numbers indicate filter size and red numbers indicate strides in x ? y ? z. Due to the anisotropy of the resolution of the images in our dataset, we design our nets so that the first convolutions are exclusively 2D while later convolutions are 3D. The field of view of a unit in the higher layers is therefore roughly cubic. To limit the number of parameters in our model, we factorize all 3D convolutions into a 2D convolution followed by a 1D convolution in z-dimension. We also use weight sharing between some convolutions at the same height. Note that the error-correcting net is a prolonged, symmetric version of the error-detecting net. For more detail of the error corrector, see the appendix. 3.2 Architecture of the error-correcting net Yet again, we implement error correction using a multiscale 3D convolutional network. The architecture is detailed in Figure 2. One difficulty with training a neural network to reconstruct the object containing the central voxel is that the desired output can change drastically as the central voxel moves between objects. We use an intermediate representation whose role is to soften this dependence on the location of the central voxel. The desired intermediate representation is a k = 6 dimensional vector v(x, y, z) at each point (x, y, z) such that points within the same object have similar vectors and points in different objects have different vectors. We transform this vector field into a binary image M representing the object overlapping the central voxel as follows:
M (x, y, z) = exp ?||v(x, y, z) ? v(0, 0, 0)||2 , where (0, 0, 0) is the central voxel. When an over-segmentation is available, we replace v(0, 0, 0) with the average of v over the supervoxel containing the central voxel. This trick makes it unnecessary to centre our windows far away from a boundary, as was necessary in [13]. Note that we backpropagate 4 Figure 3: An example of a mistake in the initial segmentation. The dendrite is missing a spine. The red overlay on the left shows the combined error map (defined in Section 2.1); the stump in the centre of the image was clearly marked as an error. Figure 4: The right shows all objects which contained a detected error in the vicinity. For clarity, each supervoxel was drawn with a different colour. The union of these objects is the binary mask which is provided as input to the error correction network. For clarity, these objects were clipped to lie within the white box representing the field of view of our error correction network. The output of the error correction network is overlaid in blue on the left. Figure 5: The supervoxels assembled in accordance with the output of the error correction network. through the transform M , so the vector representation may be seen as an implementation detail and the final output of the network is just a (soft) binary image. 4 How the error detector can ?advise? the error corrector Suppose that we would like to correct the errors in a baseline segmentation. Obviously, the errordetecting net can be used to find locations where the error-correcting net can be applied [18]. Less obviously, the error-detecting net can be used to construct the object mask that is the input to the error-correcting net. We refer to this object mask as the ?advice mask? and its construction is 5 important because the baseline object to be corrected might contain split as well as merge errors, while the object mask pruning task can correct only merge errors. The advice mask is defined is the union of the baseline object at the central pixel with all other baseline objects in the window that contain errors as judged by the error-detecting net. The advice mask is a superset of the true object overlapping the central voxel, assuming that the error-detecting net makes no mistakes. Therefore advice is suitable as an input to the object mask pruning task. The details of the above procedure are as follows. We begin with an initial baseline segmentation whose remaining errors are assumed to be sparsely distributed. During the error correction phase, we iteratively update a segmentation represented as the connected components of a graph G whose vertices are segments in a strict over-segmentation (henceforth called supervoxels). We also maintain the combined error map associated with the current segmentation. We binarize the error map by thresholding it at 0.25. Now we iteratively choose a location ` = (x, y, z) which has value 1 in the binarized combined error map. In a Px ? Py ? Pz window centred on `, we prepare an input for the error corrector by taking the union of all segments containing at least one white voxel in the error map. The error correction network produces from this input a binary image M representing the object containing the central voxel. For each supervoxel S touching the central Px /2 ? Py /2 ? Pz /2 window, let M (S) denote the average value of M inside S. If M (S) 6? [0.1, 0.9] for all S in the relevant window (i.e. the error corrector is confident in its prediction for each supervoxel), we add to G a clique on {S | M (S) > 0.9} and delete from G all edges between {S | M (S) < 0.1} and {S | M (S) > 0.9}. The effect of these updates is to change G to locally agree with M . Finally, we update the combined error map by applying the error detector at all locations where its decision could have changed. We iterate until every location is zero in the error map or has been covered by a window at least t = 2 times by the error corrector. This stopping criterion guarantees that the algorithm terminates. In practice, the segmentation converges without this auxiliary stopping condition to a state in which the error corrector fails confidence threshold everywhere. However, it is hard to certify convergence since it is possible that the error corrector could give different outputs on slightly shifted windows. Based on our validation set, increasing t beyond 2 did not measurably improve performance. Note that this algorithm deals with split and merge errors, but cannot fix errors already present at the supervoxel level. 5 5.1 Experiments Dataset Our dataset is a sample of mouse primary visual cortex (V1) acquired using serial section transmission electron microscopy at the Allen Institute for Brain Science. The voxel resolution is 3.6 nm?3.6 nm? 40 nm. Human experts used the VAST software tool [14, 2] to densely reconstruct multiple volumes that amounted to 530 Mvoxels of ground truth annotation. These volumes were used to train a neuronal boundary detection network (see the appendix for architecture). We applied the resulting boundary detector to a larger volume of size 4800 Mvoxels to produce a preliminary segmentation, which was then proofread by the tracers. This bootstrapped ground truth was used to train the error detector and corrector. A subvolume of size 910 Mvoxels was reserved for validation, and a subvolume of size 910 Mvoxels was reserved for testing. Producing the gold standard segmentation required a total of ? 560 tracer hours, while producing the bootstrapped ground truth required ? 670 tracer hours. 5.2 Baseline segmentation Our baseline segmentation was produced using a pipeline of multiscale convolutional networks for neuronal boundary detection, watershed, and mean affinity agglomeration [15]. We describe the pipeline in detail in the appendix. The segmentation performance values reported for the baseline are taken at a mean affinity agglomeration threshold of 0.23, which minimizes the variation of information error metric [17, 20] on the test volumes. 6 5.3 Training procedures Sampling procedure Here we describe our procedure for choosing a random point location in a segmentation. Uniformly random sampling is unsatisfactory since large objects such as dendritic shafts will be overrepresented. Instead, given a segmentation, we sample a location (x, y, z) with probability inversely proportional to the fraction of a window of size 128 ? 128 ? 16 centred at (x, y, z) which is occupied by the object containing the central voxel. Training of error detector An initial segmentation containing errors was produced using our baseline neuronal boundary detector combined with mean affinity agglomeration at a threshold of 0.3. Point locations were sampled according to the sampling procedure specified in 5.3. We augmented all of our data with rotations and reflections. We used a pixelwise cross-entropy loss. Training of error corrector We sampled locations in the ground truth segmentation as in 5.3. At each location ` = (x, y, z), we generated a training example as follows. Let Obj` be the ground truth object touching `. We selected a random subset of the objects in the window centred on ` including Obj` . To be specific, we chose a number p uniformly at random from [0, 1], and then selected each segment in the window with probability p in addition Obj` . The input at ` was then a binary mask representing the union of the selected objects along with the raw EM image, and the desired output was a binary mask representing only Obj` . The dataset was augmented with rotations, reflections, simulated misalignments and missing sections [15]. We used a pixelwise cross-entropy loss. Note that this training procedure uses only the ground truth segmentation and is completely independent of the error detector and the baseline segmentation. This convenient property is justified by the fact that if the error detector is perfect, the error corrector only ever receives as input unions of complete objects. 5.4 Error detection results To measure the quality of error detection, we densely sampled points in our test volume as in 5.3. In order to remove ambiguity over the precise location of errors, we filtered out points which contained an error within a surrounding window of size 80 ? 80 ? 8 but not a window of size 40 ? 40 ? 4. These locations were all unique, in that two locations in the same object were separated by at least 80, 80, 8 in x, y, z, respectively. Precision and recall simultaneously exceed 90% (see Figure 6). Empirically, many of the false positive examples come where a dendritic spine head curls back and touches its trunk. These examples locally appear to be incorrectly merged objects. We trained one error detector with access to the raw image and one without. The network?s admirable performance even without access to the image as seen in Figure 6 supports our hypothesis that error detection is a relatively easy task and can be performed using only shape cues. Merge errors qualitatively appear to be especially easy for the network to detect; an example is shown in Figure 7. 5.5 Error correction results Table 1: Comparing segmentation performance V Imerge V Isplit Rand Recall Rand Precision Baseline Without Advice With Advice 0.162 0.130 0.088 0.142 0.057 0.052 0.952 0.956 0.974 0.954 0.979 0.980 In order to demonstrate the importance of error detection to error correction, we ran two experiments: one in which the binary mask input to the error corrector was simply the union of all segments in the window (?without advice?), and one in which the binary mask was the union of all segments with a detected error (?with advice?). In the ?without advice? mode, the network is essentially asked to reconstruct the object overlapping the central voxel in one shot. Table 1 shows that advice confers a considerable advantage in performance on the error corrector. 7 1.00 0.95 0.90 Recall Experiment Obj + raw image Obj only 0.50 0.50 0.90 0.95 1.00 Precision Figure 6: Precision and recall for error detection, both with and without access to the raw image. In the test volume, there are 8248 error free locations and 944 locations with errors. In practice, we use threshold which guarantees > 95% recall and > 85% precision. Figure 7: An example of a detected error. The right shows two incorrectly merged axons, and the left shows the predicted combined error map (defined in 2.1) overlaid on the corresponding 2D image in red. Figure 8: A difficult location with missing data in one section combined with a misalignment between sections. The error-correcting net was able to trace across the missing data. It is sometimes difficult to assess the significance of an improvement in variation of information or rand score since changes can be dominated by modifications to a few large objects. Therefore, we decomposed the variation of information into a score for each object in the ground truth. Figure 9 summarizes the cumulative distribution of the values of V I(i) = V Imerge (i) + V Isplit (i) for all segments i in the ground truth. See the appendix for a precise definition of V I(i). The number of errors from the set in Sec. 5.4 that were fixed or introduced by our iterative refinement procedure is shown in 2. These numbers should be taken with a grain of salt since topologically insignificant changes could count as errors. Regardless, it is clear that our iterative refinement procedure fixed a significant fraction of the remaining errors and that ?advice? improves the error corrector. The results are qualitatively impressive as well. The error-correcting network is sometimes able to correctly merge disconnected objects, for example in Figure 8. Table 2: Number of errors fixed and introduced relative to the baseline # Errors # Errors fixed # Errors introduced Baseline Without Advice With Advice 944 474 305 547 707 8 77 68 Cumulative # of objects 1500 Method Baseline With advice Without advice 1000 500 0 0 1 2 3 VI (nats) Figure 9: Per-object VI scores for the 940 reconstructed objects in our test volume. Almost 800 objects are completely error free in our segmentation. These objects are likely all axons; almost every dendrite is missing a few spines. 5.6 Computational cost analysis Table 3 shows the computational cost of the most expensive parts of our segmentation pipeline. Boundary detection and error detection are run on the entire image, while error correction is run on roughly 10% of the possible locations in the image. Error correction is still the most costly step, but it would be 10? more costly without restricting to the locations found by the error detection network. Therefore, the cost of error detection is more than justified by the subsequent savings during the error correction phase. The number of locations requiring error correction will fall even further if the precision of the error detector increases or the error rate of the initial segmentation decreases. Table 3: Computation time for a 2048 ? 2048 ? 256 volume using a single TitanX Pascal GPU Boundary Detection 18 mins 6 Error Detection 25 mins Error Correction 55 mins Conclusion and future directions We have developed a error detector for the neuronal segmentation problem and combined it with an error correction module. In particular, we have shown that our error detectors are able to exploit priors on neuron shape, having reasonable performance even without access to the raw image. We have made significant savings in computation by applying expensive error correction procedures only where predicted necessary by the error detector. Finally, we have demonstrated that the ?advice? of error detection improves an error correction module, improving segmentation performance upon our baseline. We expect that significant improvements in the accuracy of error detection could come from aggressive data augmentation. We can mutilate a ground truth segmentation in arbitrary (or even adversarial) ways to produce unlimited examples of errors. An error detection module has many potential uses beyond the ones presented here. For example, we could use error detection to direct ground truth annotation effort toward mistakes. If sufficiently accurate, it could also be used directly as a learning signal for segmentation algorithms on unlabelled data. The idea of co-training our error-correction and error-detection networks is natural in view of recent work on generative adversarial networks [19, 16]. 9 Author contributions and acknowledgements JZ conceptualized the study and conducted most of the experiments and evaluation. IT (along with Will Silversmith) created much of the infrastructure necessary for visualization and running our algorithms at scale. KL produced the baseline segmentation. HSS helped with the writing. We are grateful to Clay Reid, Nuno da Costa, Agnes Bodor, Adam Bleckert, Dan Bumbarger, Derrick Britain, JoAnn Buchannan, and Marc Takeno for acquiring the TEM dataset at the Allen Institute for Brain Science. The ground truth annotation was created by Ben Silverman, Merlin Moore, Sarah Morejohn, Selden Koolman, Ryan Willie, Kyle Willie, and Harrison MacGowan. We thank Nico Kemnitz for proofreading a draft of this paper. We thank Jeremy Maitin-Shepard at Google and the other contributors to the neuroglancer project for creating an invaluable visualization tool. We acknowledge NVIDIA Corporation for providing us with early access to Titan X Pascal GPU used in this research, and Amazon for assistance through an AWS Research Grant. This research was supported by the Mathers Foundation, the Samsung Scholarship and the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior/ Interior Business Center (DoI/IBC) contract number D16PC0005. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI/IBC, or the U.S. Government. 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Style Transfer from Non-Parallel Text by Cross-Alignment Tianxiao Shen1 Tao Lei2 Regina Barzilay1 Tommi Jaakkola1 2 MIT CSAIL ASAPP Inc. 1 {tianxiao, regina, tommi}@csail.mit.edu 2 tao@asapp.com 1 Abstract This paper focuses on style transfer on the basis of non-parallel text. This is an instance of a broad family of problems including machine translation, decipherment, and sentiment modification. The key challenge is to separate the content from other aspects such as style. We assume a shared latent content distribution across different text corpora, and propose a method that leverages refined alignment of latent representations to perform style transfer. The transferred sentences from one style should match example sentences from the other style as a population. We demonstrate the effectiveness of this cross-alignment method on three tasks: sentiment modification, decipherment of word substitution ciphers, and recovery of word order.1 1 Introduction Using massive amounts of parallel data has been essential for recent advances in text generation tasks, such as machine translation and summarization. However, in many text generation problems, we can only assume access to non-parallel or mono-lingual data. Problems such as decipherment or style transfer are all instances of this family of tasks. In all of these problems, we must preserve the content of the source sentence but render the sentence consistent with desired presentation constraints (e.g., style, plaintext/ciphertext). The goal of controlling one aspect of a sentence such as style independently of its content requires that we can disentangle the two. However, these aspects interact in subtle ways in natural language sentences, and we can succeed in this task only approximately even in the case of parallel data. Our task is more challenging here. We merely assume access to two corpora of sentences with the same distribution of content albeit rendered in different styles. Our goal is to demonstrate that this distributional equivalence of content, if exploited carefully, suffices for us to learn to map a sentence in one style to a style-independent content vector and then decode it to a sentence with the same content but a different style. In this paper, we introduce a refined alignment of sentence representations across text corpora. We learn an encoder that takes a sentence and its original style indicator as input, and maps it to a style-independent content representation. This is then passed to a style-dependent decoder for rendering. We do not use typical VAEs for this mapping since it is imperative to keep the latent content representation rich and unperturbed. Indeed, richer latent content representations are much harder to align across the corpora and therefore they offer more informative content constraints. Moreover, we reap additional information from cross-generated (style-transferred) sentences, thereby getting two distributional alignment constraints. For example, positive sentences that are style-transferred into negative sentences should match, as a population, the given set of negative sentences. We illustrate this cross-alignment in Figure 1. 1 Our code and data are available at https://github.com/shentianxiao/language-style-transfer. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Figure 1: An overview of the proposed cross-alignment method. X1 and X2 are two sentence domains with different styles y1 and y2 , and Z is the shared latent content space. Encoder E maps a sentence to its content representation, and generator G generates the sentence back when combining with the original style. When combining with a different style, transferred X?1 is aligned with X2 and X?2 is aligned with X1 at the distributional level. To demonstrate the flexibility of the proposed model, we evaluate it on three tasks: sentiment modification, decipherment of word substitution ciphers, and recovery of word order. In all of these applications, the model is trained on non-parallel data. On the sentiment modification task, the model successfully transfers the sentiment while keeps the content for 41.5% of review sentences according to human evaluation, compared to 41.0% achieved by the control-gen model of Hu et al. (2017). It achieves strong performance on the decipherment and word order recovery tasks, reaching Bleu score of 57.4 and 26.1 respectively, obtaining 50.2 and 20.9 gap than a comparable method without cross-alignment. 2 Related work Style transfer in vision Non-parallel style transfer has been extensively studied in computer vision (Gatys et al., 2016; Zhu et al., 2017; Liu and Tuzel, 2016; Liu et al., 2017; Taigman et al., 2016; Kim et al., 2017; Yi et al., 2017). Gatys et al. (2016) explicitly extract content and style features, and then synthesize a new image by combining ?content? features of one image with ?style? features from another. More recent approaches learn generative networks directly via generative adversarial training (Goodfellow et al., 2014) from two given data domains X1 and X2 . The key computational challenge in this non-parallel setting is aligning the two domains. For example, CoupledGANs (Liu and Tuzel, 2016) employ weight-sharing between networks to learn cross-domain representation, whereas CycleGAN (Zhu et al., 2017) introduces cycle consistency which relies on transitivity to regularize the transfer functions. While our approach has a similar high-level architecture, the discreteness of natural language does not allow us to reuse these models and necessitates the development of new methods. Non-parallel transfer in natural language In natural language processing, most tasks that involve generation (e.g., translation and summarization) are trained using parallel sentences. Our work most closely relates to approaches that do not utilize parallel data, but instead guide sentence generation from an indirect training signal (Mueller et al., 2017; Hu et al., 2017). For instance, Mueller et al. (2017) manipulate the hidden representation to generate sentences that satisfy a desired property (e.g., sentiment) as measured by a corresponding classifier. However, their model does not necessarily enforce content preservation. More similar to our work, Hu et al. (2017) aims at generating sentences with controllable attributes by learning disentangled latent representations (Chen et al., 2016). Their model builds on variational auto-encoders (VAEs) and uses independency constraints to enforce that attributes can be reliably inferred back from generated sentences. While our model builds on distributional cross-alignment for the purpose of style transfer and content preservation, these constraints can be added in the same way. Adversarial training over discrete samples Recently, a wide range of techniques addresses challenges associated with adversarial training over discrete samples generated by recurrent networks (Yu et al., 2016; Lamb et al., 2016; Hjelm et al., 2017; Che et al., 2017). In our work, we employ the Professor-Forcing algorithm (Lamb et al., 2016) which was originally proposed to close the gap between teacher-forcing during training and self-feeding during testing for recurrent networks. This design fits well with our scenario of style transfer that calls for cross-alignment. By using 2 continuous relaxation to approximate the discrete sampling process (Jang et al., 2016; Maddison et al., 2016), the training procedure can be effectively optimized through back-propagation (Kusner and Hern?ndez-Lobato, 2016; Goyal et al., 2017). 3 Formulation In this section, we formalize the task of non-parallel style transfer and discuss the feasibility of the learning problem. We assume the data are generated by the following process: 1. a latent style variable y is generated from some distribution p(y); 2. a latent content variable z is generated from some distribution p(z); 3. a datapoint x is generated from conditional distribution p(x|y, z). We observe two datasets with the same content distribution but different styles y1 and y2 , where (1) (n) y1 and y2 are unknown. Specifically, the two observed datasets X1 = {x1 , ? ? ? , x1 } and X2 = (1) (m) {x2 , ? ? ? , x2 } consist of samples drawn from p(x1 |y1 ) and p(x2 |y2 ) respectively. We want to estimate the style transfer functions between them, namely p(x1 |x2 ; y1 , y2 ) and p(x2 |x1 ; y1 , y2 ). A question we must address is when this estimation problem is feasible. Essentially, we only observe the marginal distributions of x1 and x2 , yet we are going to recover their joint distribution: Z p(x1 , x2 |y1 , y2 ) = p(z)p(x1 |y1 , z)p(x2 |y2 , z)dz (1) z As we only observe p(x1 |y1 ) and p(x2 |y2 ), y1 and y2 are unknown to us. If two different y and y 0 lead to the same distribution p(x|y) = p(x|y 0 ), then given a dataset X sampled from it, its underlying style can be either y or y 0 . Consider the following two cases: (1) both datasets X1 and X2 are sampled from the same style y; (2) X1 and X2 are sampled from style y and y 0 respectively. These two scenarios have different joint distributions, but the observed marginal distributions are the same. To prevent such confusion, we constrain the underlying distributions as stated in the following proposition: Proposition 1. In the generative framework above, x1 and x2 ?s joint distribution can be recovered from their marginals only if for any different y, y 0 ? Y, distributions p(x|y) and p(x|y 0 ) are different. This proposition basically says that X generated from different styles should be ?distinct? enough, otherwise the transfer task between styles is not well defined. While this seems trivial, it may not hold even for simplified data distributions. The following examples illustrate how the transfer (and recovery) becomes feasible or infeasible under different model assumptions. As we shall see, for a certain family of styles Y, the more complex distribution for z, the more probable it is to recover the transfer function and the easier it is to search for the transfer. 3.1 Example 1: Gaussian Consider the common choice that z ? N (0, I) has a centered isotropic Gaussian distribution. Suppose a style y = (A, b) is an affine transformation, i.e. x = Az + b + , where  is a noise variable. For b = 0 and any orthogonal matrix A, Az + b ? N (0, I) and hence x has the same distribution for any such styles y = (A, 0). In this case, the effect of rotation cannot be recovered. Interestingly, if z has a more complex distribution, such as a Gaussian mixture, then affine transformations can be uniquely determined. PK Lemma 1. Let z be a mixture of Gaussians p(z) = k=1 ?k N (z; ?k , ?k ). Assume K ? 2, and there are two different ?i 6= ?j . Let Y = {(A, b)||A| = 6 0} be all invertible affine transformations, and p(x|y, z) = N (x; Az + b, 2 I), in which  is a noise. Then for all y 6= y 0 ? Y, p(x|y) and p(x|y 0 ) are different distributions. Theorem 1. If the distribution of z is a mixture of Gaussians which has more than two different components, and x1 , x2 are two affine transformations of z, then the transfer between them can be recovered given their respective marginals. 3 3.2 Example 2: Word substitution Consider here another example when z is a bi-gram language model and a style y is a vocabulary in use that maps each ?content word? onto its surface form (lexical form). If we observe two realizations x1 and x2 of the same language z, the transfer and recovery problem becomes inferring a word alignment between x1 and x2 . Note that this is a simplified version of language decipherment or translation. Nevertheless, the recovery problem is still sufficiently hard. To see this, let M1 , M2 ? Rn?n be the estimated bi-gram probability matrix of data X1 and X2 respectively. Seeking the word alignment is equivalent to finding a permutation matrix P such that P > M1 P ? M2 , which can be expressed as an optimization problem, min kP > M1 P ? M2 k2 P The same formulation applies to graph isomorphism (GI) problems given M1 and M2 as the adjacency matrices of two graphs, suggesting that determining the existence and uniqueness of P is at least GI hard. Fortunately, if M as a graph is complex enough, the search problem could be more tractable. For instance, if each vertex?s weights of incident edges as a set is unique, then finding the isomorphism can be done by simply matching the sets of edges. This assumption largely applies to our scenario where z is a complex language model. We empirically demonstrate this in the results section. The above examples suggest that z as the latent content variable should carry most complexity of data x, while y as the latent style variable should have relatively simple effects. We construct the model accordingly in the next section. 4 Method Learning the style transfer function under our generative assumption is essentially learning the conditional distribution p(x1 |x2 ; y1 , y2 ) and p(x2 |x1 ; y1 , y2 ). Unlike in vision where images are continuous and hence the transfer functions can be learned and optimized directly, the discreteness of language requires us to operate through the latent space. Since x1 and x2 are conditionally independent given the latent content variable z, Z p(x1 |x2 ; y1 , y2 ) = p(x1 , z|x2 ; y1 , y2 )dz Zz (2) = p(z|x2 , y2 ) ? p(x1 |y1 , z)dz z = Ez?p(z|x2 ,y2 ) [p(x1 |y1 , z)] This suggests us learning an auto-encoder model. Specifically, a style transfer from x2 to x1 involves two steps?an encoding step that infers x2 ?s content z ? p(z|x2 , y2 ), and a decoding step which generates the transferred counterpart from p(x1 |y1 , z). In this work, we approximate and train p(z|x, y) and p(x|y, z) using neural networks (where y ? {y1 , y2 }). Let E : X ? Y ? Z be an encoder that infers the content z for a given sentence x and a style y, and G : Y ? Z ? X be a generator that generates a sentence x from a given style y and content z. E and G form an auto-encoder when applying to the same style, and thus we have reconstruction loss, Lrec (?E , ?G ) = Ex1 ?X1 [? log pG (x1 |y1 , E(x1 , y1 ))] + Ex2 ?X2 [? log pG (x2 |y2 , E(x2 , y2 ))] (3) where ? are the parameters to estimate. In order to make a meaningful transfer by flipping the style, X1 and X2 ?s content space must coincide, as our generative framework presumed. To constrain that x1 and x2 are generated from the same latent content distribution p(z), one option is to apply a variational auto-encoder (Kingma and Welling, 2013). A VAE imposes a prior density p(z), such as z ? N (0, I), and uses a KL-divergence regularizer to align both posteriors pE (z|x1 , y1 ) and pE (z|x2 , y2 ) to it, LKL (?E ) = Ex1 ?X1 [DKL (pE (z|x1 , y1 )kp(z))] + Ex2 ?X2 [DKL (pE (z|x2 , y2 )kp(z))] 4 (4) The overall objective is to minimize Lrec + LKL , whose opposite is the variational lower bound of data likelihood. However, as we have argued in the previous section, restricting z to a simple and even distribution and pushing most complexity to the decoder may not be a good strategy for non-parallel style transfer. In contrast, a standard auto-encoder simply minimizes the reconstruction error, encouraging z to carry as much information about x as possible. On the other hand, it lowers the entropy in p(x|y, z), which helps to produce meaningful style transfer in practice as we flip between y1 and y2 . Without explicitly modeling p(z), it is still possible to force distributional alignment of p(z|y1 ) and p(z|y2 ). To this end, we introduce two constrained variants of auto-encoder. 4.1 Aligned auto-encoder Dispense with VAEs that make an explicit assumption about p(z) and align both posteriors to it, we align pE (z|y1 ) and pE (z|y2 ) with each other, which leads to the following constrained optimization problem: ? ? = arg min Lrec (?E , ?G ) ? (5) d s.t. E(x1 , y1 ) = E(x2 , y2 ) x1 ? X1 , x2 ? X2 In practice, a Lagrangian relaxation of the primal problem is instead optimized. We introduce an adversarial discriminator D to align the aggregated posterior distribution of z from different styles (Makhzani et al., 2015). D aims to distinguish between these two distributions: Ladv (?E , ?D ) = Ex1 ?X1 [? log D(E(x1 , y1 ))] + Ex2 ?X2 [? log(1 ? D(E(x2 , y2 )))] (6) The overall training objective is a min-max game played among the encoder E, generator G and discriminator D. They constitute an aligned auto-encoder: min max Lrec ? ?Ladv E,G D (7) We implement the encoder E and generator G using single-layer RNNs with GRU cell. E takes an input sentence x with initial hidden state y, and outputs the last hidden state z as its content representation. G generates a sentence x conditioned on latent state (y, z). To align the distributions of z1 = E(x1 , y1 ) and z2 = E(x2 , y2 ), the discriminator D is a feed-forward network with a single hidden layer and a sigmoid output layer. 4.2 Cross-aligned auto-encoder The second variant, cross-aligned auto-encoder, directly aligns the transfered samples from one style with the true samples from the other. Under the generative assumption, p(x2 |y2 ) = R p(x 2 |x1 ; y1 , y2 )p(x1 |y1 )dx1 , thus x2 (sampled from the left-hand side) should exhibit the x1 same distribution as transferred x1 (sampled from the right-hand side), and vice versa. Similar to our first model, the second model uses two discriminators D1 and D2 to align the populations. D1 ?s job is to distinguish between real x1 and transferred x2 , and D2 ?s job is to distinguish between real x2 and transferred x1 . Adversarial training over the discrete samples generated by G hinders gradients propagation. Although sampling-based gradient estimator such as REINFORCE (Williams, 1992) can by adopted, training with these methods can be unstable due to the high variance of the sampled gradient. Instead, we employ two recent techniques to approximate the discrete training (Hu et al., 2017; Lamb et al., 2016). First, instead of feeding a single sampled word as the input to the generator RNN, we use the softmax distribution over words instead. Specifically, during the generating process of transferred x2 from G(y1 , z2 ), suppose at time step t the output logit vector is vt . We feed its peaked distribution softmax(vt /?) as the next input, where ? ? (0, 1) is a temperature parameter. Secondly, we use Professor-Forcing (Lamb et al., 2016) to match the sequence of hidden states instead of the output words, which contains the information about outputs and is smoothly distributed. That is, the input to the discriminator D1 is the sequence of hidden states of either (1) G(y1 , z1 ) teacher-forced by a real example x1 , or (2) G(y1 , z2 ) self-fed by previous soft distributions. 5 Figure 2: Cross-aligning between x1 and transferred x2 . For x1 , G is teacher-forced by its words w1 w2 ? ? ? wt . For transfered x2 , G is self-fed by previous output logits. The sequence of hidden ? 0, ? ? ? , h ? t are passed to discriminator D1 to be aligned. Note that our first states h0 , ? ? ? , ht and h ? 0 , i.e. z1 and z2 , are variant aligned auto-encoder is a special case of this, where only h0 and h aligned. Algorithm 1 Cross-aligned auto-encoder training. The hyper-parameters are set as ? = 1, ? = 0.001 and learning rate is 0.0001 for all experiments in this paper. Input: Two corpora of different styles X1 , X2 . Lagrange multiplier ?, temperature ?. Initialize ?E , ?G , ?D1 , ?D2 repeat for p = 1, 2; q = 2, 1 do (i) Sample a mini-batch of k examples {xp }ki=1 from Xp (i) (i) Get the latent content representations zp = E(xp , yp ) (i) (i) (i) Unroll G from initial state (yp , zp ) by feeding xp , and get the hidden states sequence hp (i) Unroll G from initial state (yq , zp ) by feeding previous soft output distribution with temper? (i) ature ?, and get the transferred hidden states sequence h p end for Compute the reconstruction Lrec by Eq. (3) Compute D1 ?s (and symmetrically D2 ?s) loss: Ladv1 = ? k k 1X 1X (i) ? (i) )) log D1 (h1 ) ? log(1 ? D1 (h 2 k i=1 k i=1 (8) Update {?E , ?G } by gradient descent on loss Lrec ? ?(Ladv1 + Ladv2 ) (9) Update ?D1 and ?D2 by gradient descent on loss Ladv1 and Ladv2 respectively until convergence Output: Style transfer functions G(y2 , E(?, y1 )) : X1 ? X2 and G(y1 , E(?, y2 )) : X2 ? X1 The running procedure of our cross-aligned auto-encoder is illustrated in Figure 2. Note that crossaligning strengthens the alignment of latent variable z over the recurrent network of generator G. By aligning the whole sequence of hidden states, it prevents z1 and z2 ?s initial misalignment from propagating through the recurrent generating process, as a result of which the transferred sentence may end up somewhere far from the target domain. We implement both D1 and D2 using convolutional neural networks for sequence classification (Kim, 2014). The training algorithm is presented in Algorithm 1. 6 5 Experimental setup Sentiment modification Our first experiment focuses on text rewriting with the goal of changing the underlying sentiment, which can be regarded as ?style transfer? between negative and positive sentences. We run experiments on Yelp restaurant reviews, utilizing readily available user ratings associated with each review. Following standard practice, reviews with rating above three are considered positive, and those below three are considered negative. While our model operates at the sentence level, the sentiment annotations in our dataset are provided at the document level. We assume that all the sentences in a document have the same sentiment. This is clearly an oversimplification, since some sentences (e.g., background) are sentiment neutral. Given that such sentences are more common in long reviews, we filter out reviews that exceed 10 sentences. We further filter the remaining sentences by eliminating those that exceed 15 words. The resulting dataset has 250K negative sentences, and 350K positive ones. The vocabulary size is 10K after replacing words occurring less than 5 times with the ?<unk>? token. As a baseline model, we compare against the control-gen model of Hu et al. (2017). To quantitatively evaluate the transfered sentences, we adopt a model-based evaluation metric similar to the one used for image transfer (Isola et al., 2016). Specifically, we measure how often a transferred sentence has the correct sentiment according to a pre-trained sentiment classifier. For this purpose, we use the TextCNN model as described in Kim (2014). On our simplified dataset for style transfer, it achieves nearly perfect accuracy of 97.4%. While the quantitative evaluation provides some indication of transfer quality, it does not capture all the aspects of this generation task. Therefore, we also perform two human evaluations on 500 sentences randomly selected from the test set2 . In the first evaluation, the judges were asked to rank generated sentences in terms of their fluency and sentiment. Fluency was rated from 1 (unreadable) to 4 (perfect), while sentiment categories were ?positive?, ?negative?, or ?neither? (which could be contradictory, neutral or nonsensical). In the second evaluation, we evaluate the transfer process comparatively. The annotator was shown a source sentence and the corresponding outputs of the systems in a random order, and was asked ?Which transferred sentence is semantically equivalent to the source sentence with an opposite sentiment??. They can be both satisfactory, A/B is better, or both unsatisfactory. We collect two labels for each question. The label agreement and conflict resolution strategy can be found in the supplementary material. Note that the two evaluations are not redundant. For instance, a system that always generates the same grammatically correct sentence with the right sentiment independently of the source sentence will score high in the first evaluation setup, but low in the second one. Word substitution decipherment Our second set of experiments involves decipherment of word substitution ciphers, which has been previously explored in NLP literature (Dou and Knight, 2012; Nuhn and Ney, 2013). These ciphers replace every word in plaintext (natural language) with a cipher token according to a 1-to-1 substitution key. The decipherment task is to recover the plaintext from ciphertext. It is trivial if we have access to parallel data. However we are interested to consider a non-parallel decipherment scenario. For training, we select 200K sentences as X1 , and apply a substitution cipher f on a different set of 200K sentences to get X2 . While these sentences are nonparallel, they are drawn from the same distribution from the review dataset. The development and test sets have 100K parallel sentences D1 = {x(1) , ? ? ? , x(n) } and D2 = {f (x(1) ), ? ? ? , f (x(n) )}. We can quantitatively compare between D1 and transferred (deciphered) D2 using Bleu score (Papineni et al., 2002). Clearly, the difficulty of this decipherment task depends on the number of substituted words. Therefore, we report model performance with respect to the percentage of the substituted vocabulary. Note that the transfer models do not know that f is a word substitution function. They learn it entirely from the data distribution. In addition to having different transfer models, we introduce a simple decipherment baseline based on word frequency. Specifically, we assume that words shared between X1 and X2 do not require translation. The rest of the words are mapped based on their frequency, and ties are broken arbitrarily. Finally, to assess the difficulty of the task, we report the accuracy of a machine translation system trained on a parallel corpus (Klein et al., 2017). 2 we eliminated 37 sentences from them that were judged as neutral by human judges. 7 Method Hu et al. (2017) Variational auto-encoder Aligned auto-encoder Cross-aligned auto-encoder accuracy 83.5 23.2 48.3 78.4 Table 1: Sentiment accuracy of transferred sentences, as measured by a pretrained classifier. Method Hu et al. (2017) Cross-align sentiment 70.8 62.6 fluency 3.2 2.8 overall transfer 41.0 41.5 Table 2: Human evaluations on sentiment, fluency and overall transfer quality. Fluency rating is from 1 (unreadable) to 4 (perfect). Overall transfer quality is evaluated in a comparative manner, where the judge is shown a source sentence and two transferred sentences, and decides whether they are both good, both bad, or one is better. Word order recovery Our final experiments focus on the word ordering task, also known as bag translation (Brown et al., 1990; Schmaltz et al., 2016). By learning the style transfer functions between original English sentences X1 and shuffled English sentences X2 , the model can be used to recover the original word order of a shuffled sentence (or conversely to randomly permute a sentence). The process to construct non-parallel training data and parallel testing data is the same as in the word substitution decipherment experiment. Again the transfer models do not know that f is a shuffle function and learn it completely from data. 6 Results Sentiment modification Table 1 and Table 2 show the performance of various models for both human and automatic evaluation. The control-gen model of Hu et al. (2017) performs better in terms of sentiment accuracy in both evaluations. This is not surprising because their generation is directly guided by a sentiment classifier. Their system also achieves higher fluency score. However, these gains do not translate into improvements in terms of the overall transfer, where our model faired better. As can be seen from the examples listed in Table 3, our model is more consistent with the grammatical structure and semantic meaning of the source sentence. In contrast, their model achieves sentiment change by generating an entirely new sentence which has little overlap with the original. The discrepancy between the two experiments demonstrate the crucial importance of developing appropriate evaluation measures for comparing methods for style transfer. Word substitution decipherment Table 4 summarizes the performance of our model and the baselines on the decipherment task, at various levels of word substitution. Consistent with our intuition, the last row in this table shows that the task is trivial when the parallel data is provided. In non-parallel case, the difficulty of the task is driven by the substitution rate. Across all the testing conditions, our cross-aligned model consistently outperforms its counterparts. The difference becomes more pronounced as the task becomes harder. When the substitution rate is 20%, all methods do a reasonably good job in recovering substitutions. However, when 100% of the words are substituted (as expected in real language decipherment), the poor performance of variational autoencoder and aligned auto-encoder rules out their application for this task. Word order recovery The last column in Table 4 demonstrates the performance on the word order recovery task. Order recovery is much harder?even when trained with parallel data, the machine translation model achieves only 64.6 Bleu score. Note that some generated orderings may be completely valid (e.g., reordering conjunctions), but the models will be penalized for producing them. In this task, only the cross-aligned auto-encoder achieves grammatical reorder to a certain extent, demonstrated by its Bleu score 26.1. Other models fail this task, doing no better than no transfer. 8 From negative to positive consistently slow . consistently good . consistently fast . my goodness it was so gross . my husband ?s steak was phenomenal . my goodness was so awesome . it was super dry and had a weird taste to the entire slice . it was a great meal and the tacos were very kind of good . it was super flavorful and had a nice texture of the whole side . From positive to negative i love the ladies here ! i avoid all the time ! i hate the doctor here ! my appetizer was also very good and unique . my bf was n?t too pleased with the beans . my appetizer was also very cold and not fresh whatsoever . came here with my wife and her grandmother ! came here with my wife and hated her ! came here with my wife and her son . Table 3: Sentiment transfer samples. The first line is an input sentence, the second and third lines are the generated sentences after sentiment transfer by Hu et al. (2017) and our cross-aligned auto-encoder, respectively. Method No transfer (copy) Unigram matching Variational auto-encoder Aligned auto-encoder Cross-aligned auto-encoder Parallel translation 20% 56.4 74.3 79.8 81.0 83.8 99.0 Substitution decipher 40% 60% 80% 100% 21.4 6.3 4.5 0 48.1 17.8 10.7 1.2 59.6 44.6 34.4 0.9 68.9 50.7 45.6 7.2 79.1 74.7 66.1 57.4 98.9 98.2 98.5 97.2 Order recover 5.1 5.3 5.2 26.1 64.6 Table 4: Bleu scores of word substitution decipherment and word order recovery. 7 Conclusion Transferring languages from one style to another has been previously trained using parallel data. In this work, we formulate the task as a decipherment problem with access only to non-parallel data. The two data collections are assumed to be generated by a latent variable generative model. Through this view, our method optimizes neural networks by forcing distributional alignment (invariance) over the latent space or sentence populations. We demonstrate the effectiveness of our method on tasks that permit quantitative evaluation, such as sentiment transfer, word substitution decipherment and word ordering. The decipherment view also provides an interesting open question?when can the joint distribution p(x1 , x2 ) be recovered given only marginal distributions? We believe addressing this general question would promote the style transfer research in both vision and NLP. 9 Acknowledgments We thank Nicholas Matthews for helping to facilitate human evaluations, and Zhiting Hu for sharing his code. We also thank Jonas Mueller, Arjun Majumdar, Olga Simek, Danelle Shah, MIT NLP group and the reviewers for their helpful comments. This work was supported by MIT Lincoln Laboratory. References Peter F Brown, John Cocke, Stephen A Della Pietra, Vincent J Della Pietra, Fredrick Jelinek, John D Lafferty, Robert L Mercer, and Paul S Roossin. A statistical approach to machine translation. Computational linguistics, 16(2):79?85, 1990. 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Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. Bleu: a method for automatic evaluation of machine translation. In Proceedings of the 40th annual meeting on association for computational linguistics, pages 311?318. Association for Computational Linguistics, 2002. Allen Schmaltz, Alexander M. Rush, and Stuart Shieber. Word ordering without syntax. In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, pages 2319?2324. Association for Computational Linguistics, 2016. Yaniv Taigman, Adam Polyak, and Lior Wolf. Unsupervised cross-domain image generation. arXiv preprint arXiv:1611.02200, 2016. Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229?256, 1992. Zili Yi, Hao Zhang, Ping Tan Gong, et al. Dualgan: Unsupervised dual learning for image-to-image translation. arXiv preprint arXiv:1704.02510, 2017. Lantao Yu, Weinan Zhang, Jun Wang, and Yong Yu. Seqgan: sequence generative adversarial nets with policy gradient. arXiv preprint arXiv:1609.05473, 2016. Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. arXiv preprint arXiv:1703.10593, 2017. 11 A Proof of Lemma 1 PK Lemma 1. Let z be a mixture of Gaussians p(z) = k=1 ?k N (z; ?k , ?k ). Assume K ? 2, and there are two different ?i 6= ?j . Let Y = {(A, b)||A| = 6 0} be all invertible affine transformations, and p(x|y, z) = N (x; Az + b, 2 I), in which  is a noise. Then for all y 6= y 0 ? Y, p(x|y) and p(x|y 0 ) are different distributions. Proof. p(x|y = (A, b)) = K X ?k N (x; A?k + b, A?k A> + 2 I) k=1 For different y = (A, b) and y 0 = (A0 , b0 ), p(x|y) = p(x|y 0 ) entails that for k = 1, ? ? ? , K,  A?k + b = A0 ?k + b0 A?k A> = A0 ?k A0> Since all Y are invertible, (A?1 A0 )?k (A?1 A0 )> = ?k ?1 0 Suppose ?k = Qk Dk Q> A k is ?k ?s orthogonal diagonalization. If k = 1, all solutions for A have the form:  n o  QD 1/2 U D ?1/2 Q> U is orthogonal However, when K ? 2 and there are two different ?i 6= ?j , the only solution is A?1 A0 = I, i.e. A = A0 , and thus b = b0 . Therefore, for all y 6= y 0 , p(x|y) 6= p(x|y 0 ). 12 

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On the Non-Existence of a Universal Learning Algorithm for Recurrent Neural Networks Herbert Wiklicky Centrum voor Wiskunde en Informatica P.O.Box 4079, NL-1009 AB Amsterdam, The Netherlands? e-mail: herbert@cwi.nl Abstract We prove that the so called "loading problem" for (recurrent) neural networks is unsolvable. This extends several results which already demonstrated that training and related design problems for neural networks are (at least) NP-complete. Our result also implies that it is impossible to find or to formulate a universal training algorithm, which for any neural network architecture could determine a correct set of weights. For the simple proof of this, we will just show that the loading problem is equivalent to "Hilbert's tenth problem" which is known to be unsolvable. 1 THE NEURAL NETWORK MODEL It seems that there are relatively few commonly accepted general formal definitions of the notion of a "neural network". Although our results also hold if based on other formal definitions we will try to stay here very close to the original setting in which Judd's NP completeness result was given [Judd, 1990]. But in contrast to [Judd, 1990] we will deal here with simple recurrent networks instead of feed forward architectures. Our networks are constructed from three different types of units: .E-units compute just the sum of all incoming signals; for II -units the activation (node) function is given by the product of the incoming signals; and with E)-units - depending if the input signal is smaller or larger than a certain threshold parameter fl - the output is zero or one. Our units are connected or linked by real weighted connections and operate synchronously. Note that we could base our construction also just on one general type of units, namely what usually is called .EII -units. Furthermore, one could replace the II -units in the below 431 432 Wiklicky construction by (recurrent) modules of simple linear threshold units which had to perform unary integer multiplication. Thus, no higher order elements are actually needed. As we deal with recurrent networks, the behavior of a network now is not just given by a simple mapping from input space to output space (as with feed forward architectures). In geneml, an input pattern now is mapped to an (infinite) output sequence. But note, that if we consider as the output of a recurrent network a certain final, stable output pattern, we could return to a more static setting. 2 THE MAIN RESULT The question we will look at is how difficult it is to construct or train a neural network of the described type so that it actually exhibits a certain desired behavior, i.e. solves a given learning task. We will investigate this by the following decision problem: Decision 1 Loading Problem INSTANCE: A neural network architecture N and a learning task T . QUESTION: Is there a configuration C for N such that T is realized by C? By a network configuration we just think of a certain setting of the weights in a neural network. Our main result concerning this problem now just states that it is undecidable or unsolvable. Theorem 1 There exists no algorithm which could decide for any learning task T and any (recurrent) neural network (consisting of"?.. , TI-, and 8-units) peiformT. if the given architecture can The decision problem (as usual) gives a "lower bound" on the hardness of the related constructive problem [Garey and Johnson, 1979]. If we could construct a correct configuration for all instances, it would be trivial to decide instantly if a correct configuration exists at all. Thus we have: Corollary 2 There exists no universal learning algorithm for (recurrent) neural networks. 3 THE PROOF The proof of the above theorem is by constructing a class of neural networks for which it is impossible to decide (for all instance) if a certain learning task can be satisfied. We will refer for this to "Hilbert's tenth problem" and show that for each of its instances we can construct a neuml network, so that solutions to the loading problem would lead to solutions to the original problem (and vice versa). But as we know that Hilbert's tenth problem is unsolvable we also have to conclude that the loading problem we consider is unsolvable. 3.1 fiLBERT'S TENTH PROBLEM Our reference problem - of which we know it is unsolvable - is closely related to several famous and classical mathematical problems including for example Fermat's last theorem. On the Non-Existence of a Universal Learning Algorithm for Recurrent Neural Networks Definition 1 A diophantine equation is a polynomial D in cients. that is D(.1:J, :J:2, ... ,.1",,) =L n variables with integer coeffi- d i(3:1, .T 2, ... ,.r n ) t with each term d i of the form di( 3:1, .1:2, ... , .1: rt ) = r.i . J: i ? . J: iz .... . J : im, where the indices {i I, ?2, ... , ; rrt} are taken from {I , 2, ... , 11 } and the coefficient r.i E Z. The concrete problem, first formulated in [Hilbert, 1900] is to develop a universal algorithm how to find the integer solutions for all D, i.e. a vector (3: J, .1:2, ... ,3:,1) with .1: i E Z (or IN), such that D( 3: 1,3:2, ... , .1: rt) = O. The corresponding decision problem therefore is the following: Decision 2 Hilbert's Tenth Problem INSTANCE: Given a diophantine equation D. QUESTION: Is there an integer solutionfor D? Although this problem might seem to be quite simple - it formulation is actually the shortest among D. Hilbert's famous 23 problems - it was not until 1970 when Y. Matijasevich could prove that it is unsolvable or undecidable [Matijasevich, 1970]. There is no recursive computable predicate for diophantine equations which holds if a solution in Z (or N) exists and fails otherwise [Davis, 1973, Theorem 7.4]. 3.2 THE NETWORK ARCIDTECTURE The construction of a neural network IV for each diophantine D is now straight forward (see FigJ). It is just a three step construction. First, each variable .1: i of D is represented in IV by a small sub-network. The structure of these modules is quite simple (left side of Fig.1). Note that only the self-recurrent connection for the unit at the bottom of these modules is "weighted" by 0.0 < 'II! < 1.0. All other connection transmit their signals unaltered (i.e. w = 1.0). Second, the terms di in D are represent by Il-units in IV (as show in Fig.1). Therefore, the connections to these units from the sub-modules representing the variables .1: i of D correspond to the occurrences of these variables in each term d i. Finally, the output signals of all these Il-units is multiplied by the corresponding coefficients C:i and summed up by the ~-unit at the top. 3.3 THE SUB.MODULES The fundamental property of the networks constructed in the above way is given by the simple fact that the behavior of such a neural network IV corresponds uniquely to the evaluation of the original diophantine D. First, note that the behavior of N only depends on the weights Wi in each of the variable modules. Therefore, we will take a closer look at the behavior of these sub-modules. Suppose, that at some initial moment a signal of value 1.0 is received by each variable module. After that the signal is reset again to 0.0. 433 434 Wiklicky Wi. The "seed" signal starts circling via With each update circle this signal becomes a little bit smaller. On the other hand, the same signal is also sent to the central 8-unit, which sends a signal 1.0 to the top accumulator unit as long as the "circling" activation of the bottom unit is larger then the (preset) threshold 0,. The top unit (which also keeps track of its former activiations via a recurrent connection) therefore just counts how many updates it takes before the activiation of the bottom unit drops below 0,. The final, maximum, value which is emitted by the accumulator unit is some integer .1:, for which we have: l We thus have a correspondence between and the integer .1: = ~ I~/i where L-T J the largest integer which is smaller or equal to .1:. Given .1: i we also can construct an appropriate Wi weight i J' Wi by choosing it from the interval (exp (~~) ,exp (:r.1~!1))' 3.4 THE EQUIVALENCE To conclude the proof, we now have to demonstrate the equivalence of Hilbert's tenth problem and the loading problem for the discussed class of recurrent networks and some learning task. The learning task we will consider is the following: Map an input pattern with all signals equal to 1.0 (presented only once) to an output sequence which after afinite number of steps On the Non-Existence of a Universal Learning Algorithm for Recurrent Neural Networks is constant equal to 0.0. Note that - as discussed above - we could also consider a more static learing task where a final state, which detennines the (single) output of the network, was detennined by the condition that the outgoing signals of all 8-units had to be zero. Considering this learing task and with what we said about the behavior of the sub-modules it is now trivial to see that the constructed network just evaluates the diophantine polynomial for a set of variables ;r i corresponding to the (final) output signals of the sub-modules (which are detennined uniquely by the weight values !lii) if the input to the network is a pattern of all 1.0s. If we had a solution .1.' i of the original diophantine equation D, and if we take the corresponding values Wi (according to the above relation) as weights in the sub-modules of N, then this would also solve the loading problem for this architecture. On the other hand, if we knew the correct weights Wi for any such network N, then the corresponding integers 3:i would also solve the corresponding diophantine equation D. In particular, if it would be possible to decide if a correct set of weights Wi for N exists (for the above learning task), we could also decide if the corresponding diophantine D had a solution 3: i E :IN (and vice versa). As the whole construction was trivial, we have shown that both problems are equivalent. 4 CONCLUSIONS We demonstrated that the loading problem not only is NP-complete - as shown for simple feed fOIward architectures in [Judd, 1990], [Lin and Vitter, 1991], [Blum and Rivest, 1992], etc. - but actually unSOlvable, i.e. that the training of (recurrent) neural networks is among those problems which "indeed are intractable in an especially strong sense" [Garey and Johnson, 1979, P 12]. A related non-existence result concerning the training of higher order neural networks with integer weights was shown in [Wiklicky, 1992, WIklicky, 1994]. One should stress once again that the fact that no general algorithm exists for higher order or recurrent networks, which could solve the loading problem (for all its instances), does not imply that all instances of this problem are unsolvable or that no solutions exist. One could hope, that in most relevant cases - whatever that could mean - or, when we restrict the problem, a sub-class of problems things might become tractable. But the difference between solvable and unsolvable problems often can be very small. In particular, it is known that the problem of solving linear diophantine equations (instead of general ones) is polynomially computable, while if we go to quadratic diophantine equations the problem already becomes;V P complete [Johnson, 1990]. And for general diophantine the problem is even unsolvable. Moreover, it is also known that this problem is unsolvable if we consider only diophantine equations of maximum degree 4, and there exists a universal diophantine with only 13 variables which is unsolvable [Davis et al., 1976]. But we think, that one should interpret the "negative" results on NP-complexity as well as on undecidability of the loading problem not as restrictions for neural networks, but as related to their computational power. As it was shown that concrete neural networks can be constructed, so that they simulate a universal Turing machine [Siegelmann and Sontag, 1992, Cosnard et al., 1993]. It is mere the complexity of the problem one attempts to solve which simply cannot disappear and not some intrinsic intractability of the neural network approach. 435 436 Wiklicky Acknowledgement This work was started during the author's affiliation with the "Austrian Research Institute for Artificial Intelligence", Schottengasse 3, A-101O Wien, Austria. Further work was supported by a grant from the Austrian "Fonds zur Forderung der wissenschaftlichen Forschung" as Projekt J0828-PHY. References [Blum and Rivest, 1992] Avrim L. Blum and Ronald L. Rivest. Training a 3-node neural network is NP-complete. Neural Networks, 5:117-127,1992. [Cosnard et al. , 1993] Michael Cosnard, Max Garzon, and Pascal Koiran. Computability properties of low-dimensional dynamical systems. In Symposium on Theoretical Aspects of Computer Science (STACS '93), pages 365-373, Springer-Verlag, BerlinNew York, 1993. [Davis, 1973] Martin Davis. Hilbert's tenth problem is unsolvable. Amer. Math. Monthly, 80:233-269, March 1973. [Davis et aI., 1976] Martin Davis, Yuri Matijasevich, and Julia Robinson. Hilbert's tenth problem - diophantine equations: Positive aspects of a negative solution. In Felix E. Browder, editor, Mathematical developments arising from Hilbert, pages 323-378, American Mathematical Society, 1976. [Garey and Johnson, 1979] Michael R. Garey and David S. Johnson. Computers and Intractability -A Guide to the Theory of NP-Complete ness. W. H. Freeman, New York, 1979. [Hilbert, 1900] David Hilbert. Mathematische Probleme. Nachr. Ges. Wiss. G6ttingen, math.-phys.Kl., :253-297, 1900. [Johnson, 1990] David S. Johnson. A catalog of complexity classes. In Handbook of Theoretical Computer Science (Volume A: Algorithms and Complexity), chapter 2, pages 67-161, Elsevier - MIT Press, Amsterdam - Cambridge, Massachusetts, 1990. [Judd, 1990] J. Stephen Judd. Neural Network Design and the Complexity of Learning. MIT Press, Cambridge, Massachusetts - London, England, 1990. [Lin and Vitter, 1991] Jyh-Han Lin and Jeffrey Scott Vitter. Complexity results on learning by neural networks. Machine Learning, 6:211-230,1991. [Matijasevich, 1970] Yuri Matijasevich. Enumerable sets are diophantine. Dokl. Acad. Nauk., 191:279-282, 1970. [Siegelmann and Sontag, 1992] Hava T. Siegelmann and Eduardo D. Sontag. On the computational power of neural nets. In Fifth Workshop on Computational Learning Theory (COLT 92), pages 440-449, 1992. [Wiklicky, 1992] Herbert Wiklicky. SyntheSis and Analysis of Neural Networks - On a Framework for Artificial Neural Networks. PhD thesis, University of Vienna Technical University of Vienna, September 1992. [WIklicky, 1994] Herbert Wiklicky. The neural network loading problem is undecidable. In Euro-COLT '93 - Conference on Computational Learning Theory, page (to appear), Oxford University Press, Oxford, 1994. PART III THEORETICAL ANALYSIS: DYNAMICS AND STATISTICS 

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Cross-Spectral Factor Analysis Neil M. Gallagher*,1 , Kyle Ulrich*,2 , Austin Talbot3 , Kafui Dzirasa1,4 , Lawrence Carin2 and David E. Carlson5,6 1 Department of Neurobiology, 2 Department of Electrical and Computer Engineering, 3 Department of Statistical Science, 4 Department of Psychiatry and Behavioral Sciences, 5 Department of Civil and Environmental Engineering, 6 Department of Biostatistics and Bioinformatics , Duke University * Contributed equally to this work {neil.gallagher,austin.talbot,kafui.dzirasa, lcarin,david.carlson}@duke.edu Abstract In neuropsychiatric disorders such as schizophrenia or depression, there is often a disruption in the way that regions of the brain communicate with one another. To facilitate understanding of network-level communication between brain regions, we introduce a novel model of multisite low-frequency neural recordings, such as local field potentials (LFPs) and electroencephalograms (EEGs). The proposed model, named Cross-Spectral Factor Analysis (CSFA), breaks the observed signal into factors defined by unique spatio-spectral properties. These properties are granted to the factors via a Gaussian process formulation in a multiple kernel learning framework. In this way, the LFP signals can be mapped to a lower dimensional space in a way that retains information of relevance to neuroscientists. Critically, the factors are interpretable. The proposed approach empirically shows similar performance in classifying mouse genotype and behavioral context when compared to commonly used approaches that lack the interpretability of CSFA. CSFA provides a useful tool for understanding neural dynamics, particularly by aiding in the design of causal follow-up experiments. 1 Introduction Neuropsychiatric disorders (e.g. schizophrenia, autism spectral disorder, etc.) take an enormous toll on our society [16]. In spite of this, the underlying neural causes of many of these diseases are poorly understood and treatments are developing at a slow pace [2]. Many of these disorders have been linked to a disruption of neural dynamics and communication between brain regions [10, 34]. In recent years, tools such as optogenetics [15, 27] have facilitated the direct probing of causal relationships between neural activity in different brain regions and neural disorders [29]. Planning a well-designed experiment to study spatiotemoral dynamics in neural activity can present a challenge due to the high number of design choices, such as which region(s) to stimulate, what neuron types, and what stimulation pattern to use. In this manuscript we explore how a machine learning approach can facilitate the design of these experiments by developing interpretable and predictive methods. These two qualities are crucial because they allow exploratory experiments to be used more effectively in the design of causal studies. We explore how to construct a machine learning approach to capture neural dynamics from raw neural data during changing behavioral and state conditions. A body of literature in theoretical and experimental neuroscience has focused on linking synchronized oscillations, which are observable in LFPs and EEGs, to neural computation [19, 25]. Such oscillations are often quantified by spectral power, coherence, and phase relationships in particular frequency bands; disruption of these 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. relationships have been observed in neuropsychiatric disorders [21, 34]. There are a number of methods for quantifying synchrony between pairs of brain regions based on statistical correlation between recorded activity in those regions [37, 5], but current methods for effectively identifying such patterns on a multi-region network level, such as Independent Component Analysis (ICA), are difficult to transform to actionable hypotheses. The motivating data considered here are local field potentials (LFPs) recorded from implanted depth electrodes at multiple sites (brain regions). LFPs are believed to reflect the combined local neural activity of hundreds of thousands of neurons [9]. The unique combination of spatial and temporal precision provided by LFPs allows for accurate representation of frequency and phase relationships between activity in different brain regions. Notably, LFPs do not carry the signal precision present in spiking activity from signal neurons; however, LFP signal characteristics are more consistent between animals, meaning that information gleaned from LFPs can be used to understand population level effects, just as in fMRI or EEG studies. Our empirical results further demonstrate this phenomenon. Multi-region LFP recordings produce relatively high-dimensional datasets. Basic statistical tests typically perform poorly in such high dimensional spaces without being directed by prior knowledge due to multiple comparisons, which diminish statistical power [28]. Furthermore, typical multi-site LFP datasets are both ?big data? in the sense that there are a large number of high-dimensional measurements and ?small data? in the sense that only a few animals are used to represent the entire population. A common approach to address this issue is to describe such data by a small number of factors (e.g. dimensionality reduction), which increases the statistical power when relevant information (e.g. relationship to behavior) is captured in the factors. Many methods for reducing the dimensionality of neural datasets exist [14], but are generally either geared towards spiking data or simple general-purpose methods such as principal components analysis (PCA). Therefore, reducing the dimensionality of multi-channel LFP datasets into a set of interpretable factors can facilitate the construction of testable hypotheses regarding the role of neural dynamics in brain function. The end goal of this analysis is not simply to improve predictive performance, but to design meaningful future causal experiments. By identifying functional and interpretable networks, we can form educated hypotheses and design targeted manipulation of neural circuits. This approach has been previously successful in the field of neuroscience [10]. The choice to investigate networks that span large portions of the brain is critical, as this is the scale at which most clinical and scientific in vivo interventions are applied. Additionally, decomposing complex signatures of brain activity into contributions from individual functional networks (i.e. factors) allows for models and analyses that are more conceptually and technically tractable. Here, we introduce a new framework, denoted Cross-Spectral Factor Analysis (CSFA), which is able to accurately represent multi-region neural dynamics in a low-dimensional manifold while retaining interpretability. The model defines a set of factors, each capturing the power, coherence, and phase relationships for a distribution of neural signals. The learned parameters for each factor correspond to an interpretable representation of the network dynamics. Changes in the relative strengths of each factor can relate neural dynamics to desired variables. Empirically, CSFA discovers networks that are highly predictive of response variables (behavioral context and genotype) for recordings from mice undergoing a behavioral paradigm designed to measure an animal?s response to a challenging experience. We further show that incorporating response variables in a supervised multi-objective framework can further map relevant information into a smaller set of features, as in [31], potentially increasing statistical power. 2 Model Description Here, we describe a model to extract a low-dimensional ?brain state? representation from multichannel LFP recordings. The states in this model are defined by a set of factors, each of which describes a specific distribution of observable signals in the network. The data is segmented into time windows composed of N observations, equally spaced over time, from C distinct brain regions. We w C?N let window w be represented by Y w = [y w (see Fig 1[left]). N is determined 1 , . . . , yN ] 2 R by the sampling rate and the duration of the window. The complete dataset is represented by the set Y = {Y w }w=1,...,W . Window lengths are typically chosen to be 1-5 seconds, as this temporal resolution is assumed to be sufficient to capture the broad changes in brain state that we are interested in. We assume that window durations are short enough to make the signal approximately stationary. 2 ! "#$ !" ! "%$ ! "%& zw ? sw1 sw2 ... swL yw 1 yw 2 ... yw N W Figure 1: [left] Example of multi-site LFP data from seven brain regions, separated into time windows. [right] Visual description of the parameters of the dCSFA model. ycw : Signal from channel c in window w. zw : Task-relevant side information. sw` : Score for factor ` in window w. ?: Parameters describing CSFA model. : Parameters of side-information classifier. Shaded regions indicate observed variables and clear represent inferred variables. This assumption, while only an approximation, is appropriate because we are interested in brain state dynamics that occur on a relatively long time scale (i.e. multiple seconds). Therefore, within a single window of LFP data the observation may be represented by a stationary Gaussian process (GP). It is important to distinguish between signal dynamics, which occur on a time scale of milliseconds, and brain state dynamics, which are assumed to occur over a longer time scale. In the following, the Cross-Spectral Mixture kernel [35], a key step in the proposed model, is reviewed in Section 2.1. The formulation of the CSFA model is given in Section 2.2. Model inference is discussed in Section 2.3. In Section 2.4, a joint CSFA and classification model called discriminative CSFA (dCSFA) is introduced. Supplemental Section A discusses additional related work. Supplemental Section B gives additional mathematical background on multi-region Gaussian processes. Supplemental Section C offers an alternative formulation of the CSFA model that models the observed signal as the real component of a complex signal. For efficient calculations, computational approximations for the CSFA model are described in Supplemental Section D. 2.1 Cross-Spectral Mixture Kernel Common methods to characterize spectral relationships within and between signal channels are the power-spectral density (PSD) and cross-spectral density (CSD), respectively [30]. A set of multichannel neural recordings may be characterized by the set of PSDs for each channel and CSDs for each pair of channels, resulting in a quadratic increase in the number of parameters with the number of channels observed. In order to counteract the issues arising from many multiple comparisons, neuroscientists typically preselect channels and frequencies of interest before testing experimental hypotheses about spectral relationships in neural datasets. Instead of directly calculating each of these parameters, we use a modeling approach to estimate the PSDs and CSDs over all channels and frequency bands by using the Cross-Spectral Mixture (CSM) covariance kernel [35]. In this way we effectively reduce the number of parameters required to obtain a good representation of the PSDs and CSDs for a multi-site neural recording. The CSM multi-output kernel is given by K CSM (t, t0 ; Bq , ?q , ?q ) = Real ?P Q q=1 ? Bq kq (t, t0 ; ?q , ?q ) , (1) + j?q ? , (2) where the matrix K CSM 2 CC?C . This is the real component of a sum of Q separable kernels. Each of these kernels is given by the combination of a cross-spectral density matrix, Bq 2 CC?C , and a stationary function of two time points that defines a frequency band, kq (?). Representing ? = t t0 , as all kernels used here are stationary and depend only on the difference between the two inputs, the frequency band for each spectral kernel is defined by a spectral Gaussian kernel, kq (? ; ?q , ?q ) = exp 3 1 2 2 ?q ? which is equivalent to a Gaussian distribution in the frequency domain with variance ?q , centered at ?q . The matrix Bq is a positive semi-definite matrix with rank R. (Note: The cross-spectral density matrix Bq is also known as coregionalization matrix in spatial statistics [4]). Keeping R small for the coregionalization matrices ameliorates overfitting by reducing the overall parameter space. This relationship is maintained and Bq is updated by storing the full matrix as the outer product of a tall matrix with itself: ? qB ? ?, ? q 2 C ? R. Bq = B B (3) q Phase coherence between regions is given by the magnitudes of the complex off-diagonal entries in Bq . The phase offset is given by the complex angle of those off-diagonal entries. 2.2 Cross-Spectral Factor Analysis Our proposed model creates a low-dimensional manifold by extending the CSM framework to a multiple kernel learning framework [18]. Let tn represent the time point of the nth sample in the window and t represent [t1 , . . . , tN ]. Each window of data is modeled as w 1 yw ?w IC ), (4) n = f w (tn ) + ?n , n ? N (0, ? F w (t) = L X swl F lw (t), F w (t) = [f w (t1 ), . . . , f w (tN )], (5) l=1 where F w (t) is represented as a linear combination functions drawn from L latent factors, given by {F lw (t)}L l=1 . The l-th latent function is drawn independently for each task according to F lw (t) ? GP(0, K CSM (?; ? l )), (6) {Blq , ?lq , ?ql }Q q=1 ). where ? l is the set of parameters associated with the l factor (i.e. The GP here represents a multi-output Gaussian process due to the cross-correlation structure between the brain regions, as in [33]. Additional details on the multi-output Gaussian process formulation can be found in Supplemental Section B. th In CSFA, the latent functions {F lw (t)}L l=1 are not the same across windows; rather, the underlying cross-spectral content (power, coherence, and phase) of the signals is shared and the functional instantiation differs from window to window. A marginalization of all latent functions results in a covariance kernel that is a weighted superposition of the kernels for each latent factor, which is given mathematically as Y w ? GP(0, K CSF A (?; ?, w)) (7) K CSF A (? ; ?, w) = L X s2wl K CSM (? ; ? l ) + ? 1 ? IC. (8) l=1 Here, ? = {? 1 , . . . , ? L } is the set of parameters associated with all L factors and ? represents the Dirac delta function and constructs the additive Gaussian noise. The use of this multi-output GP formulation within the CSFA kernel means that the latent variables can be directly integrated out, facilitating inference. To address multiplicative non-identifiability, the maximum power in any frequency band is limited for each CSM kernel (i.e. max(diag(K CSM (0; ? l ))) = 1 for all l). In this way, the factor scores squared, s2wl , may now be interpreted approximately as the variance associated with factor l in window w. 2.3 Inference A maximum likelihood formulation for the zero-mean Gaussian process given by Eq. 7 is used to learn the factor scores {sw }W w=1 and CSM kernel parameters ?, given the full dataset Y. If we let N C?N C ?w be the covariance matrix obtained from the kernel K CSF A (?; ?, w) evaluated CSF A 2 C at time points t, we have ({sw }W max L(Y; {sw }W (9) w=1 , ?) = w=1 , ?) {sw }W w=1 ,? L(Y; {sw }W w=1 , ?) = W Y w=1 N (vec(Y w ); 0, ?w CSF A ), 4 (10) where vec(?) gives a column-wise vectorization of its matrix argument, and W is the total number of windows. As is common with many Gaussian processes, an analytic solution to maximize the log-likelihood does not exist. We resort to a batch gradient descent algorithm based on the Adam formulation [23]. Fast calculation of gradients is accomplished via a discrete Fourier transform (DFT) approximation for the CSM kernel [35]. This approximation alters the formulation given in Eq. 7 slightly; the modified form is given in Supplemental Section D. The hyperparameters of the model are the number of factors (L), the number of spectral Gaussians per factor (Q), the rank of the coregionalization matrix (R), and the precision of the additive white noise (?). In applications where the generative properties of the model are most important, hyperparameters should be chosen using cross-validation based on hold-out log-likelihood. In the results described below, we emphasize the predictive aspects of the model, so hyperparameters are chosen by cross-validating on predictive performance. In order to maximize the generalizability of the model to a population, validation and test sets are composed of data from complete animals/subjects that were not included in the training set. In all of the results described below, models were trained for 500 Adam iterations, with a learning rate of 0.01 and other learning parameters set to the defaults suggested in [23]. The kernel parameters ? were then fixed at their values from the 500th iteration and sufficient additional iterations were carried out until the factor scores, {sw }W w=1 , reached approximate convergence. Corresponding factor scores are learned for validation and test sets in a similar manner, by initializing the kernel parameters ? with those learned from the training set and holding them fixed while learning factor scores to convergence as outlined above. Normalization to address multiplicative identifiability, as described in Section 2.2, was applied to each model after all iterations were completed. 2.4 Discriminative CSFA We often wish to discover factors that are associated with some side information (e.g. behavioral context). More formally, given a set of labels, {z1 , . . . , zW }, we wish to maximize the ability of the factor scores, {s1 , . . . , sw }, to predict the labels. This is accomplished by modifying the objective function to include a second term related to the performance of a classifier that takes the factor scores as regressors. We term this modified model discriminative CSFA, or dCSFA. We choose the cross-entropy error of a simple logistic regression classifier to demonstrate this, giving ? ? PW P1 k sw ) W Pexp( . {{sw }W , ?} = max L(Y; {s } , ?) + 1 log 0 0 w z =k w=1 w=1 w w=1 k=0 exp( sw ) W {sw }w=1 ,? k k (11) The first term of the RHS of (11) quantifies the generative aspect of how well the model fits the data (the log-likelihood of Section 2.2). The second term is the loss function of classification. Here is a parameter that controls the relative importance of the classification loss function to the generative likelihood. It is straightforward to include alternative classifiers or side information. For example, when there are multiple classes it is desirable to set the loss function to be the cross entropy loss associated with multinomial logistic regression [24], which only involves modifying the second term of the RHS of (11). In this dCSFA formulation, and the other hyperparameters are chosen based on cross-validation of the predictive accuracy of the factors, to produce factors that are predictive as possible in a new dataset from other members of the population. The number of factors included in the classification and corresponding loss function can be limited to a number less than L. One application of dCSFA is to find a few factors predictive of side information, embedded in a full set of factors that describe a dataset [31]. In this way, the predictive factors maintain the desirable properties of a generative model, such as robustness to missing regressors. We assume that in many applications of dCSFA, the descriptive properties of the remaining factors matter only in that they provide a larger generative model to embed the discriminative factors in. In applications where the descriptive properties of the remaining factors are of major importance, hyperparameters can instead be cross-validated using the objective function from (11) applied to data from new members of the population. 2.5 Handling Missing Channels Electrode and surgical failures resulting in unusable data channels are common when collecting the multi-channel LFP datasets that motivate this work. Fortunately, accounting for missing channels is straightforward within the CSFA model by taking advantage of the marginal properties of multivariate 5 Homecage DAY 1 Open Field Tail Suspension Test Homecage DAY 2 Open Field Tail Suspension Test Figure 2: Scores for three different factors over the duration of a two-day experiment, for 6 different mice. Score trajectories are smoothed over time for visualization. Bold lines give score trajectory averaged over all 6 mice. These 6 mice were held out from the training set used to generate this dCSFA model and these factors. Homecage DAY 1 Open Field Tail Suspension Test Homecage DAY 2 Open Field Tail Suspension Test Figure 3: Scores for a single factor over the duration of a two-day experiment, for two mice each from the wild type and CLOCK 19 genotypes. Score trajectories for mice from the CLOCK 19 genetic background are in blue, and from a wild type background in red. Score trajectories for each mouse are smoothed in time. Bold lines give the average trace for each genotype. The data from those 4 mice were held out from the training set used to generate the dCSFA model resulting in this factor. (Note: some windows were thrown out due to noise contaminating the LFP signal, the remaining windows were concatenated for this figure, resulting in some ?empty? segments) Gaussian distributions. This is a standard approach in the Gaussian process literature [32]. Missing channels are handled by marginalizing the missing channel out of the covariance matrix in Eq. 7. This mechanism also allows for the application of CSFA to multiple datasets simultaneously, as long as there is some overlap in the set of regions recorded in each dataset. Similarly, the conditional properties of multivariate Gaussian distributions provide a mechanism for simulating data from missing channels. This is accomplished by finding the conditional covariance matrix for the missing channels given the original matrix (Eq. 8) and the recorded data. 3 3.1 Results Synthetic Data In order to demonstrate that CSFA is capable of accurately representing the true spectral characteristics associated with some dataset, we tested it on a synthetic dataset. The synthetic dataset was simulated 6 from a CSFA model with pre-determined kernel parameters and randomly generated score values at each window. In this way there is a known covariance matrix associated with each window of the dataset. Details of the model used to generate this data are described in Supplemental Section E and Supplemental Table 2. The cross-spectral density was learned for each window of the dataset by training a randomly initialized CSFA model and the KL-divergence compared to the true crossspectral density was computed. Hyperparameters for the learned CSFA model were chosen to match the model from which the dataset was generated. A classical issue with many factor analysis approaches, such as probabilistic PCA [7], is the assumption of a constant covariance matrix. To emphasize the point that our method captures dynamics of the covariance structure, we compare the results from CSFA to the KL-divergence from a constant estimate of the covariance matrix over all of the windows, as is assumed in traditional factor analysis approaches. CSFA had an average divergence of 5466.8 (std. dev. of 49.5) compared to 7560.2 (std. dev. of 17.9) for the mean estimate. These distributions were significantly different (p-value < 2 ? 10 308 , Wilcoxon rank sum test). This indicates that, on average, CSFA provides a much better estimate of the covariance matrix associated with a window in this synthetic dataset compared to the classical constant covariance assumption. 3.2 Mouse Data We collected a dataset of LFPs recorded from 26 mice from two different genetic backgrounds (14 wild type, 12 CLOCK 19). The CLOCK 19 line of mice have been proposed as a model of bipolar disorder [36]. There are 20 minutes of recordings for each mouse: 5 minutes occurred while the mouse was in its home cage, 5 minutes occurred during open field exploration, and 10 minutes occurred during a tail suspension test. The tail suspension test is used as an assay of response to a challenging experience [1]. We learned models for CSFA and for dCSFA in two separate classification tasks: prediction of animal genotype and of the behavioral context of the recording (i.e. home cage, open field, or tail-suspension test). Following previous applications [35], the window length was set to 5 seconds and data was downsampled to 250 Hz. Eleven distinct brain regions were recorded: Nucleus Accumbens Core, Nucleus Accumbens Shell, Basolateral Amygdala, Infralimbic Cortex, Mediodorsal Thalamus, Prelimbic Cortex, Ventral Tegmental Area, Lateral Dorsal Hippocampus, Lateral Substantia Nigra Pars Compacta, Medial Dorsal Hippocampus, and Medial Substantia Nigra Pars Compacta. Three mice of each genotype were held out as a testing set. We choose the number of factors, L, the number of spectral Gaussians per factor (i.e. factor complexity), Q, the rank of the cross-spectral density matrix, R, and the additive noise precision, ?, via 5-fold cross-validation, in which a CSFA model is trained for each combination of L 2 {10, 20, 30}, Q 2 {3, 5, 8}, R 2 {1, 2}, ? 2 {5, 20}, while leaving all data from a subset of animals out. Classification performance was used as the validation metric. L = 20, Q = 5, R = 2, ? = 20 was selected for genotype classification and L = 30, Q = 3, R = 2, ? = 5 was selected for behavioral context classification. For dCSFA, the first 3 factors were included in the classifier portion of each model. dCSFA was trained separately for the genotype classification and the behavioral classification. Logistic regression was used for genotype and multiple logistic regression was used for the behavioral paradigms. Model PCA CSFA dCSFA-3 Genotype (AUC) 0.936 0.928 0.772 Behavioral Context (Accuracy) 81.4 80.4 80.1 Table 1: Classification results for genotype binary classification and experiment task multiclass classification. PCA: Principal components of a non-parametric estimate of spectral content of signal. CSFA: CSFA factor scores. dCSFA-3: Factor scores used in discriminitve classifier for logistic-dCSFA. All numbers are reported for a regularized-logistic regression classifier. We compare our CSFA and dCSFA models to several two-stage modeling approaches that are representative of techniques commonly used in the analysis of neural oscillation data [22]. Each of these approaches begins with a method for estimating the spectral content of a signal, followed by a dimension-reducing technique. Detailed descriptions of these comparison approaches are given in Supplemental Section F. CSFA models were learned as described in Section 2.3; dCSFA models were initialized with the CSFA model reported above and trained for an additional 500 iterations. 7 Figure 4: Visual representations of a dCSFA factor. [right] Relative power-spectral (diagonal) and cross-spectral (off-diagonal) densities associated with the covariance function defining a single factor. Amplitude reported for each frequency within a power or cross-spectral density is normalized relative to the total sum of powers or coherences, respectively, at that frequency for all factors. [left] Simplified representation of the same factor. Each ?wedge? corresponds to a single brain region. Colored regions along the ?hub? of the circle represent frequency bands with significant power within that corresponding region. Colored ?spokes? represent frequency bands with significant coherence between the corresponding pair of regions. The classification accuracies comparing CSFA, dCSFA, and each of the comparison methods were calculated using the test set described above; these values are shown in Table 1. Figure 2 demonstrates that the predictive features learned from dCSFA clearly track the different behavioral paradigms. Importantly, while the predictive performance using dCSFA factor scores is not better than the CSFA model for either task, the classification requires just 3 factors, rather than 20. Compressing relevant predictive information into only a handful of factors here is desirable for a number of reasons; it reduces the necessary number of statistical tests for testing hypotheses and also offers a more interpretable situation for neuroscientists. The dCSFA factor that is most strongly associated with genotype is visualized in Figure 4. In the case of behavioral context, the 3 factors are nearly as predictive as the 30 factors from the CSFA model, indicating that dCSFA has successfully captured the relevant predictive information in those 3 factors. 3.3 Visualization The models generated by CSFA are easily visualized and interpreted in a way that allows neuroscientists to generate testable hypotheses related to brain network dynamics. Figure 4 shows one way to visualize the latent factors produced by CSFA. The right hand side shows the power and cross-spectra associated with the CSM kernel from a single factor. Together these plots define a distribution of multi-channel signals that are described by this one factor. Plots along the diagonal give power spectra for each of the 11 brain regions included in the dataset. The off diagonal plots show the cross spectra with the associated phase offset in orange. The phase offset implies that oscillations may originate in one region and travel to another, given the assumption that another (observed or 8 unobserved) region is not responsible for the observed phase offset. These assumptions are not true in general, so we emphasize that their use is in hypothesis generation. The circular plot on the bottom-left of Figure 4 visualizes the same factor in an alternative concise way. Around the edge of the circle are the names of the brain regions in the data set and a range of frequencies modeled for each region. Colored bands along the outside of the circle indicate that spectral power in the corresponding region and frequency bands is above a threshold value. Similarly, lines connecting one region to another indicate that the coherence between the two regions is above the same threshold value at the corresponding frequency band. Given the assumption that coherence implies communication between brain regions [5], this plot quickly shows which brain regions are believed to be communicating and at what frequency band in each functional network. 4 Discussion and Conclusion Multi-channel LFP datasets have enormous potential for describing brain network dynamics at the level of individual regions. The dynamic nature and high-dimensionality of such datasets makes direct interpretation quite difficult. In order to take advantage of the information in such datasets, techniques for simplifying and detecting patterns in this context are necessary. Currently available techniques for simplifying these types of high dimensional datasets into a manageable size (e.g. ICA, PCA) generally do not offer sufficient insight into the types of questions that neuroscientists are interested in. More specifically, there is evidence that neural networks produce oscillatory patterns in LFPs as signatures of network activation [19]. Methods such as CSFA, which identify and interpret these signatures at a network level, are needed to form reasonable and testable hypotheses about the dynamics of whole-brain networks. In this work, we show that CSFA detects signatures of multi-region network activity that explain variables of interest to neuroscientists (i.e. animal genotype, behavioral context). The proposed CSFA model explicitly targets known relationships of LFP data to map the highdimensional data to a low-dimensional set of features. In direct contrast to many other dimensionality reduction methods, each factor maintains a high degree of interpretability, particularly in neuroscience applications. We emphasize that CSFA captures both spectral power and coherence across brain regions, both of which have been associated with neural information processing within the brain [20]. It is important to note that this model finds temporal precedence in observed signals, rather than true causality; there are many examples where temporal precedence does not imply true causation. Therefore, we emphasize that CSFA facilitates the generation of testable hypothesis rather than demonstrating causal relationships by itself. In addition, CSFA can suggest ways of manipulating network dynamics in order to directly test their role in mental processes. Such experiments might involve closed-loop stimulation using optogenetic or transcranial magnetic stimulation to manipulate the complex temporal dynamics of neural activity captured by the learned factors. Future work will focus on making these approaches broadly applicable, computationally efficient, and reliable. It is worth noting that CSFA describes the full-cross spectral density of the data, but that there are additional signal characteristics of interest to neuroscientists that are not described, such as cross-frequency coupling [25]; another possible area of future work is the development of additional kernel formulations that could capture these additional signal characteristics. CSFA will also be generalized to include other measurement modalities (e.g. neural spiking, fMRI) to create joint generative models. In summary, we believe that CSFA fulfills three important criteria: 1. It consolidates high-dimensional data into an easily interpretable low-dimensional space. 2. It adequately represents the raw observed data. 3. It retains information from the original dataset that is relevant to neuroscience researchers. All three of these characteristics are necessary to enable neuroscience researchers to generate trustworthy hypotheses about a network-level brain dynamics. Acknowledgements In working on this project L.C. received funding from the DARPA HIST program; K.D., L.C., and D.C. received funding from the National Institutes of Health by grant R01MH099192-05S2; K.D received funding from the W.M. Keck Foundation. 9 References [1] H. M. Abelaira, G. Z. Reus, and J. Quevedo. Animal models as tools to study the pathophysiology of depression. Revista Brasileira de Psiquiatria, 2013. [2] H. Akil, S. Brenner, E. Kandel, K. S. Kendler, M.-C. King, E. Scolnick, J. D. Watson, and H. Y. Zoghbi. The future of psychiatric research: genomes and neural circuits. Science, 2010. [3] M. A. Alvarez, L. Rosasco, and N. D. Lawrence. Kernels for Vector-Valued Functions: a Review. Foundations and Trends in Machine Learning, 2012. [4] S. Banerjee, B. P. Carlin, and A. E. Gelfand. Hierarchical modeling and analysis for spatial data. Crc Press, 2014. [5] A. M. Bastos and J.-M. Schoffelen. 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Stochastic Submodular Maximization: The Case of Coverage Functions Mohammad Reza Karimi Department of Computer Science ETH Zurich mkarimi@ethz.ch Mario Lucic Department of Computer Science ETH Zurich lucic@inf.ethz.ch Hamed Hassani Department of Electrical and Systems Engineering University of Pennsylvania hassani@seas.upenn.edu Andreas Krause Department of Computer Science ETH Zurich krausea@ethz.ch Abstract Stochastic optimization of continuous objectives is at the heart of modern machine learning. However, many important problems are of discrete nature and often involve submodular objectives. We seek to unleash the power of stochastic continuous optimization, namely stochastic gradient descent and its variants, to such discrete problems. We first introduce the problem of stochastic submodular optimization, where one needs to optimize a submodular objective which is given as an expectation. Our model captures situations where the discrete objective arises as an empirical risk (e.g., in the case of exemplar-based clustering), or is given as an explicit stochastic model (e.g., in the case of influence maximization in social networks). By exploiting that common extensions act linearly on the class of submodular functions, we employ projected stochastic gradient ascent and its variants in the continuous domain, and perform rounding to obtain discrete solutions. We focus on the rich and widely used family of weighted coverage functions. We show that our approach yields solutions that are guaranteed to match the optimal approximation guarantees, while reducing the computational cost by several orders of magnitude, as we demonstrate empirically. 1 Introduction Submodular functions are discrete analogs of convex functions. They arise naturally in many areas, such as the study of graphs, matroids, covering problems, and facility location problems. These functions are extensively studied in operations research and combinatorial optimization [21]. Recently, submodular functions have proven to be key concepts in other areas such as machine learning, algorithmic game theory, and social sciences. As such, they have been applied to a host of important problems such as modeling valuation functions in combinatorial auctions, feature and variable selection [22], data summarization [25], and influence maximization [19]. Classical results in submodular optimization consider the oracle model whereby the access to the optimization objective is provided through a black box ? an oracle. However, in many applications, the objective has to be estimated from data and is subject to stochastic fluctuations. In other cases the value of the objective may only be obtained through simulation. As such, the exact computation might not be feasible due to statistical or computational constraints. As a concrete example, consider the problem of influence maximization in social networks [19]. The objective function is defined as the expectation of a stochastic process, quantifying the size of the (random) subset of nodes 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. influenced from a selected seed set. This expectation cannot be computed efficiently, and is typically approximated via random sampling, which introduces an error in the estimate of the value of a seed set. Another practical example is the exemplar-based clustering problem, which is an instance of the facility location problem. Here, the objective is the sum of similarities of all the points inside a (large) collection of data points to a selected set of centers. Given a distribution over point locations, the true objective is defined as the expected value w.r.t. this distribution, and can only be approximated as a sample average. Moreover, evaluating the function on a sample involves computation of many pairwise similarities, which is computationally prohibitive in the context of massive data sets. In this work, we provide a formalization of such stochastic submodular maximization tasks. More precisely, we consider set functions f : 2V ? R+ , defined as f (S) = E??? [f? (S)] for S ? V , where ? is an arbitrary distribution and for each realization ? ? ?, the set function f? : 2V ? R+ is monotone and submodular (hence f is monotone submodular). The goal is to maximize f subject to some constraints (e.g. the k-cardinality constraint) having access only to i.i.d. samples f??? (?). Methods for submodular maximization fall into two major categories: (i) The classic approach is to directly optimize the objective using discrete optimization methods (e.g. the G REEDY algorithm and its accelerated variants), which are state-of-the-art algorithms (both in practice and theory), at least in the case of simple constraints, and are most widely considered in the literature; (ii) The alternative is to lift the problem into a continuous domain and exploit continuous optimization techniques available therein [7]. While the continuous approaches may lead to provably good results, even for more complex constraints, their high computational complexity inhibits their practicality. In this paper we demonstrate how modern stochastic optimization techniques (such as SGD, A DA G RAD [8] and A DAM [20]), can be used to solve an important class of discrete optimization problems which can be modeled using weighted coverage functions. In particular, we show how to efficiently maximize them under matroid constraints by (i) lifting the problem into the continuous domain using the multilinear extension [35], (ii) efficiently computing a concave relaxation of the multilinear extension [30], (iii) efficiently computing an unbiased estimate of the gradient for the concave relaxation thus enabling (projected) stochastic gradient ascent-style algorithms to maximize the concave relaxation, and (iv) rounding the resulting fractional solution without loss of approximation quality [7]. In addition to providing convergence and approximation guarantees, we demonstrate that our algorithms enjoy strong empirical performance, often achieving an order of magnitude speedup with less than 1% error with respect to G REEDY. As a result, the presented approach unleashes the powerful toolkit of stochastic gradient based approaches to discrete optimization problems. Our contributions. In this paper we (i) introduce a framework for stochastic submodular optimization, (ii) provide a general methodology for constrained maximization of stochastic submodular objectives, (iii) prove that the proposed approach guarantees a (1 ? 1/e)?approximation in expectation for the class of weighted coverage functions, which is the best approximation guarantee achievable in polynomial time unless P = NP, (iv) highlight the practical benefit and efficiency of using continuous-based stochastic optimization techniques for submodular maximization, (v) demonstrate the practical utility of the proposed framework in an extensive experimental evaluation. We show for the first time that continuous optimization is a highly practical, scalable avenue for maximizing submodular set functions. 2 Background and problem formulation Let V be a ground set of n elements. A set function f : 2V ?? R+ is submodular if for every A, B ? V , it holds f (A) + f (B) ? f (A ? B) + f (A ? B). Function f is said to be monotone if f (A) ? f (B) for all A ? B ? V . We focus on maximizing f subject to some constraints on S ? V . The prototypical example is maximization under the cardinality constraint, i.e., for a given integer k, find S ? V , |S| ? k, which maximizes f . Finding an exact solution for monotone submodular functions is NP-hard [10], but a (1 ? 1/e)-approximation can be efficiently determined [28]. Going beyond the (1 ? 1/e)-approximation is NP-hard for many classes of submodular functions [28, 23]. More generally, one may consider matroid constraints, whereby (V, I) is a matroid with the family of independent sets I, and maximize f such that S ? I. The G REEDY algorithm achieves a 1/2approximation [13], but C ONTINUOUS G REEDY introduced by Vondr?k [35], Calinescu et al. [6] can achieve a (1 ? 1/e)-optimal solution in expectation. Their approach is based on the multilinear 2 extension of f , F : [0, 1]V ? R+ , defined as X Y Y F (x) = f (S) xi (1 ? xj ), S?V i?S (1) j ?S / for all x = (x1 , ? ? ? , xn ) ? [0, 1]V . In other words, F (x) is the expected value of of f over sets wherein each element i is included with probability xi independently. Then, instead of optimizing f (S) over I, we can optimize F over the matroid base polytope corresponding to (V, I): P = {x ? Rn+ | x(S) ? r(S), ?S ? V, x(V ) = r(V )}, where r(?) is the matroid?s rank function. The C ONTINUOUS G REEDY algorithm then finds a solution x ? P which provides a (1 ? 1/e)?approximation. Finally, the continuous solution x is then efficiently rounded to a feasible discrete solution without loss in objective value, using P IPAGE ROUNDING [1, 6]. The idea of converting a discrete optimization problem into a continuous one was first exploited by Lov?sz [26] in the context of submodular minimization and this approach was recently applied to a variety of problems [34, 18, 3]. Problem formulation. The aforementioned results are based on the oracle model, whereby the exact value of f (S) for any S ? V is given by an oracle. In absence of such an oracle, we face the additional challenges of evaluating f , both statistical and computational. In particular, consider set functions that are defined as expectations, i.e. for S ? V we have f (S) = E??? [f? (S)], (2) where ? is an arbitrary distribution and for each realization ? ? ?, the set function f? : 2V ? R is submodular. The goal is to efficiently maximize f subject to constraints such as the k-cardinality constraint, or more generally, a matroid constraint. As a motivating example, consider the problem of propagation of contagions through a network. The objective is to identify the most influential seed set of a given size. A propagation instance (concrete realization of a contagion) is specified by a graph G = (V, E). The influence fG (S) of a set of nodes S in instance G is the fraction of nodes reachable from S using the edges E. To handle uncertainties in the concrete realization, it is natural to introduce a probabilistic model such as the Independent Cascade [19] model which defines a distribution G over instances G ? G that share a set V of nodes. The influence of a seed set S is then the expectation f (S) = EG?G [fG (S)], which is a monotone submodular function. Hence, estimating the expected influence is computationally demanding, as it requires summing over exponentially many functions fG . Assuming f as in (2), one can easily obtain an unbiased estimate of f for a fixed set S by random sampling according to ?. The critical question is, given that the underlying function is an expectation, can we optimize it more efficiently? Our approach is based on continuous extensions that are linear operators on the class of set functions, namely, linear continuous extensions. As a specific example, considering the multilinear extension, we can write F (x) = E??? [F? (x)], where F? denotes the extension of f? . As a consequence, the value of F? (x), when ? ? ?, is an unbiased estimator for F (x) and unbiased estimates of the (sub)gradients may be obtained analogously. We explore this avenue to develop efficient algorithms for maximizing an important subclass of submodular functions that can be expressed as weighted coverage functions. Our approach harnesses a concave relaxation detailed in Section 3. Further related work. The emergence of new applications, combined with a massive increase in the amount of data has created a demand for fast algorithms for submodular optimization. A variety of approximation algorithms have been presented, ranging from submodular maximization subject to a cardinality constraint [27, 37, 4], submodular maximization subject to a matroid constraint [6], non-monotone submodular maximization [11], approximately submodular functions [16], and algorithms for submodular maximization subject to a wide variety of constraints [24, 12, 36, 17, 9]. A closely related setting to ours is online submodular maximization [33], where functions come one at a time and the goal is to provide time-dependent solutions (sets) such that a cumulative regret is minimized. In contrast, our goal is to find a single (time-independent) set that maximizes the objective (2). Another relevant setting is noisy submodular maximization, where the evaluations returned by the oracle are noisy [15, 32]. Specifically, [32] assumes a noisy but unbiased oracle (with an independent sub-Gaussian noise) which allows one to sufficiently estimate the marginal gains of items by averaging. In the context of cardinality constraints, some of these ideas can be carried to our setting by introducing additional assumptions on how the values f? (S) vary w.r.t. to their expectation f (S). However, we provide a different approach that does not rely on uniform convergence and compare sample and running time complexity comparison with variants of G REEDY in Section 3. 3 3 Stochastic Submodular Optimization We follow the general framework of [35] whereby the problem is lifted into the continuous domain, a continuous optimization algorithm is designed to maximize the transferred objective, and the resulting solution is rounded. Maximizing f subject to a matroid constraint can then be done by first maximizing its multilinear extension F over the matroid base polytope and then rounding the solution. Methods such as the projected stochastic gradient ascent can be used to maximize F over this polytope. Critically, we have to assure that the computed local optima are good in expectation. Unfortunately, the multilinear extension F lacks concavity and therefore may have bad local optima. Hence, we consider concave continuous extensions of F that are efficiently computable, and at most a constant factor away from F to ensure solution quality. As a result, such a concave extension F? could then be efficiently maximized over a polytope using projected stochastic gradient ascent which would enable the application of modern continuous optimization techniques. One class of important functions for which such an extension can be efficiently computed is the class of weighted coverage functions. The class of weighted coverage functions (WCF). Let U be a set and let g be a nonnegative P modular function on U , i.e. g(S) = u?S w(u), S ? U . Let V = {B1 , . . . , Bn } be a collection of subsets of U . The weighted coverage function f : 2V ?? R+ defined as
S ?S ? V : f (S) = g Bi ?S Bi is monotone submodular. For all u ? U , let us denote by Pu := {Bi ? V | u ? Bi } and by I(?) the indicator function. The multilinear extension of f can be expressed in a more compact way: X F (x) = ES [f (S)] = ES I(u ? Bi for some Bi ? S) ? w(u) u?U = X w(u) ? P(u ? Bi for some Bi ? S) = u?U X   Q w(u) 1 ? Bi ?Pu (1 ? xi ) (3) u?U where we used the fact that each element Bi ? V was chosen with probability xi . Concave upper bound for weighted coverage functions. To efficiently compute a concave upper bound on the multilinear extension we use the framework of Seeman and Singer [30]. Given that all the weights w(u), u ? U in (3) are non-negative, we can construct a concave upper bound for the multilinear extension F (x) using the following Lemma. Proofs can be found in the Appendix ??. Q` Lemma 1. For x ? [0, n 1]` define ?(x) := 1 ? i=1 (1 ? xi ). Then the Fenchel concave biconjugate o P` of ?(?) is ?(x) := min 1, i=1 xi . Also (1 ? 1/e) ?(x) ? ?(x) ? ?(x) ?x ? [0, 1]` . Furthermore, ? is an extension of ?, i.e. ?x ? {0, 1}` : ?(x) = ?(x). Consequently, given a weighted coverage function f with F (x) represented as in (3), we can define  X  X F? (x) := w(u) min 1, xv (4) Bv ?Pu u?U and conclude using Lemma 1 that (1 ? 1/e)F? (x) ? F (x) ? F? (x), as desired. Furthermore, F? has three interesting properties: (1) It is a concave function over [0, 1]V , (2) it is equal to f on vertices of the hypercube, i.e. for x ? {0, 1}n one has F? (x) = f ({i : xi = 1}), and (3) it can be computed efficiently and deterministically given access to the sets Pu , u ? U . In other words, we can compute the value of F? (x) using at most O(|U | ? |V |) operations. Note that F? is not the tightest concave upper bound of F , even though we use the tightest concave upper bounds for each term of F . Optimizing the concave upper bound by stochastic gradient ascent. Instead of maximizing F over a polytope P, one can now attempt to maximize F? over P. Critically, this task can be done efficiently, as F? is concave, by using projected stochastic gradient ascent. In particular, one can 4 Algorithm 1 Stochastic Submodular Maximization via concave relaxation Require: matroid M with base polytope P, ?t (step size), T (maximum # of iterations) 1: x(0) ? starting point in P 2: for t ? 0 to T ? 1 do 3: Choose gt at random from a distribution such that E[gt |x(0) , . . . , x(t) ] ? ? F? (x(t) ) 4: x(t+1/2) ? x(t) + ?t gt 5: x(t+1) ? ProjectP (x(t+1/2) ) 6: end for P ? T ? T1 Tt=1 x(t) 7: x 8: S ? R ANDOMIZED -P IPAGE -ROUND(? xT ) 9: return S such that S ? M, E[f (S)] ? (1 ? 1/e)f (OP T ) ? ?(T ). control the convergence speed by choosing from the toolbox of modern continuous optimization ? ? , and algorithms, such as S GD, A DAG RAD and A DAM. Let us denote a maximizer of F? over P by x ? also a maximizer of F over P by x . We can thus write F (? x? ) ? (1 ? 1/e)F? (? x? ) ? (1 ? 1/e)F? (x? ) ? (1 ? 1/e)F (x? ), which is the exact guarantee that previous methods give, and in general is the best near-optimality ratio that one can give in poly-time. Finally, to round the continuous solution we may apply R ANDOMIZED P IPAGE -ROUNDING [7] as the quality of the approximation is preserved in expectation. Matroid constraints. Constrained optimization can be efficiently performed by projected gradient ascent whereby after each step of the stochastic ascent, we need to project the solution back onto the feasible set. For the case of matroid constraints, it is sufficient to consider projection onto the matroid base polytope. This problem of projecting on the base polytope has been widely studied and fast algorithms exist in many cases [2, 5, 29]. While these projection algorithms were used as a key subprocedure in constrained submodular minimization, here we consider them for submodular maximization. Details of a fast projection algorithm for the problems considered in this work are presented the Appendix ??. Algorithm 1 summarizes all steps required to maximize f subject to matroid constraints. Convergence rate. Since we are maximizing a concave function F? (?) over a matroid base polytope P, convergence rate (and hence running time) depends on B := maxx?P ||x||, as well as maximum 1 gradient norm ?? (i.e. ||gt || ? ? with probability 1). In the case of the base polytope for a matroid of rank r, B is r, since each vertex of the polytope has exactly r ones. Also, from (4), one can build a rough upper bound for the norm of the gradient:
1/2 X P ||g|| ? || u?U w(u)1Pu || ? max|Pu | w(u), u?U u?U which depends on the weights w(u) as well as |Pu | and is hence problem-dependent. We will provide tighter ? upper bounds for gradient norm in our specific examples in the later sections. With ?t = B/? t, and classic results for SGD [31], we have that ? F? (x? ) ? E[F? (? xT )] ? B?/ T , ? T is the final outcome of SGD (see Algorithm 1). where T is the total number of SGD iterations and x Therefore, for a given ? > 0, after T ? B 2 ?2 /?2 iterations, we have F? (x? ) ? E[F? (? xT )] ? ?. Summing up, we will have the following theorem: Theorem 2. Let f be a weighted coverage function, P be the base polytope of a matroid M, and ? and B be as above. Then for each  > 0, Algorithm 1 after T = B 2 ?2 /?2 iterations, produces a set S ? ? M such that E[f (S ? )] ? (1 ? 1/e) maxS?M f (S) ? ?. 1 Note that the function F? is neither smooth nor strongly concave as functions such as min{1, x} are not smooth or strongly concave. 5 Remark. Indeed this approximation ratio is the best ratio one can achieve, unless P=NP [10]. A key point to make here is that our approach also works for more general constraints (in particular is efficient for simple matroids such as partition matroids). In the latter case, G REEDY only gives 1 2 -approximation and fast discrete methods like S TOCHASTIC -G REEDY [27] do not apply, whereas our method still yields an (1 ? 1/e)-optimal solution. Time Complexity. One can compute an upper bound for the running time of Algorithm 1 by estimating the time required to perform gradient computations, projection on P, and rounding. For the case of uniform matroids, projection and rounding take O(n log n) and O(n) time, respectively (see Appendix ??). Furthermore, for the applications considered in this work, namely expected influence maximization and exemplar-based clustering, we provide linear time algorithms to compute the gradients. Also when our matroid is the k-uniform matroid (i.e. k-cardinality constraint), we have ? B = k. By Theorem 2, the total computational complexity of our algorithm is O(?2 kn(log n)/?2 ). Comparison to G REEDY. Let us relate our results to the classical approach. When running the Ps G REEDY algorithm in the stochastic setting, one estimates f?(S) := 1s i=1 f?i (S) where ?1 , . . . , ?s are i.i.d. samples from ?. The following proposition bounds the sample and computational complexity of G REEDY. The proof is detailed in the Appendix ??. Proposition 3. Let f be a submodular function defined as (2). Suppose 0 ? f? (S) ? H for all S ? V and all ? ? ?. Assume S ? denotes the optimal solution for f subject to k-cardinality constraint and Sk denotes the solution computed by the greedy algorithm on f? after k steps. Then, in order to guarantee P[f (Sk ) ? (1 ? 1/e)f (S ? ) ? ?] ? 1 ? ?, it is enough to have   s ? ? H 2 (k log n + log(1/?))/?2 , i.i.d. samples from ?. The running time of G REEDY is then bounded by   2 2 O ? H nk(k log n + log(1/?))/? , where ? is an upper bound on the computation time for a single evaluation of f? (S). As an example, let us compare the worst-case complexity bound obtained for SGD (i.e. O(?2 kn(log n)/?2 )) with that of G REEDY for the influence maximization problem. Each single function evaluation for G REEDY amounts to computing the total influence of a set in a sample graph, which makes ? = O(n) (here we assume our sample graphs satisfy ? |E| = O(|V |)). Also, a crude upper bound for the size of the gradient for each sample function is H n (see Appendix ??). Hence, we can deduce that SGD can have a factor k speedup w.r.t. to G REEDY. 4 Applications We will now show how to instantiate the stochastic submodular maximization framework using several prototypical discrete optimization problems. Influence maximization. As discussed in Section 2, the Independent Cascade [19] model defines a distribution G over instances G ? G that share a set V of nodes. The influence fG (S) of a set of nodes S in instance G is the fraction of nodes reachable from S using the edges E(G). The following Lemma shows that the influence belongs to the class of WCF. Lemma 4. The influence function fG (?) is a WCF. Moreover, Q 1 X FG (x) = ES [fG (S)] = (1 ? u?Pv (1 ? xu )) (5) |V | v?V P 1 X F?G (x) = min{1, u?Pv xu }, (6) |V | v?V where Pv is the set of all nodes having a (directed) path to v. 6 We return to the problem of maximizing fG (S) = EG?G [fG (S)] given a distribution over graphs G sharing nodes V . Since fG is a weighted sum of submodular functions, it is submodular. Moreover, F (x) = ES [fG (S)] = ES [EG [fG (S)]] = EG [ES [fG (S)]] = EG [FG (x)] " # Q 1 X (1 ? u?Pv (1 ? xu )) . = EG |V | v?V Let U be the uniform distribution over vertices. Then,     P Q Q 1 F (x) = EG |V | v?V (1 ? u?Pv (1 ? xu )) = EG Ev?U [1 ? u?Pv (1 ? xu )] , (7) and the corresponding upper bound would be     P F? (x) = EG Ev?U min{1, u?Pv xu } . (8) This formulation proves to be helpful in efficient calculation of subgradients, as one can obtain a random subgradient in linear time. For more details see Appendix ??. We also provide a more efficient, biased estimator of the expectation in the Appendix. ? E) be a complete weighted bipartite graph with parts X and Y Facility location. Let G = (X ?Y, and nonnegative weights wx,y . The weights can be considered as utilities or some similarity metric. We select a subset S ? X and each y ? Y selects s ? S with the highest weight ws,y . Our goal is to maximize the average weight of these selected edges, i.e. to maximize 1 X f (S) = max ws,y (9) s?S |Y | y?Y given some constraints on S. This problem is indeed the Facility Location problem, if one takes X to be the set of facilities and Y to be the set of customers and wx,y to be the utility of facility x for customer y. Another interesting instance is the Exemplar-based Clustering problem, in which X = Y is a set of objects and wx,y is the similarity (or inverted distance) between objects x and y, and one tries to find a subset S of exemplars (i.e. centroids) for these objects. The stochastic nature of this problem is revealed when one writes (9) as the expectation f (S) = Ey?? [fy (S)], where ? is the uniform distribution over Y and fy (S) := maxs?S ws,y . One can also consider this more general case, where y?s are drawn from an unknown distribution, and one tries to maximize the aforementioned expectation. First, we claim that fy (?) for each y ? Y is again a weighted coverage function. For simplicity, let . . X = {1, . . . , n} and set mi = wi,y , with m1 ? ? ? ? ? mn and mn+1 = 0. Lemma 5. The utility function fy (?) is a WCF. Moreover, Pn Qi Fy (x) = i=1 (mi ? mi+1 )(1 ? j=1 (1 ? xj )), (10) P P n i F?y (x) = i=1 (mi ? mi+1 ) min{1, j=1 xj }. (11) We remark that the gradient of both Fy and F?y can be computed in linear time using a recursive procedure. We refer to Appendix ?? for more details. 5 Experimental Results We demonstrate the practical utility of the proposed framework and compare it to standard baselines. We compare the performance of the algorithms in terms of their wall-clock running time and the obtained utility. We consider the following problems: ? Influence Maximization for the Epinions network2 . The network consists of 75 879 nodes and 508 837 directed edges. We consider the subgraph induced by the top 10 000 nodes with the largest out-degree and use the independent cascade model [19]. The diffusion model is specified by a fixed probability for each node to influence its neighbors in the underlying graph. We set this probability p to be 0.02, and chose the number of seeds k = 50. 2 http://snap.stanford.edu/ 7 Blogs 14 CIFAR-10 ? = 0.1 2.22 ? = 0.01 12 ? = 0.01 T = 100K T = 8K 10 2.20 ? = 0.3 ? = 0.1 Utility Utility ? = 0.5 8 ? = 0.9 6 2 0 100 200 2.18 ? = 0.3 ? = 0.5 T = 8K 2.16 SSM/AdaGrad L AZY-G REEDY L AZY-S TOCH -G REEDY R ANDOM -S ELECT 4 ? = 0.9 SSM/AdaGrad L AZY-G REEDY L AZY-S TOCH -G REEDY R ANDOM -S ELECT 2.14 300 0 20 40 60 Cost (seconds) Cost (seconds) Epinions Epinions (Partition Matroid) 16.5 T = 106 0.17 ? = 0.01 ? = 0.1 15.5 0.16 Utility Utility 16.0 ? = 0.5 0.15 15.0 ? = 0.9 14.5 14.0 0 500 SSM/AdaGrad L AZY-G REEDY L AZY-S TOCH -G REEDY R ANDOM -S ELECT 1000 0.14 SSM/AdaGrad L AZY-G REEDY 0 1500 10 20 30 40 50 k (# of seeds) Cost (seconds) Figure 1: In the case of Facility location for Blog selection as well as on influence maximization on Epinions, the proposed approach reaches the same utility significantly faster. On the exemplarbased clustering of CIFAR, the proposed approach is outperformed by S TOCHASTIC -G REEDY, but nevertheless reaches 98.4% of the G REEDY utility in a few seconds (after less than 1000 iterations). On Influence Maximization over partition matroids, the proposed approach significantly outperforms G REEDY. ? Facility Location for Blog Selection. We use the data set used in [14], consisting of 45 193 blogs, and 16 551 cascades. The goal is to detect information cascades/stories spreading over the blogosphere. This dataset is heavy-tailed, hence a small random sample of the events has high variance in terms of the cascade sizes. We set k = 100. ? Exemplar-based Clustering on CIFAR-10. The data set contains 60 000 color images with resolution 32 ? 32. We use a single batch of 10 000 images and compare our algorithms to variants of G REEDY over the full data set. We use the Euclidean norm as the distance function and set k = 50. Further details about preprocessing of the data as well as formulation of the submodular function can be found in Appendix ??. Baselines. In the case of cardinality constraints, we compare our stochastic continuous optimization approach against the most efficient discrete approaches (L AZY-)G REEDY and (L AZY-)S TOCHASTIC G REEDY, which both provide optimal approximation guarantees. For S TOCHASTIC -G REEDY, we vary the parameter ? in order to explore the running time/utility tradeoff. We also report the performance of randomly selected sets. For the two facility location problems, when applying the greedy variants we can evaluate the exact objective (true expectation). In the Influence Maximization application, computing the exact expectation is intractable. Hence, we use an empirical average of s samples (cascades) from the model. We note that the number of samples suggested by Proposition 3 is overly conservative, and instead we make a practical choice of s = 103 samples. 8 Results. The results are summarized in Figure 1. On the blog selection and influence maximization applications, the proposed continuous optimization approach outperforms S TOCHASTIC -G REEDY in terms of the running time/utility tradeoff. In particular, for blog selection we can compute a solution with the same utility 26? faster than S TOCHASTIC -G REEDY with ? = 0.5. Similarly, for influence maximization on Epinions we the solution 88? faster than S TOCHASTIC -G REEDY with ? = 0.1. On the exemplar-based clustering application S TOCHASTIC -G REEDY outperforms the proposed approach. We note that the proposed approach is still competitive as it recovers 98.4% of the value after less than thousand iterations. We also include an experiment on Influence Maximization over partition matroids for the Epinions network. In this case, G REEDY only provides a 1/2 approximation guarantee and S TOCHASTIC G REEDY does not apply. To create the partition, we first sorted all the vertices by their out-degree. Using this order on the vertices, we divided the vertices into two partitions, one containing vertices with even positions, other containing the rest. Figure 1 clearly demonstrates that the proposed approach outperforms G REEDY in terms of utility (as well as running time). Acknowledgments The research was partially supported by ERC StG 307036. We would like to thank Yaron Singer for helpful comments and suggestions. References [1] Alexander A Ageev and Maxim I Sviridenko. Pipage rounding: A new method of constructing algorithms with proven performance guarantee. Journal of Combinatorial Optimization, 8(3): 307?328, 2004. [2] Francis Bach et al. Learning with submodular functions: A convex optimization perspective. R in Machine Learning, 6(2-3):145?373, 2013. Foundations and Trends [3] Francis R. Bach. 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Affinity Clustering: Hierarchical Clustering at Scale MohammadHossein Bateni Google Research bateni@google.com MohammadTaghi Hajiaghayi? University of Maryland hajiagha@cs.umd.edu Soheil Behnezhad? University of Maryland soheil@cs.umd.edu Raimondas Kiveris Google Research rkiveris@google.com Mahsa Derakhshan? University of Maryland mahsaa@cs.umd.edu Silvio Lattanzi Google Research silviol@google.com Vahab Mirrokni Google Research mirrokni@google.com Abstract Graph clustering is a fundamental task in many data-mining and machine-learning pipelines. In particular, identifying a good hierarchical structure is at the same time a fundamental and challenging problem for several applications. The amount of data to analyze is increasing at an astonishing rate each day. Hence there is a need for new solutions to efficiently compute effective hierarchical clusterings on such huge data. The main focus of this paper is on minimum spanning tree (MST) based clusterings. In particular, we propose affinity, a novel hierarchical clustering based on Bor? uvka?s MST algorithm. We prove certain theoretical guarantees for affinity (as well as some other classic algorithms) and show that in practice it is superior to several other state-of-the-art clustering algorithms. Furthermore, we present two MapReduce implementations for affinity. The first one works for the case where the input graph is dense and takes constant rounds. It is based on a Massively Parallel MST algorithm for dense graphs that improves upon the state-of-the-art algorithm of Lattanzi et al. [34]. Our second algorithm has no assumption on the density of the input graph and finds the affinity clustering in O(log n) rounds using Distributed Hash Tables (DHTs). We show experimentally that our algorithms are scalable for huge data sets, e.g., for graphs with trillions of edges. 1 Introduction Clustering is a classic unsupervised learning problem with many applications in information retrieval, data mining, and machine learning. In hierarchical clustering the goal is to detect a nested hierarchy of clusters that unveils the full clustering structure of the input data set. In this work we study the hierarchical clustering problem on real-world graphs. This problem has received a lot of attention in recent years [13, 16, 41] and new elegant formulations and algorithms have been introduced. Nevertheless many of the newly proposed techniques are sequential, hence difficult to apply on large data sets. ? Supported in part by NSF CAREER award CCF-1053605, NSF BIGDATA grant IIS-1546108, NSF AF:Medium grant CCF-1161365, DARPA GRAPHS/AFOSR grant FA9550-12-1-0423, and another DARPA SIMPLEX grant. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. With the constant increase in the size of data sets to analyze, it is crucial to design efficient large-scale solutions that can be easily implemented in distributed computing platforms (such as Spark [45] and Hadoop [43] as well as MapReduce and its extension Flume [17]), and cloud services (such as Amazon Cloud or Google Cloud). For this reason in the past decade several papers proposed new distributed algorithms for classic computer science and machine learning problems [3, 4, 7, 14, 15, 19]. Despite these efforts not much is known about distributed algorithms for hierarchical clustering. There are only two works analyzing these problems [27, 28], and neither gives any theoretical guarantees on the quality of their algorithms or on the round complexity of their solutions. In this work we propose new parallel algorithms in the MapReduce model to compute hierarchical clustering and we analyze them from both theoretical and experimental perspectives. The main idea behind our algorithms is to adapt clustering techniques based on classic minimum spanning tree algorithms such as Bor? uvka?s algorithm [11] and Kruskal?s algorithm [33] to run efficiently in parallel. Furthermore we also provide a new theoretical framework to compare different clustering algorithms based on the concept of a ?certificate? and show new interesting properties of our algorithms. We can summarize our contribution in four main points. First, we focus on the distributed implementations of two important clustering techniques based on classic minimum spanning tree algorithms. In particular we consider linkage-based clusterings inspired by Kruskal?s algorithm and a novel clustering called affinity clustering based on Bor? uvka?s algorithm. We provide new theoretical frameworks to compare different clustering algorithms based on the concept of a ?certificate? as a proof of having a good clustering and show new interesting properties of both affinity and single-linkage clustering algorithms. Then, using a connection between linkage-based clustering, affinity clustering and the minimum spanning tree problem, we present new efficient distributed algorithms for the hierarchical clustering problem in a MapReduce model. In our analysis we consider the most restrictive model for distributed computing, called Massively Parallel Communication, among previously studied MapReduce-like models [10, 23, 30]. Along the way, we obtain a constant round MapReduce algorithm for minimum spanning tree (MST) of dense graphs (in Section 5). Our algorithm for graphs with ?(n1+c ) edges ? 1+ ) space and for any given  with 0 <  < c < 1, finds the MST in dlog(c/)e+1 rounds using O(n per machine and O(nc? ) machines (i.e., optimal total space). This improves the round complexity of the state-of-the-art MST algorithm of Lattanzi et al. [34] for dense graphs which requires up to dc/e rounds using the same number of machines and space. Prior to our work, no hierarchical clustering algorithm was known in this model. Then we turn our attention to real world applications and we introduce efficient implementations of affinity clustering as well as classic single-linkage clustering that leverage Distributed Hash Tables (DHTs) [12, 31] to speed up computation for huge data sets. Last but not least, we present an experimental study where we analyze the scalability and effectiveness of our newly introduced algorithms and we observe that, in most cases, affinity clustering outperforms all state-of-the-art algorithms from both quality and scalability standpoints.2 2 Related Work Clustering and, in particular, hierarchical clustering techniques have been studied by hundreds of researchers [16, 20, 22, 32]. In social networks, detecting the hierarchical clustering structure is a basic primitive for studying the interaction between nodes [36, 39]. Other relevant applications of hierarchical clustering can be found in bioinformatics, image analysis and text classification. Our paper is closely related to two main lines of research. The first one focuses on studying theoretical properties of clustering approaches based on minimum spanning trees (MSTs). Linkagebased clusterings (often based on Kruskal?s algorithm) have been extensively studied as basic techniques for clustering datasets. The most common linkage-based clustering algorithms are singlelinkage, average-linkage and complete-linkage algorithms. In [44], Zadeh and Ben-David gave a characterization of the single-linkage algorithm. Their result has been then generalized to linkagebased algorithms in [1]. Furthermore single-linkage algorithms are known to provably recover a ground truth clustering if the similarity function has some stability properties [6]. In this paper we 2 Implementations are available at https://github.com/MahsaDerakhshan/AffinityClustering. 2 introduce a new technique to compare clustering algorithms based on ?certificates.? Furthermore we introduce and analyze a new algorithm?affinity?based on Bor? uvka?s well-known algorithm. We show that affinity is not only scalable for huge data sets but also its performance is superior to several uvka?s algorithm is a state-of-the-art clustering algorithms. To the best of our knowledge though Bor? well-known and classic algorithm, not many clustering algorithms have been considered based on Bor? uvka?s. The second line of work is closely related to distributed algorithms for clustering problems. Several models of MapReduce computation have been introduced in the past few years [10, 23, 30]. The first paper that studied clustering problems in these models is by Ene et al. [18], where the authors prove that any ? approximation algorithm for the k-center or k-median problems can produce 4? + 2 and 10? + 3 approximation factors, respectively, for the k-center or k-median problems in the MapReduce model. Subsequently several papers [5, 7, 8] studied similar problems in the MapReduce model. A lot of efforts also went into studying efficient algorithms on graphs [3, 4, 7, 15, 14, 19]. However the problem of hierarchical clustering did not receive a lot of attention. To the best of our knowledge there are only two papers [27, 28] on this topic, and neither analyzes the problem formally or proves any guarantee in any MapReduce model. 3 Minimum Spanning Tree-Based Clusterings We begin by going over two famous algorithms for minimum spanning tree and define the corresponding algorithms for clustering. uvka?s algorithm [11], first published in 1926, is Bor? uvka?s algorithm and affinity clustering: Bor? an algorithm for finding a minimum spanning tree (MST)3 . The algorithm was rediscovered a few times, in particular by Sollin [42] in 1965 in the parallel computing literature. Initially each vertex forms a group (cluster) by itself. The algorithm begins by picking the cheapest edge going out of each cluster, in each round (in parallel) joins these clusters to form larger clusters and continues joining in a similar manner until a tree spanning all vertices is formed. Since the size of the smallest cluster at least doubles each time, the number of rounds is at most O(log n). In affinity clustering, we uvka?s algorithm after r > 0 rounds when for the first time we have at most k clusters for a stop Bor? desired number k > 0. In case the number of clusters is strictly less than k, we delete the edges that we added in the last round in a non-increasing order (i.e., we delete the edge with the highest weight first) to obtain exactly k clusters. To the best of our knowledge, although Bor? uvka?s algorithm is a well-known and classic algorithm, clustering algorithms based on it have not been considered much. uvka?s algorithm: each cluster here A natural hierarchy of nodes can be obtained by continuing Bor? will be a subset of future clusters. We call this hierarchical affinity clustering. We present distributed implementations of Bor? uvka/affinity in Section 5 and show its scalability even for huge graphs. We also show affinity clustering, in most cases, works much better than several well-known clustering algorithms in Section 6. Kruskal?s algorithm and single-linkage clustering: Kruskal?s algorithm [33] first introduced in 1956 is another famous algorithm for finding MST. The algorithm is highly sequential and iteratively picks an edge of the least possible weight that connects any two trees (clusters) in the forest.4 Though the number of iterations in Kruskal?s algorithm is n ? 1 (the number of edges of any tree on n nodes), the algorithm can be implemented in O(m log n) time with simple data structures (m is the number of edges) and in O(ma(n)) time using a more sophisticated disjoint-set data structure, where a(.) is the extremely slowly growing inverse of the single-valued Ackermann function. In single-linkage clustering, we stop Kruskal?s algorithm when we have at least k clusters (trees) for a desired number k > 0. Again if we desire to obtain a corresponding hierarchical single-linkage clustering, by adding further edges which will be added in Kruskal?s algorithm later, we can obtain a natural hierarchical clustering (each cluster here will be a subset of future clusters). As mentioned above, Kruskal?s Algorithm and single-linkage clustering are highly sequential, however as we show in Section 5 thinking backward once we have an efficient implementation of Bor? uvka?s 3 More precisely the algorithm works when there is a unique MST, in particular, when all edge weights are distinct; however this can be easily achieved by either perturbing the edge weights by an  > 0 amount or have a tie-breaking ordering for edges with the same weights 4 Unlike Bor? uvka?s method, this greedy algorithm has no limitations on the distinctness of edge weights. 3 (or any MST algorithm) in Map-Reduce and using Distributed Hash Tables (DHTs), we can achieve an efficient parallel implementation of single-linkage clustering as well. We show scalability of this implementation even for huge graphs in Section 5 and its performance in experiments in Section 6. 4 Guaranteed Properties of Clustering Algorithms An important property of affinity clustering is that it produces clusters that are roughly of the same size. This is intuitively correct since at each round of the algorithm, each cluster is merged to at least one other cluster and as a result, the size of even the smallest cluster is at least doubled. In fact linkage based algorithms (and specially single linkage) are often criticized for producing uneven clusters; therefore it is tempting to give a theoretical guarantee for the size ratio of the clusters that affinity produces. Unfortunately, as it is illustrated in Figure 1, we cannot give any worst case bounds since even in one round we may end up having a cluster of size ?(n) and another cluster of size O(1). As the first property, we show that at least in the first round, this does not happen when the observations are randomly distributed. Our empirical results on real world data sets in Section 6.1, further confirm this property for all rounds, and on real data sets. Figure 1: An example of how affinity may produce a large component in one round. We start by defining the nearest neighbor graph. Definition 1 (Nearest Neighbor Graph). Let S be a set of points in a metric space. The nearest neighbor graph of S, denoted by GS , has |S| vertices, each corresponding to an element in S and if a ? S is the nearest element to b ? S in S, graph GS contains an edge between the corresponding vertices of a and b. At each round of affinity clustering, all the vertices that are in the same connected component of the nearest neighbor graph will be merged together5 . Thus, it suffices to bound the connected components? size. For a random model of points, consider a Poisson point process X in Rd (d ? 1) with density 1. It has two main properties. First, the number of points in any finite region of volume V is Poisson distributed with mean V . Second, the number of points in any two disjoint regions are independent of each other. Theorem 1 (H?ggstr?m et al. [38]). For any d ? 2, consider the (Euclidean distance) nearest neighbor graph G of a realization of a Poisson point process in Rd with density 1. All connected components of G are finite almost surely. Theorem 1 implies that the size of the maximum connected component of the points within any finite region in Rd is bounded by almost a constant number. This is a very surprising result compared to the worst case scenario of having a connected component that contains all the points. Note that although the aforementioned bound holds for the first round of affinity, after the connected components are contracted, we cannot necessarily assume that the new points are Poisson distributed and the same argument cannot be used for the rest of the rounds. Next we present further properties of affinity clustering. Let us begin by introducing the concept of ?cost? for a clustering solution to be able to compare clustering algorithms. Definition 2. The cost of a cluster is the sum of edge lengths (weights) of a minimum Steiner tree connecting all vertices inside the cluster. The cost of a clustering is the sum of the costs of its clusters. Finally a non-singleton clustering of a graph is a partition of its vertices into clusters of size at least two. Even one round of affinity clustering often produces good solutions for several applications. Now we are ready to present the following extra property of the result of the first round of affinity clustering. 5 Depending on the variant of affinity that we use, the distance function will be updated. 4 Theorem 2. The cost of any non-singleton clustering is at least half of that of the clustering obtained after the first round of affinity clustering. Before presenting the proof of Theorem 2, we need to demonstrate the concept of disc painting introduced previously in [29, 2, 21, 9, 25]. In this setting, we consider a topological structure of a graph metric in which each edge is a curve connecting its endpoints whose length is equal to its weight. We assume each vertex has its own color. A disc painting is simply a set of disjoint disks centered at terminals (with the same colors of the center vertices). A disk of radius r centered at vertex v paints all edges (or portions) of them which are at distance r from vertex v with the color of v. Thus we paint (portions of) edges by different disks each corresponding to a vertex and each edge can be painted by at most two disks. With this definition of disk painting, we now demonstrate the proof of Theorem 2. Next we turn our focus to obtain structural properties for single-linkage clustering. We denote by Fk the set of edges added after k iterations of Kruskal, i.e., when we have n ? k clusters in single-linkage clustering. Note that Fk is a forest, i.e., a set of edges with no cycle. First we start with an important observation whose proof comes directly from the description of the single-linkage algorithm. Proposition 3. Suppose we run single-linkage clustering until we have n ? k clusters. Let doutside be the minimum distance between any two clusters and dinside be the maximum distance of any edge added to forest Fk . Then doutside ? dinside . We note that Proposition 3 demonstrates the following important property of single-linkage clustering: Each vertex of a cluster at any time has a neighbor inside to which is closer than any other vertex outside of its clusters. Next we define another criterion for desirability of a clustering algorithm. This generalizes Proposition 3. Definition 3. An ?-certificate for a clustering algorithm, where ? ? 1, is an assignment of shares to each vertex of the graph with the following two properties: (1) The cost of each cluster is at most ? times the sum of shares of vertices inside the cluster; (2) For any set S of vertices containing at most one from each cluster in our solution, the imaginary cluster S costs at least the sum of shares of vertices in S. Note that intuitively the first property guarantees that vertices inside each cluster can pay the cost of their corresponding cluster and that there is no free-rider. The second property intuitively implies we cannot find any better clustering by combining vertices from different clusters in our solution. Next we show that there always exists a 2-certificate for single-linkage clustering guaranteeing its worst-case performance. Theorem 4. Single-linkage always produces a clustering solution that has a 2-certificate. 5 5.1 Distributed Algorithms Constant Round Algorithm For Dense Graphs Unsurprisingly, finding the affinity clustering of a given graph G is closely related to the problem of finding its Minimum Spanning Tree (MST). In fact, we show the data that is encoded in the MST of G is sufficient for finding its affinity clustering (Theorem 9). This property is also known to be true for single linkage [24]. For MapReduce algorithms this is particularly useful because the MST requires a substantially smaller space than the original graph and can be stored in one machine. Therefore, once we have the MST, we can obtain affinity or single linkage in one round. The main contribution of this section is an algorithm for finding the MST (and therefore the affinity clustering) of dense graphs in constant rounds of MapReduce which improves upon prior known dense graph MST algorithms of Karloff et al. [30] and Lattanzi et al. [34]. Theoretical Model. Let N denote the input size. There are a total number of M machines and each of them has a space of size S. Both S and M must be substantially sublinear in N . In each round, the machines can run an arbitrary polynomial time algorithm on their local data. No communication is allowed during the rounds but any two machines can communicate with each other between the rounds as long as the total communication size of each machine does not exceed its memory size. 5 Algorithm 1 MST of Dense Graphs Input: A weighted graph G Output: The minimum spanning tree of G 1: function MST(G = (V, E), ) 2: c ? logn (m/n) . Since G is assumed to be dense we know c > 0. 3: while |E| > O(n1+ ) do 4: R EDUCE E DGES(G, c) 5: c ? (c ? )/2 6: Move all the edges to one machine and find MST of G in there. 7: function R EDUCE E DGES(G = (V, E), c) 8: k ? n(c?)/2 9: Independently and u.a.r. partition V into k subsets {V1 , . . . , Vk }. 10: Independently and u.a.r. partition V into k subsets {U1 , . . . , Uk }. 11: Let Gi,j be a subgraph of G with vertex set Vi ? Uj containing any edge (v, u) ? E(G) where v ? Vi and u ? Uj . 12: for any i, j ? {1, . . . , k} do 13: Send all the edges of Gi,j to the same machine and find its MST in there. 14: Remove an edge e from E(G) , if e ? Gi,j and it is not in MST of Gi,j . This model is called Massively Parallel Communication (MPC) in the literature and is ?arguably the most popular one? [26] among MapReduce like models. Theorem 5. Let G = (V, E) be a graph with n vertices and n1+c edges for any constant c > 0 and let w : E 7? R+ be its edge weights. For any given  such that 0 <  < c, there exists a randomized algorithm for finding the MST of G that runs in at most dlog (c/)e + 1 rounds of MPC where ? 1+ ) with high probability and the total number of required every machine uses a space of size O(n machines is O(nc? ). ? 1+c )) on all machines to store the input. Our algorithm, therefore, uses only enough total space (O(n The following observation is mainly used by Algorithm 1 to iteratively remove the edges that are not part of the final MST. Lemma 6. Let G0 = (V 0 , E 0 ) be a (not necessarily connected) subgraph of the input graph G. If an edge e ? E 0 is not in the MST of G0 , then it is not in the MST of G either. To be more specific, we iteratively divide G into its subgraphs, such that each edge of G is at least in one subgraph. Then, we handle each subgraph in one machine and throw away the edges that are not in their MST. We repeat this until there are only O(n1+ ) edges left in G. Then we can handle all these edges in one machine and find the MST of G. Algorithm 1 formalizes this process. Lemma 7. Algorithm 1 correctly finds the MST of the input graph in dlog (c/)e + 1 rounds. By Lemma 6 we know any edge that is removed from is not part of the MST therefore it suffices to prove the while loop in Algorithm 1 takes dlog (c/)e + 1 iterations. ? 1+ ) with high probability. Lemma 8. In Algorithm 1, every machine uses a space of size O(n The combination of Lemma 7 and Lemma 8 implies that Algorithm 1 is indeed in MPC and Theorem 5 holds. See supplementary material for omitted proofs. The next step is to prove all the information that is required for affinity clustering is indeed contained in the MST. Theorem 9. Let G = (V, E) denote an arbitrary graph, and let G0 = (V, E 0 ) denote the minimum spanning tree of G. Running affinity clustering algorithm on G gives the same clustering of V as running this algorithm on G0 . By combining the MST algorithm given for Theorem 5 and the sufficiency of MST for computing affinity clustering (Theorem 9) and single linkage ([24]) we get the following corollary. Corollary 10. Let G = (V, E) be a graph with n vertices and n1+c edges for any constant c > 0 and let w : E 7? R+ be its edge weights. For any given  such that 0 <  < c, there exists a 6 randomized algorithm for affinity clustering and single linkage that runs in dlog (c/)e + 1 rounds of ? 1+ ) with high probability and the total number MPC where every machine uses a space of size O(n c? of required machines is O(n ). 5.2 Logarithmic Round Algorithm For Sparse Graphs Consider a graph G(V, E) on n = |V | vertices, with edge weights w : E 7? R. We assume that the edge weights denote distances. (The discussion applies mutatis mutandis to the case where edge weights signify similarity.) The algorithm works for a fixed number of synchronous rounds, or until no further progress is made, say, by reaching a single cluster of all vertices. Each round consists of two steps: First, every vertex picks its best edge (i.e., that of the minimum weight) at each round; and then the graph is contracted along the selected edges. (See Algorithm 2 in the appendix.) For a connected graph, the algorithm continues until a single cluster of all vertices is obtained. The supernodes at different rounds can be thought of as a hierarchical clustering of the vertices. While the first step of each round has a trivial implementation in MapReduce, the latter might take ?(log n) MapReduce rounds to implement, as it is an instance of the connected components problem. Using a DHT was shown to significantly improve the running time here, by implementing the operation in one round of MapReduce [31]. Basically we have a read-only random-access table mapping each vertex to its best neighbor. Repeated lookups in the table allows each vertex to follow the chain of best neighbors until a loop (of length two) is encountered. This assigns a unique name for each connected component; then all the vertices in the same component are reduced into a supernode. Theorem 11. The affinity clustering algorithm runs in O(log n) rounds of MapReduce when we have access to a distributed hash table (DHT). Without the DHT, the algorithm takes O(log2 n) rounds. 6 Experiments 6.1 Quality Analysis In this section, we compare well known hierarchical and flat clustering algorithms, such as k-means, single linkage, complete linkage and average linkage with different variants of affinity clustering, such as single affinity, complete affinity and average affinity. We run our experiments on several data sets from the UCI database [37] and use Euclidean distance6 . To evaluate the outputs of these algorithms we use Rand index which is defined as follows. Definition 4 (Rand index [40]). Given a set V = {v1 , . . . , vn } of n points and two clusterings X = {X1 , . . . , Xr } and Y = {Y1 , . . . , Ys } of V . Define the following. ? a: the number of pairs in V that are in the same cluster in X and in the same cluster in Y . ? b: the number of pairs in V that are in different clusters in X and in different clusters in Y .
the Rand index r(X, Y ) is defined to be (a + b)/ n2 . By having the ground truth clustering T of a data set, we define the Rand index score of a clustering X, to be r(X, T ). The Rand index based scores are in range [0, 1] and a higher number implies a better clustering. For a hierarchical clustering, the level of its corresponding tree with the highest score is used for evaluations. Figure 2 (a) compares the Rand index score of different clustering algorithms for different data sets. We observe that single affinity generally performs really well and is among the top two algorithms for most of the datasets (all except Glass). Average affinity also seems to perform well and in some cases (e.g., for Soybean data set) it produces a very high quality clustering compared to others. To summarize, linkage based algorithms do not seem to be as good as affinity based algorithms but in some cases k-means could be close. 6 We consider Iris, Wine, Soybean, Digits and Glass data sets. 7 1.0 0.8 Single Affinity 0.8 Clusters' Size Ratio Rand Index Score Algorithm Average Affinity Complete Affinity Complete Linkage 0.6 Average Linkage Single Linkage 0.4 Algorithm Single Affinity 0.6 Average Affinity Complete Affinity 0.4 Complete Linkage Average Linkage 0.2 Single Linkage k-Means k-Means 0.0 Iris Soybean Wine Glass Iris Digits Datasets Soybean Wine Glass Digits Datasets (a) (b) Figure 2: Comparison of clustering algorithms based on their Rand index score (a) and clusters size ratio (b). Table 1: Statistics about datasets used. (Numbers for ImageGraph are approximate.) The fifth column shows the relative running time of affinity clustering, and the last column is the speedup obtained by a ten-fold increase in parallelism. Dataset LiveJournal Orkut Friendster ImageGraph # nodes 4,846,609 3,072,441 65,608,366 2 ? 1010 # edges 7,861,383,690 42,687,055,644 1,092,793,541,014 1012 max degree 444,522 893,056 2,151,462 14000 running time 1.0 2.4 54 142 speedup 4.3 9.2 5.9 4.1 Another property of the algorithms that we measure is the clusters? size ratio. Let X = {X1 , . . . , Xr } be a clustering. We define the size ratio of X to be mini,j?[r] |Xi |/|Xj |. As it is visualized in Figure 2 (b), affinity based algorithms have a much higher size ratio (i.e., the clusters are more balanced) compared to linkage based algorithms. This confirms the property that we proved for Poisson distributions in Section 4 for real world data sets. Hence we believe affinity clustering is superior to (or at least as good as) the other methods when the dataset under consideration is not extremely unbalanced. 6.2 Scalability Here we demonstrate the scalability of our implementation of affinity clustering. A collection of public and private graphs of varying sizes are studied. These graphs have between 4 million and 20 billion vertices and from 4 billion to one trillion edges. The first three graphs in Table 1 are based on public graphs [35]. As most public graphs are unweighted, we use the number of common neighbors between a pair of nodes as the weight of the edge between them. (This is computed for all pairs, whether they form a pair in the original graph or not, and then new edges of weight zero are removed.) The last graph is based on (a subset of) an internal corpus of public images found on the web and their similarities. We note that we use the ?maximum? spanning tree variant of affinity clustering; here edge weights denote similarity rather than distance. While we cannot reveal the exact running times and number of machines used in the experiments, we report these quantities in ?normalized form.? We only run one round of affinity clustering (consisting of a ?Find Best Neighbors? and a ?Contract Graph? step). Two settings are used in the experiments. We once use W MapReduce workers and D machines for the DHT, and compare this to the case with 10W MapReduce workers and D machines for the DHT. This ten-fold increase in the number of MapReduce workers leads to four- to ten-fold decrease in the total running time for different datasets. Each running time is itself the average over three runs to reduce the effect of external network events. Table 1 also shows how the running time changes with the size of the graph. With a modest number of MapReduce workers, affinity clustering runs in less than an hour for all the graphs. 8 References [1] Margareta Ackerman, Shai Ben-David, and David Loker. Characterization of linkage-based clustering. In COLT 2010 - The 23rd Conference on Learning Theory, Haifa, Israel, June 27-29, 2010, pages 270?281, 2010. [2] Ajit Agrawal, Philip N. 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Unsupervised Transformation Learning via Convex Relaxations Tatsunori B. Hashimoto John C. Duchi Percy Liang Stanford University Stanford, CA 94305 {thashim,jduchi,pliang}@cs.stanford.edu Abstract Our goal is to extract meaningful transformations from raw images, such as varying the thickness of lines in handwriting or the lighting in a portrait. We propose an unsupervised approach to learn such transformations by attempting to reconstruct an image from a linear combination of transformations of its nearest neighbors. On handwritten digits and celebrity portraits, we show that even with linear transformations, our method generates visually high-quality modified images. Moreover, since our method is semiparametric and does not model the data distribution, the learned transformations extrapolate off the training data and can be applied to new types of images. 1 Introduction Transformations (e.g, rotating or varying the thickness of a handwritten digit) capture important invariances in data, which can be useful for dimensionality reduction [7], improving generative models through data augmentation [2], and removing nuisance variables in discriminative tasks [3]. However, current methods for learning transformations have two limitations. First, they rely on explicit transformation pairs?for example, given pairs of image patches undergoing rotation [12]. Second, improvements in transformation learning have focused on problems with known transformation classes, such as orthogonal or rotational groups [3, 4], while algorithms for general transformations require solving a difficult, nonconvex objective [12]. To tackle the above challenges, we propose a semiparametric approach for unsupervised transformation learning. Specifically, given data points x1 , . . . , xn , we find K linear transformations A1 . . . AK such that the vector from each xi to its nearest neighbor lies near the span of A1 xi . . . AK xi . The idea of using nearest neighbors for unsupervised learning has been explored in manifold learning [1, 7], but unlike these approaches and more recent work on representation learning [2, 13], we do not seek to model the full data distribution. Thus, even with relatively few parameters, the transformations we learn naturally extrapolate off the training distribution and can be applied to novel types of points (e.g., new types of images). Our contribution is to express transformation matrices as a sum of rank-one matrices based on samples of the data. This new objective is convex, thus avoiding local minima (which we show to be a problem in practice), scales to real-world problems beyond the 10 ? 10 image patches considered in past work, and allows us to derive disentangled transformations through a trace norm penalty. Empirically, we show our method is fast and effective at recovering known disentangled transformations, improving on past baseline methods based on gradient descent and expectation maximization [11]. On the handwritten digits (MNIST) and celebrity faces (CelebA) datasets, our method finds interpretable and disentangled transformations?for handwritten digits, the thickness of lines and the size of loops in digits such as 0 and 9; and for celebrity faces, the degree of a smile. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. 2 Problem statement Given a data point x ? Rd (e.g., an image) and strength scalar t ? R, a transformation is a smooth function f : Rd ? R ? Rd . For example, f (x, t) may be a rotated image. For a collection {fk }K k=1 of transformations, we consider entangled transformations, defined for a vector of strengths t ? RK PK by f (x, t) := k=1 fk (x, tk ). We consider the problem of estimating a collection of transformations P K f ? := k=1 fk? given random observations as follows: let pX be a distribution on points x and pT on transformation strength vectors t ? RK , where the components tk are independent under pT . Then iid iid for x ?i ? pX and ti ? pT , i = 1, . . . , n, we observe the transformations xi = f ? (? xi , ti ), while x ?i ? and ti are unobserved. Our goal is to estimate the K functions f1? , . . . , fK . 2.1 Learning transformations based on matrix Lie groups In this paper, we consider the subset of generic transformations defined via matrix Lie groups. These are natural as they map Rd ? Rd and form a family of invertible transformations that we can parameterize by an exponential map. We begin by giving a simple example (rotation of points in two dimensions) and using this to establish the idea of the exponential map and its linear approximation. We then use these linear approximations for transformation learning. A matrix Lie group is a set of invertible matrices closed under multiplication and inversion. In the example of rotation in two dimensions, theset of all rotationsis parameterized by the angle ?, and cos(?) ? sin(?) any rotation by ? has representation R? = . The set of rotation matrices form a sin(?) cos(?) Lie group, as R? R?? = I and the rotations are closed under composition. Linear approximation. In our context, the important property of matrix Lie groups is that for transformations near the identity, they have local linear approximations (tangent spaces, the associated Lie algebra), and these local linearizations map back into the Lie group via the exponential map [9]. As a simple example, consider the rotation R? , which satisfies R? = I + ?A + O(?2 ), where  0 ?1 A= , and R? = exp(?A) for all ? (here exp is the matrix exponential). The infinitesimal 1 0 structure of Lie groups means that such relationships hold more generally through the exponential map: for any matrix Lie group G ? Rd?d , there exists ? > 0P such that for all R ? G with kR ? Ik ? ?, there is an A ? Rd?d such that R = exp(A) = I + m?1 Am /m!. In the case that G is a one-dimensional Lie group, we have more: for each R near I, there is a t ? R satisfying R = exp(tA) = I + ? X t m Am . m! m=1 The matrix tA = log R in the exponential map is the derivative of our transformation (as A ? (R ? I)/t for R ? I small) and is analogous to locally linear neighborhoods in manifold learning [10]. The exponential map states that for transformations close to the identity, a linear approximation is accurate. For any matrix A, we can also generate a collection of associated 1-dimensional manifolds as follows: letting x ? Rd , the set Mx = {exp(tA)x | t ? R} is a manifold containing x. Given two nearby points xt = exp(tA)x and xs = exp(sA)x, the local linearity of the exponential map shows that xt = exp((t ? s)A)xs = xs + (t ? s)Axs + O((t ? s)2 ) ? xs + (t ? s)Axs . (1) Single transformation learning. The approximation (1) suggests a learning algorithm for finding a transformation from points on a one-dimensional manifold M : given points x1 , . . . , xn sampled from M , pair each point xi with its nearest neighbor xi . Then we attempt to learn a transformation matrix A satisfying xi ? xi + ti Axi for some small ti for each of these nearest neighbor pairs. As nearest neighbor distances kxi ? xi k ? 0 as n ? ? [6], the linear approximation (1) eventually holds. For a one-dimensional manifold and transformation, we could then solve the problem minimize {ti },A n X ||ti Axi ? (xi ? xi )||2 . i=1 2 (2) If instead of using nearest neighbors, the pairs (xi , xi ) were given directly as supervision, then this objective would be a form of first-order matrix Lie group learning [12]. Sampling and extrapolation. The learning problem (2) is semiparametric: our goal is to learn a transformation matrix A while considering the density of points x as a nonparametric nuisance variable. By focusing on the modeling differences between nearby (x, x) pairs, we avoid having to specify the density of x, which results in two advantages: first, the parametric nature of the model means that the transformations A are defined beyond the support of the training data; and second, by not modeling the full density of x, we can learn the transformation A even when the data comes from highly non-smooth distributions with arbitrary cluster structure. 3 Convex learning of transformations The problem (2) makes sense only for one-dimensional manifolds without superposition of transformations, so we now extend the ideas (using the exponential map and its linear approximation) to a full matrix Lie group learning problem. We shall derive a natural objective function for this problem and provide a few theoretical results about it. 3.1 Problem setup As real-world data contains multiple degrees of freedom, we learn several one-dimensional transformations, giving us the following multiple Lie group learning problem: Definition 3.1. Given data x1 . . . xn ? Rd with xi ? Rd as the nearest neighbor of xi , the nonconvex transformation learning problem objective is n X K X minimize (3) tik Ak xi ? (xi ? xi ) . t?Rd?K ,A?Rd?d i=1 k=1 2 This problem is nonconvex, and prior authors have commented on the difficulty of optimizing similar objectives [11, 14]. To avoid this difficulty, we will construct a convex relaxation. Define a matrix 2 Z ? Rn?d , where row Zi is an unrolling of the transformation that approximately takes any xi to x ?i . Then Eq. (3) can be written as min rank(Z)=K n X kmat(Zi )xi ? (xi ? xi )k2 , (4) i=1 2 where mat : Rd ? Rd?d is the matricization operator. Note the rank of Z is at most K, the number of transformations. We then relax the rank constraint to a trace norm penalty as min n X kmat(Zi )xi ? (xi ? xi )k2 + ? kZk? . (5) i=1 2 However, the matrix Z ? Rn?d is too large to handle for real-world problems. Therefore, we propose approximating the objective function by modeling the transformation matrices as weighted sums of observed transformation pairs. This idea of using sampled pairs is similar to a kernel method: we will show that the true transformation matrices A?k can be written as a linear combination of 1 rank-one matrices (xi ? xi )x> i . As intuition, assume that we are given a single point xi ? Rd and xi = ti A? xi + xi , where ti ? R is unobserved. If we approximate A? via the rank-one approximation A = (xi ? xi )x> i , then ?2 kxj k2 Axi + xi = xi . This shows that A captures the behavior of A? on a single point xi . By sampling sufficiently many examples and appropriately weighting each example, we can construct an accurate approximation over all points. 1 Section 9 of the supplemental material introduces a kernelized version that extends this idea to general manifolds. 3 Let us subsample x1 , . . . , xr (WLOG, these are the first r points). Given these samples, let us write a n?r transformation A as a weighted sum of r rank-one matrices (xj ? xj )x> . j with weights ? ? R We then optimize these weights: n X X r > (6) min ?ij (xj ? xj )xj xi ? (xi ? xi ) + ? k?k? . ? i=1 j=1 2 Next we show that with high probability, the weighted sum of O(K 2 d) samples is close in operator norm to the true transformation matrix A? (Lemma 3.2 and Theorem 3.3). 3.2 Learning one transformation via subsampling We begin by giving the intuition behind the sampling based objective in the one-transformation case. The correctness of rank-one reconstruction is obvious for the special case where the number of samples r = d, and for each i we define xi = ei , where ei is the i-th canonical basis vector. In this case xi = ti A? ei + ei for some unknown tP can easily reconstruct A? with a weighted i ? R. Thus weP ?1 ? > combination of rank-one samples as A = i A ei ei = i ?i (xi ? xi )x> i with ?i = ti . In the general case, we observe the effects of A? on a non-orthogonal set of vectors x1 . . . xr as xi ? xi = ti A? xi . A similar argument follows by changing our basis to make ti xi the i-th canonical basis vector and reconstructing A? in this new basis. The change of basis matrix for this case is the Pr ?1/2 map ? where ? = i=1 xi x> i /r. Our lemma below makes the intuition precise and shows that given r > d samples, there exists P ?1 weights ? ? Rd such that A? = i ?i (xi ? xi )x> , where ? is the inner product matrix from i ? above. This justifies our objective in Eq. (6), since we can whiten x to ensure ? = I, and there exists weights ?ij which minimizes the objective by reconstructing A? . Lemma 3.2. Given x1 . . . xr drawn i.i.d. from a density with full-rank covariance, and neighboring points xi . . . xr defined by xi = ti A? xi + xi for some unknown ti 6= 0 and A? ? Rd?d . If r ? d, then there exists weights ? ? Rr which recover the unknown A? as A? = r X ?1 ?i (xi ? xi )x> , i ? i=1 where ?i = 1/(rti ) and ? = Pr > i=1 xi xi /r. Proof. The identity xi = ti A? xi + xi implies ti (??1/2 A? ?1/2 )??1/2 xi = ??1/2 (xi ? xi ). ?1/2 > Summing both sides with weights ?i and multiplying by x> ) yields i (? r X ?1/2 > ?i ??1/2 (xi ? xi )x> ) = i (? i=1 r X ?1/2 > ?i ti (??1/2 A? ?1/2 )??1/2 xi x> ) i (? i=1 ?1/2 =? ? A ? 1/2 r X ?1/2 > ?i ti ??1/2 xi x> ) . i (? i=1 Pr ?1/2 > By construction of ??1/2 and ?i = 1/(ti r), i=1 ?i ti ??1/2 xi x> ) = I. Therefore, i (? Pr ?1/2 > ?1/2 > ?1/2 ? 1/2 d ?1/2 ? ? (x ? x )x (? ) = ? A ? . When x spans R , ? is both invertible i i i i i=1 and symmetric giving the theorem statement. 3.3 Learning multiple transformations In the case of multiple transformations, the definition of recovering any single transformation matrix A?k is ambiguous since given transformations A?1 and A?2 , the matrices A?1 + A?2 and A?1 ? A?2 both locally generate the same family of transformations. We will refer to the transformations A? ? RK?d?d and strengths t ? Rn?K as disentangled if t> t/r = ? 2 I for a scalar ? 2 > 0. This criterion implies that the activation strengths are uncorrelated across the observed data. We will later 4 show in section 3.4 that this definition of disentangling captures our intuition, has a closed form estimate, and is closely connected to our optimization problem. We show an analogous result to the one-transformation case (Lemma 3.2) which shows that given r?k r > K 2 samples we can find weights ? ? which reconstruct any of the K disentangled PR r transformation matrices A?k as A?k ? Ak = i=1 ?ik (xi ? xi )x> i . This implies that minimization over ? leads to estimates of A? . In contrast to Lemma 3.2, the multiple transformation recovery guarantee is probabilistic and inexact. This is because each summand (xi ? xi )x> i contains effects from all K transformations, and there is no weighting scheme which exactly isolates the effects of a single transformation A?k . Instead, we utilize the randomness in t to estimate A?k by approximately canceling the contributions from the K ? 1 other transformations. Theorem 3.3. Let x1 . . . xr ? Rd be i.i.d isotropic random variables and for each k ? [K], define t1,k . . . tr,k ? R as i.i.d draws from a symmetric random variable with t> t/r = ? 2 I ? Rd?d , tik < C1 , and kxi k2 < C2 with probability one. PK Given x1 . . . xr , and neighbors x1 . . . xr defined as xi = k=1 tik A?k xi + xi for some A?k ? Rd?d , there exists ? ? Rr?K such that for all k ? [K], ! ! r ?2 ?r?2 supk kA?k k ? X > P Ak ? ?ik (xi ? xi )xi > ? < Kd exp . ?1 2K 2 (2C12 C22 (1 + K ?1 supk kA?k k ?) i=1 Proof. We give a proof sketch and defer the details to the supplement (Section 7). We claim that for any k, ?ik = ?tik 2 r satisfies the theorem statement. Following the one-dimensional case, we can expand the outer product in terms of the transformation A? as Ak = r X ?ik (xi ? xi )x> i = K X k0 =1 i=1 A?k0 r X ?ik tik0 xi x> i . i=1 Pr k As before, we must now control the inner terms Zkk0 = i=1 ?ik tik0 xi x> i . We want Zk0 to be close tik 0 0 to the identity when k = k and near zero when k 6= k. Our choice of ?ik = ?2 r does this since if k 0 6= k then ?ik tik0 are zero mean with random sign, resulting in Rademacher concentration bounds near zero, and if k 0 = k then Bernstein bounds show that Zkk ? I since E[?ik ti ] = 1. 3.4 Disentangling transformations Given K estimated transformations A1 . . . AK ? Rd?d and strengths t ? Rn?K , any invertible P matrix W ? RK?K can be used to find an equivalent family of transformations A?i = k Wik Ak P ?1 and t?ik = j Wkj tij . Despite this unidentifiability, there is a choice of A?1 . . . A?K and t? which is equivalent to A1 . . . AK but disentangled, meaning that across the observed transformation pairs {(xi , xi )}ni=1 , the strengths for any two pairs of transformations are uncorrelated t?> t?/n = I. This is a necessary condition to captures the intuition that two disentangled transformations will have independent strength distributions. For example, given a set of images generated by changing lighting conditions and sharpness, we expect the sharpness of an image to be uncorrelated to lighting condition. Formally, we will define a set of A? such that: t??j and t??i are uncorrelated over the observed data, and any pair of transformations A?i x and A?j x generate decorrelated outputs. In contrast to mutual information based approaches to finding disentangled representations, our approach only seeks to control second moments, but enforces decorrelation both in the latent space (tik ) as well as the observed space (A?i x). P 2 Theorem 3.4. Given Ak ? Rd?d , t ? Rn?k with i tik = 0, define Z = U SV > ? Rn?d as the PK SVD of Z, where each row is Zi = k=1 tik vec(Ak ). The transformation A?k = Sk,k mat(Vk> ) and strengths t?ik = Uik fulfils the following properties: P P ? ? ? k tik Ak xi = k tik Ak xi (correct behavior), 5 ? t?> t? = I (uncorrelated in latent space), ? E[hA?i X, A?j Xi] = 0 for any i 6= j and random variable X with E[XX > ] = I (uncorrelated in observed space). Proof. The first property follows since Z is rank-K by construction, and the rank-K SVD preserves P > k tik Ak exactly. The second property follows from the SVD, U U = I. The last property follows > > ? ? from V V = I, implying tr(Ai Aj ) = 0 for i 6= j. By linearity of trace: E[hA?i X, A?j Xi] = Si,i Sj,j tr(mat(Vi )mat(Vj )> ) = 0. Interestingly, this SVD appears in both the convex and subsampling algorithm (Eq. 6) as part of the proximal step for the trace norm optimization. Thus the rank sparsity induced by the trace norm naturally favors a small number of disentangled transformations. 4 Experiments We evaluate the effectiveness of our sampling-based convex relaxation for learning transformations in two ways. In section 4.1, we check whether we can recover a known set of rotation / translation transformations applied to a downsampled celebrity face image dataset. Next, in section 4.2 we perform a qualitative evaluation of learning transformations over raw celebrity faces (CelebA) and MNIST digits, following recent evaluations of disentangling in adversarial networks [2]. 4.1 Recovering known transformations We validate our convex relaxation and sampling procedure by recovering synthetic data generated from known transformations, and compare these to existing approaches for learning linear transformations. Our experiment consists of recovering synthetic transformations applied to 50 image subsets of a downsampled version (18 ? 18) of CelebA. The resolution and dataset size restrictions were due to runtime restrictions from the baseline methods. We compare two versions of our matrix Lie group learning algorithm against two baselines. For our method, we implement and compare convex relaxation with sampling (Eq. 6) and convex relaxation and sampling followed by gradient descent. This second method ensures that we achieve exactly the desired number of transformations K, since trace norm regularization cannot guarantee a fixed rank constraint. The full convex relaxation (Eq. 5) is not covered here, since it is too slow to run on even the smallest of our experiments. As baselines, we compare to gradient descent with restarts on the nonconvex objective (Eq. 3) and the EM algorithm from Miao and Rao [11] run for 20 iterations and augmented with the SVD based disentangling method (Theorem 3.4). These two methods represent the two classes of existing approaches to estimating general linear transformations from pairwise data [11]. Optimization for our methods and gradient descent use minibatch proximal gradient descent with Adagrad [8], where the proximal step for trace norm penalties use subsampling down to five thousand points and randomized SVD. All learned transformations were disentangled using the SVD method unless otherwise noted (Theorem 3.4). Figures 1a and b show the results of recovering a single horizontal translation transformation with error measured in operator norm. Convex relaxation plus gradient descent (Convex+Gradient) achieves the same low error across all sampled 50 image subsets. Without the gradient descent, convex relaxation alone does not achieve low error, since the trace norm penalty does not produce exactly rank-one results. Gradient descent on the other hand gets stuck in local minima even with stepsize tuning and restarts as indicated by the wide variance in error across runs. All methods outperform EM while using substantially less time. Next, we test disentangling and multiple-transformation recovery for random rotations, horizontal translations, and vertical translations (Figure 1c). In this experiment, we apply the three types of transformations to the downsampled CelebA images, and evaluate the outputs by measuring the minimum-cost matching for the operator norm error between learned transformation matrices and 6 the ground truth. Minimizing this metric requires recovering the true transformations up to label permutation. We find results consistent with the one-transform recovery case, where convex relaxation with gradient descent outperforms the baselines. We additionally find SVD based disentangling to be critical to recovering multiple transformations. We find that removing SVD from the nonconvex gradient descent baseline leads to substantially worse results (Figure 1c). (a) Operator norm error for recovering a single translation transform (b) Sampled convex relaxations are faster than baselines (c) Multiple transformations can be recovered using SVD based disentangling Figure 1: Sampled convex relaxation with gradient descent achieves lower error on recovering a single known transformation (panel a), runs faster than baselines (panel b) and recovers multiple disentangled transformations accurately (panel c). 4.2 Qualitative outputs We now test convex relaxation with sampling on MNIST and celebrity faces. We show a subset of learned transformations here and include the full set in the supplemental Jupyter notebook. (a) Thickness (b) Blur (c) Loop size (d) Angle Figure 2: Matrix transformations learned on MNIST (top rows) and extrapolating on Kannada handwriting (bottom row). Center column is the original digit, flanking columns are generated by applying the transformation matrix. On MNIST digits we trained a five-dimensional linear transformation model over a 20,000 example subset of the data, which took 10 minutes. The components extracted by our approach represent coherent stylistic features identified by earlier work using neural networks [2] such as thickness, rotation as well as some new transformations loop size and blur. Examples of images generated from these learned transformations are shown in figure 2. The center column is the original image and all other images are generated by repeatedly applying transformation matrices). We also found that the transformations could also sometimes extrapolate to other handwritten symbols, such as Kannada handwriting [5] (last row, figure 2). Finally, we visualize the learned transformations by summing the estimated transformation strength for each transformation across the minimum spanning tree on the observed data (See supplement section 9 for details). This visualization demonstrates that the learned representation of the data captures the style of the digit, such as thickness and loop size and ignores the digit identity. This is a highly desirable trait for the algorithm, as it means that we can extract continuous factors of variations such as digit thickness without explicitly specifying and removing cluster structure in the data (Figure 3). 7 (a) PCA (b) InfoGAN Figure 3: Embedding of MNIST digits based on two transformations: thickness and loop size. The learned transformations captures extracts continuous, stylistic features which apply across multiple clusters despite being given no cluster information. Figure 4: Baselines applied to the same MNIST data often entangle digit identity and style. In contrast to our method, many baseline methods inadvertently capture digit identity as part of the learned transformation. For example, the first component of PCA simply adds a zero to every image (Figure 4), while the first component of InfoGAN has higher fidelity in exchange for training instability, which often results in mixing digit identity and multiple transformations (Figure 4). Finally, we apply our method to the celebrity faces dataset and find that we are able to extract high-level transformations using only linear models. We trained a our model on a 1000-dimensional PCA projection of CelebA constructed from the original 116412 dimensions with K = 20, and found both global scene transformation such as sharpness and contrast (Figure 5a) and more high level-transformations such as adding a smile (Figure 5b). (a) Contrast / Sharpness (b) Smiling / Skin tone Figure 5: Learned transformations for celebrity faces capture both simple (sharpness) and high-level (smiling) transformations. For each panel, the center column is the original image, and columns to the left and right were generated by repeatedly applying the learnt transformation. 8 5 Related Work and Discussion Learning transformation matrices, also known as Lie group learning, has a long history with the closest work to ours being Miao and Rao [11] and Rao and Ruderman [12]. These earlier methods use a Taylor approximation to learn a set of small (< 10 ? 10) transformation matrices given pairs of image patches undergoing a small transformation. In contrast, our work does not require supervision in the form of transformation pairs and provides a scalable new convex objective function. There have been improvements to Rao and Ruderman [12] focusing on removing the Taylor approximation in order to learn transformations from distant examples: Cohen and Welling [3, 4] learned commutative and 3d-rotation Lie groups under a strong assumption of uniform density over rotations. Sohl-Dickstein et al. [14] learn commutative transformations generated by normal matrices using eigendecompositions and supervision in the form of successive 17 ? 17 image patches in a video. Our work differs because we seek to learn multiple, general transformation matrices from large, high-dimensional datasets. Because of this difference, our algorithm focuses on scalability and avoiding local minima at the expense of utilizing a less accurate first-order Taylor approximation. This approximation is reasonable, since we fit our model to nearest neighbor pairs which are by definition close to each other. Empirically, we find that these approximations result in a scalable algorithm for unsupervised recovery of transformations. Learning to transform between neighbors on a nonlinear manifold has been explored in Doll?r et al. [7] and Bengio and Monperrus [1]. Both works model a manifold by predicting the linear neighborhoods around points using nonlinear functions (radial basis functions in Doll?r et al. [7] and a one-layer neural net in Bengio and Monperrus [1]). In contrast to these methods, which begin with the goal of learning all manifolds, we focus on a class of linear transformations, and treat the general manifold problem as a special kernelization. This has three benefits: first, we avoid the high model complexity necessary for general manifold learning. Second, extrapolation beyond the training data occurs explicitly from the linear parametric form of our model (e.g., from digits to Kannada). Finally, linearity leads to a definition of disentangling based on correlations and a SVD based method for recovering disentangled representations. In summary, we have presented an unsupervised approach for learning disentangled representations via linear Lie groups. We demonstrated that for image data, even a linear model is surprisingly effective at learning semantically meaningful transformations. Our results suggest that these semi-parametric transformation models are promising for identifying semantically meaningful low-dimensional continuous structures from high-dimensional real-world data. Acknowledgements. We thank Arun Chaganty for helpful discussions and comments. This work was supported by NSF-CAREER award 1553086, DARPA (Grant N66001-14-2-4055), and the DAIS ITA program (W911NF-16-3-0001). Reproducibility. Code, data, and experiments can be found on Codalab Worksheets (http://bit.ly/2Aj5tti). 9 

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A Sharp Error Analysis for the Fused Lasso, with Application to Approximate Changepoint Screening Kevin Lin Carnegie Mellon University Pittsburgh, PA 15213 kevinl1@andrew.cmu.edu James Sharpnack University of California, Davis Davis, CA 95616 jsharpna@ucdavis.edu Alessandro Rinaldo Carnegie Mellon University Pittsburgh, PA 15213 arinaldo@stat.cmu.edu Ryan J. Tibshirani Carnegie Mellon University Pittsburgh, PA 15213 ryantibs@stat.cmu.edu Abstract In the 1-dimensional multiple changepoint detection problem, we derive a new fast error rate for the fused lasso estimator, under the assumption that the mean vector has a sparse number of changepoints. This rate is seen to be suboptimal (compared to the minimax rate) by only a factor of log log n. Our proof technique is centered around a novel construction that we call a lower interpolant. We extend our results to misspecified models and exponential family distributions. We also describe the implications of our error analysis for the approximate screening of changepoints. 1 Introduction Consider the 1-dimensional multiple changepoint model yi = ?0,i + i , i = 1, . . . , n, (1) where i , i = 1, . . . , n are i.i.d. errors, and ?0,i , i = 1, . . . , n is a piecewise constant mean sequence, having a set of changepoints  S0 = i ? {1, . . . , n ? 1} : ?0,i 6= ?0,i+1 . (2) This is a well-studied setting, and there is a large body of literature on estimation of the piecewise constant mean vector ?0 ? Rn and its changepoints S0 using various estimators; refer, e.g., to the surveys Brodsky and Darkhovski (1993); Chen and Gupta (2000); Eckley et al. (2011). In this work, we consider the 1-dimensional fused lasso (also called 1d fused lasso, or simply fused lasso) estimator, which, given a data vector y ? Rn from a model as in (1), is defined by n n?1 X 1X ?b = argmin (yi ? ?i )2 + ? |?i ? ?i+1 |, 2 i=1 ??Rn i=1 (3) where ? ? 0 serves as a tuning parameter. This was proposed and named by Tibshirani et al. (2005), but the same idea was proposed earlier in signal processing, under the name total variation denoising, by Rudin et al. (1992). Variants of the fused lasso have been used in biology to detect regions where two genomic samples differ due to genetic variations (Tibshirani and Wang, 2008), in finance to detect shifts in the stock market (Chan et al., 2014), and in neuroscience to detect changes in stationary behaviors of the brain (Aston and Kirch, 2012). Popularity of the fused lasso can be attributed in part to its computational scalability, the optimization problem in (3) being convex and highly structured. There has also been plenty of supporting statistical theory developed for the fused lasso, which we review in Section 2. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Notation. We will make use of the following quantities that are defined in terms of the mean ?0 in (1) and its changepoint set S0 in (2). We denote the size of the changepoint set by s0 = |S0 |. We enumerate S0 = {t1 , . . . , ts0 }, where 1 ? t1 < . . . < ts0 < n, and for convenience we set t0 = 0, ts0 +1 = n. The smallest distance between changepoints in ?0 is denoted by Wn = min i=0,1...,s0 (ti+1 ? ti ), (4) and the smallest distance between consecutive levels of ?0 by Hn = min |?0,i+1 ? ?0,i |. i?S0 We use D ? R(n?1)?n to denote the difference operator ? ?1 1 0 ... 1 ... ? 0 ?1 D=? .. .. .. ? . . . 0 0 . . . ?1 ? 0 0 ? ?. ? (5) (6) 1 Note that s0 = kD?0 k0 . We write DS to extract rows of D indexed by a subset S ? {1, . . . , n ? 1}, and D?S to extract the rows in S c = {1, . . . , n ? 1} \ S. For a vector x ? Rn , we use kxk2n = kxk22 /n to denote its length-scaled `2 norm. For sequences an , bn , we use standard asymptotic notation: an = O(bn ) to denote that an /bn is bounded for large enough n, an = ?(bn ) to denote that bn /an is bounded for large enough n, an = ?(bn ) to denote that both an = O(bn ) and an = ?(bn ), an = o(bn ) to denote that an /bn ? 0, and an = ?(bn ) to denote that bn /an ? 0. For random sequences An , Bn , we write An = OP (Bn ) to denote that An /Bn is bounded in probability. A random variable Z is said to have a sub-Gaussian distribution provided that E(Z) = 0 and P(|Z| > t) ? 2 exp(?t2 /(2? 2 )) for all t ? 0, and a constant ? > 0. Summary of results. Our main focus is on deriving a sharp estimation error bound for the fused lasso, parametrized by the number of changepoints s0 in ?0 . We also study several consequences of our error bound and its analysis. A summary of our contributions is as follows. ? New error analysis for the fused lasso. In Section 3, we develop a new error analysis for the fused lasso, in the model (1) with sub-Gaussian errors. Our analysis leverages a novel quantity that we call a lower interpolant to approximate the fused lasso estimate (once it has been orthogonalized with respect to the changepoint structure of the mean ?0 ) with 2s0 + 2 monotonic segments, which allows for finer control of the empirical process term. When s0 = O(1), and the changepoint locations in S0 are (asymptotically) evenly spaced, our main result implies Ek?b ? ?0 k2n = O(log n(log log n)/n) for the fused lasso estimator ?b in (3). This is slower than the minimax rate by a log log n factor. Our result improves on previously established results from Dalalyan et al. (2017), and after the completion of this paper, was itself improved upon by Guntuboyina et al. (2017) (who are able to remove the extraneous log log n factor). ? Extension to misspecified and exponential family models. In Section 4, we extend our error analysis to cover a mean vector ?0 that is not necessarily piecewise constant (or in other words, has potentially many changepoints). In Section 5, we extend our analysis to exponential family models. The latter extension, especially, is of practical importance, as many applications, e.g., CNV data analysis, call for changepoint detection on count data. ? Application to approximate screening and recovery. In Section 6, we establish that the maximum distance between any true changepoint and its nearest estimated changepoint is OP (log n(log log n)/Hn2 ) using the fused lasso, when s0 = O(1) and all changepoints are (asymptotically) evenly spaced. After applying simple post-processing step, we show that the maximum distance between any estimated changepoint and its nearest true changepoint is of the same order. Our proof technique relies only on the estimation error rate of the fused lasso, and therefore immediately generalizes to any estimator of ?0 , where the distance (for approximate changepoint screening and recovery) is a function of the inherent error rate. The supplementary document gives numerical simulations that support the theory in this paper. 2 2 Preliminary review of existing theory We begin by describing known results on the quantity k?b ? ?0 k2n , the estimation error between the fused lasso estimate ?b in (3) and the mean ?0 in (1). Early results on the fused lasso are found in Mammen and van de Geer (1997) (see also Tibshirani (2014) for a translation to a setting more consistent with that of the current paper). These authors study what may be called the weak sparsity case, in which it is that assumed kD?0 k1 ? Cn , with D being the difference operator in (6). Assuming additionally that the errors in (1) are sub-Gaussian, ?1/3 Mammen and van de Geer (1997) show that for a choice of tuning parameter ? = ?(n1/3 Cn ), the fused lasso estimate ?b in (3) satisfies k?b ? ?0 k2n = OP (n?2/3 Cn2/3 ). (7) The weak sparsity setting is not the focus of our paper, but we still recall the above result to give a sense of the difference between the weak and strong sparsity settings, the latter being the setting in which we assume control over s0 = kD?0 k0 , as we do in the current paper. Prior to this paper, the strongest result in the strong sparsity setting pwas given by Dalalyan et al. (2017), who assume N (0, ? 2 ) errors in (1), and show that for ? = ? 2n log(n/?), the fused lasso estimate satisfies   n 2 2 s0 log(n/?) b log n + k? ? ?0 kn ? C? , (8) n Wn with probability at least 1 ? 2?, for large enough n, and a constant C > 0, where recall Wn is the minimum distance between changepoints in ?0 , as in (4). Our main result in Theorem 1 improves upon (8) in two ways: by reducing the p first log n term inside the brackets to log s0 + log log n, and reducing the second n/Wn term to n/Wn . After our paper was completed, Guntuboyina et al. (2017) gave an even sharper error rate for the fused lasso (and more broadly, for trend the family of higher-order filtering estimates as defined in Steidl et al. (2006); Kim et al. (2009); Tibshirani (2014)). Again assuming N (0, ? 2 ) errors in (1), as well as Wn ? cn/(s0 + 1) for some constant c ? 1, these authors show that the family of fused lasso estimates {?b? , ? ? 0} (using subscripts here to explicitly denote the dependence on the tuning parameter ?) satisfies   en s0 + 1 4? 2 ? log , (9) inf k?b? ? ?0 k2n ? C? 2 + ??0 n s0 + 1 n with probability at least 1 ? exp(??), for large enough n, and a constant p C > 0. The above bound is sharper than ours in Theorem 1 in that (log s0 + log log n) log n + n/Wn is replaced essentially by log Wn . (Also, the result in (9) does not actually require Wn ? cn/(s0 + 1), but only requires the distance between changepoints where jumps alternate in sign to be larger than cn/(s0 + 1), which is another improvement.) Further comparisons will be made in Remark 1 following Theorem 1. There are numerous other estimators, e.g., based on segmentation techniques or wavelets, that admit estimation results comparable to those above. These are described in Remark 2 following Theorem 1. Lastly, it can be seen the minimax estimation error over the class of signals ?0 with s0 changepoints, assuming N (0, ? 2 ) errors in (1), satisfies   n s0 , (10) inf sup Ek?b ? ?0 k2n ? C? 2 log b n s 0 ? kD?0 k0 ?s0 for large enough n, and a constant C > 0. This says that one cannot hope to improve the rate in (9). The minimax result in (10) follows from standard minimax theory for sparse normal means problems, as in, e.g., Johnstone (2015); for a proof, see Padilla et al. (2016). 3 Sharp error analysis for the fused lasso estimator Here we derive a sharper error bound for the fused lasso, improving upon the previously established result of Dalalyan et al. (2017) as stated in (8). Our proof is based on a concept that we call a lower interpolant, which as far as we can tell, is a new idea that may be of interest in its own right. 3 Theorem 1. Assume the data model in (1), with errors i , i = 1, . . . , n i.i.d. from a sub-Gaussian distribution. Then under a choice of tuning parameter ? = (nWn )1/4 , the fused lasso estimate ?b in (3) satisfies ! r s n 0 k?b ? ?0 k2n ? ? 2 c (log s0 + log log n) log n + , n Wn with probability at least 1 ? exp(?C?), for all ? > 1 and n ? N , where c, C, N > 0 are constants that depend on only ? (the parameter appearing in the sub-Gaussian distribution of the errors). An immediate corollary is as follows. Corollary 1. Under the same assumptions as in Theorem 1, we have s0 Ek?b ? ?0 k2n ? c (log s0 + log log n) log n + n r ! n , Wn for some constant c > 0. We give some remarks comparing Theorem 1 to related results in the literature. Remark 1 (Comparison to Dalalyan et al. (2017); Guntuboyina et al. (2017)). We can see that the result in Theorem p 1 is sharper than that in (8) from Dalalyan et al. (2017) for any s0 , Wn , as log s0 ? log n and n/Wn ? n/Wn . Moreover, when s0 = O(1) and Wn = ?(n), the rates are log2 n/n and log n(log log n)/n from Theorem 1 and (8), respectively. Comparing the result in Theorem 1 to that in (9) from Guntuboyina et al. (2017), the latter is sharper p in that it reduces the factor of (log s0 + log log n) log n + n/Wn to a single term of log Wn . In the case s0 = O(1) and Wn = ?(n), the rates are log n(log log n)/n and log n/n from Theorem 1 and (8), respectively, and the latter rate cannot be improved, owing to the minimax lower bound in (10). Similar to our expectation bound in Corollary 1, Guntuboyina et al. (2017) establish   en 2 2 s0 + 1 b inf Ek?? ? ?0 kn ? C? log , (11) ??0 n s0 + 1 for the family of fused lasso estimates {?b? , ? ? 0}, for large enough n, and a constant C > 0. Like their high probability result in (9), their expectation result in (11) is stated in terms of an infimum over ? ? 0, and does not provide an explicit value of ? that attains the bound. (Inspection of their proofs suggests that it is not at all easy to make such a value of ? explicit.) Meanwhile, Theorem 1 and Corollary 1 have the advantage this choice is made explicit, as in ? = (nWn )1/4 . Remark 2 (Comparison to other estimators). Various other estimators obtain comparable estimation error rates. In what follows, all results are stated in the case s0 = O(1).P The Potts estimator, Pn?1 n?1 defined by replacing the `1 penalty i=1 |?i ? ?i+1 | in (3) with the `0 penalty i=1 1{?i 6= ?i+1 }, Potts Potts 2 b b and denoted say by ? , satisfies a bound k? ? ?0 kn = O(log n/n) a.s. as shown by Boysen et al. (2009). Wavelet denoising (placing weak conditions on the wavelet basis), denoted by ?bwav , satisfies Ek?bwav ? ?0 k2n = O(log2 n/n) as shown by Donoho and Johnstone (1994). Pairing unbalanced Haar (UH) wavelets with a basis selection method, Fryzlewicz (2007) developed an estimator ?bUH with Ek?bUH ? ?0 k2n = O(log2 n/n). Though they are not written in this form, the results in Fryzlewicz (2016) imply that his ?tail-greedy? unbalanced Haar (TGUH) estimator, ?bTGUH , satisfies k?bTGUH ? ?0 k2n = O(log2 n/n) with probability tending to 1. Here is an overview of the proof of Theorem 1. The full proof is deferred until the supplement, as with all proofs in this paper. We begin by deriving a basic inequality (stemming from the optimality of the fused lasso estimate ?b in (3)):
b1 . k?b ? ?0 k22 ? 2> (?b ? ?0 ) + 2? kD?0 k1 ? kD?k (12) To precisely control the empirical process term > (?b ? ?0 ), we consider a decomposition > (?b ? ?0 ) = > ?b + > x b, b Here P0 is the projection matrix onto the piecewise where we define ?b = P0 (?b ? ?0 ) and x b = P1 ?. constant structure inherent in ?0 , and P1 = I ? P0 . More precisely, writing S0 = {t1 , . . . , ts0 } for the set of ordered changepoints in ?0 , we define Bj = {tj + 1, . . . , tj+1 }, and denote by 1Bj ? Rn 4 the indicator of block Bj , for j = 0, . . . , s0 . In this notation, P0 is the projection onto the (s0 + 1)dimensional linear subspace R = span{1B0 , . . . , 1Bs0 }. The parameter ?b lies in an low-dimensional subspace, which makes bounding the term > ?b relatively easy. Bounding the term > x b requires a much more intricate argument, which is spelled out in the following lemmas. Lemma 1 is a deterministic result ensuring the existence of what we call a lower interpolant zb to x b. This interpolant approximates x b using 2s0 + 2 monotonic segments, and its empirical process term > zb can be finely controlled, as shown in Lemma 2. The residual from the interpolant approximation, denoted w b=x b ? zb, has an empirical process term > w b that is more crudely controlled, in Lemma 3. Put together, as in > x b = > zb + > w, b gives the final control on > x b. Before stating Lemma 1, we define the class of vectors containing the lower interpolant. Given any collection of changepoints t1 < . . . < ts0 (and t0 = 0, ts0 +1 = n), let M be the set of ?piecewise monotonic? vectors z ? Rn , with the following properties, for each i = 0, . . . , s0 : (i) there exists a point t0i such that ti + 1 ? t0i ? ti+1 , and such that the absolute value |zj | is nonincreasing over the segment j ? {ti + 1, . . . , t0i }, and nondecreasing over the segment j ? {t0i , . . . , ti+1 }; (ii) the signs remain constant on the monotone pieces, sign(zti ) ? sign(zj ) ? 0, sign(zti+1 ) ? sign(zj ) ? 0, j = ti + 1, . . . , t0i , j = t0i + 1, . . . , ti+1 . Now we state our lemma that characterizes the lower interpolant. Lemma 1. Given changepoints t0 < . . . < ts0 +1 , and any x ? Rn , there exists a vector z ? M (not necessarily unique), such that the following statements hold: kD?S0 xk1 = kD?S0 zk1 + kD?S0 (x ? z)k1 , ? 4 s0 kzk2 , kDS0 xk1 = kDS0 zk1 ? kD?S0 zk1 + ? Wn kzk2 ? kxk2 and kx ? zk2 ? kxk2 , (13) (14) (15) where D ? R(n?1)?n is the difference matrix in (6). We call a vector z with these properties a lower interpolant to x. Loosely speaking, the lower interpolant zb can be visualized by taking a string that lies initially on top of x b, is nailed down at the changepoints t0 , . . . ts0 +1 , and then pulled taut while maintaining that it is not greater (elementwise) than x b, in magnitude. Here ?pulling taut? means that kDb z k1 is made small. Figure 1 provides illustrations of the interpolant zb to x b for a few examples. Note that zb consists of 2s0 + 2 monotonic pieces. This special structure leads to a sharp concentration inequality. The next lemma is the primary contributor to the fast rate given in Theorem 1. Lemma 2. Given changepoints t1 < . . . < ts0 , there exists constants cI , CI , NI > 0 such that when  ? Rn has i.i.d. sub-Gaussian components, ! p
|> z| P sup > ?cI (log s0 + log log n)s0 log n ? 2 exp ? CI ? 2 c2I (log s0 + log log n) , z?M kzk2 for any ? > 1, and n ? NI . Finally, the following lemma controls the residuals, w b=x b ? zb. Lemma 3. Given changepoints t1 < . . . < ts0 , there exists constants cR , CR > 0 such that when  ? Rn has i.i.d. sub-Gaussian components,   ? |> w| P sup p > ?cR (ns0 )1/4 ? 2 exp(?CR ? 2 c2R s0 ), kD?S0 wk1 kwk2 w?R? for any ? > 1, where R? is the orthogonal complement of R = span{1B0 , . . . , 1Bs0 }. 5 ? 8 10 0 200 400 600 6 4 2 0 ?2 800 0 Index 200 400 600 800 600 800 Index ?1.5 ?3 ?2 ?0.5 0.0 ?1 0.5 0 1.0 ? ?? ?? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ????? ??? ??? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? ? ? ? ? ??? ??? ? ? ? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? ?? ?? ??? ? ? ? ? ? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? ? ???? ? ? ???? ? ??? ? ?? ? ? ? ??? ? ? ? ? ?? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ??? ? ?? ? ? ????? ? ? ? ? ?? ? ? ? ? ? ?? ?? ? ? ?? ? ? ? ? ?? ??? ? ?? ? ? ?? ? ? ? ?? ? ?? ? ? ?? ?? ?? ?? ?? ? ? ?? ? ??? ? ? ? ?? ? ? ? ? ??? ? ? ? ? ? ? ? ? ? ? ? ??? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ? ? ? ? ? ?? ? ?? ? ? ?? ? ? ???? ? ? ? ?? ? ? ? ? ? ?? ?? ? ? ? ??? ?? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ??? ? ? ? ?? ??? ?? ? ?? ??? ? ??? ? ? ??? ? ? ? ?? ?? ?? ????? ? ?? ? ?? ? ?? ?? ? ? ? ? ?? ? ?? ? ? ? ?? ??? ?? ? ? ?? ? ??? ?? ? ? ? ? ? ?? ? ??? ? ? ? ?? ? ?? ?? ?? ??? ?? ? ??? ? ? ? ? ?? ?? ? ?? ? ???? ?? ?? ? ? ? ?? ?? ? ? ?? ? ? ??? ? ??? ?? ?? ? ? ?? ? ? ? ???? ? ?? ?? ?? ?? ? ? ??? ?? ? ??? ? ? ? ? ?????? ? ? ? ? ? ? ? ??? ? ? ?? ???? ??? ?? ?? ? ? ?? ?? ?? ? ? ? ? 1 5 0 ?5 ? ?? ? ??? ?? ? ? ?? ?? ? ???? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ??? ? ??? ?? ??????? ?? ???? ?? ? ? ? ? ??? ? ? ? ?? ?? ? ?? ? ?? ? ??? ??? ? ?? ? ? ?? ?? ? ? ? ?? ? ?? ??? ? ??? ? ?????? ??? ?? ? ? ?? ?? ?? ? ?? ?? ???? ? ? ? ??? ? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ??? ? ???? ? ? ? ? ? ? ? ? ????? ? ?? ? ? ?? ? ?? ?? ? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ????? ? ?? ? ?? ? ? ? ??? ? ? ?? ? ? ? ???? ? ?? ??? ? ? ?? ??? ? ? ?? ?? ? ? ?? ?? ? ? ? ? ??? ?? ?? ??? ? ??? ? ? ? ?? ? ?? ? ? ? ? ?? ? ?? ? ? ? ? ?? ? ?? ??? ?? ??? ?? ?? ?? ?? ? ? ? ?? ? ? ? ?? ? ? ??? ? ? ? ? ? ? ? ? ? ?? ? ??? ? ? ? ?? ? ?? ??? ? ??? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ? ? ???? ?? ? ? ? ?? ? ? ?? ? ? ?? ?? ???? ? ? ???? ? ? ?? ?? ???? ? ? ?? ? ? ?? ??? ? ??? ?? ??? ?? ? ? ?? ?? ??? ? ?? ? ? ?? ? ? ? ?? ?? ??? ?? ? ??? ? ?? ?? ?? ?? ? ?? ?? ?? ? ? ?? ????? ? ?? ?? ? ? ? ?? ? ?? ? ?? ?? ??? ?? ???? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ??? ?? ? ? ?? ?? ?? ? ??? ?? ?? ?? ? ? ? ? ? ?? ? ? ? ? ? ? 0 200 400 600 800 0 200 400 Figure 1: The lower interpolants for two examples (in the left and right columns), each with n = 800 points. In the top row, the data y (in gray) and underlying signal ?0 (red) are plotted across the locations 1, . . . , n. Also shown is the fused lasso estimate ?b (blue). In the bottom row, the error vector x b = P1 ?b is plotted (blue) as well as the interpolant (black), and the dotted vertical lines (red) denote the changepoints t1 , . . . ts0 of ?0 . 4 Extension to misspecified models We consider data from the model in (1) but where the mean ?0 is not necessarily piecewise constant (i.e., where s0 is potentially large). Let us define ?0 (s) = argmin k?0 ? ?k22 subject to kD?k0 ? s, (16) ??Rn which we call the best s-approximation to ?0 . We now present an extension of Theorem 1. Theorem 2. Assume the data model in (1), with errors i , i = 1, . . . , n i.i.d. from a sub-Gaussian distribution. For any s, consider the best s-approximation ?0 (s) to ?0 , as in (16), and let Wn (s) be the minimum distance between the s changepoints in ?0 (s). Then under a choice of tuning parameter ? = (nWn (s))1/4 , the fused lasso estimate ?b in (3) satisfies ! r s n 2 2 2 k?b ? ?0 kn ? k?0 (s) ? ?0 kn + ? c (log s + log log n) log n + , (17) n Wn (s) with probability at least 1 ? exp(?C?), for all ? > 1 and n ? N , where c, C, N > 0 are constants b 0 ? s on an event E, then that depend on only ?. Further, if ? is chosen large enough so that kD?k ! 2 ? n s 2 2 (log s + log log n) log n + + k?b ? ?0 (s)kn ? ? c , (18) n Wn (s) ?2 on E intersected with an event of probability at least 1 ? exp(?C?), for all ? > 1, n ? N , where c, C, N > 0 are the same constants as above. The first result in (17) in Theorem 2 is a standard oracle inequality. It provides a bound on the error of the fused lasso estimator that decomposes into two parts, the first term being the approximation error, determined by the proximity of ?0 (s) to ?0 , and second term being the usual bound we would encounter if the mean truly had s changepoints. The second result in (18) in the theorem is a direct bound on the estimation error k?b ? ?0 (s)k2n . We see that the estimation error can be small, apparently regardless of the size of k?0 (s) ? ?0 k2n , if we take ? to be large enough for ?b to itself have s changepoints. But the rate worsens as ? grows larger, so implicitly, the proximity of ?0 (s) to ?0 does play an role (if ?0 were actually far away from a signal with s changepoints, then we may have to take ? very large to ensure that ?b has s changepoints). 6 Remark 3 (Comparison to other results). Dalalyan et al. (2017); Guntuboyina et al. (2017) also provide oracle inequalities and their results could be adapted to take forms as in Theorem 2. It is not clear to us that previous results on other estimators, such as those from Remark 2, adapt as easily. 5 Extension to exponential family models We consider data y = (y1 , . . . , yn ) ? Rn with independent components distributed according to
p(yi ; ?0,i ) = h(yi ) exp yi ?0,i ? ?(?0,i ) , i = 1, . . . , n. (19) Here, for each i = 1, . . . , n, the parameter ?0,i is the natural parameter in the exponential family and ? is the cumulant generating function. As before, in the location model, we are mainly interested in the case in which the natural parameter vector ?0 is piecewise constant (with s0 denoting its number of changepoints, as before). Estimation is now based on penalization of the negative log-likelihood: n n X X
?b = argmin ? yi ?i + ?(?i ) + ? |?i ? ?i+1 |, (20) ??Rn i=1 i=1 Since the cumulant generating function ? is always convex in exponential families, the above is a convex optimization problem. We present an estimation error bound the present setting. Theorem 3. Assume the data model in (19), with a strictly convex, twice continuously differentiable cumulant generating function ?. Assume that ?0,i ? [l, u], i = 1, . . . , n for constants l, u ? R, and add the constraints ?i ? [l, u], i = 1, . . . , n in the optimization problem in (20). Finally, assume that the random variables yi ? E(yi ), i = 1, . . . , n obey a sub-Gaussian distribution, with parameter ?. Then under a choice of tuning parameter ? = (nWn )1/4 , the exponential family fused lasso estimate ?b in (20) (subject to the additional boundedness constraints) satisfies ! r n 2 2 s0 b (log s0 + log log n) log n + , k? ? ?0 kn ? ? c n Wn with probability at least 1 ? exp(?C?), for all ? > 1 and n ? N , where c, C, N > 0 are constants that depend on only l, u, ?. Remark 4 (Roles of l, u). The restriction of ?0,i and the optimization parameters in (20) to [l, u], for i = 1, . . . , n, is used to ensure that the second derivative of ? is bounded away from zero. (The same property could be accomplished by instead adding a small squared `2 penalty on ? in (20).) A more refined analysis could alleviate the need for this bounded domain (or extra squared `2 penalty) but we do not pursue this for simplicity. Remark 5 (Sub-Gaussianity in exponential families). When are the random variables yi ? E(yi ), i = 1, . . . , n sub-Gaussian, in an exponential family model (19)? A simple sufficient condition (not specific to exponential families, in fact) is that these centered variates are bounded. This covers the binomial model yi ? Bin(k, ?(?0,i )), where ?(?0,i ) = 1/(1 + e??0,i ), i = 1, . . . , n, and k is a fixed constant. Hence Theorem 3 applies to binomial data. For Poisson data yi ? Pois(?(?0,i )), where ?(?0,i ) = e?0,i , i = 1, . . . , n, we now give two options for the analysis. The first is to assume a maximum achieveable count (which may be reasonable in CNV data) and then apply Theorem 3 owing again to boundedness. The second is to invoke the fact that Poisson random variables have sub-exponential (rather than sub-Gaussian) tails, and then use a truncation argument, to show that for the Poisson fused lasso estimate ?b in (20) (under the additional boundedness constraints), with ? = log n(nWn )1/4 , ! r s log n n 0 2 2 k?b ? ?0 kn ? ? c (log s0 + log log n) log n + , (21) n Wn with probability at least 1 ? exp(?C?) ? 1/n, for all ? > 1 and n ? N , where c, C, N > 0 are constants depending on l, u. This is slower than the rate in Theorem 3 by a factor of log n. Remark 6 (Comparison to other results). The results in Dalalyan et al. (2017); Guntuboyina et al. (2017) assume normal errors. It seems believable to us that the results of Dalalyan et al. (2017) could be extended to sub-Gaussian errors and hence exponential family data, in a manner similar to what we have done above in Theorem 3. To us, this is less clear for the results of Guntuboyina et al. (2017), which rely on some technical calculations involving Gaussian widths. It is even less clear to us how results from other estimators, as in Remark 2, extend to exponential family data. 7 6 Approximate changepoint screening and recovery In many applications of changepoint detection, one may be interested in estimation of the changepoint locations in ?0 , rather than the mean vector ?0 as a whole. In this section, we show that estimation of the changepoint locations and of ?0 itself are two very closely linked problems, in the following sense: any procedure with guarantees on its error in estimating ?0 automatically has certain approximate changepoint detection guarantees, and not surprisingly, a faster error rate (in estimating ?0 ) translates into a stronger statement about approximate changepoint detection. We use this general link to prove new approximate changepoint screening results for the fused lasso. We also show that in general a simple post-processing step may be used to discard spurious detected changepoints, and again apply this to the fused lasso to yield new approximate changepoint recovery results. It helps to introduce some additional notation. For a vector ? ? Rn , we write S(?) for the set of its changepoint indices, i.e.,  S(?) = i ? {1, . . . , n ? 1} : ?i 6= ?i+1 . Recall, we abbreviate S0 = S(?0 ) for the changepoints of the underlying mean ?0 . For two discrete sets A, B, we define the metrics  d(A|B) = max min |a ? b| and dH (A, B) = max d(A|B), d(B|A)}. b?B a?A The first metric above can be seen as a one-sided screening distance from B to A, measuring the furthest distance of an element in B to its closest element in A. The second metric above is known as the Hausdorff distance between A and B. Approximate changepoint screening. We present our general theorem on changepoint screening. The basic idea behind the result is quite simple: if an estimator misses a (large) changepoint in ?0 , then its estimation error must suffer, and we can use this fact to bound the screening distance. Theorem 4. Let ?e ? Rn be an estimator such that k?e ? ?0 k2n = OP (Rn ). Assume that nRn /Hn2 = o(Wn ), where, recall, Hn is the minimum gap between adjacent levels of ?0 , defined in (5), and Wn is the minimum distance between adjacent changepoints of ?0 , defined in (4). Then  
nRn e d S(?) | S0 = OP . Hn2 Remark 7 (Generic setting: no specific data model, and no assumptions on estimator). Importantly, Theorem 4 assumes no data model whatsoever, and treats ?e as a generic estimator of ?0 . (Of course, through the statement k?e ? ?0 k2n = OP (Rn ), one can see that ?e is random, constructed from e data that depends on ?0 , but no specific data model is required, nor are any specific properties of ?, other than its error rate.) This flexibility allows for the result to be applied in any problem setting in which one has control of the error in estimating a piecewise constant parameter ?0 (in some cases this may be easier to obtain, compared to direct analysis of detection properties). A similar idea was used (concurrently and independently) by Fryzlewicz (2016) in the analysis of the TGUH estimator. Combining the above theorem with known error rates for the fused lasso estimator?(7) in the weak sparsity case, and Theorem 1 in the strong sparsity case?gives the following result. Corollary 2. Assume the data model in (1), with errors i , i = 1, . . . ,? n i.i.d. from a sub-Gaussian 1/3 distribution. Let Cn = kD?0 k1 , and assume that Hn = ?(n1/6 Cn / Wn ). Then the fused lasso ?1/3 estimator ?b in (3) with ? = ?(n1/3 Cn ) satisfies  1/3 2/3 
b | S0 = OP n Cn d S(?) . (22) Hn2 p Alternatively, assume ? s0 = O(1), Wn = ?(n), and Hn = ?( log n(log log n)/n). Then the fused lasso with ? = ?( n) satisfies  
log n(log log n) b d S(?) | S0 = OP . (23) Hn2 p Remark 8 (Changepoint detection limit). The restriction Hn = ?( ?log n(log log n)/n) for (23) in Corollary 2 is very close to the optimal detection limit of Hn = ?(1/ n): Duembgen and Walther (2008) showed that in Gaussian changepoint model with a single elevated region, and Wn ? = ?(n), there is no test for detecting a changepoint that has asymptotic power 1 unless Hn = ?(1/ n). 8 Combining Theorem 4 with (21) gives the following (a similar result holds for the binomial model). for i = 1, . . . , n, and assume k?0 k? = O(1), Corollary 3. Assume yi ? Pois(e?0,i ), independently, p s0 = O(1), Wn = ?(n), Hn = ?(log n log log n/n). Then for the Poisson?fused lasso estimator ?b in (20) (subject to appropriate boundedness constraints) with ? = ?(log n n), we have   2
b | S0 = OP log n(log log n) . d S(?) Hn2 Approximate changepoint recovery. We present a post-processing procedure for the estimated e to eliminate changepoints of ?e that lie far away from changepoints of ?0 . Our changepoints in ?, procedure is based on convolving ?e with a filter that resembles the mother Haar wavelet. Consider i+b i Xn X 1 e = 1 Fi (?) ?ej ? ?ej , for i = bn , . . . , n ? bn , (24) bn j=i+1 bn j=i?bn +1 e at all locations i = bn , . . . , n ? bn , for an integral bandwidth bn > 0. By evaluating the filter Fi (?) and retaining only locations at which the filter value is large (in magnitude), we can approximately recovery the changepoints of ?0 , in the Hausdorff metric. Theorem 5. Let ?e ? Rn be such that k?e ? ?0 k2n = OP (Rn ). Consider the following procedure: we evaluate the filter in (24) with bandwidth bn at locations in n o e = i ? {bn , . . . , n ? bn } : i ? S(?), e or i + bn ? S(?), e or i ? bn ? S(?) e ? {bn , n ? bn }, IF (?) e = {i ? IF (?) e : |Fi (?)| e ? ?n }, for a threshold level ?n . If and define a set of filtered points SF (?) 2 bn , ?n satisfy bn = ?(nRn /Hn ), 2bn ? Wn , and ?n /Hn ? ? ? (0, 1) as n ? ?, then  
e S0 ? 2bn ? 1 as n ? ?. P dH SF (?), e in Theorem 5 is not necessarily of a subset of the original Note that the set of filtered points |SF (?)| e e ? 3|S(?)| e + 2. set of estimated changepoints S(?), but it has the property |SF (?)| We finish with corollaries for the fused lasso. For space reasons, remarks comparing them to related approximate recovery results in the literature are deferred to the supplement. Corollary 4. Assume the data model in (1), with errors i , i = 1, . . . , n i.i.d. from a sub-Gaussian distribution. Let Cn = kD?0 k1 . If we apply the post-processing procedure in Theorem 5 to the fused ?1/3 2/3 lasso estimator ?b in (3) with ? = ?(n1/3 Cn ), bn = bn1/3 Cn ?n2 /Hn2 c ? Wn /2 for a sequence ?n ? ?, and ?n /Hn ? ? ? (0, 1), then  
2n1/3 Cn2/3 ?n2 b P dH SF (?), S0 ? ? 1 as n ? ?. (25) Hn2 Alternatively, assuming s0 =?O(1), Wn = ?(n), if we apply the same post-processing procedure to the fused lasso with ? = ?( n), bn = blog n(log log n)?n2 /Hn2 c ? Wn /2 for a sequence ?n ? ?, and ?n /Hn ? ? ? (0, 1), then   2
b S0 ? 2 log n(log log n)?n ? 1 as n ? ?. P dH SF (?), (26) Hn2 Corollary 5. Assume yi ? Pois(e?0,i ), independently, for i = 1, . . . , n, and assume k?0 k? = O(1), s0 = O(1), Wn = ?(n). If we apply the post-processing method in Theorem 5 to the Poisson fused ? lasso estimator ?b in (20) (subject to appropriate boundedness constraints) with ? = ?(log n n), 2 2 2 bn = blog n(log log n)?n /Hn c ? Wn /2 for a sequence ?n ? ?, and ?n /Hn ? ? ? (0, 1), then   2 2
b S0 ? 2 log n(log log n)?n ? 1 as n ? ?. P dH SF (?), Hn2 7 Summary We gave a new error analysis for the fused lasso, with extensions to misspecified models and data from exponential families. We showed that error bounds for general changepoint estimators lead to approximate changepoint screening results, and after post-processing, approximate recovery results. Acknolwedgements. JS was supported by NSF Grant DMS-1712996. RT was supported by NSF Grant DMS-1554123. 9 References John AD Aston and Claudia Kirch. Evaluating stationarity via change-point alternatives with applications to fMRI data. The Annals of Applied Statistics, 6(4):1906?1948, 2012. Leif Boysen, Angela Kempe, Volkmar Liebscher, Axel Munk, and Olaf Wittich. Consistencies and rates of convergence of jump-penalized least squares estimators. The Annals of Statistics, 37(1): 157?183, 2009. Boris Brodsky and Boris Darkhovski. Nonparametric Methods in Change-Point Problems. Springer, 1993. Ngai Hang Chan, Chun Yip Yau, and Rong-Mao Zhang. Group lasso for structural break time series. Journal of the American Statistical Association, 109(506):590?599, 2014. Jie Chen and Arjun Gupta. Parametric Statistical Change Point Analysis. Birkhauser, 2000. Arnak S. Dalalyan, Mohamed Hebiri, and Johannes Lederer. On the prediction performance of the lasso. Bernoulli, 23(1):552?581, 2017. David L Donoho and Iain M Johnstone. Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(3):425?455, 1994. Lutz Duembgen and Guenther Walther. Multiscale inference about a density. The Annals of Statistics, 36(4):1758?1785, 2008. Idris Eckley, Paul Fearnhead, and Rebecca Killick. Analysis of changepoint models. In David Barber, Taylan Cemgil, and Silvia Chiappa, editors, Bayesian Time Series Models, chapter 10, pages 205?224. Cambridge University Press, Cambridge, 2011. Piotr Fryzlewicz. Unbalanced Haar technique for nonparametric function estimation. Journal of the American Statistical Association, 102(480):1318?1327, 2007. Piotr Fryzlewicz. Tail-greedy bottom-up data decompositions and fast multiple change-point detection. 2016. URL http://stats.lse.ac.uk/fryzlewicz/tguh/tguh.pdf. Adityanand Guntuboyina, Donovan Lieu, Sabyasachi Chatterjee, and Bodhisattva Sen. Spatial adaptation in trend filtering. arXiv preprint arXiv:1702.05113, 2017. Iain M. Johnstone. Gaussian Estimation: Sequence and Wavelet Models. Cambridge University Press, 2015. Draft version. Seung-Jean Kim, Kwangmoo Koh, Stephen Boyd, and Dimitry Gorinevsky. `1 trend filtering. SIAM Review, 51(2):339?360, 2009. Enno Mammen and Sara van de Geer. Locally adaptive regression splines. The Annals of Statistics, 25(1):387?413, 1997. Oscar Hernan Madrid Padilla, James Sharpnack, James Scott, , and Ryan J. Tibshirani. The DFS fused lasso: Linear-time denoising over general graphs. arXiv preprint arXiv:1608.03384, 2016. Leonid Rudin, Stanley Osher, and Emad Faterni. Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 60(1?4):259?268, 1992. Gabriel Steidl, Stephan Didas, and Julia Neumann. Splines in higher order TV regularization. International Journal of Computer Vision, 70(3):214?255, 2006. Robert Tibshirani and Pei Wang. Spatial smoothing and hot spot detection for cgh data using the fused lasso. Biostatistics, 9(1):18?29, 2008. Robert Tibshirani, Michael Saunders, Saharon Rosset, Ji Zhu, and Keith Knight. Sparsity and smoothness via the fused Lasso. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(1):91?108, 2005. Ryan J. Tibshirani. Adaptive piecewise polynomial estimation via trend filtering. The Annals of Statistics, 42(1):285?323, 2014. 10 

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Linear Time Computation of Moments in Sum-Product Networks Geoff Gordon Machine Learning Department Carnegie Mellon University Pittsburgh, PA 15213 ggordon@cs.cmu.edu Han Zhao Machine Learning Department Carnegie Mellon University Pittsburgh, PA 15213 han.zhao@cs.cmu.edu Abstract Bayesian online algorithms for Sum-Product Networks (SPNs) need to update their posterior distribution after seeing one single additional instance. To do so, they must compute moments of the model parameters under this distribution. The best existing method for computing such moments scales quadratically in the size of the SPN, although it scales linearly for trees. This unfortunate scaling makes Bayesian online algorithms prohibitively expensive, except for small or tree-structured SPNs. We propose an optimal linear-time algorithm that works even when the SPN is a general directed acyclic graph (DAG), which significantly broadens the applicability of Bayesian online algorithms for SPNs. There are three key ingredients in the design and analysis of our algorithm: 1). For each edge in the graph, we construct a linear time reduction from the moment computation problem to a joint inference problem in SPNs. 2). Using the property that each SPN computes a multilinear polynomial, we give an efficient procedure for polynomial evaluation by differentiation without expanding the network that may contain exponentially many monomials. 3). We propose a dynamic programming method to further reduce the computation of the moments of all the edges in the graph from quadratic to linear. We demonstrate the usefulness of our linear time algorithm by applying it to develop a linear time assume density filter (ADF) for SPNs. 1 Introduction Sum-Product Networks (SPNs) have recently attracted some interest because of their flexibility in modeling complex distributions as well as the tractability of performing exact marginal inference [11, 5, 6, 9, 16?18, 10]. They are general-purpose inference machines over which one can perform exact joint, marginal and conditional queries in linear time in the size of the network. It has been shown that discrete SPNs are equivalent to arithmetic circuits (ACs) [3, 8] in the sense that one can transform each SPN into an equivalent AC and vice versa in linear time and space with respect to the network size [13]. SPNs are also closely connected to probabilistic graphical models: by interpreting each sum node in the network as a hidden variable and each product node as a rule encoding context-specific conditional independence [1], every SPN can be equivalently converted into a Bayesian network where compact data structures are used to represent the local probability distributions [16]. This relationship characterizes the probabilistic semantics encoded by the network structure and allows practitioners to design principled and efficient parameter learning algorithms for SPNs [17, 18]. Most existing batch learning algorithms for SPNs can be straightforwardly adapted to the online setting, where the network updates its parameters after it receives one instance at each time step. This online learning setting makes SPNs more widely applicable in various real-world scenarios. This includes the case where either the data set is too large to store at once, or the network needs to adapt to the change of external data distributions. Recently Rashwan et al. [12] proposed an 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. online Bayesian Moment Matching (BMM) algorithm to learn the probability distribution of the model parameters of SPNs based on the method of moments. Later Jaini et al. [7] extended this algorithm to the continuous case where the leaf nodes in the network are assumed to be Gaussian distributions. At a high level BMM can be understood as an instance of the general assumed density filtering framework [14] where the algorithm finds an approximate posterior distribution within a tractable family of distributions by the method of moments. Specifically, BMM for SPNs works by matching the first and second order moments of the approximate tractable posterior distribution to the exact but intractable posterior. An essential sub-routine of the above two algorithms [12, 7] is to efficiently compute the exact first and second order moments of the one-step update posterior distribution (cf. 3.2). Rashwan et al. [12] designed a recursive algorithm to achieve this goal in linear time when the underlying network structure is a tree, and this algorithm is also used by Jaini et al. [7] in the continuous case. However, the algorithm only works when the underlying network structure is a tree, and a naive computation of such moments in a DAG will scale quadratically w.r.t. the network size. Often this quadratic computation is prohibitively expensive even for SPNs with moderate sizes. In this paper we propose a linear time (and space) algorithm that is able to compute any moments of all the network parameters simultaneously even when the underlying network structure is a DAG. There are three key ingredients in the design and analysis of our algorithm: 1). A linear time reduction from the moment computation problem to the joint inference problem in SPNs, 2). A succinct evaluation procedure of polynomial by differentiation without expanding it, and 3). A dynamic programming method to further reduce the quadratic computation to linear. The differential approach [3] used for polynomial evaluation can also be applied for exact inference in Bayesian networks. This technique has also been implicitly used in the recent development of a concave-convex procedure (CCCP) for optimizing the weights of SPNs [18]. Essentially, by reducing the moment computation problem to a joint inference problem in SPNs, we are able to exploit the fact that the network polynomial of an SPN computes a multilinear function in the model parameters, so we can efficiently evaluate this polynomial by differentiation even if the polynomial may contain exponentially many monomials, provided that the polynomial admits a tractable circuit complexity. Dynamic programming can be further used to trade off a constant factor in space complexity (using two additional copies of the network) to reduce the quadratic time complexity to linear so that all the edge moments can be computed simultaneously in two passes of the network. To demonstrate the usefulness of our linear time sub-routine for computing moments, we apply it to design an efficient assumed density filter [14] to learn the parameters of SPNs in an online fashion. ADF runs in linear time and space due to our efficient sub-routine. As an additional contribution, we also show that ADF and BMM can both be understood under a general framework of moment matching, where the only difference lies in the moments chosen to be matched and how to match the chosen moments. 2 Preliminaries We use [n] to abbreviate {1, 2, . . . , n}, and we reserve S to represent an SPN, and use |S| to mean the size of an SPN, i.e., the number of edges plus the number of nodes in the graph. 2.1 Sum-Product Networks A sum-product network S is a computational circuit over a set of random variables X = {X1 , . . . , Xn }. It is a rooted directed acyclic graph. The internal nodes of S are sums or products and the leaves are univariate distributions over Xi . In its simplest form, the leaves of S are indicator variables IX=x , which can also be understood as categorical distributions whose entire probability mass is on a single value. Edges from sum nodes are parameterized with positive weights. Sum node computes a weighted sum of its children and product node computes the product of its children. If we interpret each node in an SPN as a function of leaf nodes, then the scope of a node in SPN is defined as the set of variables that appear in this function. More formally, for any node v in an SPN, if v is a terminal node, say, an indicator variable over X, then scope(v) = {X}, else scope(v) = ?v??Ch(v) scope(? v ). An SPN is complete iff each sum node has children with the same scope, and is decomposable iff for every product node v, scope(vi ) ? scope(vj ) = ? for every pair (vi , vj ) of children of v. It has been shown that every valid SPN can be converted into a complete and decomposable SPN with at most a quadratic increase in size [16] without changing the underlying distribution. As a result, in this work we assume that all the SPNs we discuss are complete and decomposable. 2 Let x be an instantiation of the random vector X. We associate an unnormalized probability Vk (x; w) with each node k when the input to the network is x with network weights set to be w: ? ? = xi ) if k is a leaf node over Xi ?p(X Q i if k is a product node Vk (x; w) = (1) j?Ch(k) Vj (x; w) ? ?P w V (x; w) if k is a sum node j?Ch(k) k,j j where Ch(k) is the child list of node k in the graph and wk,j is the edge weight associated with sum node k and its child node j. The probability of a joint assignment X = x is computed by the value at the root of S with input x divided by a normalization constant Vroot (1; w): p(x) = Vroot (x; w)/Vroot (1; w), where Vroot (1; w) is the value of the root node when all the values of leaf nodes are set to be 1. This essentially corresponds to marginalizing out the random vector X, which will ensure p(x) defines a proper probability distribution. Remarkably, all queries w.r.t. x, including joint, marginal, and conditional, can be answered in linear time in the size of the network. 2.2 Bayesian Networks and Mixture Models We provide two alternative interpretations of SPNs that will be useful later to design our linear time moment computation algorithm. The first one relates SPNs with Bayesian networks (BNs). Informally, any complete and decomposable SPN S over X = {X1 , . . . , Xn } can be converted into a bipartite BN with O(n|S|) size [16]. In this construction, each internal sum node in S corresponds to one latent variable in the constructed BN, and each leaf distribution node corresponds to one observable variable in the BN. Furthermore, the constructed BN will be a simple bipartite graph with one layer of local latent variables pointing to one layer of observable variables X. An observable variable is a child of a local latent variable if and only if the observable variable appears as a descendant of the latent variable (sum node) in the original SPN. This means that the SPN S can be understood as a BN where the number of latent variables per instance is O(|S|). The second perspective is to view an SPN S as a mixture model with exponentially many mixture components [4, 18]. More specifically, we can decompose each complete and decomposable SPN S into a sum of induced trees, where each tree corresponds to a product of univariate distributions. To proceed, we first formally define what we called induced trees: Definition 1 (Induced tree SPN). Given a complete and decomposable SPN S over X = {X1 , . . . , Xn }, T = (TV , TE ) is called an induced tree SPN from S if 1). Root(S) ? TV ; 2). If v ? TV is a sum node, then exactly one child of v in S is in TV , and the corresponding edge is in TE ; 3). If v ? TV is a product node, then all the children of v in S are in TV , and the corresponding edges are in TE . It has been shown that Def. 1 produces subgraphs of S that are trees as long as the original SPN S is complete and decomposable [4, 18]. One useful result based on the concept of induced trees is: Theorem 1 ([18]). Let ?S = P Vroot (1; 1). ?S counts Q the number of unique induced trees in S, and ?S Q n Vroot (x; w) can be written as t=1 w (k,j)?TtE k,j i=1 pt (Xi = xi ), where Tt is the tth unique induced tree of S and pt (Xi ) is a univariate distribution over Xi in Tt as a leaf node. Thm. 1 shows that ?S = Vroot (1; 1) can also be computed efficiently by setting all the edge weights to be 1. In general counting problems are in the #P complexity class [15], and the fact that both probabilistic inference and counting problem are tractable in SPNs also implies that SPNs work on subsets of distributions that have succinct/efficient circuit representation. Without loss of generality assuming that sum layers alternate with product layers in S, we have ?S = ?(2H(S) ), where H(S) is the height of S. Hence the mixture model represented by S has number of mixture components that is exponential in the height of S. Thm. 1 characterizes both the number of components and the form of each component in the mixture model, as well as their mixture weights. For the convenience of later discussion, we call Vroot (x; w) the network polynomial of S. Corollary 1. The network polynomial Vroot (x; w) is a multilinear function of w with positive coefficients on each monomial. Corollary 1 holds since each monomial corresponds to an induced tree and each edge appears at most once in the tree. This property will be crucial and useful in our derivation of a linear time algorithm for moment computation in SPNs. 3 3 Linear Time Exact Moment Computation 3.1 Exact Posterior Has Exponentially Many Modes Let m be theQ number of sum nodes in S. Suppose we are given a fully factorized prior distribution m p0 (w; ? ) = k=1 p0 (wk ; ?k ) over w. It is worth pointing out the fully factorized prior distribution is well justified by the bipartite graph structure of the equivalent BN we introduced in section 2.2. We are interested in computing the moments of the posterior distribution after we receive one observation from the world. Essentially, this is the Bayesian online learning setting where we update the belief about the distribution of model parameters as we observe data from the world sequentially. Note that wk corresponds to the weight vector associated with sum node k, so wk is a vector that satisfies wk > 0 and 1T wk = 1. Let us assume that the prior distribution for each wk is Dirichlet, i.e., P m m Y Y ?( j ?k,j ) Y ?k,j ?1 Q p0 (w; ? ) = Dir(wk ; ?k ) = wk,j j ?(?k,j ) j k=1 k=1 After observing one instance x, we have the exact posterior distribution to be: p(w | x) = p0 (w; ? )p(x | w)/p(x). Let Zx , p(x) and realize that the network polynomial also computes the likelihood p(x | w). Plugging the expression for the prior distribution as well as the network polynomial into the above Bayes formula, we have ?S Y n m Y Y 1 X wk,j pt (xi ) p(w | x) = Dir(wk ; ?k ) Zx t=1 i=1 k=1 (k,j)?TtE Since Dirichlet is a conjugate distribution to the multinomial, each term in the summation is an updated Dirichlet with a multiplicative constant. So, the above equation suggests that the exact posterior distribution becomes a mixture of ?S Dirichlets after one observation. In a data set of D instances, the exact posterior will become a mixture of ?SD components, which is intractable to maintain since ?S = ?(2H(S) ). The hardness of maintaining the exact posterior distribution appeals for an approximate scheme where we can sequentially update our belief about the distribution while at the same time efficiently maintain the approximation. Assumed density filtering [14] is such a framework: the algorithm chooses an approximate distribution from a tractable family of distributions after observing each instance. A typical choice is to match the moments of an approximation to the exact posterior. 3.2 The Hardness of Computing Moments In order to find an approximate distribution to match the moments of the exact posterior, we need to be able to compute those moments under the exact posterior. This is not a problem for traditional mixture models including mixture of Gaussians, latent Dirichlet allocation, etc., since the number of mixture components in those models are assumed to be small constants. However, this is not the case for SPNs, where the effective number of mixture components is ?S = ?(2H(S) ), which also depends on the input network S. Qn 1 To i=1 pt (xi ) and ut , R simplify Q the notation, for each t ? [?S ], we define ct , p (w) (k,j)?TtE wk,j dw. That is, ct corresponds to the product of leaf distributions in the tth w 0 Q induced tree Tt , and ut is the moment of (k,j)?TtE wk,j , i.e., the product of tree edges, under the prior distribution p0 (w). Realizing that the posterior distribution needs to satisfy the normalization constraint, we have: Z ?S ?S X Y X ct p0 (w) wk,j dw = ct ut = Zx (2) t=1 w t=1 (k,j)?TtE Note that the prior distribution for a sum node is a Dirichlet distribution. In this case we can compute a closed form expression for ut as: Y Z Y Y ? P k,j ut = p0 (wk )wk,j dwk = Ep0 (wk ) [wk,j ] = (3) 0 wk j 0 ?k,j (k,j)?TtE 1 (k,j)?TtE (k,j)?TtE For ease of notation, we omit the explicit dependency of ct on the instance x . 4 More generally, let f (?) be a function applied to each edge weight in an SPN. We use the notation Mp (f ) to mean the moment of function f evaluated under distribution p. We are interested in computing Mp (f ) where p = p(w | x), which we call the one-step update posterior distribution. More specifically, for each edge weight wk,j , we would like to compute the following quantity: Z Z ?S Y 1 X ct wk0 ,j 0 dw (4) Mp (f (wk,j )) = f (wk,j )p(w | x) dw = p0 (w)f (wk,j ) Zx t=1 w w 0 0 (k ,j )?TtE We note that (4) is not trivial to compute as it involves ?S = ?(2H(S) ) terms. Furthermore, in order to conduct moment matching, we need to compute the above moment for each edge (k, j) from a sum node. A naive computation will lead to a total time complexity ?(|S| ? 2H(S) ). A linear time algorithm to compute these moments has been designed by Rashwan et al. [12] when the underlying structure of S is a tree. This algorithm recursively computes the moments in a top-down fashion along the tree. However, this algorithm breaks down when the graph is a DAG. In what follows we will present a O(|S|) time and space algorithm that is able to compute all the moments simultaneously for general SPNs with DAG structures. We will first show a linear time reduction from the moment computation in (4) to a joint inference problem in S, and then proceed to use the differential trick to efficiently compute (4) for each edge in the graph. The final component will be a dynamic program to simultaneously compute (4) for all edges wk,j in the graph by trading constant factors of space complexity to reduce time complexity. 3.3 Linear Time Reduction from Moment Computation to Joint Inference Let us first compute (4) for a fixed edge (k, j). Our strategy is to partition all the induced trees based on whether they contain the tree edge (k, j) or not. Define TF = {Tt | (k, j) 6? Tt , t ? [?S ]} and TT = {Tt | (k, j) ? Tt , t ? [?S ]}. In other words, TF corresponds to the set of trees that do not contain edge (k, j) and TT corresponds to the set of trees that contain edge (k, j). Then, Z Y 1 X Mp (f (wk,j )) = ct p0 (w)f (wk,j ) wk0 ,j 0 dw Zx w Tt ?TT (k0 ,j 0 )?TtE Z Y 1 X + ct p0 (w)f (wk,j ) wk0 ,j 0 dw (5) Zx w 0 0 Tt ?TF (k ,j )?TtE For the induced trees that contain edge (k, j), we have Z Y 1 X 1 X wk0 ,j 0 dw = ct p0 (w)f (wk,j ) ct ut Mp00,k (f (wk,j )) Zx Zx w 0 0 Tt ?TT (6) Tt ?TT (k ,j )?TtE where p00,k is the one-step update posterior Dirichlet distribution for sum node k after absorbing the term wk,j . Similarly, for the induced trees that do not contain the edge (k, j): Z Y 1 X 1 X ct p0 (w)f (wk,j ) wk0 ,j 0 dw = ct ut Mp0,k (f (wk,j )) (7) Zx Zx w 0 0 Tt ?TF Tt ?TF (k ,j )?TtE where p0,k is the prior Dirichlet distribution for sum node k. The aboveQ equation holds by changing the order of integration and realize that since (k, j) is not in tree Tt , (k0 ,j 0 )?TtE wk0 ,j 0 does not contain the term wk,j . Note that both Mp0,k (f (wk,j )) and Mp00,k (f (wk,j )) are independent of specific induced trees, so we can combine the above two parts to express Mp (f (wk,j )) as: ! ! 1 X 1 X Mp (f (wk,j )) = ct ut Mp0,k (f (wk,j )) + ct ut Mp00,k (f (wk,j )) (8) Zx Zx Tt ?TF Tt ?TT From (2) we have ?S 1 X ct ut = 1 and Zx t=1 ?S X ct ut = X Tt ?TT t=1 5 ct ut + X Tt ?TF ct ut This implies that Mp (f ) is in fact a convex combination of Mp0,k (f ) and Mp00,k (f ). In other words, since both Mp0,k (f ) and Mp00,k (f ) can be computed in closed form for each edge (k, j), so in order to compute (4), we only need to be able to compute the two Recall that for Qcoefficients efficiently. P each induced tree Tt , we have the expression of ut as ut = (k,j)?TtE ?k,j / j 0 ?k,j 0 . So the term P?S t=1 ct ut can thus be expressed as: ?S X ct ut = t=1 ?S X n ?k,j Y pt (xi ) 0 j 0 ?k,j i=1 Y (9) P t=1 (k,j)?TtE The key observation that allows us to find the linear time reduction lies in the fact that (9) shares exactly the same functional form as the network polynomial, with the only difference being the specification of edge weights in the network. The following lemma formalizes our argument. P?S Lemma 1. t=1 ct ut can be computed in O(|S|) time and space in a bottom-up evaluation of S. Proof. Compare the form of (9) to the network polynomial: p(x | w) = Vroot (x; w) = ?S X Y wk,j t=1 (k,j)?TtE n Y pt (xi ) (10) i=1 Clearly (9) and (10) share the P same functional form and the only difference lies in that the edge weight used in (9) is given by ?k,j / j 0 ?k,j 0 while the edge weight used in (10) is given by wk,j , both of which are constrained to be positive and locally normalized. This means P that in order to compute the value of (9), we can replace all the edge weights wk,j with ?k,j / j 0 ?k,j 0 , and a bottom-up pass evaluation of S will give us the desired result at the root of the network. The linear time and space complexity then follows from the linear time and space inference complexity of SPNs.  In other words, we reduce the original moment computation problem for edge (k, j) to a joint inference problem in S with a set of weights determined by ? . 3.4 Efficient Polynomial Evaluation by Differentiation P To evaluate (8), we also need to compute Tt ?TT ct ut efficiently, where the sum is over a subset of induced trees that contain edge (k, j). Again, due to the exponential lower bound on the number of unique induced trees, a brute force computation is infeasible in the worst case. The key observation is P?S that we can use the differential trick P to solve this problem by realizing the fact that Zx = t=1 ct ut is a multilinear function in ?k,j / j 0 ?k,j 0 , ?k, j and it has a tractable circuit representation since it shares the same network structure with S. P P?S Lemma 2. Tt ?TT ct ut = wk,j (? t=1 ct ut /?wk,j ), and it can be computed in O(|S|) time and space in a top-down differentiation of S. Proof. Define wk,j , ?k,j / X Tt ?TT ct ut = P ?k,j 0 , then Y wk0 ,j 0 X j0 Tt ?TT (k0 ,j 0 )?TtE = wk,j X Y Tt ?TT (k0 ,j 0 )?TtE (k0 ,j 0 )6=(k,j) = wk,j n Y pt (xi ) i=1 wk0 ,j 0 n Y i=1 pt (xi ) + 0 ? X X ? ? ct ut + ct ut ?wk,j ?wk,j Tt ?TT Tt ?TF X ct ut Tt ?TF ! = wk,j ?S ? X ct ut ?wk,j t=1 ! where the second equality is by Corollary 1 that the network polynomial is a multilinear function of wk,j and the third equality holds because TF is the set of trees that do not contain wk,j . The last equality follows by simple algebraic transformations. In summary, the above lemma holds because of the fact that differential operator applied to a multilinear function acts as a selector for all the 6 P P ?S P ct ut ? Tt ?TT ct ut can monomials containing a specific variable. Hence, Tt ?TF ct ut = t=1 also be computed. To show the linear time and space complexity, recall that the differentiation w.r.t.wk,j can P?Sbe efficiently computed by back-propagation in a top-down pass of S once we have computed t=1 ct ut in a bottom-up pass of S.  Remark. The fact that we can compute the differentiation w.r.t. wk,j using the original circuit without expanding it underlies many recent advances in the algorithmic design of SPNs. Zhao et al. [18, 17] used the above differential trick to design linear time collapsed variational algorithm and the concave-convex produce for parameter estimation in SPNs. A different but related approach, where the differential operator is taken w.r.t. input indicators, not model parameters, is applied in computing the marginal probability in Bayesian networks and junction trees [3, 8]. We finish this discussion by concluding that when the polynomial computed by the network is a multilinear function in terms of model parameters or input indicators (such as in SPNs), then the differential operator w.r.t. a variable can be used as an efficient way to compute the sum of the subset of monomials that contain the specific variable. 3.5 Dynamic Programming: from Quadratic to Linear P?S Define Dk (x; w) = ?Vroot (x; w)/?Vk (x; w). Then the differentiation term ? t=1 ct ut /?wk,j in Lemma 2 can be computed via back-propagation in a top-down pass of the network as follows: P?S ? t=1 ct ut ?Vroot (x; w) ?Vk (x; w) = = Dk (x; w)Vj (x; w) (11) ?wk,j ?Vk (x; w) ?wk,j Let ?k,j = (wk,j Vj (x; w)Dk (x; w)) /Vroot (x; w) and fk,j = f (wk,j ), then the final formula for computing the moment of edge weight wk,j under the one-step update posterior p is given by Mp (fk,j ) = (1 ? ?k,j ) Mp0 (fk,j ) + ?k,j Mp00 (fk,j ) (12) Corollary 2. For each edges (k, j), (8) can be computed in O(|S|) time and space. The corollary simply follows from Lemma 1 and Lemma 2 with the assumption that the moments under the prior has closed formPsolution. By definition, we also have ?k,j = Tt ?TT ct ut /Zx , Dk (x; w) + hence 0 ? ?k,j ? 1, ?(k, j). This formula wk,j shows that ?k,j computes the ratio of all the induced trees that contain edge (k, j) to the network. Roughly speaking, this measures how important the contribution of a specific edge is Vj (x; w) ? ? ? to the whole network polynomial. As a result, we can interpret (12) as follows: the more important the edge is, the more portion of the moment comes from the new observation. We visualize our moment computation method for a single edge (k, j) in Fig. 1. Figure 1: The moment computation only needs Remark. CCCP for SPNs was originally de- three quantities: the forward evaluation value at rived using a sequential convex relaxation tech- node j, the backward differentiation value node k, nique, where in each iteration a concave surro- and the weight of edge (k, j). gate function is constructed and optimized. The key update in each iteration of CCCP ([18], (7)) is 0 given as follows: wk,j ? wk,j Vj (x; w)Dk (x; w)/Vroot (x; w), where the R.H.S. is exactly the same as ?k,j defined above. From this perspective, CCCP can also be understood as implicitly applying the differential trick to compute ?k,j , i.e., the relative importance of edge (k, j), and then take updates according to this importance measure. In order to compute the moments of all the edge weights wk,j , a naive computation would scale O(|S|2 ) because there are O(|S|) edges in the graph and from Cor. 2 each such computation takes O(|S|) time. The key observation that allows us to further reduce the complexity to linear comes from the structure of ?k,j : ?k,j only depends on three terms, i.e., the forward evaluation value 7 Vj (x; w), the backward differentiation value Dk (x; w) and the original weight of the edge wk,j . This implies that we can use dynamic programming to cache both Vj (x; w) and Dk (x; w) in a bottom-up evaluation pass and a top-down differentiation pass, respectively. At a high level, we trade off a constant factor in space complexity (using two additional copies of the network) to reduce the quadratic time complexity to linear. Theorem 2. For all edges (k, j), (8) can be computed in O(|S|) time and space. Proof. During the bottom-up evaluation pass, in order to compute the value Vroot (x; w) at the root of S, we will also obtain all the values Vj (x; w) at each node j in the graph. So instead of discarding these intermediate Vj (x; w), we cache them by allocating additional space at each node j. So after one bottom-up evaluation pass of the network, we will also have all the Vj (x; w) for each node j, at the cost of one additional copy of the network. Similarly, during the top-down differentiation pass of the network, because of the chain rule, we will also obtain all the intermediate Dk (x; w) at each node k. Again, we cache them. Once we have both Vj (x; w) and Dk (x; w) for each edge (k, j), from (12), we can get all the moments for all the weighted edges in S simultaneously. Because the whole process only requires one bottom-up evaluation pass and one top-down differentiation pass of S, the time complexity is 2|S|. Since we use two additional copies of S, the space complexity is 3|S|.  We summarize the linear time algorithm for moment computation in Alg. 1. Algorithm 1 Linear Time Exact Moment Computation Input: Prior p0 (w | ? ), moment f , SPN S and input x. Output: Mp (f (wk,j P )), ?(k, j). 1: wk,j ? ?k,j / j 0 ?k,j 0 , ?(k, j). 2: Compute Mp0 (f (wk,j )) and Mp00 (f (wk,j )), ?(k, j). 3: Bottom-up evaluation pass of S with input x. Record Vk (x; w) at each node k. 4: Top-down differentiation pass of S with input x. Record Dk (x; w) at each node k. 5: Compute the exact moment for each (k, j): Mp (fk,j ) = (1 ? ?k,j ) Mp0 (fk,j ) + ?k,j Mp00 (fk,j ). 4 Applications in Online Moment Matching In this section we use Alg. 1 as a sub-routine to develop a new Bayesian online learning algorithm for SPNs based on assumed density filtering [14]. To do so, we find an approximate distribution by minimizing the KLQ divergence between the one-step update posterior and the approximate distribution. m Let P = {q | q = k=1 Dir(wk ; ?k )}, i.e., P is the space of product of Dirichlet densities that are decomposable over all the sum nodes in S. Note that since p0 (w; ? ) is fully decomposable, we have p0 ? P. One natural choice is to try to find an approximate distribution q ? P such that q minimizes the KL-divergence between p(w|x) and q, i.e., p? = arg min KL(p(w | x) || q) q?P It is not hard to show that when q is an exponential family distribution, which is the case in our setting, the minimization problem corresponds to solving the following moment matching equation: Ep(w|x) [T (wk )] = Eq(w) [T (wk )] (13) where T (wk ) is the vector of sufficient statistics of q(wk ). When q(?) is a Dirichlet, we have T (wk ) = log wk , where the log is understood to be taken elementwise. This principle of finding an approximate distribution is also known as reverse information projection in the literature of information theory [2]. As a comparison, information projection corresponds to minimizing KL(q || p(w | x)) within the same family of distributions q ? P. By utilizing our efficient linear time algorithm for exact moment computation, we propose a Bayesian online learning algorithm for SPNs based on the above moment matching principle, called assumed density filtering (ADF). The pseudocode is shown in Alg. 2. In the ADF algorithm, for each edge wk,j the above moment matching equation amounts to solving the following equation: X ?(?k,j ) ? ?( ?k,j 0 ) = Ep(w|x) [log wk,j ] j0 8 where ?(?) is the digamma function. This is a system of nonlinear equations about ? where the R.H.S. of the above equation can be computed using Alg. 1 in O(|S|) time for all the edges (k, j). To efficiently solve it, we take exp(?) at both sides of the equation and approximate the L.H.S. using the fact that exp(?(?k,j )) ? ?k,j ? 12 for ?k,j > 1. Expanding the R.H.S. of the above equation using the identity from (12), we have: ? ? X
exp ??(?k,j ) ? ?( ?w,j 0 )? = exp Ep(w|x) [log wk,j ] j0 !(1??k,j ) !?k,j ?k,j ? 12 ?k,j + 12 ?k,j ? 12 P ? P (14) ?P 1 = 1 1 0 0 0 j 0 ?k,j ? 2 j 0 ?k,j ? 2 j 0 ?k,j + 2 P Note that (?k,j ? 0.5)/( j 0 ?k,j 0 ? 0.5) is approximately the mean of the prior Dirichlet under p0 P and (?k,j + 0.5)/( j 0 ?k,j 0 + 0.5) is approximately the mean of p00 , where p00 is the posterior by adding one pseudo-count to wk,j . So (14) is essentially finding a posterior with hyperparameter ? such that the posterior mean is approximately the weighted geometric mean of the means given by p0 and p00 , weighted by ?k,j . Instead of matching the moments given by the sufficient statistics, also known as the natural moments, BMM tries to find an approximate distribution q by matching the first order moments, i.e., the mean of the prior and the one-step update posterior. Using the same notation, we want q to match the following equation: ?k,j ?k,j ?k,j + 1 Eq(w) [wk ] = Ep(w|x) [wk ] ? P = (1 ? ?k,j ) P + ?k,j P (15) 0 0 0 ? ? j 0 k,j j 0 k,j j 0 ?k,j + 1 Again, we can interpret the above equation as to find the posterior hyperparameter ? such that the posterior mean is given by the weighted arithmetic mean of the means given by p0 and p00 , weighted by ?k,j . Notice that due to the normalization constraint, we cannot solve for ? directly from the above equations, and in order to solve for ? we will need one more equation to be added into the system. However, from line 1 of Alg. 1, what we need in the next iteration of the algorithm is not ?, but only its normalized version. So we can get rid of the additional equation and use (15) as the update formula directly in our algorithm. Using Alg. 1 as a sub-routine, both ADF and BMM enjoy linear running time, sharing the same order of time complexity as CCCP. However, since CCCP directly optimizes over the data log-likelihood, in practice we observe that CCCP often outperforms ADF and BMM in log-likelihood scores. Algorithm 2 Assumed Density Filtering for SPN Input: Prior p0 (w | ? ), SPN S and input {xi }? i=1 . 1: p(w) ? p0 (w | ? ) 2: for i = 1, . . . , ? do 3: Apply Alg. 1 to compute Ep(w|xi ) [log wk,j ] for all edges (k, j). 4: Find p? = arg minq?P KL(p(w | xi ) || q) by solving the moment matching equation (13). 5: p(w) ? p?(w). 6: end for 5 Conclusion We propose an optimal linear time algorithm to efficiently compute the moments of model parameters in SPNs under online settings. The key techniques used in the design of our algorithm include the liner time reduction from moment computation to joint inference, the differential trick that is able to efficiently evaluate a multilinear function, and the dynamic programming to further reduce redundant computations. Using the proposed algorithm as a sub-routine, we are able to improve the time complexity of BMM from quadratic to linear on general SPNs with DAG structures. We also use the proposed algorithm as a sub-routine to design a new online algorithm, ADF. As a future direction, we hope to apply the proposed moment computation algorithm in the design of efficient structure learning algorithms for SPNs. We also expect that the analysis techniques we develop might find other uses for learning SPNs. 9 Acknowledgements HZ thanks Pascal Poupart for providing insightful comments. HZ and GG are supported in part by ONR award N000141512365. References [1] C. Boutilier, N. Friedman, M. Goldszmidt, and D. Koller. Context-specific independence in Bayesian networks. In Proceedings of the Twelfth international conference on Uncertainty in artificial intelligence, pages 115?123. Morgan Kaufmann Publishers Inc., 1996. [2] I. Csisz?r and F. Matus. Information projections revisited. IEEE Transactions on Information Theory, 49(6):1474?1490, 2003. [3] A. Darwiche. A differential approach to inference in Bayesian networks. Journal of the ACM (JACM), 50(3):280?305, 2003. [4] A. Dennis and D. Ventura. Greedy structure search for sum-product networks. In International Joint Conference on Artificial Intelligence, volume 24, 2015. [5] R. Gens and P. Domingos. Discriminative learning of sum-product networks. In Advances in Neural Information Processing Systems, pages 3248?3256, 2012. [6] R. Gens and P. Domingos. Learning the structure of sum-product networks. In Proceedings of The 30th International Conference on Machine Learning, pages 873?880, 2013. [7] P. Jaini, A. Rashwan, H. Zhao, Y. Liu, E. Banijamali, Z. Chen, and P. Poupart. Online algorithms for sum-product networks with continuous variables. In Proceedings of the Eighth International Conference on Probabilistic Graphical Models, pages 228?239, 2016. [8] J. D. Park and A. Darwiche. A differential semantics for jointree algorithms. Artificial Intelligence, 156(2):197?216, 2004. [9] R. Peharz, S. Tschiatschek, F. Pernkopf, and P. Domingos. On theoretical properties of sumproduct networks. In AISTATS, 2015. [10] R. Peharz, R. Gens, F. Pernkopf, and P. Domingos. On the latent variable interpretation in sum-product networks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 39 (10):2030?2044, 2017. [11] H. Poon and P. Domingos. Sum-product networks: A new deep architecture. In Proc. 12th Conf. on Uncertainty in Artificial Intelligence, pages 2551?2558, 2011. [12] A. Rashwan, H. Zhao, and P. Poupart. Online and distributed bayesian moment matching for parameter learning in sum-product networks. In Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, pages 1469?1477, 2016. [13] A. Rooshenas and D. Lowd. Learning sum-product networks with direct and indirect variable interactions. In ICML, 2014. [14] H. W. Sorenson and A. R. Stubberud. Non-linear filtering by approximation of the a posteriori density. International Journal of Control, 8(1):33?51, 1968. [15] L. G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8 (2):189?201, 1979. [16] H. Zhao, M. Melibari, and P. Poupart. On the relationship between sum-product networks and bayesian networks. In ICML, 2015. [17] H. Zhao, T. Adel, G. Gordon, and B. Amos. Collapsed variational inference for sum-product networks. In ICML, 2016. [18] H. Zhao, P. Poupart, and G. Gordon. A unified approach for learning the parameters of sum-product networks. NIPS, 2016. 10 

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A Meta-Learning Perspective on Cold-Start Recommendations for Items Manasi Vartak? Massachusetts Institute of Technology mvartak@csail.mit.edu Jeshua Bratman Twitter Inc. jbratman@twitter.com Arvind Thiagarajan Twitter Inc. arvindt@twitter.com Conrado Miranda Twitter Inc. cmiranda@twitter.com Hugo Larochelle? Google Brain hugolarochelle@google.com Abstract Matrix factorization (MF) is one of the most popular techniques for product recommendation, but is known to suffer from serious cold-start problems. Item cold-start problems are particularly acute in settings such as Tweet recommendation where new items arrive continuously. In this paper, we present a meta-learning strategy to address item cold-start when new items arrive continuously. We propose two deep neural network architectures that implement our meta-learning strategy. The first architecture learns a linear classifier whose weights are determined by the item history while the second architecture learns a neural network whose biases are instead adjusted. We evaluate our techniques on the real-world problem of Tweet recommendation. On production data at Twitter, we demonstrate that our proposed techniques significantly beat the MF baseline and also outperform production models for Tweet recommendation. 1 Introduction The problem of recommending items to users ? whether in the form of products, Tweets, or ads ? is a ubiquitous one. Recommendation algorithms in each of these settings seek to identify patterns of individual user interest and use these patterns to recommend items. Matrix factorization (MF) techniques [19], have been shown to work extremely well in settings where many users rate the same items. MF works by learning separate vector embeddings (in the form of lookup tables) for each user and each item. However, these techniques are well known for facing serious challenges when making cold-start recommendations, i.e. when having to deal with a new user or item for which a vector embedding hasn?t been learned. Cold-start problems related to items (as opposed to users) are particularly acute in settings where new items arrive continuously. A prime example of this scenario is Tweet recommendation in the Twitter Home Timeline [1]. Hundreds of millions of Tweets are sent on Twitter everyday. To ensure freshness of content, the Twitter timeline must continually rank the latest Tweets and recommend relevant Tweets to each user. In the absence of user-item rating data for the millions of new Tweets, traditional matrix factorization approaches that depend on ratings cannot be used. Similar challenges related to item cold-start arise when making recommendations for news [20], other types of social media, and streaming data applications. In this paper, we consider the problem of item cold-start (ICS) recommendation, focusing specifically on providing recommendations when new items arrive continuously. Various techniques [3, 14, 27, 17] ? ? Work done as an intern at Twitter Work done while at Twitter 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. have been proposed in the literature to extend MF (traditional and probabilistic) to address cold-start problems. Majority of these extensions for item cold-start involve the incorporation of item-specific features based on item description, content, or intrinsic value. From these, a model is prescribed that can parametrically (as opposed to a lookup table) infer a vector embedding for that item. Such item embeddings can then be compared with the embeddings from the user lookup table to perform recommendation of new items to these users. However, in a setting where new items arrive continuously, we posit that relying on user embeddings trained offline into a lookup table is sub-optimal. Indeed, this approach cannot capture substantial variations in user interests occurring on short timescales, a common phenomenon with continuously produced content. This problem is only partially addressed when user embeddings are retrained periodically or incrementally adjusted online. Alternatively, recommendations could be made by performing content-based filtering [21], where we compare each new item to other items the user has rated in the recent past. However, a pure content-based filtering approach does not let us share and transfer information between users. Instead, we would like a method that performs akin to content filtering using a user?s item history, but shares information across users through some form of transfer learning between recommendation tasks across users. In other words, we would like to learn a common procedure that takes as input a set of items from any user?s history and produces a scoring function that can be applied to new test items and indicate how likely this user is to prefer that item. In this formulation, we notice that the recommendation problem is equivalent to a meta-learning problem [28] where the objective is to learn a learning algorithm that can take as input a (small) set of labeled examples (a user?s history) and will output a model (scoring function) that can be applied to new examples (new items). In meta-learning, training takes place in an episodic manner, where a training set is presented along with a test example that must be correctly labeled. In our setting, an episode is equivalent to presenting a set of historical items (and ratings) for a particular user along with test items that must be correctly rated for that user. The meta-learning perspective is appealing for a few reasons. First, we are no longer tied to the MF model where a rating is usually the inner product of the user and item embeddings; instead, we can explore a variety of ways to combine user and item information. Second, it enables us to take advantage of deep neural networks to learn non-linear embeddings. And third, it specifies an effective way to perform transfer learning across users (by means of shared parameters), thus enabling us to cope with limited amount of data per user. A key part of designing a meta-learning algorithm is the specification of how a model is produced for different tasks. In this work, we propose and evaluate two strategies for conditioning the model based on task. The first, called linear weight adaptation, is a light-weight method that builds a linear classifier and adapts weights of the classifier based on the task information. The second, called non-linear bias adaptation, builds a neural network classifier that uses task information to adapt the biases of the neural network while sharing weights across all tasks. Thus our contributions are: (1) we introduce a new hybrid recommendation method for the item cold-start setting that is motivated by limitations in current MF extensions for ICS; (2) we introduce a meta-learning formulation of the recommendation problem and elaborate why a meta-learning perspective is justified in this setting; (3) we propose two key architectures for meta-learning in this recommendation context; and (4) we evaluate our techniques on a production Twitter dataset and demonstrate that they outperform an approach based on lookup table embeddings as well as state-of-the-art industrial models. 2 Problem Formulation Similar to other large-scale recommender systems that must address the ICS problem [6], we view recommendation as a binary classification problem as opposed to a regression problem. Specifically, for an item ti and user uj , the outcome eij ? {0, 1} indicates whether the user engaged with the item. Engagement can correspond to different actions in different settings. For example, in video recommendation, engagement can be defined as a user viewing the video; in ad-click prediction, engagement can be defined as clicking on an ad; and on Twitter, engagement can be an action related to a Tweet such as liking, Retweeting or replying. Our goal in this context is to predict the probability 2 that uj will engage with ti : Pr(eij =1|ti , uj ) . (1) Once the engagement probability has been computed, it can be combined with other signals to produce a ranked list of recommended items. As discussed in Section 1, in casting recommendations as a form of meta-learning, we view the problem of making predictions for one user as an individual task or episode. Let Tj be the set of items to which a user uj has been exposed (e.g. Tweets recommended to uj ). We represent each user in terms of their item history, i.e., the set of items to which they have been exposed and their engagement for each of these items. Specifically, user uj is represented by their item history Vj = {(tm , emj )} : tm ? Tj . Note that we limit item history to only those items that were seen before ti . We then model the probability of Eq. 1 as the output of a model f (ti ; ?) where the parameters ? are produced from the item history Vj of user uj : Pr(eij =1|ti , uj ) = f (ti ; H(Vj )) (2) Thus meta-learning consists of learning the function H(Vj ) that takes history Vj and produces parameters of the model f (ti ; ?). In this paper, we propose two neural network architectures for learning H(Vj ), depicted in Fig. 1. The first approach, called Linear Weight Adaptation (LWA) and shown in Fig. 1a, assumes f (ti ; ?) is a linear classifier on top of non-linear representations of items and uses the user history to adapt classifier weights. The second approach, called Non-Linear Bias Adaptation (NLBA) and shown in Fig. 1b, assumes f (ti ; ?) to be a neural network classifier on top of item representations and uses the user history to adapt the biases of the units in the classifier. In the following subsections, we describe the two architectures and differences in how they model the classification of a new item ti from its representation F(ti ). 3 Proposed Architectures As shown in Fig. 1, both architectures we propose take as input (a) the items to which a user has been exposed along with the rating (i.e. class) assigned to each item by this user (positive, i.e., 1, or negative, i.e., 0), and (b) a test item for which we seek to predict a rating. Each of our architectures in turn consists of two sub-networks. The first sub-network learns a common representation of items (historical and new), which we note F(t). In our implementation, item representations F(t) are learned by a deep feed-forward neural network. We then compute a single representative per class by applying an aggregating function G to the representations of items tm ? Tj from each class. A simple example of G is the unweighted mean, while more complicated functions may order items by recency and learn to weigh individual embeddings differently. Thus, the class representative embedding for class c ? {0, 1} can be expressed as shown below. Rjc = G({F(tm )} : tm ? Tj ? (emj = c)) (3) Once we have obtained class representatives, the second sub-network applies the LWA and NLBA approaches to adapt the learned model based on item histories. 3.1 Linear Classifier with Task-dependent weights Our first approach to conditioning predictions on users? item histories has parallels to latent factor models and is appealing due to its simplicity: we learn a linear classifier (for new items) whose weights are determined by the user?s history Vj . Given the two class-representative embeddings Rj0 , Rj1 described above, LWA provides the bias (first term) and weights (second term) of a linear logistic regression classifier as follows: [b, (w0 Rj0 + w1 Rj1 )] = H(Vj ) (4) where scalars b, w0 , w1 are trainable parameters. More concretely, with f (ti ; ?) being logistic regression, Eq. 2 becomes: Pr(eij =1|ti , uj ) = ?(b + F(ti ) ? (w0 Rj0 + w1 Rj1 )) 3 (5) User 1 User 2 { }G }{ F(t1 ) F (t1 ) G F(t3 ) positive class F(t5 ) F(t7 ) positive class F(t8 ) { }G F (t2 ) G F(t4 ) negative class } { F(t4 ) F(t6 ) negative class (a) Linear Classifier with Weight Adaptation. Changes in the shading of each connection with the output unit for two users illustrates that the weights of the classifier vary based on each user?s item history. The output bias indicated by the shades of the circles however remains the same. User 1 User 2 { }G G }{ { }G G F(t1 ) F (t1 ) F(t3 ) positive class F(t7 ) positive class F (t2 ) F(t4 ) F(t5 ) F(t8 ) negative class } { F(t4 ) F(t6 ) negative class (b) Non-linear Classifier with Bias Adaptation. Changes in the shading of each unit between two users illustrates that the biases of these units vary based on each user?s item history. The weights however remain the same. Figure 1: Proposed meta-learning architectures While bias b of the classifier is constant across users, its weight vector varies with user-specific item histories (i.e., based on the representative embeddings for classes). This means that different dimensions of F(ti ), such as properties of item ti , get weighted differently depending on the user. While simple, the LWA method can be quite effective (see Section 5). Moreover, in some cases, it may be preferred over more complex methods because it allows significant amount of computation to be performed offline. For example, in Eq. 5, the only quantities that are unknown at prediction time are F(ti ). All the rest, including Rjc , can be pre-computed, reducing the cost of prediction to the computation of one dot product and one sigmoid. 3.2 Non-linear Classifier with Task-dependent Bias Our first meta-learning strategy is simple and is reminiscent of MF with non-linear embeddings. However, it limits the effect of task information, specifically Rj0 and Rj1 , on the final prediction. Our second strategy, NLBA, learns a neural network classifier with H hidden-layers where the bias (first term) and weights (second term) of the output, as well as the biases (third term) and weights (fourth term) of all hidden layers are determined as follows: H [v0 Rj0 + v1 Rj1 , w, {Vl0 Rj0 + Vl1 Rj1 }H l=1 , {Wl }l=1 ] = H(Vj ) 4 (6) 1 H H Here, the vectors v0 , v1 , w and matrices {Vl0 }H l=1 , {Vl }l=1 , {Wl }l=1 are all trainable parameters. In contrast to LWA, all weights (output and hidden) in NLBA are constant across users, while the biases of output and hidden units are adapted per user. One can think of this approach as learning a shared pool of hidden units whose activation can be modulated depending on the user (e.g. a unit could be entirely shot down for a user with a large negative bias). Compared to LWA, NLBA produces a non-linear classifier of the item representations F(ti ) and can model complex interactions between classes and also between the classes and the test item. For example, interactions allow NLBA to explore a different part of the classifier function space that is not accessible to LWA (e.g., ratio of the k th dimension of the class representatives). We find empirically that NLBA significantly improves model performance compared to LWA (Section 5). Selecting Historical Impressions. A natural question with our meta-learning formulation is the minimum item history size required to make accurate predictions. In general, item history size depends on the problem and variability within items, and must be empirically determined. Often, due to the long tail of users, item histories can be very large (e.g., consider a bot which likes every item). Therefore, we recommend setting an upper limit on item history size. Further, for any user, the number of items with positive engagement (eij =1) can be quite different from those with negative engagement (eij =0). Therefore, in our experiments, we choose to independently sample histories for the two classes up to a maximum size k for each class. Note that while this sampling strategy makes the problem more tractable, this sampling throws off certain features (e.g. user click through rate) that would benefit from having the true proportions of positive and negative engagements. 4 Related Work Algorithms for recommendation broadly fall into two categories: content-filtering and collaborativefiltering. Content filtering [21] uses information about items (e.g. product categories, item content, reviews, price) and users (e.g. age, gender, location) to match users to items. In contrast, collaborative filtering [19, 23] uses past user-item ratings to predict future ratings. The most popular technique for performing collaborative filtering is via latent factor models where items and users are represented in a common latent space and ratings are computed as the inner product between the user and item representations. Matrix factorization (MF) is the most popular instantiation of latent factor models and has been used for large scale recommendations of products [19], movies [15]) and news [7]. A significant problem with traditional MF approaches is that they suffer from cold-start problems, i.e., they cannot be applied to new items or users. In order to address the cold-start problem, work such as [3, 14, 27, 17] has extended the MF model so that user- and item-specific terms can be included in their respective representations. These methods are called hybrid methods. Given the power of deep neural networks to learn representations of images and text, many of the new hybrid methods such as [30] and [12] use deep neural networks to learn item representations. Deep learning models based on ID embeddings (as opposed to content embeddings) have also been used for performing large scale video recommendation in [6]. The work in [5, 30] operates in a problem setting similar to ours where new scientific articles must be recommended to users based on other articles in their library. In these settings, users are represented in terms of scientific papers in their ?libraries?. Note that unlike our setting where we have positive as well as negative information, there are no negative examples present in this setting. [9, 10] propose RNN architecture for a similar problem, namely that of recommending items during short-lived web sessions. [11] propose co-factorization machines to jointly model topics in Tweets while making recommendations. In this paper, we propose to view recommendations from a meta-learning perspective [28, 18]. Recently, meta-learning has been explored as a popular strategy for learning from a small number of items (also called few-shot learning [16, 13]). Successful applications of meta-learning include MatchingNets [29] in which an episodic scheme is used to train a meta-learner to classify images given very few examples belonging to each classes. In particular, MatchingNets use LSTMs to learn attention weights over all points in the support set and use a weighted sum to make predictions for the test item. Similarly, in [24], the authors propose an LSTM-based meta-learner to learn another learner that performs few-shot classification. [25] proposes a memory-augmented neural network for meta-learning. The key idea is that the network can slowly learn useful representations of data through weight updates while the external memory can cache new data for rapid learning. Most 5 recently, [4] proposes to learn active learning algorithms via a technique based on MatchingNets. While the above state-of-the-art meta-learning techniques are powerful and potentially useful for recommendations, they do not scale to large datasets with hundreds of millions of examples. Our approach of computing a mean representative per class is similar to [26] and [22] in terms of learning class representative embeddings. Our work also has parallels to the recent work on DeepSets [31] where the authors propose a general strategy for performing machine learning tasks on sets. The authors propose to learn an embedding per item and then use a permutation invariant operation, usually a sumpool or maxpool, to learn a single representation that is then passed to another neural network for performing the final classification or regression. Our techniques differ in that our input sets are not homogeneous as assumed in DeepSets and therefore we need to learn multiple representatives, and unlike DeepSets, our network must work for variable size histories and therefore a weighted sum is more effective than the unweighted sum. 5 5.1 Evaluation Recommending Tweets on Twitter Home Timeline We evaluated our proposed techniques on the real-world problem of Tweet recommendation. The Twitter Home timeline is the primary means by which users on Twitter consume Tweets from their networks [1]. 300+ million monthly active users on Twitter send hundreds of millions of Tweets per day. Every time a user visits Twitter, the timeline ranking algorithm scores Tweets from the accounts they follow and identifies the most relevant Tweets for that user. We model the timeline ranking problem as one of engagement prediction as described in Section 2. Given a Tweet ti and a user uj , the task is to predict the probability of uj engaging with ti , i.e., Pr(eij =1|ti , uj ; ?). Engagement can be any action related to the Tweet such as Retweeting, liking or replying. For the purpose of this paper, we will limit our analysis to prediction of one kind of engagement, namely that of liking a Tweet. Because hundreds of millions of new Tweets arrive every day, as discussed in Section 1, traditional matrix factorization schemes suffer from acute cold-start problems and cannot be used for Tweet recommendation. In this work, we cast the problem in terms of meta-learning and adopt the formulation developed in Eq. 2. Dataset. We used production data regarding Tweet engagement to perform an offline evaluation of our techniques. Specifically, the training data was generated as follows: for a particular day d, we collect data for all Tweet impressions (i.e., Tweets shown to a user) generated during that day. Each data point consists of a Tweet ti , the user uj to whom it was shown, and the engagement outcome eij . We then join impression data with item histories (per user) that are computed using impressions from the week prior to d. As discussed in Section 2, there are different strategies for selecting items to build the item history. For this problem, we independently sample impressions with positive engagement and negative engagement, up to a maximum of k engagements in each class. We experimented with different values of k and chose the smallest one that did not produce a significant drop in performance. After applying other typical filtering operations, our training dataset consists of hundreds of millions of examples (i.e., ti , uj pairs) for day d. The test and validation sets were similarly constructed, but for different days. For feature preprocessing, we scale and discretize continuous features and one-hot-encode categorical features. Baseline Models. We implemented different architectural variations of the two meta-learning approaches proposed in Section 2. Along with comparisons within architectures, we compare our models against three external models: (a) first, an industrial baseline (PROD-BASE) not using meta-learning; (b) the industrial baseline augmented with a latent factor model for users (MF); and (c) the state-of-the-art production model for engagement prediction (PROD-BEST). PROD-BASE is a deep feed-forward neural network that uses information pertaining only to the current Tweet in order to predict engagement. This information includes features of the current Tweet ti (e.g. its recency, whether it contains a photo, number of likes), features about the user uj (e.g. how heavily the user uses Twitter, their network), and the Tweet?s author (e.g. strength of connection between the user and author, past interactions). Note that this network uses no historical information about the user. This baseline learns a combined item-user embedding (due to user features present in the input) and performs classification based on this embedding. While relatively simple, this model presents a very strong baseline due to the presence of high-quality features. 6 Model AUROC AUROC (w/CTR) MF (shallow) +0.22% +1.32% MF (deep) +0.55% +1.87% PROD-BEST +2.54% +2.54% LWA +1.76% +2.43% LWA? +1.98% +2.31% Table 1: Performance with LWA Model AUROC AUROC (w/CTR) MF (shallow) +0.22% +1.32% MF (shallow) +0.55% +1.87% PROD-BEST +2.54% +2.54% NLBA +2.65% +2.76% Table 2: Performance with NBLA To mimic latent factor models in cold-start settings, in the second baseline, MF we augmented PRODBASE to learn a latent-space representations of users based on ratings. MF uses an independently learned user representation and a current Tweet representation whose inner product is used to make the classification. We evaluate two versions of MF, one that uses a shallow network (1 layer) for learning the representations and another than uses a deep network (5 layers) to learn representations. PROD-BEST is the production model for engagement prediction based on deep neural networks. PROD-BEST uses features not only for the current Tweet but historical features as described in [8]. PROD-BEST is a highly tuned model and represents the state-of-art in Tweet engagement prediction. Experimental Setup. All models were implemented in the Twitter Deep Learning platform [2]. Models were trained to minimize cross-entropy loss and SGD was used for optimization. We use AUROC as the evaluation metric in our experiments. All performance numbers denote percent AUROC improvement relative to the production baseline model, PROD-BASE. For every model, we performed hyperparameter tuning on a validation set using random search and report results for the best performing model. 5.2 Results Linear Classifier with Weight Adaptation. We evaluated two key instantiations of the LWA approach. First, we test the basic architecture described in Section 3.1 where we calculate one representative embedding from the positive and negative class, and take a linear combination of the dot products of the new item with the respective embeddings. We note this architecture LWA (refer Fig. 1.a). When learning class representatives, we use a deep feed-forward neural network (F in Fig. 1.a) to first compute embeddings and then use a weighted average to learn class representatives (G in Fig. 1.a). We also evaluate a model variation where we augment LWA with a network that uses only the new item embedding to produce a prediction that is then combined with the prediction of LWA to produce the final probability. The intuition behind this model, called LWA? , is that the linear weight adaptation approach works well when there are non-zero items in the two engagement classes. In cases where one of the classes is empty, the model can fall back on the current item to predict engagement. We show the performance of all LWA variants in Table 1. Note that for all models, we also test a variant where we explicitly pass a user-specific click-throughrate (CTR) to the model. The reason is that the CTR provides a strong prior on the probability p(eij = 1) but cannot be easily learned from our architectures because the ratio of positive to negative engagements in the item history is not proportional to the CTR. User-specific CTR can be thought of as being part of the bias term from Eq. 2. We find that the simplest classifier adaptation model, LWA, already improves upon the production baseline (PROD-BASE) by >1.5 percent. Adding the bias in the form of CTR, improves the AUROC even further. Because learning a good class representative embedding is key for our metalearning approaches, we performed experiments varying only the architectures used to learn class representative embeddings (i.e., architectures for F, G in Fig. 1a). The top-level classifier was kept constant at LWA but we varied the aggregation function and depth of the feed forward network used to learn F. Results of this experimentation are shown in Table 3. We find that deep networks work better than shallow networks but a model with 9 hidden layers performs worse than a 5 layer network, possibly due to over-fitting. We also find that weighted combinations of embeddings (when items are sorted by time) perform significantly better than simple averages. A likely reason for the effectiveness of weighted combinations is that item histories can be variable sized; therefore, weighing non-zero entries higher produces better representatives. Non-Linear Classifier with Bias Adaptation. As with LWA, we evaluated different variations of the non-linear bias adaptation approach. Results of this evaluation are shown in Table 2. We use a 7 Hidden AVG Weighted Layers AVG 1 +1.8% +2.31% 5 +2.20% +2.42% 9 +2.09% +1.65% Table 3: Learning a representative per class Engagements AUROC Used POS/NEG +2.54% POS-ONLY +1.76% NEG-ONLY +1.87% Table 4: Effect of different engagements weighted mean for computing class representatives in NLBA. We see that this network immediately beats PROD-BASE by a large margin. Moreover, it also readily beats the state-of-the-art model, PROD-BEST. Augmenting NLBA with user-specific CTR further allows the network to cleanly beat the highly optimized PROD-BEST. For NLBA architectures, we also evaluated the impact on model AUROC when examples of only one class (eij =0 or 1) are present in the item history. These architectures replicate the strategy of only using one class of items to make predictions. These numbers approximate the gain that could be achieved by using a DeepSets [31]-like approach. The results of this evaluation are shown in Table 4. As expected, we find that dropping either type of engagement reduces performance significantly. Summary of Results. We find that both our proposed approaches improve on the baseline production model (PROD-BASE) by up to 2.76% and NLBA readily beats the state-of-the-art production model (PROD-BEST). As discussed in Sec. 3.2, we find that NLBA clearly outperforms LWA because of the non-linear classifier and access to a larger space of functions. A breakdown of NLBA performance by overall user engagement identifies that NLBA shows large gains for highly engaged users. For both techniques, although the improvements in AUROC may appear small numerically, they have large product impact because they translate to significantly higher number of engagements. This gain is particularly noteworthy when compared to the highly optimized and tuned PROD-BEST model. 6 Discussion In this paper, we proposed to view recommendations from a meta-learning perspective and proposed two architectures for building meta-learning models for recommendation. While our techniques show clear wins over state-of-the-art models, there are several avenues for improving the model and operationalizing it. First, our model does not explicitly model the time-varying aspect of engagements. While weighting impressions differently is a way of modeling time dependencies (e.g., more recent impressions get more weight) scalable versions of sequence models such as [29, 10] could be used to explicitly model time dynamics. Second, while we chose a balanced sampling strategy for producing item histories, we believe that different strategies may be more appropriate in different recommendation settings, and thus merit further exploration. Third, producing and storing item histories for every user can be expensive with respect to computational as well as storage cost. Therefore, we can explore the computation of representative embeddings in an online fashion such that at any given time, the system tracks the k most representative embeddings. Finally, we believe that there is room for future work exploring effective visualizations of learned embeddings when input items are not easy to interpret (i.e., beyond images and text). 7 Conclusions In this paper, we study the recommendation problem when new items arrive continuously. We propose to view item cold-start through the lens of meta-learning where making recommendations for one user is considered to be one task and our goal is to learn across many such tasks. We formally define the meta-learning problem and propose two distinct approaches to condition the recommendation model on the task. The linear weight adaptation approach adapts weights of a linear classifier depending on task information while the non-linear bias adaptation approach learns task specific item representations and adapts biases of a neural network based on task information. We perform an empirical evaluation of our techniques on the Tweet recommendation problem. On production Twitter data, we show that our meta-learning approaches comfortably beat the state-of-the-art production models for Tweet recommendation. We show that the non-linear bias adaptation approach outperforms the linear weight adaptation approach. We thus demonstrate that recommendation through meta-learning is effective for item cold-start recommendations and may be extended to other recommendation settings as well. 8 Addressing Reviewer Comments We thank the anonymous reviewers for their feedback on the paper. We have incorporated responses to reviewer comments in the paper text. 9 Acknowledgement We would like to thank all members of the Twitter Timelines Quality team as well as the Twitter Cortex team for their help and guidance with this work. References [1] "never miss important tweets from people you follow". never-miss-important-tweets-from-people-you-follow. Accessed: 2017-05-03. https://blog.twitter.com/2016/ [2] Using deep learning at scale in twitter?s timelines. using-deep-learning-at-scale-in-twitter-s-timelines. Accessed: 2017-05-09. https://blog.twitter.com/2017/ [3] D. Agarwal and B.-C. Chen. Regression-based latent factor models. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 19?28. ACM, 2009. [4] P. Bachman, A. Sordoni, and A. Trischler. Learning algorithms for active learning. In D. Precup and Y. W. 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Predicting Scene Parsing and Motion Dynamics in the Future Xiaojie Jin1 , Huaxin Xiao2 , Xiaohui Shen3 , Jimei Yang3 , Zhe Lin3 Yunpeng Chen2 , Zequn Jie4 , Jiashi Feng2 , Shuicheng Yan5,2 1 2 NUS Graduate School for Integrative Science and Engineering (NGS), NUS Department of ECE, NUS 3 Adobe Research 4 Tencent AI Lab 5 Qihoo 360 AI Institute Abstract The ability of predicting the future is important for intelligent systems, e.g. autonomous vehicles and robots to plan early and make decisions accordingly. Future scene parsing and optical flow estimation are two key tasks that help agents better understand their environments as the former provides dense semantic information, i.e. what objects will be present and where they will appear, while the latter provides dense motion information, i.e. how the objects will move. In this paper, we propose a novel model to simultaneously predict scene parsing and optical flow in unobserved future video frames. To our best knowledge, this is the first attempt in jointly predicting scene parsing and motion dynamics. In particular, scene parsing enables structured motion prediction by decomposing optical flow into different groups while optical flow estimation brings reliable pixel-wise correspondence to scene parsing. By exploiting this mutually beneficial relationship, our model shows significantly better parsing and motion prediction results when compared to well-established baselines and individual prediction models on the large-scale Cityscapes dataset. In addition, we also demonstrate that our model can be used to predict the steering angle of the vehicles, which further verifies the ability of our model to learn latent representations of scene dynamics. 1 Introduction Future prediction is an important problem for artificial intelligence. To enable intelligent systems like autonomous vehicles and robots to react to their environments, it is necessary to endow them with the ability of predicting what will happen in the near future and plan accordingly, which still remains an open challenge for modern artificial vision systems. In a practical visual navigation system, scene parsing and dense motion estimation are two essential components for understanding the scene environment. The former provides pixel-wise prediction of semantic categories (thus the system understands what and where the objects are) and the latter describes dense motion trajectories (thus the system learns how the objects move). The visual system becomes ?smarter? by leveraging the prediction of these two types of information, e.g. predicting how the car coming from the opposite direction moves to plan the path ahead of time and predict/control the steering angle of the vehicle. Despite numerous models have been proposed on scene parsing [4, 7, 17, 26, 28, 30, 15] and motion estimation [2, 9, 21], most of them focus on processing observed images, rather than predicting in unobserved future scenes. Recently, a few works [22, 16, 3] explore how to anticipate the scene parsing or motion dynamics, but they all tackle these two tasks separately and fail to utilize the benefits that one task brings to the other. In this paper, we try to close this research gap by presenting a novel model for jointly predicting scene parsing and motion dynamics (in terms of the dense optical flow) for future frames. More importantly, we leverage one task as the auxiliary of the other in a mutually boosting way. See Figure 1 for 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Input Xt ? 4 Input Xt ?3 Input Xt ? 2 Input X t ?1 Output ?t Output ?t ?d ? ? Input St ? 4 Input St ? 3 Input St ? 2 Input St ?1 Output St Output St ?d Figure 1: Our task. The proposed model jointly predicts scene parsing and optical flow in the future. Top: Future flow (highlighted in red) anticipated using preceding frames. Bottom: Future scene parsing (highlighted in red) anticipated using preceding scene parsing results. We use the flow field color coding from [2]. an illustration of our task. For the task of predictive scene parsing, we use the discriminative and temporally consistent features learned in motion prediction to produce parsing prediction with more fine details. For the motion prediction task, we utilize the semantic segmentations produced by predictive parsing to separately estimate motion for pixels with different categories. In order to perform the results for multiple time steps, we take the predictions as input and iterate the model to predict subsequent frames. The proposed model has a generic framework which is agnostic to backbone deep networks and can be conveniently trained in an end-to-end manner. Taking Cityscapes [5] as testbed, we conduct extensive experiments to verify the effectiveness of our model in future prediction. Our model significantly improves mIoU of parsing predictions and reduces the endpoint error (EPE) of flow predictions compared to strongly competitive baselines including a warping method based on optical flow, standalone parsing prediction or flow prediction and other state-of-the-arts methods [22]. We also present how to predict steering angles using the proposed model. 2 Related work For the general field of classic flow (motion) estimation and image semantic segmentation, which is out of this paper?s scope, we refer the readers to comprehensive review articles [2, 10]. Below we mainly review existing works that focus on predictive tasks. Flow and scene parsing prediction The research on predictive scene parsing or motion prediction is still relatively under-explored. All existing works in this direction tackle the parsing prediction and flow prediction as independent tasks. With regards to motion prediction, Luo et al. [19] employed a convolutional LSTM architecture to predict sequences of 3D optical flow. Walker et al. [35] made long-term motion and appearance prediction via a transition and context model. [31] trained CNN for predicting motion of handwritten characters in a synthetic dataset. [36] predicted future optical flow given a static image. Different from above works, our model not only predicts the flow but also scene parsing at the same time, which definitely provides richer information to visual systems. There are also only a handful number of works exploring the prediction of scene parsing in future frames. Jin et al. [16] trained a deep model to predict the segmentations of the next frame from preceding input frames, which is shown to be beneficial for still-image parsing task. Based on the network proposed in [20], Natalia et al. [22] predicted longer-term parsing maps for future frames using the preceding frames? parsing maps. Different from [22], we simultaneously predict optical flows for future frames. Benefited from the discriminative local features learned from flow prediction, the model produces more accurate parsing results. Another related work to ours is [24] which employed an RNN to predict the optical flow and used the flow to warp preceding segmentations. Rather than simply producing the future parsing map through warping, our model predicts flow and scene parsing jointly using learning methods. More importantly, we leverage the benefit that each task brings to the other to produce better results for both flow prediction and parsing prediction. Predictive learning While there are few works specifically on predictive scene parsing or dense motion prediction, learning to prediction in general has received a significant attention from the 2 Flow Anticipating Network 2 Res. Blocks Up-sampling Conv MOV - OBJ Lflow X t ?1 ? - OBJ LSTA flow CNN1 Xt ? 4 - OBJ LOTH flow Parsing Anticipating Network Transform Layer St ?1 ? Lseg CNN2 St ? 4 Figure 2: The framework of our model for predicting future scene parsing and optical flow for one time step ahead. Our model is motivated by the assumption that flow and parsing prediction are mutually beneficial. We design the architecture to promote such mutual benefits. The model consists of two module networks, i.e. the flow anticipating network (blue) which takes preceding frames: Xt?4:t?1 as input and predicts future flow and the parsing anticipating network (yellow) which takes the preceding parsing results: St?4:t?1 as input and predicts future scene parsing. By providing pixel-level class information (i.e. St?1 ), the parsing anticipating network benefits the flow anticipating network to enable the latter to semantically distinguish different pixels (i.e. moving/static/other objects) and predict their flows more accurately in the corresponding branch. Through the transform layer, the discriminative local features learned by the flow anticipating network are combined with the parsing anticipating network to facilitate parsing over small objects and avoid over-smooth in parsing predictions. When predicting multiple time-steps ahead, the prediction of the parsing network in a time-step is used as the input in the next time-step. research community in recent years. Research in this area has explored different aspects of this problem. [37] focused on predicting the trajectory of objects given input image. [13] predicted the action class in the future frames. Generative adversarial networks (GAN) are firstly introduced in [11] to generate natural images from random noise, and have been widely used in many fields including image synthesis [11], future prediction [18, 20, 34, 36, 32, 33] and semantic inpainting [23]. Different from above methods, our model explores a new predictive task, i.e. predicting the scene parsing and motion dynamics in the future simultaneously. Multi-task learning Multi-task learning [1, 6] aims to solve multiple tasks jointly by taking advantage of the shared domain knowledge in related tasks. Our work is partially related to multi-task learning in that both the parsing results and motion dynamics are predicted jointly in a single model. However, we note that predicting parsing and motion ?in the future? is a novel and challenging task which cannot be straightforwardly tackled by conventional multi-task learning methods. To our best knowledge, our work is the first solution to this challenging task. 3 Predicting scene parsing and motion dynamics in the future In this section, we first propose our model for predicting semantics and motion dynamics one time step ahead, and then extend our model to perform predictions for multiple time steps. Due to high cost of acquiring dense human annotations of optical flow and scene parsing for natural scene videos, only subset of frames are labeled for scene parsing in the current datasets. Following [22], to circumvent the need for datasets with dense annotations, we train an adapted Res101 model (denoted as Res101-FCN, more details are given in Sec. 4.1) for scene parsing to produce the target semantic segmentations for frames without human annotations. Similarly, to obtain the dense flow map for each frame, we use the output of the state-of-the-art epicflow [25] as our target optical flow. Note that our model is orthogonal to specific flow methods since they are only used to produce the target flow for training the flow anticipating network. Notations used in the following text are as follows. Xi denotes the i-th frame of a video and Xt?k:t?1 denotes the sequence of frames with length k from Xt?k to Xt?1 . The semantic segmentation of Xt is denoted as St , which is the 3 output of the penultimate layer of Res101-FCN. St has the same spatial size as Xt and is a vector of length C at each location, where C is the number of semantic classes. We denote Ot as the pixel-wise optical flow map from Xt?1 to Xt , which is estimated via epicflow [25]. Correspondingly, S?t and ? t denote the predicted semantic segmentation and optical flow. O 3.1 Prediction for one time step ahead Model overview The key idea of our approach is to model flow prediction and parsing prediction jointly, which are potentially mutually beneficial. As illustrated in Figure 2, the proposed model consists of two module networks that are trained jointly, i.e. the flow anticipating network that takes preceding frames Xt?k:t?1 as input to output the pixelwise flow prediction for Ot (from Xt?1 to Xt ), and the parsing anticipating network that takes the segmentation of preceding frames St?k:t?1 as input to output pixelwise semantic prediction for an unobserved frame Xt . The mutual influences of each network on the other are exploited in two aspects. First, the last segmentations St?1 produced by the parsing anticipating network convey pixel-wise class labels, which are used by the flow anticipating network to predict optical flow values for each pixel according to its belonging object group, e.g. moving objects or static objects. Second, the parsing anticipating network combines the discriminative local feature learned by the flow anticipating network to produce sharper and more accurate parsing predictions. Since both parsing prediction and flow prediction are essentially both the dense classification problem, we use the same deep architecture (Res101-FCN) for predicting parsing results and optical flow. Note the Res101-FCN used in this paper can be replaced by any CNNs. We adjust the input/output layers of these two networks according to the different channels of their input/output. The features extracted by feature encoders (CNN1 and CNN2 ) are spatially enlarged via up-sampling layers and finally fed to a convolutional layer to produce pixel-wise predictions which have the same spatial size as input. Flow anticipating network In videos captured for autonomous driving or navigation, regions with different class labels have different motion patterns. For example, the motion of static objects like road is only caused by the motion of the camera while the motion of moving objects is a combination of motions from both the camera and objects themselves. Therefore compared to methods that predict all pixels? optical flow in a single output layer, it would largely reduce the difficulty of feature learning by separately modeling the motion of regions with different classes. Following [29], we assign each class into one of three pre-defined object groups, i.e. G = {moving objects (MOV-OBJ), static objects (STA-OBJ), other objects (OTH-OBJ)} in which MOV-OBJ includes pedestrians, truck, etc., STA-OBJ includes sky, road, etc., and OTH-OBJ includes vegetation and buildings, etc. which have diverse motion patterns and shapes. We append a small network (consisting of two residual blocks) to the feature encoder (CNN1 ) for each object group to learn specified motion representations. During training, the loss for each pixel is only generated at the branch that corresponds to the object group to which the pixel belongs. Similarly, in testing, the flow prediction for each pixel is generated by the corresponding branch. The loss function between ? t and target output Ot is the model output O ? t , Ot ) = Lflow (O X Lgflow ; Lgflow = g?G 1 |Ng | X i,j ? ti,j Ot ? O (i,j)?Ng 2 (1) where (i, j) index the pixel in the region Ng . Parsing anticipating network The input of the parsing anticipating network is a sequence of preceding segmentations St?k:t?1 . We also explore other input space alternatives, including preceding frames Xt?k:t?1 , and the combination of preceding frames and corresponding segmentations Xt?k:t?1 St?k:t?1 , and we observe that the input St?k:t?1 achieves the best prediction performance. We conjecture it is easier to learn the mapping between variables in the same domain (i.e. both are semantic segmentations). However, there are two drawbacks brought by this strategy. Firstly, St?k:t?1 lose the discriminative local features e.g. color, texture and shape etc., leading to the missing of small objects in predictions, as illustrated in Figure 3 (see yellow boxes). The flow prediction network may learn such features from the input frames. Secondly, due to the lack of local features in St?k:t?1 , it is difficult to learn accurate pixel-wise correspondence in the parsing anticipating 4 network, which causes the predicted labeling maps to be over-smooth, as shown in Figure 3. The flow prediction network can provide reliable dense pixel-wise correspondence by regressing to the target optical flow. Therefore, we integrate the features learned by the flow anticipating network with the parsing prediction network through a transform layer (a shallow CNN) to improve the quality of predicted labeling maps. Depending on whether human annotations are available, the loss function is defined as ? S) = Lseg (S, ? ?? P log(S?ti,j (c)), Xt has human annotation, (i,j)?Xt ?L (S, ? S) + Lgdl (S, ? S), otherwise `1 (2) where c is the ground truth class for the pixel at location (i, j). It is a conventional pixel-wise cross-entropy loss when Xt has human annotations. L`1 and Lgdl are `1 loss and gradient difference loss [20] which are defined as ? S) = L`1 (S, X    i,j i,j  St ? S?t , (i,j)?Xt Lgdl = X      i,j    i?1,j | ? |S?ti,j ? S?ti?1,j | + |Sti,j?1 ? Sti,j | ? |S?ti,j?1 ? S?ti,j | . |St ? St (i,j)?Xt The `1 loss encourages predictions to regress to the target values while the gradient difference loss produces large errors in the gradients of the target and predictions. The reason for using different losses for human and non-human annotated frames in Eq. 2 is that the automatically produced parsing ground-truth (by the pre-trained Res101-FCN) of the latter may contain wrong annotations. The cross-entropy loss using one-hot vectors as labels is sensitive to the wrong annotations. Comparatively, the ground-truth labels used in the combined loss (L`1 + Lgdl ) are inputs of the softmax layer (ref. Sec. 3) which allow for non-zero values in more than one category, thus our model can learn useful information from the correct category even if the annotation is wrong. We find replacing L`1 + Lgdl with the cross-entropy loss reduces the mIoU of the baseline S2S (i.e. the parsing participating network) by 1.5 from 66.1 when predicting the results one time-step ahead. Now we proceed to explain the role of the transform layer which transforms the features of CNN1 before combining them with those of CNN2 . Compared with naively combining the features from two networks (e.g., concatenation), the transform layer brings the following two advantages: 1) naturally normalize the feature maps to proper scales; 2) align the features of semantic meaning such that the integrated features are more powerful for parsing prediction. Effectiveness of this transform layer is clearly validated in the ablation study in Sec. 4.2.1. The final objective of our model is to minimize the combination of losses from the flow anticipating network and the parsing anticipating network as follows ? t , S?t ) = Lflow (O ? t , Ot ) + Lseg (S, ? S). L(Xt?k:t?1 , St?k:t?1 , X 3.2 Prediction for multiple time steps ahead Based on the above model which predicts scene parsing and flow for the single future time step, we explore two ways to predict further into the future. Firstly, we iteratively apply the model to predict one more time step into the future by treating the prediction as input in a recursive way. Specifically, for predicting multiple time steps in the flow anticipating network, we warp the most recent frame ? t to get the X ? t which is then combined with Xt?k?1:t?1 to feed Xt?1 using the output prediction O ? t+1 , and so forth. For the parsing anticipating network, we the flow anticipating network to generate O combine the predicted parsing map S?t with St?k?1:t?1 as the input to generate the parsing prediction at t + 1. This scheme is easy to implement and allows us to predict arbitrarily far into the future without increasing training complexity w.r.t. with the number of time-steps we want to predict. Secondly, we fine-tune our model by taking into account the influence that the recurrence has on prediction for multiple time steps. We apply our model recurrently as described above to predict two time steps ahead and apply the back propagation through time (BPTT) [14] to update the weight. We have verified through experiments that the fine-tuning approach can further improve the performance as it models longer temporal dynamics during training. 5 Figure 3: Two examples of prediction results for predicting one time step ahead. Odd row: The images from left to right are Xt?2 , Xt?1 , the target optical flow map Ot , the flow predictions from PredFlow and the flow predictions from our model. Even row: The images from left to right are St?2 , St?1 , the ground truth semantic annotations at the time t, the parsing prediction from S2S and the parsing prediction from our model. The flow predictions from our model show clearer object boundaries and predict more accurate values for moving objects (see black boxes) compared to PredFlow. Our model is superior to S2S by being more discriminative to the small objects in parsing predictions (see yellow boxes). Figure 4: An example of prediction results for predicting ten time steps ahead. Top (from left to right): Xt?11 , Xt?10 , the target optical flow map Ot , the flow prediction from PredFlow and the flow prediction from our model. Bottom (from left to right): St?11 , St?10 , the ground truth semantic annotation at the time t, the parsing prediction from S2S and the parsing prediction from our model. Our model outputs better prediction compared to PredFlow (see black boxes) and S2S (see yellow boxes). 4 4.1 Experiment Experimental settings Datasets We verify our model on the large scale Cityscapes [5] dataset which contains 2,975/500 train/val video sequences with 19 semantic classes. Each video sequence lasts for 1.8s and contains 30 frames, among which the 20th frame has fine human annotations. Every frame in Cityscapes has a resolution of 1,024 ? 2,048 pixels. Evaluation criteria We use the mean IoU (mIoU) for evaluating the performance of predicted parsing results on those 500 frames in the val set with human annotations. For evaluating the performance of flow prediction, we use the average endpoint error (EPE) [2] following conventions [8] p which is defined as N1 (u ? uGT )2 + (v ? vGT )2 where N is the number of pixels per-frame, and u and v are the components of optical flow along x and y directions, respectively. To be consistent with mIoU, EPEs are also reported on the 20th frame in each val sequence. Baselines To fully demonstrate the advantages of our model on producing better predictions, we compare our model against the following baseline methods: 6 Table 1: The performance of parsing prediction on Cityscapes val set. For each competing model, we list the mIoU/EPE when predicting one time step ahead. Best results in bold. Table 2: The performance of motion prediction on Cityscapes val set. For each model, we list the mIoU/EPE when predicting one time step ahead. Best results in bold. Model mIoU EPE Model mIoU EPE Copy last input Warp last input PredFlow S2S [22] 59.7 61.3 61.3 62.6 3.03 3.03 2.71 - Copy last input Warp last input PredFlow S2S [22] 41.3 42.0 43.6 50.8 9.40 9.40 8.10 - ours (w/o Trans. layer) ours 64.7 66.1 2.42 2.30 ours (w/o Recur. FT) ours 52.6 53.9 6.63 6.31 ? Copy last input Copy the last optical flow (Ot?1 ) and parsing map (St?1 ) at time t ? 1 as predictions at time t. ? Warp last input Warp the last segmentation St?1 using Ot?1 to get the parsing prediction at the next time step. In order to make flow applicable to the correct locations, we also warp the flow field using the optical flow in each time step. ? PredFlow Perform flow prediction without the object masks generated from segmentations. The architecture is the same as the flow prediction net in Figure 2 which generates pixel-wise flow prediction in a single layer, instead of multiple branches. For fair comparison with our joint model, in the following we report the average result of two independent PredFlow with different random initializations. When predicting the segmentations at time t, we use the flow prediction output by PredFlow at time t to warp the segmentations at time t ? 1. This baseline aims to verify the advantages brought by parsing prediction when predicting flow. ? S2S [22] Use only parsing anticipating network. The difference is that the former does not leverage features learned by the flow anticipating network to produce parsing predictions. We replace the backbone network in the original S2S as the same one of ours, i.e. Res101FCN and retrain S2S with the same configurations as those of ours. Similar to the PredFlow, the average performance of two randomly initialized S2S is reported. This baseline aims to verify the advantages brought by flow prediction when predicting parsing. Implementation details Throughout the experiments, we set the length of the input sequence as 4 frames, i.e. k = 4 in Xt?k:t?1 and St?k:t?1 (ref. Sec. 3). The original frames are firstly downsampled to the resolution of 256 ? 512 to accelerate training. In the flow anticipating network, we assign 19 semantic classes into three object groups which are defined as follows: MOV-OBJ including person, rider, car, truck, bus, train, motorcycle and bicycle, STA-OBJ including road, sidewalk, sky, pole, traffic light and traffic sign and OTH-OBJ including building, wall, fence, terrain and vegetation. For data augmentation, we randomly crop a patch with the size of 256 ? 256 and perform random mirror for all networks. All results of our model are based on single-model singlescale testing. For other hyperparameters including weight decay, learning rate, batch size and epoch number etc., please refer to the supplementary material. All of our experiments are carried out on NVIDIA Titan X GPUs using the Caffe library. 4.2 Results and analysis Examples of the flow predictions and parsing predictions output by our model for one-time step and ten-time steps are illustrated in Figure 3 and Figure 4 respectively. Compared to baseline models, our model produces more visually convincing prediction results. 4.2.1 One-time step anticipation Table 1 lists the performance of parsing and flow prediction on the 20th frame in the val set which has ground truth semantic annotations. It can be observed that our model achieves the best performance on both tasks, demonstrating the effectiveness on learning the latent representations for future prediction. Based on the results, we analyze the effect of each component in our model as follows. 7 The effect of flow prediction on parsing prediction Compared with S2S which does not leverage flow predictions, our model improves the mIoU with a large margin (3.5%). As shown in Figure 3, compared to S2S, our model performs better on localizing the small objects in the predictions e.g. pedestrian and traffic sign, because it combines the discriminative local features learned in the flow anticipating network. These results clearly demonstrate the benefit of flow prediction for parsing prediction. The effect of parsing prediction on flow prediction Compared with the baseline PredFlow which has no access to the semantic information when predicting the flow, our model reduces the average EPE from 2.71 to 2.30 (a 15% improvement), which demonstrates parsing prediction is beneficial to flow prediction. As illustrated in Figure 3, the improvement our model makes upon PredFlow comes from two aspects. First, since the segmentations provide boundary information of objects, the flow map predicted by our model has clearer object boundaries while the flow map predicted by PredFlow is mostly blurry. Second, our model shows more accurate flow predictions on the moving objects (ref. Sec. 4.1 for the list of moving objects). We calculate the average EPE for only the moving objects, which is 2.45 for our model and 3.06 for PredFlow. By modeling the motion of different objects separately, our model learns better representation for each motion mode. If all motions are predicted in one layer as in PredFlow, then the moving objects which have large displacement than other regions are prone to smoothness. Benefits of the transform layer As introduced in Sec. 3.1, the transform layer improves the performance of our model by learning the latent feature space transformations from CNN1 to CNN2 . In our experiments, the transform layer contains one residual block [12] which has been widely used due to its good performance and easy optimization. Details of the residual block used in our experiments are included in the supplementary material. Compared to the variant of our model w/o the transform layer, adding the transform layer improves the mIoU by 1.4 and reduces EPE by 0.12. We observe that stacking more residual blocks only leads to marginal improvements at larger computational costs. 4.2.2 Longer duration prediction The comparison of the prediction performance among all methods for ten time steps ahead is listed in Table 2, from which one can observe that our model performs the best in this challenging task. The effect of each component in our model is also verified in this experiment. Specifically, compared with S2S, our model improves the mIoU by 3.1% due to the synergy with the flow anticipating network. The parsing prediction helps reducing the EPE of PredFlow by 1.79. Qualitative results are illustrated in Figure 4. The effect of recurrent fine-tuning As explained in Sec. 3.2, it helps our model to capture long term video dynamics by fine-tuning the weights when recurrently applying the model to predict the next time step in the future. As shown in Table 2, compared to the variant w/o recurrent ft, our model w/ recurrent fine-tuning improves the mIoU by 1.3% and reduces the EPE by 0.32, therefore verifying the effect of recurrent fine-tuning. 4.3 Application for predicting the steering angle of a vehicle With the parsing prediction and flow prediction available, one can enable the moving agent to be more alert about the environments and get ?smarter?. Here, we investigate one application: predicting the steering angle of the vehicle. The intuition is it is convenient to infer the steering angle given the predicted flow of static objects, e.g. road and sky, the motion of which is only caused by ego-motion of the camera mounted on the vehicle. Specifically, we append a fully connected layer to take the features learned in the STA-OBJ branch in the flow anticipating network as input and perform regression to steering angles. We test our model on the dataset from Comma.ai [27] which consists of 11 videos 1 https://github.com/commaai/research 8 Table 3: Comparison results of steering angle prediction on a dataset from Comma.ai [27]. The criteria is the mean square error (MSE, in degree2 ) between the prediction and groud truth. Model MSE (degrees2 ) Copy last prediction Comma.ai1 [27] 4.81 ?4 ours 2.96 amounting to about 7 hours. The data of steering angles have been recorded for each frame captured at 20Hz with the resolution of 160 ? 320. We randomly sample 50K/5K frames from the train set for training and validation purpose. Since there are videos captured at night, we normalize all training frames to [0, 255]. Similar to Cityscapes, we use epicflow and Res101-FCN to produce the target output for flow prediction and parsing prediction, respectively. We first train our model following Sec. 3 and then fine-tune the whole model with the MSE loss after adding the fully connected layer for steering angle prediction. During training, random crop with the size of 160 ? 160 and random mirror are employed and other hyperparameter settings follow Sec. 4.1. The testing results are listed in Table 3. Compared to the model from Comma.ai which uses a five-layer CNN to estimate the steering angle from a single frame and is trained end-to-end on all the training frames (396K), our model achieves much better performance (2.84 versus ?4 in degrees2 ). Although we do not push the performance by using more training data and more complex prediction models (only a fully connected layer is used in our model for output steering angle), this preliminary experiment still verifies the advantage of our model in learning the underlying latent parameters. We think it is just an initial attempt in validating the dense prediction results through applications, which hopefully can stimulate researchers to explore other interesting ways to utilize the parsing prediction and flow prediction. 5 Conclusion In this paper, we proposed a novel model to predict the future scene parsing and motion dynamics. To our best knowledge, this is the first research attempt to anticipate visual dynamics for building intelligent agents. The model consists of two networks: the flow anticipating network and the parsing anticipating network which are jointly trained and benefit each other. On the large scale Cityscapes dataset, the experimental results demonstrate that the proposed model generates more accurate prediction than well-established baselines both on single time step prediction and multiple time prediction. In addition, we also presented a method to predict the steering angle of a vehicle using our model and achieve promising preliminary results on the task. Acknowledgements The work of Jiashi Feng was partially supported by National University of Singapore startup grant R-263-000-C08-133, Ministry of Education of Singapore AcRF Tier One grant R-263-000-C21-112 and NUS IDS grant R-263-000-C67-646. References [1] Andreas Argyriou, Theodoros Evgeniou, and Massimiliano Pontil. Multi-task feature learning. In NIPS, 2007. [2] Simon Baker, Daniel Scharstein, JP Lewis, Stefan Roth, Michael J Black, and Richard Szeliski. A database and evaluation methodology for optical flow. International Journal of Computer Vision, 92(1):1?31, 2011. 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[8] Philipp Fischer, Alexey Dosovitskiy, Eddy Ilg, Philip H?usser, Caner Haz?rba?s, Vladimir Golkov, Patrick van der Smagt, Daniel Cremers, and Thomas Brox. Flownet: Learning optical flow with convolutional networks. arXiv preprint arXiv:1504.06852, 2015. [9] David F Fouhey and C Lawrence Zitnick. Predicting object dynamics in scenes. In CVPR, 2014. [10] Alberto Garcia-Garcia, Sergio Orts-Escolano, Sergiu Oprea, Victor Villena-Martinez, and Jose GarciaRodriguez. A review on deep learning techniques applied to semantic segmentation. arXiv preprint arXiv:1704.06857, 2017. [11] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In NIPS, 2014. [12] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, 2016. 9 [13] Minh Hoai and Fernando De la Torre. Max-margin early event detectors. International Journal of Computer Vision, 107(2):191?202, 2014. [14] Herbert Jaeger. Tutorial on training recurrent neural networks, covering BPPT, RTRL, EKF and the" echo state network" approach, volume 5. GMD-Forschungszentrum Informationstechnik, 2002. [15] Xiaojie Jin, Yunpeng Chen, Jiashi Feng, Zequn Jie, and Shuicheng Yan. Multi-path feedback recurrent neural network for scene parsing. In AAAI, 2017. [16] Xiaojie Jin, Xin Li, Huaxin Xiao, Xiaohui Shen, Zhe Lin, Jimei Yang, Yunpeng Chen, Jian Dong, Luoqi Liu, Zequn Jie, et al. Video scene parsing with predictive feature learning. In ICCV, 2017. [17] Jonathan Long, Evan Shelhamer, and Trevor Darrell. Fully convolutional networks for semantic segmentation. In CVPR, 2015. [18] William Lotter, Gabriel Kreiman, and David Cox. Unsupervised learning of visual structure using predictive generative networks. arXiv preprint arXiv:1511.06380, 2015. [19] Zelun Luo, Boya Peng, De-An Huang, Alexandre Alahi, and Li Fei-Fei. Unsupervised learning of long-term motion dynamics for videos. arXiv preprint arXiv:1701.01821, 2017. [20] Michael Mathieu, Camille Couprie, and Yann LeCun. Deep multi-scale video prediction beyond mean square error. arXiv preprint arXiv:1511.05440, 2015. [21] Rudolf Mester. Motion estimation revisited: an estimation-theoretic approach. In Image Analysis and Interpretation (SSIAI), 2014 IEEE Southwest Symposium on, pages 113?116. IEEE, 2014. [22] Natalia Neverova, Pauline Luc, Camille Couprie, Jakob Verbeek, and Yann LeCun. Predicting deeper into the future of semantic segmentation. arXiv preprint arXiv:1703.07684, 2017. [23] Deepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and Alexei A Efros. Context encoders: Feature learning by inpainting. In CVPR, 2016. [24] Viorica Patraucean, Ankur Handa, and Roberto Cipolla. Spatio-temporal video autoencoder with differentiable memory. arXiv preprint arXiv:1511.06309, 2015. [25] Jerome Revaud, Philippe Weinzaepfel, Zaid Harchaoui, and Cordelia Schmid. Epicflow: Edge-preserving interpolation of correspondences for optical flow. In CVPR, 2015. [26] Anirban Roy and Sinisa Todorovic. Scene labeling using beam search under mutex constraints. In CVPR, 2014. [27] Eder Santana and George Hotz. Learning a driving simulator. CoRR, abs/1608.01230, 2016. [28] Alexander G Schwing and Raquel Urtasun. Fully connected deep structured networks. arXiv preprint arXiv:1503.02351, 2015. [29] Laura Sevilla-Lara, Deqing Sun, Varun Jampani, and Michael J Black. Optical flow with semantic segmentation and localized layers. In CVPR, 2016. [30] Richard Socher, Cliff C Lin, Chris Manning, and Andrew Y Ng. Parsing natural scenes and natural language with recursive neural networks. In ICML, 2011. [31] Nitish Srivastava, Elman Mansimov, and Ruslan Salakhudinov. Unsupervised learning of video representations using lstms. In ICML, 2015. [32] Ruben Villegas, Jimei Yang, Seunghoon Hong, Xunyu Lin, and Honglak Lee. Decomposing motion and content for natural video sequence prediction. In ICLR, 2017. [33] Ruben Villegas, Jimei Yang, Yuliang Zou, Sungryull Sohn, Xunyu Lin, and Honglak Lee. Learning to generate long-term future via hierarchical prediction. In ICML, 2017. [34] Carl Vondrick, Hamed Pirsiavash, and Antonio Torralba. Generating videos with scene dynamics. In NIPS, 2016. [35] Jacob Walker, Abhinav Gupta, and Martial Hebert. Patch to the future: Unsupervised visual prediction. In CVPR, 2014. [36] Jacob Walker, Abhinav Gupta, and Martial Hebert. Dense optical flow prediction from a static image. In ICCV, 2015. [37] Jenny Yuen and Antonio Torralba. A data-driven approach for event prediction. In ECCV, 2010. 10 

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Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference Geoffrey Roeder University of Toronto roeder@cs.toronto.edu Yuhuai Wu University of Toronto ywu@cs.toronto.edu David Duvenaud University of Toronto duvenaud@cs.toronto.edu Abstract We propose a simple and general variant of the standard reparameterized gradient estimator for the variational evidence lower bound. Specifically, we remove a part of the total derivative with respect to the variational parameters that corresponds to the score function. Removing this term produces an unbiased gradient estimator whose variance approaches zero as the approximate posterior approaches the exact posterior. We analyze the behavior of this gradient estimator theoretically and empirically, and generalize it to more complex variational distributions such as mixtures and importance-weighted posteriors. Introduction Recent advances in variational inference have begun to make approximate inference practical in large-scale latent variable models. One of the main recent advances has been the development of variational autoencoders along with the reparameterization trick [Kingma and Welling, 2013, Rezende et al., 2014]. The reparameterization trick is applicable to most continuous latent-variable models, and usually provides lower-variance gradient estimates than the more general REINFORCE gradient estimator [Williams, 1992]. KL( ?init k ?true ) 1 Optimization using: Path Derivative Total Derivative 400 600 800 1000 1200 Intuitively, the reparameterization trick provides more informative gradients by exposing the dependence of samIterations pled latent variables z on variational parameters ?. In contrast, the REINFORCE gradient estimate only de- Figure 1: Fitting a 100-dimensional variapends on the relationship between the density function tional posterior to another Gaussian, using log q? (z|x, ?) and its parameters. standard gradient versus our proposed path derivative gradient estimator. Surprisingly, even the reparameterized gradient estimate contains the score function?a special case of the REINFORCE gradient estimator. We show that this term can easily be removed, and that doing so gives even lower-variance gradient estimates in many circumstances. In particular, as the variational posterior approaches the true posterior, this gradient estimator approaches zero variance faster, making stochastic gradient-based optimization converge and "stick" to the true variational parameters, as seen in figure 1. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. 1.1 Contributions ? We present a novel unbiased estimator for the variational evidence lower bound (ELBO) that has zero variance when the variational approximation is exact. ? We provide a simple and general implementation of this trick in terms of a single change to the computation graph operated on by standard automatic differentiation packages. ? We generalize our gradient estimator to mixture and importance-weighted lower bounds, and discuss extensions to flow-based approximate posteriors. This change takes a single function call using automatic differentiation packages. ? We demonstrate the efficacy of this trick through experimental results on MNIST and Omniglot datasets using variational and importance-weighted autoencoders. 1.2 Background Making predictions or computing expectations using latent variable models requires approximating the posterior distribution p(z|x). Calculating these quantities in turn amounts to using Bayes? rule: p(z|x) = p(x|z)p(z)/p(x). Variational inference approximates p(z|x) with a tractable distribution q? (z|x) parameterized by ? that is close in KL-divergence to the exact posterior. Minimizing the KL-divergence is equivalent to maximizing the evidence lower bound (ELBO): L(?) = Ez?q [log p(x, z) ? log q? (z | x)] (ELBO) An unbiased approximation of the gradient of the ELBO allows stochastic gradient descent to scalably learn parametric models. Stochastic gradients of the ELBO can be formed from the REINFORCEstyle gradient, which applies to any continuous or discrete model, or a reparameterized gradient, which requires the latent variables to be modeled as continuous. Our variance reduction trick applies to the reparameterized gradient of the evidence lower bound. 2 Estimators of the variational lower bound In this section, we analyze the gradient of the ELBO with respect to the variational parameters to show a source of variance that depends on the complexity of the approximate distribution. When the joint distribution p(x, z) can be evaluated by p(x|z) and p(z) separately, the ELBO can be written in the following three equivalent forms: L(?) = Ez?q [log p(x|z) + log p(z) ? log q? (z|x)] = Ez?q [log p(x|z) + log p(z))] + H[q? ] = Ez?q [log p(x|z)] ? KL(q? (z|x)||p(z)) (1) (2) (3) Which ELBO estimator is best? When p(z) and q? (z|x) are multivariate Gaussians, using equation (3) is appealing because it analytically integrates out terms that would otherwise have to be estimated by Monte Carlo. Intuitively, we might expect that using exact integrals wherever possible will give lower-variance estimators by reducing the number of terms to be estimated by Monte Carlo methods. Surprisingly, even when analytic forms of the entropy or KL divergence are available, sometimes it is better to use (1) because it will have lower variance. Specifically, this occurs when q? (z|x) = p(z|x), i.e. the variational approximation is exact. Then, the variance of the full Monte Carlo estimator L?M C iid is exactly zero. Its value is a constant, independent of z ? q? (z|x). This follows from the assumption q? (z|x) = p(z|x): L?M C (?) = log p(x, z) ? log q? (z|x) = log p(z|x) + log p(x) ? log p(z|x) = log p(x), (4) This suggests that using equation (1) should be preferred when we believe that q? (z|x) ? p(z|x). Another reason to prefer the ELBO estimator given by equation (1) is that it is the most generally applicable, requiring a closed form only for q? (z|x). This makes it suitable for highly flexible approximate distributions such as normalizing flows [Jimenez Rezende and Mohamed, 2015], Real NVP [Dinh et al., 2016], or Inverse Autoregressive Flows [Kingma et al., 2016]. 2 Estimators of the lower bound gradient What about estimating the gradient of the evidence lower bound? Perhaps surprisingly, the variance of the gradient of the fully Monte Carlo estimator (1) with respect to the variational parameters is not zero, even when the variational parameters exactly capture the true posterior, i.e., q? (z|x) = p(z|x). This phenomenon can be understood by decomposing the gradient of the evidence lower bound. Using the reparameterization trick, we can express a sample z from a parametric distribution q? (z) as a deterministic function of a random variable  with some fixed distribution and the parameters ? of q? , i.e., z = t(, ?). For example, if q? is a diagonal Gaussian, then for  ? N (0, I), z = ? + ? is a sample from q? . Under such a parameterization of z, we can decompose the total derivative (TD) of the integrand of estimator (1) w.r.t. the trainable parameters ? as ? TD (, ?) = ?? [log p(x|z) + log p(z) ? log q? (z|x)] ? = ?? [log p(z|x) + log p(x) ? log q? (z|x)] = ?z [log p(z|x) ? log q? (z|x)] ?? t(, ?) ? ?? log q? (z|x), | {z } | {z } path derivative When q? (z|x) = p(z|x) for all z, the path derivative component of equation (7) is identically zero for all z. However, the score function component is not necessarily zero for any z in some finite sample, meaning that the total derivative gradient estimator (7) will have nonzero variance even when q matches the exact posterior everywhere. This variance is induced by the Monte Carlo sampling procedure itself. Figure 3 depicts this phenomenon through the loss surface of log p(x, z) ? log q? (z|x) for a Mixture of Gaussians approximate and true posterior. (7) score function log p(x, z) ? log q? (z|x) Surface Along Trajectory through True ? ELBO log p(x, z) ? log q? (z|x) The reparameterized gradient estimator w.r.t. ? decomposes into two parts. We call these the path derivative and score function components. The path derivative measures dependence on ? only through the sample z. The score function measures the dependence on log q? directly, without considering how the sample z changes as a function of ?. (5) (6) La ten tV ar ia ble (z) q? (z|x) = p(z|x) (?) eters aram nal P riatio Va Figure 2: The evidence lower bound is a function of the sampled latent variables z and the variational parameters ?. As the variational distribution approaches the true posterior, the gradient with respect to the sampled z (blue) vanishes. Path derivative of the ELBO Could we remove the high-variance score function term from the gradient estimate? For stochastic gradient descent to converge, we require that our gradient estimate is unbiased. By construction, the gradient estimate given by equation (7) is unbiased. Fortunately, the problematic score function term has expectation zero. If we simply remove that term, we maintain an unbiased estimator of the true gradient: ( ( ? PD (, ?) = ?z [log p(z|x) ? log q? (z|x)] ?? t(, ?) ? ??(log (8) ? q?( (z|x). ( (( This estimator, which we call the path derivative gradient estimator due to its dependence on the gradient flow only through the path variables z to update ?, is equivalent to the standard gradient estimate with the score function term removed. The path derivative estimator has the desirable property that as q? (z|x) approaches p(z|x), the variance of this estimator goes to zero. When to prefer the path derivative estimator Does eliminating the score function term from the gradient yield lower variance in all cases? It might seem that its removal can only have a variance reduction effect on the gradient estimator. Interestingly, the variance of the path derivative gradient estimator may actually be higher in some cases. This will be true when the score function is positively correlated with the remaining terms in the total derivative estimator. In this case, the score function acts as a control variate: a zero-expectation term added to an estimator in order to reduce variance. 3 Alg. 1 Standard ELBO Gradient Alg. 2 Path Derivative ELBO Gradient Input: Variational parameters ?t , Data x t ? p() def L?t (?): zt ? sample_q(?, t ) Input: Variational parameters ?t , Data x t ? p() def L?t (?): zt ? sample_q(?, t ) ?0 ? stop_gradient(?) return log p(x, zt ) - log q(zt |x, ?0 ) return ?? L?t (?t ) return log p(x, zt ) - log q(zt |x, ?) return ?? L?t (?t ) Control variates are usually scaled by an adaptive constant c? , which modifies the magnitude and direction of the control variate to optimally reduce variance, as in Ranganath et al. [2014]. In the preceding discussion, we have shown that cb? = 1 is optimal when the variational approximation is exact, since that choice yields analytically zero variance. When the variational approximation is not exact, an estimate of c? based on the current minibatch will change sign and magnitude depending on the positive or negative correlation of the score function with the path derivative. Optimal scale estimation procedures is particularly important when the variance of an estimator is so large that convergence is unlikely. However, in the present case of reparameterized gradients, where the variance is already low, estimating a scaling constant introduces another source of variance. Indeed, we can only recover the true optimal scale when the variational approximation is exact in the regime of infinite samples during Monte Carlo integration. Moreover, the score function must be independently estimated in order to scale it. Estimating the gradient of the score function independent of automatic reverse-mode differentiation can be a challenging engineering task for many flexible approximate posterior distributions such as Normalizing Flows [Jimenez Rezende and Mohamed, 2015], Real NVP [Dinh et al., 2016], or IAF [Kingma et al., 2016]. By contrast, in section 6 we show improved performance on the MNIST and Omniglot density estimation benchmarks by approximating the optimal scale with 1 throughout optimization. This technique is easy to implement using existing automatic differentiation software packages. However, if estimating the score function independently is computationally feasible, and a practitioner has evidence that the variance induced by Monte Carlo integration will reduce the overall variance away from the optimum point, we recommend establishing an annealling schedule for the optimal scaling constant that converges to 1. 3 Implementation Details In this section, we introduce algorithms 1 and 2 in relation to reverse-mode automatic differentiation, and discuss how to implement the new gradient estimator in Theano, Autograd, Torch or Tensorflow Bergstra et al. [2010], Maclaurin et al. [2015], Collobert et al. [2002], Abadi et al. [2015]. Algorithm 1 shows the standard reparameterized gradient for the ELBO. We require three function definitions: q_sample to generate a reparameterized sample from the variational approximation, and functions that implement log p(x, z) and log q(z|x, ?). Once the loss L?t is defined, we can leverage automatic differentiation to return the standard gradient evaluated at ?t . This yields equation (7). Algorithm 2 shows the path derivative gradient for the ELBO. The only difference from algorithm 1 is the application of the stop_gradient function to the variational parameters inside L?t . Table 1 indicates the names of stop_gradient in popular software packages. Theano: Autograd: TensorFlow: Torch: T.gradient.disconnected_grad autograd.core.getval tf.stop_gradient torch-autograd.util.get_value Table 1: Functions that implement stop_gradient 4 Alg. 4 IWAE ELBO Gradient Alg. 3 Path Derivative Mixture ELBO Gradient Input: Params ?t , Data x 1 , 2 , . . . , K ? p() ?0t ? stop_gradient(?t ) def wi (?, i ): zi ? sample_q(?, i ) p(x,zi ) return q(z 0 i |x,?t )
PK 1 return ?? log k i=1 wi (?, i ) {?tj }K j=1 , ?t Input: Params ?t = = {?it }K i=1 , Data x t ? p() ?0t , ?t0 ? stop_gradient(?t , ?t ) def L?ct (?): zct ? sample_q(?, t ) PK return log p(x, zct ) - log c=1 ?t0c q(zct |x, ?0t )
PK c ?c c return ??,? c=1 ?t Lt (?t ) This simple modification to algorithm 1 generates a copy of the parameter variable that is treated as a constant with respect to the computation graph generated for automatic differentiation. The copied variational parameters are used to evaluate variational the density log q? at z. Recall that the variational parameters ? are used both to generate z through some deterministic function of an independent random variable , and to evaluate the density of z through log q? . By blocking the gradient through variational parameters in the density function, we eliminate the score function term that appears in equation (7). Per-iteration updates to the variational parameters ? rely on the z channel only, e.g., the path derivative component of the gradient of the loss function L?t . This yields the gradient estimator corresponding to equation (8). 4 Extensions to Richer Variational Families Trace Norm of Covariance Matrix Mixture Distributions In this section, we discuss extensions of the path derivative gradient estimator to richer variational approximations to the true posterior. Using a mixture distribution as an approximate posterior in an otherwise differentiable estimator introduces a problematic, non-differentiable random variable ? ? Cat(?). We solve this by integrating out the discrete mixture choice from both the ELBO and the mixture distribution. In this section, we show that such a gradient estimator is unbiased, and introduce an extended algorithm to handle mixture variational families. For any mixture of K base distributions q? (z|x), a mixture variational family can be defined by PK q?M (z|x) = c=1 ?c q?c (z|x), where ?M = {?1 , ..., ?k , ?1 , ..., ?k } are variational parameters, e.g., the weights and distributional parameters for each component. Then, the mixture ELBO LM is given by: K X c=1 True Posterior Variational Approximation Total Derivative Estimator Path Derivative Estimator 8.0e+05 6.0e+05 4.0e+05 2.0e+05 0.0e+00 Variational Parameters ?init ? ?true Figure 3: Fitting a mixture of 5 Gaussians as a variational approximation to a posterior that is also a mixture of 5 Gaussians. Path derivative and score function gradient components were measured 1000 times. The path derivative goes to 0 as the variational approximation becomes exact, along an arbitrarily chosen path  ?c Ezc ?q?c log p(x, zc ) ? log X K k=1  ?k q?k (zc |x) , where the outer sum integrates over the choice of mixture component for each sample from q?M , and the inner sum evaluates the density. Applying the new gradient estimator to the mixture ELBO involves applying it to each q?k (zc |x) in the inner marginalization. Algorithm 3 implements the gradient estimator of (8) in the context of a continuous mixture distribution. Like algorithm 2, the new gradient estimator of 3 differs from the vanilla gradient estimator only in the application of stop_gradient to the variational parameters. This eliminates the gradient of the score function from the gradient of any mixture distribution. 5 Importance-Weighted Autoencoder We also explore the effect of our new gradient estimator on the IWAE bound Burda et al. [2015], defined as   X  K 1 p(x, zi ) ? LK = Ez1 ,...,zK ?q(z|x) log (9) k i=1 q(zi |x) with gradient ?? L?K = Ez1 ,...,zK ?q(z|x) X K i=1  ? i ?? log wi w (10) P ? i := wi / ki=1 wi . Since ?? log wi is the same gradient as where wi := p(x, zi )/q(zi |x) and w the Monte Carlo estimator of the ELBO (equation (7)), we can again apply our trick to get a new estimator. However, it is not obvious whether this new gradient estimator is unbiased. In the unmodified IWAE bound, when q = p, the gradient with respect to the variational parameters reduces to:  X  k ? i ?? log q? (zi |x) . Ez1 ,...,zk ?q(z|x) ? w (11) i=1 ? i and the partial derivative term. Hence, we cannot Each sample zi is used to evaluate both w simply appeal to the linearity of expectation to show that this gradient is 0. Nevertheless, a natural extension of the variance reduction technique in equation (8) is to apply our variance reduction to each importance-weighted gradient sample. See algorithm 4 for how to implement the path derivative estimator in this form. We present empirical validation of the idea in our experimental results section, which shows markedly improved results using our gradient estimator. We observe a strong improvement in many cases, supporting our conjecture that the gradient estimator is unbiased as in the mixture and multi-sample ELBO cases. Flow Distributions Flow-based approximate posteriors such as Kingma et al. [2016], Dinh et al. [2016], Jimenez Rezende and Mohamed [2015] are a powerful and flexible framework for fitting approximate posterior distributions in variational inference. Flow-based variational inference samples an initial z0 from a simple base distribution with known density, then learns a chain of invertible, parameterized maps fk (zk?1 ) that warp z0 into zK = fK ? fK?1 ? ... ? f1 (z0 ). The endpoint zK represents  a sample  from a more flexible distribution with density log qK (zK ) = log q0 (z0 ) ? PK ?fk   k=1 log det ?zk?1 . We expect our gradient estimator to improve the performance of flow-based stochastic variational inference. However, due to the chain composition used to learn zK , we cannot straightforwardly apply our trick as described in algorithm 2. This is because each intermediate zj , 1 ? j ? K contributes to the path derivative component in equation (8). The log-Jacobian terms used in the evaluation of log q(zk ), however, require this gradient information to calculate the correct estimator. By applying stop_gradient to the variational parameters used to generate each intermediate zi and passing only the endpoint zK to a log density function, we would lose necessary gradient information at each intermediate step needed for the gradient estimator to be correct. At time of writing, the requisite software engineering to track and expose intermediate steps during backpropagation is not implemented in the packages listed in Table 1, and so we leave this to future work. 5 Related Work Our modification of the standard reparameterized gradient estimate can be interpreted as adding a control variate, and in fact Ranganath et al. [2014] investigated the use of the score function as a control variate in the context of non-reparameterized variational inference. The variance-reduction effect we use to motivate our general gradient estimator has been noted in the special cases of Gaussian distributions with sparse precision matrices and Gaussian copula inference in Tan and Nott [2017] and Han et al. [2016] respectively. In particular, Tan and Nott [2017] observes that by 6 MNIST VAE stochastic layers Omniglot IWAE VAE IWAE k Total Path Total Path Total Path Total Path 1 1 5 50 86.76 86.47 86.35 86.40 86.33 86.48 86.76 85.54 84.78 86.40 85.20 84.45 108.11 107.62 107.80 107.39 107.40 107.42 108.11 106.12 104.67 107.39 105.42 104.16 2 1 5 50 85.33 85.01 84.78 84.77 84.68 84.33 85.33 83.89 82.90 84.77 83.57 83.16 107.58 106.31 106.30 105.22 104.87 105.70 107.56 104.79 103.38 105.22 103.59 102.86 Table 2: Results on variational (VAE) and importance-weighted (IWAE) autoencoders using the total derivative estimator, equation (7), versus the path derivative estimator, equation (8) (ours). eliminating certain terms from a gradient estimator for Gaussian families parameterized by sparse precision matrices, multiple lower-variance unbiased gradient estimators may be derived. Our work is a generalization to any continuous variational family. This provides a framework for easily implementing the technique in existing software packages that provide automatic differentiation. By expressing the general technique in terms of automatic differentiation, we eliminate the need for case-by-case analysis of the gradient of the variational lower bound as in Tan and Nott [2017] and Han et al. [2016]. An innovation by Ruiz et al. [2016] introduces the generalized reparameterization gradient (GRG) which unifies the REINFORCE-style and reparameterization gradients. GRG employs a weaker form of reparameterization that requires only the first moment to have no dependence on the latent variables, as opposed to complete independence as in Kingma and Welling [2013]. GRG improves on the variance of the score-function gradient estimator in BBVI without the use of Rao-Blackwellization as in Ranganath et al. [2014]. A term in their estimator also behaves like a control variate. The present study, in contrast, develops a simple drop-in variance reduction technique through an analysis of the functional form of the reparameterized evidence lower bound gradient. Our technique is developed outside of the framework of GRG but can strongly improve the performance of existing algorithms, as demonstrated in section 6. Our technique can be applied alongside GRG. In the python toolkit Edward [Tran et al., 2016], efforts are ongoing to develop algorithms that implement stochastic variational inference in general as a black-box method. In cases where an analytic form of the entropy or KL-divergence is known, the score function term can be avoided using Edward. This is equivalent to using equations (2) or (3) respectively to estimate the ELBO. As of release 1.2.4 of Edward, the total derivative gradient estimator corresponding to (7) is used for reparameterized stochastic variational inference. 6 Experiments Experimental Setup Because we follow the experimental setup of Burda et al. [2015], we review it briefly here. Both benchmark datasets are composed of 28 ? 28 binarized images. The MNIST dataset was split into 60, 000 training and 10, 000 test examples. The Omniglot dataset was split into 24, 345 training and 8070 test examples. Each model used Xavier initialization [Glorot and Bengio, 2010] and trained using Adam with parameters ?1 = 0.9, ?2 = 0.999, and  = 1e?4 with 20 observations per minibatch [Kingma and Ba, 2015]. We compared against both architectures reported in Burda et al. [2015]. The first has one stochastic layer with 50 hidden units, encoded using two fully-connected layers of 200 neurons each, using a tanh nonlinearity throughout. The second architecture is two stochastic layers: the first stochastic layer encodes the observations, with two fully-connected layers of 200 hidden units each, into 100 dimensional outputs. The output is used as the parameters of diagonal Gaussian. The second layer takes samples from this Gaussian and passes them through two fully-connected layers of 100 hidden units each into 50 dimensions. See table 2 for NLL scores estimated as the mean of equation (9) with k=5000 on the test set. We can see that the path derivative gradient estimator improves over the original gradient estimator in all but two cases. 7 Benchmark Datasets We evaluate our path derivative estimator using two benchmark datasets: MNIST, a dataset of handwritten digits [LeCun et al., 1998], and Omniglot, a dataset of handwritten characters from many different alphabets [Lake, 2014]. To underscore both the easy implementation of this technique and the improvement it offers over existing approaches, we have empirically evaluated our new gradient estimator by a simple modification of existing code1 [Burda et al., 2015]. Omniglot Results For a two-stochastic-layer VAE using the multi-sample ELBO with gradient corresponding to equation (8) improves over the results in Burda et al. [2015] by 2.36, 1.44, and 0.6 nats for k={1, 5, 50} respectively. For a one-stochastic-layer VAE, the improvements are more modest: 0.72, 0.22, and 0.38 nats lower for k={1, 5, 50} respectively. A VAE with a deep recognition network appears to benefit more from our path derivative estimator than one with a shallow recognition network. For comparison, a VAE using the path derivative estimator with k=5 samples performs only 0.08 nats worse than an IWAE using the total derivative gradient estimator (7) and 5 samples. By contrast, using the total derivative (vanilla) estimator for both models, IWAE otherwise outperforms VAE for k=5 samples by 1.52 nats. By increasing the accuracy of the ELBO gradient estimator, we may also increase the risk of overfitting. Burda et al. [2015] report that they didn?t notice any significant problems with overfitting, as the training log likelihood was usually 2 nats lower than the test log likelihood. With our gradient estimator, we observe only 0.77 nats worse performance for a VAE with k=50 compared to k=5 in the two-layer experiments. IWAE using equation (8) markedly outperforms IWAE using equation (7) on Omniglot. For a 2-layer IWAE, we observe an improvement of 2.34, 1.2, and 0.52 nats for k={1, 5, 50} respectively. For a 1-layer IWAE, the improvements are 0.72, 0.7, and 0.51 for k={1, 5, 50} respectively. Just as in the VAE Omniglot results, a deeper recognition network for an IWAE model benefits more from the improved gradient estimator than a shallow recognition network. MNIST Results For all but one experiment, a VAE with our path derivative estimator outperforms a vanilla VAE on MNIST data. For k=50 with one stochastic layer, our gradient estimator underperforms a vanilla VAE by 0.13 nats. Interestingly, the training NLL for this run is 86.11, only 0.37 nats different than the test NLL. The similar magnitude of the two numbers suggests that training for longer than Burda et al. [2015] would improve the performance of our gradient estimator. We hypothesize that the worse performance using the path derivative estimator is a consequence of fine-tuning towards the characteristics of the total derivative estimator. For a two-stochastic-layer VAE on MNIST, the improvements are 0.56, 0.33 and 0.45 for k={1, 5, 50} respectively. In a one-stochastic-layer VAE on MNIST, the improvements are 0.36 and 0.14 for k={1, 5} respectively. The improvements on IWAE are of a similar magnitude. For k=50 in a two-layer path-derivative IWAE, we perform 0.26 nats worse than with a vanilla IWAE. The training loss for the k=50 run is 82.74, only 0.42 nats different. As in the other failure case, this suggests we have room to improve these results by fine-tuning over our method. For a two stochastic layer IWAE, the improvements are 0.66 and 0.22 for k=1 and 5 respectively. In a one stochastic layer IWAE, the improvements are 0.36, 0.34, and 0.33 for k={1, 5, 50} respectively. 7 Conclusions and Future Work We demonstrated that even when the reparameterization trick is applicable, further reductions in gradient variance are possible. We presented our variance reduction method in a general way by expressing it as a modification of the computation graph used for automatic differentiation. The gain from using our method grows with the complexity of the approximate posterior, making it complementary to the development of non-Gaussian posterior families. Although the proposed method is specific to variational inference, we suspect that similar unbiased but high-variance terms might exist in other stochastic optimization settings, such as in reinforcement learning, or gradient-based Markov Chain Monte Carlo. 1 See https://github.com/geoffroeder/iwae 8 References Mart?n Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. 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Efficient Approximation Algorithms for String Kernel Based Sequence Classification Muhammad Farhan Department of Computer Science School of Science and Engineering Lahore University of Management Sciences Lahore, Pakistan 14030031@lums.edu.pk Juvaria Tariq Department of Mathematics School of Science and Engineering Lahore University of Management Sciences Lahore, Pakistan jtariq@emory.edu Arif Zaman Department of Computer Science School of Science and Engineering Lahore University of Management Sciences Lahore, Pakistan arifz@lums.edu.pk Mudassir Shabbir Department of Computer Science Information Technology University Lahore, Pakistan mudassir.shabbir@itu.edu.pk Imdad Ullah Khan Department of Computer Science School of Science and Engineering Lahore University of Management Sciences Lahore, Pakistan imdad.khan@lums.edu.pk Abstract Sequence classification algorithms, such as SVM, require a definition of distance (similarity) measure between two sequences. A commonly used notion of similarity is the number of matches between k-mers (k-length subsequences) in the two sequences. Extending this definition, by considering two k-mers to match if their distance is at most m, yields better classification performance. This, however, makes the problem computationally much more complex. Known algorithms to compute this similarity have computational complexity that render them applicable only for small values of k and m. In this work, we develop novel techniques to efficiently and accurately estimate the pairwise similarity score, which enables us to use much larger values of k and m, and get higher predictive accuracy. This opens up a broad avenue of applying this classification approach to audio, images, and text sequences. Our algorithm achieves excellent approximation performance with theoretical guarantees. In the process we solve an open combinatorial problem, which was posed as a major hindrance to the scalability of existing solutions. We give analytical bounds on quality and runtime of our algorithm and report its empirical performance on real world biological and music sequences datasets. 1 Introduction Sequence classification is a fundamental task in pattern recognition, machine learning, and data mining with numerous applications in bioinformatics, text mining, and natural language processing. Detecting proteins homology (shared ancestry measured from similarity of their sequences of amino 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. acids) and predicting proteins fold (functional three dimensional structure) are essential tasks in bioinformatics. Sequence classification algorithms have been applied to both of these problems with great success [3, 10, 13, 18, 19, 20, 25]. Music data, a real valued signal when discretized using vector quantization of MFCC features is another flavor of sequential data [26]. Sequence classification has been used for recognizing genres of music sequences with no annotation and identifying artists from albums [12, 13, 14]. Text documents can also be considered as sequences of words from a language lexicon. Categorizing texts into classes based on their topics is another application domain of sequence classification [11, 15]. While general purpose classification methods may be applicable to sequence classification, huge lengths of sequences, large alphabet sizes, and large scale datasets prove to be rather challenging for such techniques. Furthermore, we cannot directly apply classification algorithms devised for vectors in metric spaces because in almost all practical scenarios sequences have varying lengths unless some mapping is done beforehand. In one of the more successful approaches, the variable-length sequences are represented as fixed dimensional feature vectors. A feature vector typically is the spectra (counts) of all k-length substrings (k-mers) present exactly [18] or inexactly (with up to m mismatches) [19] within a sequence. A kernel function is then defined that takes as input a pair of feature vectors and returns a real-valued similarity score between the pair (typically inner-product of the respective spectra?s). The matrix of pairwise similarity scores (the kernel matrix) thus computed is used as input to a standard support vector machine (SVM) [5, 27] classifier resulting in excellent classification performance in many applications [19]. In this setting k (the length of substrings used as bases of feature map) and m (the mismatch parameter) are independent variables directly related to classification accuracy and time complexity of the algorithm. It has been established that using larger values of k and m improve classification performance [11, 13]. On the other hand, the runtime of kernel computation by the efficient trie-based algorithm [19, 24] is O(k m+1 |?|m (|X| + |Y |)) for two sequences X and Y over alphabet ?. Computation of mismatch kernel between two sequences X and Y reduces to the following two problems. i) Given two k-mers ? and ? that are at Hamming distance d from each other, determine the size of intersection of m-mismatch neighborhoods of ? and ? (k-mers that are at distance at most m from both of them). ii) For 0 ? d ? min{2m, k} determine the number of pairs of k-mers (?, ?) ? X ? Y such that Hamming distance between ? and ? is d. In the best known algorithm [13] the former problem is addressed by precomputing the intersection size in constant time for m ? 2 only. While a sorting and enumeration based technique is proposed for the latter problem that has computational complexity O(2k (|X| + |Y |), which makes it applicable for moderately large values of k (of course limited to m ? 2 only). In this paper, we completely resolve the combinatorial problem (problem i) for all values of m. We prove a closed form expression for the size of intersection of m-mismatch neighborhoods that lets us precompute these values in O(m3 ) time (independent of |?|, k, lengths and number of sequences). For the latter problem we devise an efficient approximation scheme inspired by the theory of locality sensitive hashing to accurately estimate the number of k-mer pairs between the two sequences that are at distance d. Combining the above two we design a polynomial time approximation algorithm for kernel computation. We provide probabilistic guarantees on the quality of our algorithm and analytical bounds on its runtime. Furthermore, we test our algorithm on several real world datasets with large values of k and m to demonstrate that we achieve excellent predictive performance. Note that string kernel based sequence classification was previously not feasible for this range of parameters. 2 Related Work In the computational biology community pairwise alignment similarity scores were used traditionally as basis for classification, like the local and global alignment [5, 29]. String kernel based classification was introduced in [30, 9]. Extending this idea, [30] defined the gappy n-gram kernel and used it in conjunction with SVM [27] for text classification. The main drawback of this approach is that runtime for kernel evaluations depends quadratically on lengths of the sequences. An alternative model of string kernels represents sequences as fixed dimensional vectors of counts of occurrences of k-mers in them. These include k-spectrum [18] and substring [28] kernels. This notion is extended to count inexact occurrences of patterns in sequences as in mismatch [19] and profile [10] kernels. In this transformed feature space SVM is used to learn class boundaries. This approach 2 yields excellent classification accuracies [13] but computational complexity of kernel evaluation remains a daunting challenge [11]. The exponential dimensions (|?|k ) of the feature space for both the k-spectrum kernel and k, mmismatch kernel make explicit transformation of strings computationally prohibitive. SVM does not require the feature vectors explicitly; it only uses pairwise dot products between them. A trie-based strategy to implicitly compute kernel values for pairs of sequences was proposed in [18] and [19]. A (k, m)-mismatch tree is introduced which is a rooted |?|-ary tree of depth k, where each internal node has a child corresponding to each symbol in ? and every leaf corresponds to a k-mer in ?k . The runtime for computing the k, m mismatch kernel value between two sequences X and Y , under this trie-based framework, is O((|X| + |Y |)k m+1 |?|m ), where |X| and |Y | are lengths of sequences. This makes the algorithm only feasible for small alphabet sizes and very small number of allowed mismatches. The k-mer based kernel framework has been extended in several ways by defining different string kernels such as restricted gappy kernel, substitution kernel, wildcard kernel [20], cluster kernel [32], sparse spatial kernel [12], abstraction-augmented kernel [16], and generalized similarity kernel [14]. For literature on large scale kernel learning and kernel approximation see [34, 1, 7, 22, 23, 33] and references therein. 3 Algorithm for Kernel Computation In this section we formulate the problem, describe our algorithm and analyze it?s runtime and quality. k-spectrum and k, m-mismatch kernel: Given a sequence X over alphabet ?, the k, m-mismatch spectrum of X is a |?|k -dimensional vector, ?k,m (X) of number of times each possible k-mer occurs in X with at most m mismatches. Formally, ! X ?k,m (X) = (?k,m (X)[?])???k = Im (?, ?) , (1) ??X ???k where Im (?, ?) = 1, if ? belongs to the set of k-mers that differ from ? by at most m mismatches, i.e. the Hamming distance between ? and ?, d(?, ?) ? m. Note that for m = 0, it is known as k-spectrum of X. The k, m-mismatch kernel value for two sequences X and Y (the mismatch spectrum similarity score) [19] is defined as: X K(X, Y |k, m) = h?k,m (X), ?k,m (Y )i = ?k,m (X)[?]?k,m (Y )[?] ???k = X X Im (?, ?) ???k ??X X Im (?, ?) = X X X Im (?, ?)Im (?, ?). (2) ??X ??Y ???k ??Y For a k-mer ?, let Nk,m (?) = {? ? ?k : d(?, ?) ? m} be the m-mutational neighborhood of ?. Then for a pair of sequences X and Y , the k, m-mismatch kernel given in eq (2) can be equivalently computed as follows [13]: X X X K(X, Y |k, m) = Im (?, ?)Im (?, ?) ??X ??Y ???k = X X |Nk,m (?) ? Nk,m (?)| = ??X ??Y X X Im (?, ?), (3) ??X ??Y where Im (?, ?) = |Nk,m (?) ? Nk,m (?)| is the size of intersection of m-mutational neighborhoods of ? and ?. We use the following two facts. Fact 3.1. Im (?, ?), the size of the intersection of m-mismatch neighborhoods of ? and ?, is a function of k, m, |?| and d(?, ?) and is independent of the actual k-mers ? and ? or the actual positions where they differ. (See section 3.1) Fact 3.2. If d(?, ?) > 2m, then Im (?, ?) = 0. In view of the above two facts we can rewrite the kernel value (3) as min{2m,k} K(X, Y |k, m) = X X Im (?, ?) = X i=0 ??X ??Y 3 Mi ? I i , (4) where Ii = Im (?, ?) when d(?, ?) = i and Mi is the number of pairs of k-mers (?, ?) such that d(?, ?) = i, where ? ? X and ? ? Y . Note that bounds on the last summation follows from Fact 3.2 and the fact that the Hamming distance between two k-mers is at most k. Hence the problem of kernel evaluation is reduced to computing Mi ?s and evaluating Ii ?s. 3.1 Closed form for Intersection Size Let Nk,m (?, ?) be the intersection of m-mismatch neighborhoods of ? and ? i.e. Nk,m (?, ?) = Nk,m (?) ? Nk,m (?). As defined earlier |Nk,m (?, ?)| = Im (?, ?). Let Nq (?) = {? ? ?k : d(?, ?) = q} be the set of k-mers that differ with ? in exactly q indices. Note that Nq (?) ? Nr (?) = ? for all q 6= r. Using this and defining nqr (?, ?) = |Nq (?) ? Nr (?)|, Nk,m (?, ?) = m [ m [ Nq (?) ? Nr (?) and q=0 r=0 Im (?, ?) = m X m X nqr (?, ?). q=0 r=0 Hence we give a formula to compute nij (?, ?). Let s = |?|. Theorem 3.3. Given two k-mers ? and ? such that d(?, ?) = d, we have that i+j?d 2 ij n (?, ?) =    X 2d ? i ? j + 2t d k?d (s ? 2)i+j?2t?d (s ? 1)t . i + j ? 2t ? d t d ? (i ? t) t=0 Proof. nij (?, ?) can be interpreted as the number of ways to make i changes in ? and j changes in ? to get the same string. For clarity, we first deal with the case when we have d(?, ?) = 0, i.e both strings are identical. We wish to find nij (?, ?) = |Ni (?) ? Nj (?)|. It is clear that in this case i = j, otherwise making i and j changes to the same string will not result in the same string.
Hence nij = ki (s ? 1)i . Second we consider ?, ? such that d(?, ?) = k. Clearly k ? i and k ? j. Moreover, since both strings do not agree at any index, character at every index has to be changed in at least one of ? or ?. This gives k ? i + j. Now for a particular index p, ?[p] and ?[p] can go through any one of the following three changes. Let ?[p] = x, ?[p] = y. (I) Both ?[p] and ?[p] may change from x and y respectively to some character z. Let l1 be the count of indices going through this type of change. (II) ?[p] changes from x to y, call the count of these l2 . (III) ?[p] changes from y to x, let this count be l3 . It follows that i = l1 + l2 , j = l1 + l3 , , l1 + l2 + l3 = k. This results in l1 = i + j ? k. Since l1 is the count of indices at which
characters of both strings k change, we have s ? 2 character choices for each such index and i+j?k possible combinations of indices for l . From the remaining l + l = 2k ? i ? j indices, we choose l2 = k ? j indices in 1 2 3
2k?i?j ways and change the characters at these indices of ? to characters of ? at respective indices. k?j Finally, we are left with only l3 remaining indices and we change them according to the definition of l3 . Thus the total number of strings we get after making i changes in ? and j changes in ? is    k 2k ? i ? j i+j?k (s ? 2) . i+j?k k?j Now we consider general strings ? and ? of length k with d(?, ?) = d. Without loss of generality assume that they differ in the first d indices. We parameterize the system in terms of the number of changes that occur in the last k ? d indices of the strings i.e let t be the number of indices that go through a change in last k ? d indices. Number of possible such changes is   k?d (s ? 1)t . (5) t Lets call the first d-length substrings of both strings ?0 and ? 0 . There are i ? t characters to be changed in ?0 and j ? t in ? 0 . As reasoned above, we have d ? (i ? t) + (j ? t) =? t ? i+j?d . 2 4 In this setup we get i ? t = l1 + l2 , j ? t = l1 + l3 , l1 + l2 + l3 = d and l1 = (i ? t) + (j ? t) ? d. We immediately get that for a fixed t, the total number of resultant strings after making i ? t changes in ?0 and j ? t changes in ? 0 is    2d ? (i ? t) ? (j ? t) d (s ? 2)(i?t)+(j?t)?d . (6) d ? (i ? t) (i ? t) + (j ? t) ? d For a fixed t, every substring counted in (5), every substring counted in (6) gives a required string obtained after i and j changes in ? and ? respectively. The statement of the theorem follows. Corollary 3.4. Runtime of computing Id is O(m3 ), independent of k and |?|. This is so, because if d(?, ?) = d, Id = m P m P nqr (?, ?) and nqr (?, ?) can be computed in O(m). q=0 r=0 3.2 Computing Mi Recall that given two sequences X and Y , Mi is the number of pairs of k-mers (?, ?) such that d(?, ?) = i, where ? ? X and ? ? Y . Formally, the problem of computing Mi is as follows: Problem 3.5. Given k, m, and two sets of k-mers SX and SY (set of k-mers extracted from the sequences X and Y respectively) with |SX | = nX and |SY | = nY . Compute Mi = |{(?, ?) ? SX ? SY : d(?, ?) = i}| for 0 ? i ? min{2m, k}. Note that the brute force approach to compute Mi requires O(nX ? nY ? k) comparisons. Let Qk (j) denote the set of all j-sets of {1, . . . , k} (subsets of indices). For ? ? Qk (j) and a k-mer ?, let ?|? be the j-mer obtained by selecting the characters at the j indices in ?. Let f? (X, Y ) be the number of pairs of k-mers in SX ? SY as follows; f? (X, Y ) = |{(?, ?) ? SX ? SY : d(?|? , ?|? ) = 0}|. We use the following important observations about f? . Fact 3.6. For 0 ? i ? k and ? ? Qk (k ? i), if d(?|? , ?|? ) = 0, then d(?, ?) ? i. Fact 3.7. For 0 ? i ? k and ? ? Qk (k ? i), f? (X, Y ) can be computed in O(kn log n) time. This can be done by first lexicographically sorting the k-mers in each of SX and SY by the indices in ?. The pairs in SX ? SY that are the same at indices in ? can then be enumerated in one linear scan over the sorted lists. Let n = nX + nY , runtime of this computation is O(k(n + |?|)) if we use counting sort (as in [13]) or O(kn log n) for mergesort (since ? has O(k) indices.) Since this procedure is repeated many times, we refer to this as the SORT-ENUMERATE subroutine. We define X Fi (X, Y ) = f? (X, Y ). (7) ??Qk (k?i) Lemma 3.8. Fi (X, Y ) =  i  X k?j Mj . k?i j=0 (8) Proof. Let (?, ?) be a pair that contributes to Mj , i.e. d(?, ?) = j. Then for every ? ? Qk (k ? i) that has all indices within the k ? j positions where ? and ? agree,
the pair (?, ?) is counted in
f? (X, Y ). The number of such ??s are k?j , hence Mj is counted k?j k?i k?i times in Fi (X, Y ), yielding the required equality. Corollary 3.9. Mi can readily be computed as: Mi = Fi (X, Y ) ? i?1 P j=0 k?j k?i
Mj .

k By definition, Fi (X, Y ) can be computed with k?i = ki f? computations. Let t = min{2m, k}. K(X, Y |k, m) can be evaluated by (4) after computing Mi (by (8)) and Ii (by Corollary 3.4) for 0 ? i ? t. The overall complexity of this strategy thus is ! t   X k (k ? i)(n log n + n) + O(n) = O(k ? 2k?1 ? (n log n)). i i=0 5 Algorithm 1 : Approximate-Kernel(SX ,SY ,k,m,,?,B) 1: I, M 0 ??ZEROS(t + 1) 2: ? ?  ? ? 3: Populate I using Corollary 3.4 4: for i = 0 to t do 5: ?F ? 0 6: iter ? 1 7: varF ? ? 8: while varF > ? 2 ? iter
< B do k 9: ? ? RANDOM( k?i ) ?F ? (iter ? 1) + SORT- ENUMERATE(SX , SY , k, ?) 10: ?F ? . Application of Fact 3.7 iter 11: varF ? VARIANCE(?F , varF , iter) . Compute online variance 12: iter ? iter +
1 k 13: F 0 [i] ? ?F ? k?i 0 0 14: M [i] ? F [i] 15: for j = 0 to i ? 1 do . Application of Corollary 3.9
0 16: M 0 [i] ? M 0 [i] ? k?j ? M [j] k?i 17: K 0 ? SUMPRODUCT(M 0 , I) 18: return K 0 . Applying Equation (4) We give our algorithm to approximate K(X, Y |k, m), it?s explanation followed by it?s analysis. Algorithm 1 takes , ? ? (0, 1), and B ? Z+ as input parameters; the first two controls the accuracy of estimate while B is an upper bound on the sample size. We use (7) to estimate Fi = Fi (X, Y ) with an online sampling algorithm, where we choose ? ? Qk (k ? i) uniformly at random and compute the online mean and variance of the estimate for Fi . We continue to sample until the variance is below the threshold (? 2 = 2 ?) or the sample size reaches the upper bound B. We scale up our estimate by the population size and use it to compute Mi0 (estimates of Mi ) using Corollary 3.9. These Mi0 ?s together with the precomputed exact values of Ii ?s are used to compute our estimate, K 0 (X, Y |k, m, ?, ?, B), for the kernel value using (4). First we give an analytical bound on the runtime of Algorithm 1 then we provide guarantees on it?s performance. Theorem 3.10. Runtime of Algorithm 1 is bounded above by O(k 2 n log n). Proof. Observe that throughout the execution of the algorithm there are at most tB computations of f? , which by Fact 3.7 needs O(kn log n) time. Since B is an absolute constant and t ? k, we get that the total runtime of the algorithm is O(k 2 n log n). Note that in practice the while loop in line 8 is rarely executed for B iterations; the deviation is within the desired range much earlier. Let K 0 = K 0 (X, Y |k, m, , ?, B) be our estimate (output of Algorithm 1) for K = K(X, Y |k, m). Theorem 3.11. K 0 is an unbiased estimator of the true kernel value, i.e. E(K 0 ) = K. Proof. For this we need the following result, whose proof is deferred. Lemma 3.12. E(Mi0 ) = Mi . Pt By Line 17 of Algorithm 1, E(K 0 ) = E( i=0 Ii Mi0 ). Using the fact that Ii ?s are constants and Lemma 3.12 we get that E(K 0 ) = t X min{2m,k} Ii E(Mi0 ) = i=0 X Ii Mi = K. i=0 Theorem 3.13. For any 0 < , ? < 1, Algorithm 1 is an (Imax , ?)?additive approximation algorithm, i.e. P r(|K ? K 0 | ? Imax ) < ?, where Imax = maxi {Ii }. 6 Note that these are very loose bounds, in practice we get approximation far better than these bounds. Furthermore, though Imax could be large, but it is only a fraction of one of the terms in summation for the kernel value K(X, Y |k, m). Proof. Let Fi0 be our estimate for Fi (X, Y ) = Fi . We use the following bound on the variance of K 0 that is proved later. Lemma 3.14. V ar(K 0 ) ? ?( ? Imax )2 . 0 0 By Lemma 3.12 we have E(K p ) = K, hence by Lemma 3.14, P r[|K ? K|] ? Imax is equivalent 1 0 0 0 to P r[|K ? E(K )|] ? ?? V ar(K ). By the Chebychev?s inequality, this latter probability is at most ?. Therefore, Algorithm 1 is an (Imax , ?)?additive approximation algorithm. Proof. (Proof of Lemma 3.12) We prove it by induction on i. The base case (i = 0) is true as we compute M 0 [0] exactly, i.e. M 0 [0] = M [0]. Suppose E(Mj0 ) = Mj for 0 ? j ? i ? 1. Let iter be the number of iterations for i, after execution of Line 10 we get   Piter   f? (X, Y ) k k F 0 [i] = ?F = r=1 r , k?i k?i iter where ?r is the random (k ? i)-set chosen in the rth iteration of the while loop. Since ?r is chosen uniformly at random we get that       k k k Fi (X, Y ) 0 = E(f?r (X, Y )) = . (9) E(F [i]) = E(?F )
k k?i k?i k?i k?i After the loop on Line 15 is executed we get that E(M 0 [i]) = Fi (X, Y ) ? i?1 P j=0 k?j k?i
E(Mj0 ). Using E(Mj0 ) = Mj (inductive hypothesis) in (8) we get that E(Mi0 ) = Mi . Proof. (Proof of Lemma 3.14) After execution of the while loop in Algorithm 1, we have Fi0 = i
0 P k?j k?i Mj . We use the following fact that follows from basic calculations. j=0 Fact 3.15. Suppose X0 , . . . , Xt are random variables and let S = are constants. Then V ar(S) = t X a2i V ar(Xi ) + 2 i=0 t t X X Pt i=0 ai Xi , where a0 , . . . , at ai aj Cov(Xi , Xj ). i=0 j=i+1 Using fact 3.15 and definitions of Imax and ? we get that V ar(K 0 ) = t X Ii 2 V ar(Mi0 ) + 2 i=0 t t X X Ii Ij Cov(Mi0 , Mj0 ) i=0 j=i+1 ? ? t t t X X X 2 2 2 ? Cov(Mi0 , Mj0 )? ? Imax V ar(Ft0 ) ? Imax ? 2 = ?(?Imax )2 . ? Imax V ar(Mi0 ) + 2 i=0 i=0 j=i+1 The last inequality follows from the following relation derived from definition of Fi0 and Fact 3.15. V ar(Ft0 ) = 2 t  X k?i i=0 k?t V ar(Mi0 ) + 2    t t X X k?i k?j Cov(Mi0 , Mj0 ). k ? t k ? t i=0 j=i+1 7 (10) 4 Evaluation We study the performance of our algorithm in terms of runtime, quality of kernel estimates and predictive accuracies on standard benchmark sequences datasets (Table 1) . For the range of parameters feasible for existing solutions, we generated kernel matrices both by algorithm of [13] (exact) and our algorithm (approximate). These experiments are performed on an Intel Xeon machine with (8 Cores, 2.1 GHz and 32 GB RAM) using the same experimental settings as in [13, 15, 17]. Since our algorithm is applicable for significantly wider range of k and m, we also report classification performance with large k and m. For our algorithm we used B ? {300, 500} and ? ? {0.25, 0.5} with no significant difference in results as implied by the theoretical analysis. In all reported results B = 300 and ? = 0.5. In order to perform comparisons, for a few combinations of parameters we generated exact kernel matrices of each dataset on a much more powerful machine (a cluster of 20 nodes, each having 24 CPU?s with 2.5 GHz speed and 128GB RAM). Sources for datasets and source code are available at 1 . Table 1: Datasets description Name Ding-Dubchak [6] SCOP [4, 31] Music [21, 26] Artist20 [8, 17] ISMIR [17] Task protein fold recognition protein homology detection music genre recognition artist identification music genre recognition Classes 27 54 10 20 6 Seq. 694 7329 1000 1413 729 Av.Len. 169 308 2368 9854 10137 Evaluation 10-fold CV 54 binary class. 5-fold CV 6-fold CV 5-fold CV Running Times: We report difference in running times for kernels generation in Figure 1. Exact kernels are generated using code provided by authors of [13, 14] for 8 ? k ? 16 and m = 2 only. We achieve significant speedups for large values of k (for k = 16 we get one order of magnitude gains in computational efficiency on all datasets). The running times for these algorithms are O(2k n) and O(k 2 n log n) respectively. We can use larger values of k without an exponential penalty, which is visible in the fact that in all graphs, as k increases the growth of running time of the exact algorithm is linear (on the log-scale), while that of our algorithm tends to taper off. Figure 1: Log scaled plot of running time of approximate and exact kernel generation for m = 2 Exact 10000 Running Time (sec) Approximate 1000 100 10 1 k 8 10 12 14 DingDubchak 16 8 10 12 SCOP 14 16 8 10 12 14 16 8 MusicGenre 10 12 14 ISMIR2004 16 8 10 12 14 16 Artist20 Kernel Error Analysis: We show that despite reduction in runtimes, we get excellent approximation of kernel matrices. In Table 2 we report point-to-point error analysis of the approximate kernel matrices. We compare our estimates with exact kernels for m = 2. For m > 2 we report statistical error analyses. More precisely, we evaluate differences with principal submatrices of the exact kernel matrix. These principal submatrices are selected by randomly sampling 50 sequences and computing their pairwise kernel values. We report errors for four datasets; the fifth one, not included for space reasons, showed no difference in error. From Table 2 it is evident that our empirical performance is significantly more precise than the theoretical bounds proved on errors in our estimates. 1 https://github.com/mufarhan/sequence_class_NIPS_2017 8 Table 2: Mean absolute error (MAE) and root mean squared error (RMSE) of approximate kernels. For m > 2 we report average MAE and RMSE of three random principal submatrices of size 50 ? 50 (k, m) (10, 2) (12, 2) (14, 2) (16, 2) (12, 6) Music Genre RMSE MAE 0 0 0 0 2.0E?8 0 1.3E?8 0 1.97E?5 8.5E?7 ISMIR RMSE MAE 0 0 0 0 2.0E?8 0 4.0E?8 3.3E?9 Artist20 RMSE MAE 0 0 0 0 3.3E?8 1.3E?8 SCOP RMSE MAE 1.3E?6 9.0E?8 1.4E?6 1.0E?8 2.9E?6 1.3E?8 2.9E?6 1.0E?8 2.4E?4 1.8E?5 Prediction Accuracies: We compare the outputs of SVM on the exact and approximate kernels using the publicly available SVM implementation LIBSVM [2]. We computed exact kernel matrices by brute force algorithm for a few combinations of parameters for each dataset on the much more powerful machine. Generating these kernels took days; we only generated to compare classification performance of our algorithm with the exact one. We demonstrate that our predictive accuracies are sufficiently close to that with exact kernels in Table 3 (bio-sequences) and Table 4 (music). The parameters used for reporting classification performance are chosen in order to maintain comparability with previous studies. Similarly all measurements are made as in [13, 14], for instance for music genre classification we report results of 10-fold cross-validation (see Table 1). For our algorithm we used B = 300 and ? = 0.5 and we take an average of performances over three independent runs. Table 3: Classification performance comparisons on SCOP (ROC) and Ding-Dubchak (Accuracy) k, m 8, 2 10, 2 12, 2 14, 2 16, 2 10, 5 10, 7 12, 8 SCOP Exact Approx ROC ROC50 ROC ROC50 88.09 38.71 88.05 38.60 81.65 28.18 80.56 26.72 71.31 23.27 66.93 11.04 67.91 7.78 63.67 6.66 64.45 6.89 61.64 5.76 91.60 53.77 91.67 54.1 90.27 48.18 90.30 48.44 91.44 50.54 90.97 52.08 Ding-Dubchak Exact Approx Accuracy 34.01 31.65 28.1 26.9 27.23 26.66 25.5 25.5 25.94 25.03 45.1 43.80 58.21 57.20 58.21 57.83 Table 4: Classification error comparisons on music datasets exact and estimated kernels k, m 10, 2 14, 2 16, 2 10, 7 12, 6 12, 8 5 Music Genre Exact Estimate 61.30 ? 3.3 61.30 ? 3.3 71.70 ? 3.0 71.70 ? 3.0 73.90 ? 1.9 73.90 ? 1.9 37.00 ? 3.5 37.00 ? 3.5 54.20 ? 2.7 54.13 ? 2.9 43.70 ? 3.2 44.20 ? 3.2 ISMIR Exact Estimate 54.32 ? 1.6 54.32 ? 1.6 55.14 ? 1.1 55.14 ? 1.1 54.73 ? 1.5 54.73 ? 1.5 27.16 ? 1.6 52.12 ? 2.0 52.08 ? 1.5 47.03 ? 2.6 47.41 ? 2.4 Artist20 Exact Estimate 82.10 ? 2.2 82.10 ? 2.2 86.84 ? 1.8 86.84 ? 1.8 87.56 ? 1.8 87.56 ? 1.8 55.75 ? 4.7 55.75 ? 4.7 79.57 ? 2.4 80.00 ? 2.6 67.57 ? 3.6 Conclusion In this work we devised an efficient algorithm for evaluation of string kernels based on inexact matching of subsequences (k-mers). We derived a closed form expression for the size of intersection of m-mismatch neighborhoods of two k-mers. Another significant contribution of this work is a novel statistical estimate of the number of k-mer pairs at a fixed distance between two sequences. Although large values of the parameters k and m were known to yield better classification results, known algorithms are not feasible even for moderately large values. Using the two above mentioned results our algorithm efficiently approximate kernel matrices with probabilistic bounds on the accuracy. Evaluation on several challenging benchmark datasets for large k and m, show that we achieve state of the art classification performance, with an order of magnitude speedup over existing solutions. 9 References [1] F. R. Bach and M. I. Jordan. Predictive low-rank decomposition for kernel methods. In International Conference on Machine Learning, ICML, pages 33?40, 2005. [2] C.-C. Chang and C.-J. Lin. LIBSVM: A library for support vector machines. ACM Transactions on Intelligent Systems and Technology, 2:27:1?27:27, 2011. [3] J. Cheng and P. Baldi. A machine learning information retrieval approach to protein fold recognition. Bioinformatics, 22(12):1456?1463, 2006. [4] L. Conte, B. Ailey, T. Hubbard, S. Brenner, A. Murzin, and C. Chothia. Scop: A structural classification of proteins database. Nucleic Acids Research, 28(1):257?259, 2000. [5] N. Cristianini and J. Shawe-Taylor. An introduction to support vector machines and other kernel-based learning methods. Cambridge university press, 2000. [6] C. Ding and I. Dubchak. Multi-class protein fold recognition using support vector machines and neural networks. Bioinformatics, 17(4):349?358, 2001. [7] P. Drineas and M. W. Mahoney. On the nystr?m method for approximating a gram matrix for improved kernel-based learning. The Journal of Machine Learning Research, 6:2153?2175, 2005. [8] D. P. Ellis. Classifying music audio with timbral and chroma features. In ISMIR, volume 7, pages 339?340, 2007. [9] D. Haussler. Convolution kernels on discrete structures. Technical Report UCS-CRL-99-10, University of California at Santa Cruz, 1999. [10] R. Kuang, E. Ie, K. Wang, K. Wang, M. Siddiqi, Y. Freund, and C. Leslie. Profile-based string kernels for remote homology detection and motif extraction. Journal of Bioinformatics and Computational Biology, 3(3):527?550, 2005. [11] P. Kuksa. Scalable kernel methods and algorithms for general sequence analysis. PhD thesis, Department of Computer Science, Rutgers, The State University of New Jersey, 2011. [12] P. Kuksa, P.-H. Huang, and V. Pavlovic. Fast protein homology and fold detection with sparse spatial sample kernels. In 19th International Conference on Pattern Recognition, ICPR, pages 1?4. IEEE, 2008. [13] P. Kuksa, P.-H. Huang, and V. Pavlovic. Scalable algorithms for string kernels with inexact matching. In Advances in Neural Information Processing Systems, NIPS, pages 881?888. MIT Press, 2009. [14] P. Kuksa, I. Khan, and V. Pavlovic. Generalized similarity kernels for efficient sequence classification. In SIAM International Conference on Data Mining, SDM, pages 873?882. SIAM, 2012. [15] P. Kuksa and V. Pavlovic. Spatial representation for efficient sequence classification. In 20th International Conference on Pattern Recognition, ICPR, pages 3320?3323. IEEE, 2010. [16] P. Kuksa, Y. Qi, B. Bai, R. Collobert, J. Weston, V. Pavlovic, and X. Ning. Semi-supervised abstraction-augmented string kernel for multi-level bio-relation extraction. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, ECML-PKDD, pages 128?144. Springer, 2010. [17] P. P. Kuksa. Efficient multivariate sequence classification. In CoRR abs/1409.8211, 2013. [18] C. Leslie, E. Eskin, and W. Noble. The spectrum kernel: A string kernel for svm protein classification. In Pacific Symposium on Biocomputing, volume 7 of PSB, pages 566?575, 2002. [19] C. Leslie, E. Eskin, J. Weston, and W. Noble. Mismatch string kernels for svm protein classification. In Advances in Neural Information Processing Systems, NIPS, pages 1441?1448. MIT Press, 2003. 10 [20] C. Leslie and R. Kuang. Fast string kernels using inexact matching for protein sequences. Journal of Machine Learning Research, 5:1435?1455, 2004. [21] T. Li, M. Ogihara, and Q. Li. A comparative study on content-based music genre classification. In 26th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, ACM/SIGIR, pages 282?289. ACM, 2003. [22] A. Rahimi and B. Recht. Random features for large-scale kernel machines. In Advances in Neural Information Processing Systems, NIPS, pages 1177?1184, 2007. [23] A. Rahimi and B. Recht. Weighted sums of random kitchen sinks: Replacing minimization with randomization in learning. In Advances in Neural Information Processing Systems, NIPS, pages 1313?1320, 2008. [24] J. Shawe-Taylor and N. Cristianini. Kernel methods for pattern analysis. Cambridge university press, 2004. [25] S. Sonnenburg, G. R?tsch, and B. Sch?lkopf. Large scale genomic sequence svm classifiers. In 22nd International Conference on Machine Learning, ICML, pages 848?855. ACM, 2005. [26] G. Tzanetakis and P. Cook. Musical genre classification of audio signals. IEEE Transactions on Speech and Audio Processing, 10(5):293?302, 2002. [27] V. Vapnik. Statistical learning theory, volume 1. Wiley New York, 1998. [28] S. Vishwanathan and A. Smola. Fast kernels for string and tree matching. In Advances in Neural Information Processing Systems, NIPS, pages 585?592, 2002. [29] M. Waterman, J. Joyce, and M. Eggert. Computer alignment of sequences. Phylogenetic analysis of DNA sequences, pages 59?72. [30] C. Watkins. Dynamic alignment kernels. In Advances in Large Margin Classifiers, pages 39?50. 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VLSI Phase Locking Architectures for Feature Linking in Multiple Target Tracking Systems Andreas G. Andreou andreou@jhunix.hcf.jhu.edu Department of Electrical and Computer Engineering The Johns Hopkins University Baltimore, MD 21218 Thomas G. Edwards tedwards@src.umd.edu Department of Electrical Engineering The University of Maryland College Park, MD 20722 Abstract Recent physiological research has shown that synchronization of oscillatory responses in striate cortex may code for relationships between visual features of objects. A VLSI circuit has been designed to provide rapid phase-locking synchronization of multiple oscillators to allow for further exploration of this neural mechanism. By exploiting the intrinsic random transistor mismatch of devices operated in subthreshold, large groups of phase-locked oscillators can be readily partitioned into smaller phase-locked groups. A mUltiple target tracker for binary images is described utilizing this phase-locking architecture. A VLSI chip has been fabricated and tested to verify the architecture. The chip employs Pulse Amplitude Modulation (PAM) to encode the output at the periphery of the system. 1 Introduction In striate cortex, visual information coming from the retina (via the lateral geniculate nuclei) is processed to extract retinotopic maps of visual features. Some cells in cortex are receptive to lines of particular orientation, length, and/or movement direction (Hubel, 1988). A fundamental problem of visual processing is how to 866 VLSI Phase Locking Architectures for Feature Linking in Multiple Target Tracking Systems associate certain groups of features together to form coherent representations of objects. Since there is an almost infinite number of possible feature combinations, it seems unlikely that there are dedicated "grandmother" cells which code for every possible feature combination. There probably exists a type of adaptive and transitory method to "bind" these features together. The Binding Problem (Crick, 1990) is the problem of making neural elements which are receptive to these visual features temporarily become active as a group that codes for a particular object, yet maintaining the group's specificity towards that object, even when there are several different interleaved objects in the visual field. Temporal correlation of neural response is one solution to the binding problem (von der Malsburg, 1986). Response from neurons (or neural oscillating circuits) which are receptive to a particular visual feature are required to have high temporal correlation with responses to other visual features that correspond to the same object. This would require that there is stimulus-driven oscillation in visual cortex, and that there is also a degree of oscillation synchronization between neural circuits receptive to the same object. Both of these requirements have been found in visual cortex (Gray, 1987; Gray, 1989). Furthermore, there have been several computer simulations of the synchronization phenomena and related visual processing tasks (Baldi, 1990; Eckhorn, 1990). This paper describes a phase-locking architecture for a circuit which performs a multiple-target tracking problem. It will accomplish this task by establishing a zero valued phase difference between oscillators that are receptive to those features to be "bound" together to form an object. Each object will then be recognized as a group of synchronous oscillators, and oscillators that correspond to different objects will be identified due to their lack of synchronization. We assume these oscillators have low duty-cycle pulsed outputs, and the oscillators which correspond to the same object will all pulse high at the same time. Target location will be communicated to the periphery by Pulse Amplitude Modulation (PAM). 2 The Neural Oscillator The oscillator for the target tracker must have two qualities. It needs to be capable of producing a fairly smooth phase representation so that it is easy to compare the difference between oscillator phases to allow for robust phase-locking. It is also useful to have a pulsed output present so that one group of oscillators can be easily discerned from another group of oscillators when their outputs are examined over time. The self-resetting neuron circuit (Mead, 1989) provides both of these outputs (Figure 1). Current lin provided by FET Ql charges capacitor Cl until positive feedback though the non-inverting CMOS amplifier and capacitor C2 brings V p ha.ge all the way to Vdd. This causes the output voltage to go high, which turns on Q2 thus draining charge from Cl by I reret through Q3 and lowering V p ha.ge. When the Vpha"e is brought low enough, positive feedback brings both Vpha"e and the output voltage down to Vu. This turns transistor Q2 off, and the cycle repeats. The duration of output pulses is inversely proportional to I relet - lin, and the time between output pulses is inversely proportional to lin. Figure 2 is a plot of the pulse output voltage and Vphale vs. time. 867 868 Andreou and Edwards Vphase f - - -..... ---~ Pulse Output Figure 1: Self-Resetting Neural Oscillator Oscillator Output 5 ,..... ,..... \ \ I \ I \ I \ , \ I I I , I o I I I I I I I I I I I I v I I I J I I I I I I I I / \ \ \ I I I \ I I I \ \ I I I ,..... I \ \ I \ \ \ \ I ,..... I J v I I -1+---~--~---r--~--~----~--+---~---r--~ -1 o 1 2 3 4 5 6 7 8 9 (lO-6 sec ) Figure 2: Plot of Pulse Output (line) and Phase Voltage (dashed) vs. Time for Neural Oscillator VLSI Phase Locking Architectures for Feature Linking in Multiple Target Tracking Systems 3 Phase Locking To achieve stable and reliable performance, the Comparator Model (Kammen, 1990) of phase-locking was used. Oscillator phase is adjusted according to . ) aO(x,t) (1~ at =w(x)+! ;f;;t0("t)-O(x,t) . Where O( x, t) is the phase of oscillator x at time t, w( x) is the intrinsic phase advance of oscillator x, n is the total number of oscillators, and ! is a sigmoid sq uashing-function. Each object in the visual field requires one averaging circuit to achieve phase-locking of its receptive oscillators. But at any time we do not know the number of objects which will be in the visual field. Therefore, instead of having a pool of monolithic averaging circuits, it is preferable to distribute the averaging function over all the oscillator cells in a way which allows partitioning of the visual field into multiple phase-locked groups of oscillators. The follower-aggregator circuit (Mead, 1989) can be used to develop the average phase information using current-mode computation. It consists of transconductance amplifiers connected as voltage-followers with all outputs tied together to form the average of all input voltages. The phase averaging circuitry can be distributed among the oscillators by placing one transconductance amplifier in each oscillator cell, and linking those oscillators to be phase-locked by a common line. The visual field can be partitioned into multiple phase-locked groups with separate average phases by using FETs to gate whether or not the averaging information can pass through an oscillator cell to its neighbors. To lock an oscillator in phase with the rest of the oscillators which are attached to the averaging line, extra current is provided to the oscillator by a transconductance amplifier to slightly speed up or slow down the oscillator to match its phase to the average phase of the oscillators in the group. Figure 3 shows t.he circuit for a complete phase-locking oscillator cell. Computer simulations of this phase-locking system were carried out using the Analog circuit simulator. Figure 4 shows the result of a simulation of two oscillators. Vgate is the voltage controlling the NFET of the transmission gate which links the phase averaging lines of the two oscillators together (the PFETs are controlled complementary). As soon as the Vgate is brought high, the oscillat.ors rapidly phase lock. 4 Target Location We will assume that the input to a visual tracking chip is a binary image projected onto the die. Phototransistors detect the brightness of each pixel, and if it is above a threshold level, the pixel control circuitry will turn the pixel's oscillator on. If a pixel oscillator is turned on, gating circuitry will allow the propagation of the phase averaging line through the pixel's oscillator cell to its nearest-neighbors. Illuminated 869 870 Andreou and Edwards To top neighbor v??~~ To left - _ 4 neighbor . - - - - - - - - - To right neighbor T. VgateP~ ,,-+--~< VgateN Vf>-1 To bottom neighbor Figure 3: Phase-Locking Oscillator Cell Pulse Output VLSI Phase Locking Architectures for Feature Linking in Multiple Target Tracking Systems Sp1ke2~-...J I--------i -5 10 sec Figure 4: Phase-Locking Simulation nearest-neighbor connected pixels will thus have their oscillators turned on and will become phase-locked. The follower-aggregator circuit can be modified to determine linear position (Maher, 1989) by using voltage taps off of a resistive line as inputs to the transconductance amplifiers, and biasing the amplifiers by currents that correspond to the pulsing outputs of the oscillators (see Figure 5). During the time that a group of oscillators are spiking, the output of the tracking circuitry will yield a location corresponding to the average position of the distribution of those oscillators. There can be many different nearest-neighbor connected objects projected onto the die, and the position of the center of each object is communicated to the periphery via PAM. Thus, we can use multiplexing in time to simplify connectivity of communication with the periphery of the chip. 5 Test Chip A chip to test the Comparator Model phase-locking method and multiple-target tracking system was fabricated by the MOSIS service in 2.0 J1.m feature size CMOS. To keep this test chip simple, the oscillators were arranged in a one-dimensional chain, and voltage inputs to the chip were used to control whether or not a pixel was considered "illuminated." A polysilicon resistive line was used to provide linear position information to the tracking system. All transistors used were minimum size (6 J1.m wide and 4 J1.m long). The test chip was able to rapidly and robustly phase-lock groups of nearest-neighbor 871 872 Andreou and Edwards Postional Voltage Line Vpulse(n) Vpulse(n-l) Vpulse(n+l) Average Position Line Figure 5: Circuit to determine object location connected oscillators. This phase-locking could occur with oscillator frequencies set from 10 Hz to 4 KHz. A phase-locked group of oscillators would almost instantly split into two separate phase-locked groups with little temporal correlation between them when a connected chain of on oscillators was severed by turning off an oscillator in the middle of the original group . Mismatch in the transconductances of the oscillator transistors provided easy desynchronization. Position tracking was measured by examining the resistive-line aggregator output during the time a certain phase-locked group of oscillators was pulsing . When multiple phase-locked groups of oscillators were active, it was still quite easy to make out the positional PAM voltage associated with each group by triggering an oscilloscope off of the pulsing output of an oscillator in that group. While there are occasional instances of two or more groups pulsing at the same time, if the duty cycle of the spiking oscillator is kept relatively small, there is little interference on average. 6 Discussion It is becoming obvious that oscillation and synchronization phenomena in cortex may play an important role in neural information processing. In addition to striate cortex, the olfactory bulb also has oscillatory neural circuits which may be important in neural information processing (Freeman, 1988) . It has been suggested that temporal correlation may be used for pattern segmentation in associative memories (Wang, 1990), and correlations between multiple oscillators may be used for storing time intervals (Miall, 1989). We have described a circuit which performs Comparator Model phase-locking. The distributed and partitionable qualities of this circuit make it attractive as a possible physiological model. The PAM representation of object position shows one way that connectivity requirements can be minimized for communication in a neuromorphic VLSI Phase Locking Architectures for Feature Linking in Multiple Target Tracking Systems system. The chip has been fabricated using subthreshold CMOS technology, and thus uses little power. Acknowledgements The authors are pleased to acknowledge helpful discussion with C. Koch and J. Lazzaro. Chip fabrication was provided by the MOSIS service. References P. Baldi & R. Meir. (1990) Computing with arrays of coupled oscillators: an application to preattentive texture discrimination. Neural Computation 2, 458-47l. F . Crick & C. Koch. (1990) Towards a neurobiological theory of consciousness . Seminars in the Neurosciences 2, 263-275. R. Eckhorn, H. J. Reitboek, M. Arndt & P. Dicke. (1990) Feature linking via synchronization among distributed assemblies: simulations of results from cat visual cortex. Neural Computation 2, 293-307. W. J. Freeman, Y. Yao, & B. Burke. (1988) Central pattern generating and recognizing in olfactory bulb: a correlation learning rule. Neural Networks 1, 277-288 . C. M. Gray & W. Singer. (1987) Stimulus-specific neuronal oscillations in the cat visual cortex: A cortical functional unit. Soc. Neurosci. Abstr. 13(404.3) C. M. Gray, P. Konig, A. K. Engel & W. Singer. (1989) Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature (London) 338, 334-337. D. H. Hubel. (1988) Eye, Brain, and Vision. New York, NY: Scientific American Library. D. M. Kammen, C . Koch & P. J . Holmes. (1990) Collective oscillations in the visual cortex. In D. S. Touretzky (ed.) Advances in Neural Information Processing Systems 2. San Mateo, CA: Morgan Kaufman Publishers . C. A. Mead. (1989)Analog VLSI and Neural Systems. Reading, MA: AddisonWesley. M. A. Maher, S. P. Deweerth, M. A. Mahowald & C. A. Mead. (1989) Implementing neural architectures using analog VLSI circuits. IEEE Trans. Cire. Sys. 36, 643652. C. Miall. (1989) The storage of time intervals using oscillating neurons . Neural Computation 1, 359-37l. C. von der Malsburg. (1986) A neural cocktail-party processor. Biological Cybernetics. 54,29-40. D. Wang, J. Buhmann & C. von der Malsburg . (1990) Pattern segmentation in associative memory. Neural Computation 2, 95-106. 873 

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Kernel Feature Selection via Conditional Covariance Minimization Jianbo Chen? University of California, Berkeley jianbochen@berkeley.edu Martin J. Wainwright University of California, Berkeley wainwrig@berkeley.edu Mitchell Stern? University of California, Berkeley mitchell@berkeley.edu Michael I. Jordan University of California, Berkeley jordan@berkeley.edu Abstract We propose a method for feature selection that employs kernel-based measures of independence to find a subset of covariates that is maximally predictive of the response. Building on past work in kernel dimension reduction, we show how to perform feature selection via a constrained optimization problem involving the trace of the conditional covariance operator. We prove various consistency results for this procedure, and also demonstrate that our method compares favorably with other state-of-the-art algorithms on a variety of synthetic and real data sets. 1 Introduction Feature selection is an important issue in statistical machine learning, leading to both computational benefits (lower storage and faster computation) and statistical benefits, including increased model interpretability. With large data sets becoming ever more prevalent, feature selection has seen widespread usage across a variety of real-world tasks in recent years, including text classification, gene selection from microarray data, and face recognition [3, 14, 17]. In this work, we consider the supervised variant of feature selection, which entails finding a subset of the input features that explains the output well. This practice can reduce the computational expense of downstream learning by removing features that are redundant or noisy, while simultaneously providing insight into the data through the features that remain. Feature selection algorithms can generally be divided into two groups: those which are agnostic to the choice of learning algorithm, and those which attempt to find features that optimize the performance of a specific learning algorithm.1 Kernel methods have been successfully applied under each of these paradigms in recent work; for instance, see the papers [1, 8, 16, 19, 23, 25, 26, 29]. Kernel feature selection methods have the advantage of capturing nonlinear relationships between the features and the labels. Many previous approaches are filter methods based on the Hilbert-Schmidt Independence Criterion (HSIC), as proposed by Gretton et al. [13] as a measure of dependence. For instance, Song et al. [24, 25] proposed to optimize HSIC with greedy algorithms on features. Masaeli et al. [19] proposed Hilbert-Schmidt Feature Selection (HSFS), which optimizes HSIC with a continuous relaxation. In later work, Yamada et al. [29] proposed the HSIC-LASSO, in which the dual augmented Lagrangian can be used to find a global optimum. There are also wrapper methods ? Equal contribution. Feature selection algorithms that operate independently of the choice of predictor are referred to as filter methods. Algorithms tailored to specific predictors can be further divided into wrapper methods, which use learning algorithms to evaluate features based on their predictive power, and embedded methods, which combine feature selection and learning into a single problem [14]. 1 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. and embedded methods using kernels. Most of the methods add weights to features and optimize the original kernelized loss function together with a penalty on the weights [1, 5, 8, 11, 12, 27, 28]. For example, Cao et al. [5] proposed margin-based algorithms for SVMs to select features in the kernel space. Lastly, Allen [1] proposed an embedded method suitable for kernel SVMs and kernel ridge regression. In this paper, we propose to use the trace of the conditional covariance operator as a criterion for feature selection. We offer theoretical motivation for this choice and show that our method can be interpreted both as a filter method and as a wrapper method for a certain class of learning algorithms. We also show that the empirical estimate of the criterion is consistent as the sample size increases. Finally, we conclude with an empirical demonstration that our algorithm is comparable to or better than several other popular feature selection algorithms on both synthetic and real-world tasks. 2 Formulating feature selection Let X ? Rd be the domain of covariates X, and let Y be the domain of responses Y . Given n independent and identically distributed (i.i.d.) samples {(xi , yi ), i = 1, 2, . . . , n} generated from an unknown joint distribution PX,Y together with an integer m ? d, our goal is to select m of the d total features X1 , X2 , . . . , Xd which best predict Y . Let S be the full set of features, and let T ? S denote a subset of features. For ease of notation, we identify S = {X1 , . . . , Xd } with [d] = {1, . . . , d}, and also identify XT with T . We formulate the problem of supervised feature selection from two perspectives below. The first perspective motivates our algorithm as a filter method. The second perspective offers an interpretation as a wrapper method. 2.1 From a dependence perspective Viewing the problem from the perspective of dependence, we would ideally like to identify a subset of features T of size m such that the remaining features S \ T are conditionally independent of the responses given T . However, this may not be achievable when the number of allowable features m is small. We therefore quantify the extent of the remaining conditional dependence using some metric Q, and aim to minimize Q over all subsets T of the appropriate size. More formally, let Q : 2[d] ! [0, 1) be a function mapping subsets of [d] to the non-negative reals that satisfies the following properties: ? For a subset of features T , we have Q(T ) = 0 if and only if XS\T and Y are conditionally independent given XT . ? The function Q is non-decreasing, meaning that Q(T ) ? Q(S) whenever T ? S. Hence, the function Q achieves its maximum for the full feature set T = [d]. Given a fixed integer m, the problem of supervised feature selection can then be posed as min Q(T ). T :|T |=m (1) This formulation can be taken as a filter method for feature selection. 2.2 From a prediction perspective An alternative perspective aims at characterizing how well XT can predict Y directly within the context of a specific learning problem. Formally, we define the error of prediction as EF (X) = inf EX,Y L(Y, f (X)), (2) f 2F where F is a class of functions from X to Y, and L is a loss function specified by the user. For example, in a univariate regression problem, the function class F might be the set of all linear functions, and the loss function might be the squared error L(Y, f (X)) = (Y f (X))2 . We then hope to solve the following problem: min EF (XT ) = min T :|T |?m inf EX,Y L(Y, f (XT )), T :|T |?m f 2Fm m where Fm is a class of functions supported on R . That is, we aim to find the subset of m features that minimizes the prediction error. This formulation thus falls within the scope of wrapper methods for feature selection. 2 3 Conditional Covariance Operator The conditional covariance operator provides a measure of conditional dependence for random variables. It was first proposed by Baker [2], and was further studied and used for sufficient dimension reduction by Fukumizu et al. [9, 10]. We provide a brief overview of this operator and some of its key properties here. Let (HX , kX ) and (HY , kY ) denote reproducing kernel Hilbert spaces (RKHSs) of functions on spaces X and Y, respectively. Also let (X, Y ) be a random vector on X ? Y with joint distribution PX,Y . Assume the kernels kX and kY are bounded in expectation: EX [kX (X, X)] < 1 and EY [kY (Y, Y )] < 1. (3) The cross-covariance operator associated with the pair (X, Y ) is the mapping ?Y X : HX ! HY defined by the relations hg, ?Y X f iHY = EX,Y [(f (X) EX [f (X)])(g(Y ) EY [g(Y )])] for all f 2 HX and g 2 HY . (4) Baker [2] showed there exists a unique bounded operator VY X such that 1/2 1/2 (5a) ?Y X = ?Y Y VY X ?XX . The conditional covariance operator is then defined as ?Y Y |X = ?Y Y 1/2 1/2 (5b) ?Y Y VY X VXY ?Y Y . Among other results, Fukumizu et al. [9, 10] showed that the conditional covariance operator captures the conditional variance of X given Y . More precisely, if the sum HX + R is dense in L2 (PX ), where L2 (PX ) is the space of all square-integrable functions on X , then we have hg, ?Y Y |X giHY = EX [varY |X [g(Y )|X]] for any g 2 HY . (6) From Proposition 2 in the paper [10], we also know the residual error of g(Y ) with g 2 HY can be characterized by the conditional covariance operator. More formally, for any g 2 HY , we have hg, ?Y Y |X giHY = inf EX,Y ((g(Y ) EY [g(Y )]) f 2HX 4 (f (X) EX [f (X)]))2 . (7) Proposed method In this section, we describe our method for feature selection. 4.1 Criterion with its derivation Let (H1 , k1 ) denote an RKHS supported on X ? Rd . Let T ? [d] be a subset of features with cardinality m ? d, and for all x 2 Rd , take xT 2 Rd to be the vector with components xTi = xi if i 2 T or 0 otherwise. We define the kernel k1T by k1T (x, x ?) = k1 (xT , x ?T ) for all x, x ? 2 X . Suppose further that the kernel k1 is permutation-invariant. That is, for any x, x ? 2 X and permutation ?, denoting (x?(1) , . . . , x?(d) ) as x? , we have k1 (x, x ?) = k1 (x? , x ?? ). (Note that this property holds for many common kernels, including the linear, polynomial, Gaussian, and Laplacian kernels.) Then for every T of cardinality m, k1T generates the same RKHS supported on Rm . We call this ? 1, e RKHS (H k1 ). We will show the trace of the conditional covariance operator trace(?Y Y |X ) can be interpreted as a dependence measure, as long as the RKHS H1 is large enough. We say that an RKHS (H, k) is characteristic if the map P ! EP [k(X, ?)] 2 H is one-to-one. If k is bounded, this is equivalent to saying that H + R is dense in L2 (P ) for any probability measure P [10]. We have the following lemma, whose proof is given in the appendix: Lemma 1. If k1 is bounded and characteristic, then e k1 is also characteristic. Let (H2 , k2 ) denote an RKHS supported on Y. Based on the above lemma, we have the following theorem, which is a parallel version of Theorem 4 in [10]: 3 Theorem 2. If (H1 , k1 ) and (H2 , k2 ) are characteristic, we have ?Y Y |X holding if and only if Y ? ? X|XT . ?Y Y |XT with equality The proof is postponed to the appendix. With this generic result in place, we now narrow our focus to problems with univariate responses, including univariate regression, binary classification and multi-class classification. In the case of regression, we assume H2 is supported on R, and we take k2 to be the linear kernel: (8) k2 (y, y?) = y y? for all y, y? 2 R. For binary or multi-class classification, we take k2 to be the Kronecker delta function: ? 1 if y = y?, k2 (y, y?) = (y, y?) = (9) 0 otherwise. This can be equivalently interpreted as P a linear kernel k(y, y?) = hy, y?i assuming a one-hot encoding of Y , namely that Y = {y 2 {0, 1}k : i yi = 1} ? Rk , where k is the number of classes. P When Y is R or {y 2 {0, 1}k : i yi = 1} ? Rk , we obtain the following corollary of Theorem 2: P Corollary 3. If (H1 , k1 ) is characteristic, Y is R or {y 2 {0, 1}k : i yi = 1} ? Rk , and (H2 , k2 ) includes the identity function on Y, then we have Tr(?Y Y |X ) ? Tr(?Y Y |XT ) for any subset T of features. Moreover, the equality Tr(?Y Y |X ) = Tr(?Y Y |XT ) holds if and only if Y ? ? X|XT . Hence, in the univariate case, the problem of supervised feature selection reduces to minimizing the trace of the conditional covariance operator over subsets of features with controlled cardinality: (10) min Q(T ) := Tr(?Y Y |XT ). T :|T |=m In the regression setting, Equation (7) implies the residual error of regression can also be characterized by the trace of the conditional covariance operator when using the linear kernel on Y. More formally, we have the following observation: ? 1, e Corollary 4. Let ?Y Y |XT denote the conditional covariance operator of (XT , Y ) in (H k1 ). m Define the space of functions Fm from R to Y as ? 1 + R := {f + c : f 2 H ? 1 , c 2 R}. Fm = H Then we have (11) f (XT ))2 . Tr(?Y Y |XT ) = EFm (XT ) = inf EX,Y (Y f 2Fm (12) Given the fact that the trace of the conditional covariance operator can characterize the dependence and the prediction error in regression, we will use the empirical estimate of it as our objective. Given n samples {(x1 , y1 ), . . . , (xn , yn )}, the empirical estimate is given by [10]: ? (n) ? (n) trace(? Y Y |XT ) := trace[?Y Y ? (n) (? ? (n) ? Y XT XT XT + "n I) = "n trace[GY (GXT + n"n In ) 1 1 ? (n) ?X T Y ], ]. ? T (n) , ? ? T (n) and ? ? (n) are the covariance operators defined with respect to the empirical where ? YX XT X YY distribution and GXT and GY are the centralized kernel matrices, respectively. Concretely, we define GXT : = (In 1 n T )KXT (In 1 n T ) and GY : = (In 1 n T )KY (In 1 n T ). The (i, j)th entry of the kernel matrix KXT is k?1 (xiT , xjT ), with xiT denoting the ith sample with only features in T . As the kernel k2 on the space of responses is linear, we have KY = YYT , where Y is the n ? k matrix with each row being a sample response. Without loss of generality, we assume each column of Y is zero-mean, so that GY = KY = YYT . Our objective then becomes: trace[GY (GXT + n"n In ) 1 ] = trace[YYT (GXT + n"n In ) 4 1 ] = trace[YT (GXT + n"n In ) 1 Y]. (13) For simplicity, we only consider univariate regression and binary classification where k = 1, but our discussion carries over to the multi-class setting with minimal modification. The objective becomes ? (n) (T ) := yT (GX + n"n In ) min Q T 1 |T |=m (14) y, where y = (y1 , . . . , yn )T is an n-dimensional vector. We show the global optimal of the problem (14) is consistent. More formally, we have the following theorem: Theorem 5 (Feature Selection Consistency). Let the set A = argmin|T |?m Q(T ) be the set of all the ? (n) (T ) be a global optimal of (14). If "n ! 0 optimal solutions to (12) and T?(n) 2 argmin|T |?m Q and "n n ! 1 as n ! 1, we have P (T?(n) 2 A) ! 1. (15) Our proof is provided in the appendix. A comparable result is given in Fukumizu et al. [10] for the consistency of their dimension reduction estimator, but as our minimization takes place over a finite set, our proof is considerably simpler. 5 Optimization Finding a global optimum for (14) is NP-hard for generic kernels [28], and exhaustive search is computationally intractable if the number of features is large. We therefore approximate the problem of interest via continuous relaxation, as has previously been done in past work on feature selection [4, 27, 28]. 5.1 Initial relaxation We begin by introducing a binary vector w 2 {0, 1}d to indicate which features are active. This allows us to rephrase the optimization problem from (14) as min w yT (Gw X + n"n In ) 1 y (16) subject to wi 2 {0, 1}, i = 1, . . . , d, T w = m, where denotes the Hadamard product between two vectors and Gw matrix of Kw X with (Kw X )ij = k1 (w xi , w xj ). X is the centralized kernel We then approximate the problem (16) by relaxing the domain of w to the unit hypercube [0, 1]d and replacing the equality constraint with an inequality constraint: min w yT (Gw X + n"n In ) 1 y subject to 0 ? wi ? 1, i = 1, . . . , d, T (17) w ? m. This objective can be optimized using projected gradient descent, and represents our first tractable approximation. A solution to the relaxed problem is converted back into a solution for the original problem by setting the m largest values of w to 1 and remaining values to 0. We initialize w to the uniform vector (m/d)[1, 1, . . . , 1]T in order to avoid the corners of the constraint set during the early stages of optimization. 5.2 Computational issues The optimization problem can be approximated and manipulated in a number of ways so as to reduce the computational complexity, and we discuss a few here. Removing the inequality constrant. The hard constraint T w ? m requires a nontrivial projection step, such as the one detailed in Duchi et al. [6]. We can instead replace it with a soft constraint 5 and move it to the objective. Letting problem min w 0 be a hyperparameter, this gives rise to the modified 1 yT (Gw X + n"n In ) 1 y+ 1( T w m) subject to 0 ? wi ? 1, i = 1, . . . , d. (18) Removing the matrix inverse. The matrix inverse in the objective function is an expensive operation. In light of this, we first define an auxiliary variable ? 2 Rn , add the equality constraint ? = (Gw X +n"n In ) 1 y, and rewrite the objective as ?T y. We then note that we may multiply both sides of the constraint by the centered kernel matrix to obtain the relation (Gw X + n"n In )? = y. Letting 2 0 be a hyperparameter, we finally replace this relation by a soft `2 constraint to obtain min w,? yT ? + 2 k(Gw X + n"n In )? subject to 0 ? wi ? 1, i = 1, . . . , d, T yk22 (19) w ? m. Using a kernel approximation. Rahimi and Recht [22] propose a method for approximating kernel evaluations by inner products of random feature vectors, so that k(x, x ?) ? z(x)T z(? x) for a random T map z depending on the choice of kernel k. Let Kw ? Uw Uw be such a decomposition, where T Uw 2 Rn?D for some D < n. Then, defining Vw = (I /n)Uw , we similarly have that the centered kernel matrix can be written as Gw ? Vw VwT . By the Woodbury matrix identity, we may write 1 1 1 T (Gw X + n"n In ) 1 ? I Vw (ID + V Vw ) 1 VwT 2 2 "n n "n n "n n w (20) 1 T 1 T = (I Vw (Vw Vw + "n nID ) Vw ). "n n Substituting this into our objective function, scaling by ?n n, and removing the constant term yT y resulting from the identity matrix gives a new approximate optimization problem. This modification reduces the complexity of each optimization step from O(n2 d + n3 ) to O(n2 D + D3 + nDd). Choice of formulation. We remark that each of the three approximations beyond the initial relaxation may be independently used or omitted, allowing for a number of possible objectives and constraint sets. We explore some of these configurations in the experimental section below. 6 Experiments In this section, we evaluate our approach on both synthetic and real-world data sets. We compare with several strong existing algorithms, including recursive feature elimination (RFE) [15], Minimum Redundancy Maximum Relevance (mRMR) [21], BAHSIC [24, 25], and filter methods using mutual information (MI) and Pearson?s correlation (PC). We use the author?s implementation for BAHSIC2 and use Scikit-learn [20] and Scikit-feature [17] packages for the rest of the algorithms. 6.1 Synthetic data We begin with experiments on the following synthetic data sets: ? Binary classification (Friedman et al. [7]). Given Y = 1, (X1 , . . . , X10 ) ? N (0, I10 ). P4 Given Y = 1, X1 through X4 are standard normal conditioned on 9 ? j=1 Xj2 ? 16, and (X5 , . . . , X10 ) ? N (0, I6 ). ? 3-dimensional XOR as 4-way classification. Consider the 8 corners of the 3-dimensional hypercube (v1 , v2 , v3 ) 2 { 1, 1}3 , and group them by the tuples (v1 v3 , v2 v3 ), leaving 4 sets of vectors paired with their negations {v (i) , v (i) }. Given a class i, a point is generated from the mixture distribution (1/2)N (v (i) , 0.5I3 ) + (1/2)N ( v (i) , 0.5I3 ). Each example additionally has 7 standard normal noise features for a total of 10 dimensions. 2 http://www.cc.gatech.edu/~lsong/code.html 6 Figure 1: The above plots show the median rank (y-axis) of the true features as a function of sample size (x-axis) for the simulated data sets. Lower median ranks are better. The dotted line indicates the optimal median rank. ? Additive nonlinear regression: Y = 2 sin(2X1 ) + max(X2 , 0) + X3 + exp( X4 ) + ", where (X1 , . . . , X10 ) ? N (0, I10 ) and " ? N (0, 1). The first data set represents a standard nonlinear binary classification task. The second data set is a multi-class classification task where each feature is independent of Y by itself but a combination of three features has a joint effect on Y . The third data set arises from an additive model for nonlinear regression. Each data set has d = 10 dimensions in total, but only m = 3 or 4 true features. Since the identity of these features is known, we can evaluate the performance of a feature selection algorithm by computing the median rank it assigns to the real features, with lower median ranks indicating better performance. Given enough samples, we would expect this value to come close to the optimal lower bound of (m + 1)/2. Our experimental setup is as follows. We generate 10 independent copies of each data set with sample sizes ranging from 10 to 100, and record the median ranks assigned to the true features by each algorithm. This process is repeated a total of 100 times, and the results are averaged across trials. For kernel-based methods, we use a Gaussian kernel k(x, x ?) = exp( kx x ?k2 /(2 2 )) on T X and a linear kernelpk(y, y?) = y y? on Y . We take to be the median pairwise distance between samples scaled by 1/ 2. Since the number of true features is known, we provide this as an input to algorithms that require it. Our initial experiments use the basic version of our algorithm from Section 5.1. When the number of desired features m is fixed, only the regularization parameter " needs to be chosen. We use " = 0.001 for the classification tasks and " = 0.1 for the regression task, selecting these values from {0.001, 0.01, 0.1} using cross-validation. Our results are shown in Figure 1. On the binary and 4-way classification tasks, our method outperforms all other algorithms, succeeding in identifying the true features using fewer than 50 samples where others require close to 100 or even fail to converge. On the additive nonlinear model, several algorithms perform well, and our method is on par with the best of them across all sample sizes. These experiments show that our algorithm is comparable to or better than several widely-used feature selection techniques on a selection of synthetic tasks, and is adept at capturing several kinds of nonlinear relationships between the covariates and the responses. When compared in particular to its closest relative BAHSIC, a backward-elimination algorithm based on the Hilbert?Schmidt independence criterion, we see that our algorithm often produces higher quality results with fewer samples, and even succeeds in the non-additive problem where BAHSIC fails to converge. We also rerun these experiments separately for each of the first two approximations described in Section 5.2 above, selecting 1 from {0.001, 0.01, 0.1} and 2 from {1, 10, 100} using crossvalidation. We find that comparable results can be attained with either approximate objective, but note that the algorithm is more robust to changes in 1 than 2 . 7 Samples Features Classes ALLAML CLL-SUB-111 glass ORL orlraws10P pixraw10P TOX-171 vowel warpAR10P warpPIE10P wine Yale 72 111 214 400 100 100 171 990 130 210 178 165 7,129 11,340 10 1,024 10,304 10,000 5,784 10 2,400 2,420 13 1,024 2 3 6 40 10 10 4 11 10 10 3 15 Table 1: Summary of the benchmark data sets we use for our empirical evaluation. Figure 2: The above plots show classification accuracy (y-axis) versus number of selected features (x-axis) for our real-world benchmark data sets. Higher accuracies are better. 6.2 Real-world data In the previous section, we found that our method for feature selection excelled in identifying nonlinear relationships on a variety of synthetic data sets. We now turn our attention to a collection of real-word tasks, studying the performance of our method and other nonlinear approaches when used in conjunction with a kernel SVM for downstream classification. We carry out experiments on 12 standard benchmark tasks from the ASU feature selection website [17] and the UCI repository [18]. A summary of our data sets is provided in Table 1. The data sets are drawn from several domains including gene data, image data, and voice data, and span both the low-dimensional and high-dimensional regimes. For every task, we run each algorithm being evaluated to obtain ranks for all features. Performance is then measured by training a kernel SVM on the top m features and computing the resulting accuracy as measured by 5-fold cross-validation. This is done for m 2 {5, 10, . . . , 100} if the total number of features d is larger than 100, or m 2 {1, 2, . . . , d} otherwise. In all cases we fix the regularization constant of the SVM to C = 1 and use a Gaussian kernel with set as in the previous section over the selected features. For our own algorithm, we fix " = 0.001 across all experiments and set the number of desired features to m = 100 if d > 100 or dd/5e otherwise. Our results are shown in Figure 2. Compared with three other popular methods for nonlinear feature selection, i.e. mRMR, BAHSIC, and MI, we find that our method is the strongest performer in the large majority of cases, sometimes by a substantial margin as in the case of TOX-171. While our method is occasionally outperformed in the beginning when the number of selected features is small, it either ties or overtakes the leading method by the end in all but one instance. We remark that our method consistently improves upon the performance of the related BAHSIC method, suggesting that our objective based on conditional covariance may be more powerful than one based on the Hilbert-Schmidt independence criterion. 8 7 Conclusion In this paper, we proposed an approach to feature selection based on minimizing the trace of the conditional covariance operator. The idea is to select the features that maximally account for the dependence of the response on the covariates. We do so by relaxing from an intractable discrete formulation of the problem to a continuous approximation suitable for gradient-based optimization. We demonstrate the effectiveness of our approach on multiple synthetic and real-world experiments, finding that it often outperforms other state-of-the-art approaches, including another competitive kernel feature selection method based on the Hilbert-Schmidt independence criterion. References [1] Genevera I Allen. 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Convergence of Gradient EM on Multi-component Mixture of Gaussians Bowei Yan University of Texas at Austin boweiy@utexas.edu Mingzhang Yin University of Texas at Austin mzyin@utexas.edu Purnamrita Sarkar University of Texas at Austin purna.sarkar@austin.utexas.edu Abstract In this paper, we study convergence properties of the gradient variant of Expectation-Maximization algorithm [11] for Gaussian Mixture Models for arbitrary number of clusters and mixing coefficients. We derive the convergence rate depending on the mixing coefficients, minimum and maximum pairwise distances between the true centers, dimensionality and number of components; and obtain a near-optimal local contraction radius. While there have been some recent notable works that derive local convergence rates for EM in the two symmetric mixture of Gaussians, in the more general case, the derivations need structurally different and non-trivial arguments. We use recent tools from learning theory and empirical processes to achieve our theoretical results. 1 Introduction Proposed by [7] in 1977, the Expectation-Maximization (EM) algorithm is a powerful tool for statistical inference in latent variable models. A famous example is the parameter estimation problem under parametric mixture models. In such models, data is generated from a mixture of a known family of parametric distributions. The mixture component from which a datapoint is generated from can be thought of as a latent variable. Typically the marginal data log-likelihood (which integrates the latent variables out) is hard to optimize, and hence EM iteratively optimizes a lower bound of it and obtains a sequence of estimators. This consists of two steps. In the expectation step (E-step) one computes the expectation of the complete data likelihood with respect to the posterior distribution of the unobserved mixture memberships evaluated at the current parameter estimates. In the maximization step (M-step) one this expectation is maximized to obtain new estimators. EM always improves the objective function. While it is established in [4] that the true parameter vector is the global maximizer of the log-likelihood function, there has been much effort to understand the behavior of the local optima obtained via EM. When the exact M-step is burdensome, a popular variant of EM, named Gradient EM is widely used. The idea here is to take a gradient step towards the maxima of the expectation computed in the E-step. [11] introduces a gradient algorithm using one iteration of Newton?s method and shows the local properties of the gradient EM are almost identical with those of the EM. Early literature [22, 24] mostly focuses on the convergence to the stationary points or local optima. In [22] it is proven that the sequence of estimators in EM converges to stationary point when the lower bound function from E-step is continuous. In addition, some conditions are derived under 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. which EM converges to local maxima instead of saddle points; but these are typically hard to check. A link between EM and gradient methods is forged in [24] via a projection matrix and the local convergence rate of EM is obtained. In particular, it is shown that for GMM with well-separated centers, the EM achieves faster convergence rates comparable to a quasi-Newton algorithm. While the convergence of EM deteriorates under worse separations, it is observed in [20] that the mixture density determined by estimator sequence of EM reflects the sample data well. In recent years, there has been a renewed wave of interest in studying the behavior of EM especially in GMMs. The global convergence of EM for a mixture of two equal-proportion Gaussian distributions is fully characterized in [23]. For more than two clusters, a negative result on EM and gradient EM being trapped in local minima arbitrarily far away from the global optimum is shown in [9]. For high dimensional GMMs with M components, the parameters are learned via reducing the dimensionality via a random projection in [5]. In [6] the two-round method is proposed, where one first initializes with more than M points, then prune to get one point in every cluster. It is pointed out in this paper that in high dimensional space, when the clusters are well separated, the mixing weight will go to either 0 or 1 after one single update. It is showed in [25, 17] that one can cluster high dimensional sub-gaussian mixtures by semi-definite programming relaxations. For the convergence rate of EM algorithm, it is observed in [19] that a very small mixing proportion for one mixture component compared to others leads to slow convergence. [2] gives non-asymptotic convergence guarantees in isotropic, balanced, two-component GMM; their result proves the linear convergence of EM if the center is initialized in a small neighborhood of the true parameters. The local convergence result in this paper has a sub-optimal contraction region. K-means clustering is another widely used clustering method. Lloyd?s algorithm for k-means clustering has a similar flavor as EM. At each step, it recomputes the centroids of each cluster and updates the membership assignments alternatively. While EM does soft clustering at each step, Lloyd?s algorithm obtains hard clustering. The clustering error of Lloyd?s algorithm for arbitrary number of clusters is studied in [13]. The authors also show local convergence results where the contraction region is less restrictive than [2]. We would like to point out that there are many notable algorithms [10, 1, 21] with provable guarantees for estimating mixture models. In [14, 8] the authors propose polynomial time algorithms which achieve epsilon approximation to the k-means loss. A spectral algorithm for learning mixtures of gaussians is proposed in [21]. We want to point out that our aim is not to come up with a new algorithm for mixture models, but to understand the interplay of model parameters in the convergence of gradient EM for a mixture of Gaussians with M components. As we discuss later, our work also immediately leads to convergence guarantees of Stochastic Gradient EM. Another important difference is that the aim of these works is recovering the hidden mixture component memberships, whereas our goal is completely different: we are interested in understanding how well EM can estimate the mean parameters under a good initialization. In this paper, we study the convergence rate and local contraction radius of gradient EM under GMM with arbitrary number of clusters and mixing weights which are assumed to be known. For simplicity, we assume that the components share the same covariance matrix, which is known. Thus it suffices to carry out our analysis for isotropic GMMs with identity as the shared covariance matrix. We obtain a near-optimal condition on the contraction region in contrast to [2]?s contraction radius for the mixture of two equal weight Gaussians. We want to point out that, while the authors of [2] provide a general set of conditions to establish local convergence for a broad class of mixture models, the derivation of specific results and conditions on local convergence are tailored to the balance and symmetry of the model. We follow the same general route: first we obtain conditions for population gradient EM, where all sample averages are replaced by their expected counterpart. Then we translate the population version to the sample one. While the first part is conceptually similar, the general setting calls for more involved analysis. The second step typically makes use of concepts from empirical processes, by pairing up Ledoux-Talagrand contraction type arguments with well established symmetrization results. However, in our case, the function is not a contraction like in the symmetric two component case, since it involves the cluster estimates of all M components. Furthermore, the standard analysis of concentration inequalities by McDiarmid?s inequality gets complicated because the bounded difference condition is not satisfied in our setting. We overcome these difficulties by taking advan- 2 tage of recent tools in Rademacher averaging for vector valued function classes, and variants of McDiarmid type inequalities for functions which have bounded difference with high probability. The rest of the paper is organized as follows. In Section 2, we state the problem and the notations. In Section 3, we provide the main results in local convergence rate and region for both population and sample-based gradient EM in GMMs. Section 4 and 5 provide the proof sketches of population and sample-based theoretical results, followed by the numerical result in Section 6. We conclude the paper with some discussions. 2 Problem Setup and Notations Consider a GMM with M clusters in d dimensional space, with weights ? = (?P 1 , ? ? ? , ?M ). Let ?i ? Rd be the mean of cluster i. Without loss of generality, we assume EX = i ?i ?i = 0 and the known covariance matrix for all components is Id . Let ? ? RM d be the vector stacking the ?i s vertically. We represent the mixture as X ? GMM(?, ?, Id ), which has the density function PM p(x|?) = i=1 ?i ?(x|?i , Id ). where ?(x; P ?, ?) is the PDF of  N (?, ?). Then the population logM likelihood function as L(?) = EX log ? ?(X|? , I ) i d . The Maximum Likelihood Estimai=1 i ? ML = arg max p(X|?). EM algorithm P tor is then defined as ? is based on using an auxiliary function to lower bound the log likelihood. Define Q(?|?t ) = EX [ i p(Z = i|X; ?t ) log ?(X; ?i , Id )], where Z denote the unobserved component membership of data point X. The standard EM update is ?t+1 = arg max? Q(?|?t ). Define ?i ?(X|?i , Id ) wi (X; ?) = PM j=1 ?j ?(X|?j , Id ) The update step for gradient EM, defined via the gradient operator G(?t ) : RM d ? RM d , is   G(?t )(i) := ?t+1 = ?ti + s[?Q(?t |?t )]i = ?ti + sEX ?i wi (X; ?t )(X ? ?ti ) . i (1) (2) where s > 0 is the step size and (.)(i) denotes the part of the stacked vector corresponding to the ith mixture component. We will also use Gn (?) to denote the empirical counterpart of the population gradient operator G(?) defined in Eq (2). We assume we are given an initialization ?0i and the true mixing weight ?i for each component. 2.1 Notations Define Rmax and Rmin as the largest and smallest distance between cluster centers i.e., Rmax = maxi6=j k??i ? ??j k, Rmin = mini6=j k??i ? ??j k. Let ?max and ?min be the maximal and . Standard complexity analysis notation minimal cluster weights, and define ? as ? = ??max min ? o(?), O(?), ?(?), ?(?) will be used. f (n) = ?(g(n)) is short for ?(g(n)) ignoring logarithmic factors, equivalent to f (n) ? Cg(n) logk (g(n)), similar for others. We use ? to represent the kronecker product. 3 Main Results Despite being a non-convex problem, EM and gradient EM algorithms have been shown to exhibit good convergence behavior in practice, especially with good initializations. However, existing local convergence theory only applies for two-cluster equal-weight GMM. In this section, we present our main result in two parts. First we show the convergence rate and present a near-optimal radius for contraction region for population gradient EM. Then in the second part we connect the population version to finite sample results using concepts from empirical processes and learning theory. 3.1 Local contraction for population gradient EM Intuitively, when ?t equals the ground truth ?? , then the Q(?|?? ) function will be well-behaved. This function is a key ingredient in [2], where the curvature of the Q(?|?) function is shown to be close to the curvature of Q(?|?? ) when the ? is close to ?? . This is a local property that only requires the gradient to be stable at one point. 3 Definition 1 (Gradient Stability). The Gradient Stability (GS) condition, denoted by GS(?, a), is satisfied if there exists ? > 0, such that for ?ti ? B(??i , a) with some a > 0, for ?i ? [M ]. k?Q(?t |?? ) ? ?Q(?t |?t )k ? ?k?t ? ?? k The GS condition is used to prove contraction of the sequence of estimators produced by population gradient EM. However, for most latent variable models, it is typically challenging to verify the GS condition and obtain a tight bound on the parameter ?. We derive the GS condition under milder conditions (see Theorem 4 in Section 4), which bounds the deviation of the partial gradient evaluated at ?ti uniformly over all i ? [M ]. This immediately implies the global GS condition defined in Definition 1. Equipped with this result, we achieve a nearly optimal local convergence radius for general GMMs in Theorem 1. The proof of this theorem can be found in Appendix B.2. ? ? d0 ), Theorem 1 (Convergence for Population gradient EM). Let d0 := min{d, M }. If Rmin = ?( 0 0 ? with initialization ? satisfying, k?i ? ?i k ? a, ?i ? [M ], where s   2 ! Rmin p M ? a? ? d0 O log max , Rmax , d0 2 ?min then the Population EM converges: k?t ? ?? k ? ? t k?0 ? ?? k,  2 where ? = M 2 (2? + 4) (2Rmax + d0 ) exp ? ?max ? ?min + 2? <1 ?max + ?min 
2 ? ?a d0 /8 < ?min . ?= Rmin 2 Remark 1. The local contraction radius is largely improved compared to that in [2], which has Rmin /8 in the two equal sized symmetric GMM setting. It can be seen that in Theorem 1, a/Rmin goes to 12 as the signal to noise ratio goes to infinity. We will show in simulations that when initialized from some point that lies Rmin /2 away from the true center, gradient EM only converges to a stationary point which is not a global optimum. More discussion can be found in Section 6. 3.2 Finite sample bound for gradient EM In the finite sample setting, as long as the deviation of the sample gradient from the population gradient is uniformly bounded, the convergence in the population setting implies the convergence in finite sample scenario. Thus the key ingredient in the proof is to get this uniform bound over all (i) parameters in the contraction region A, i.e. bound sup??A kG(i) (?) ? Gn (?)k, where G and Gn are defined in Section 2. To prove the result, we expand the difference and define the following function for i ? [M ], where u is a unit vector on a d dimensional sphere S d?1 . This appears because we can write the Euclidean norm of any vector B, as kBk = supu?S d?1 hB, ui. n 1X w1 (Xi ; ?)hXi ? ?1 , ui ? Ew1 (X; ?)hX ? ?1 , ui. ??A n i=1 giu (X) = sup (3) We will drop the super and subscript and prove results for g1u without loss of generality. The outline of the proof is to show that g(X) is close to its expectation. This expectation can be further bounded via the Rademacher complexity of the corresponding function class (defined below in Eq (4)) by the tools like the symmetrization lemma [18]. Consider the following class of functions indexed by ? and some unit vector on d dimensional sphere u ? S d?1 : Fiu = {f i : X ? R|f i (X; ?, u) = wi (X; ?)hX ? ?i , ui} (4) We need to bound the M functions classes separately for each mixture. Given a finite n-sample (X1 , ? ? ? , Xn ), for each class, we define the Rademacher complexity as the expectation of empirical 4 Rademacher complexity. ? ? n X 1 ? n (Fiu ) = E ? sup i f i (Xj ; ?, u)? ; R ??A n j=1 ? n (Fiu ) Rn (Fiu ) = EX R where i ?s are the i.i.d. Rademacher random variables. For many function classes, the computation of the empirical Rademacher complexity can be hard. For complicated functions which are Lipschitz w.r.t functions from a simpler function class, one can use Ledoux-Talagrand type contraction results [12]. In order to use the Ledoux-Talagrand contraction, one needs a 1-Lipschitz function, which we do not have, because our function involves ?i , i ? [M ]. Also, the weight functions wi are not separable in terms of the ?i ?s. Therefore the classical contraction lemma does not apply. In our analysis, we need to introduce a vector-valued function, with each element involving only one ?i , and apply a recent result of vector-versioned contraction lemma [15]. With some careful analysis, we get the following. The details are deferred to Section 5. Proposition 1. Let Fiu as defined in Eq. (4) for ?i ? [M ], then for some universal constant c, ? cM 3/2 (1 + Rmax )3 d max{1, log(?)} u ? Rn (Fi ) ? n After getting the Rademacher complexity, one needs to use concentration results like McDiarmid?s inequality [16] to achieve the finite-sample bound. Unfortunately for the functions defined in Eq. (4), the martingale difference sequence does not have bounded differences. Hence it is difficult to apply McDiarmid?s inequality in its classical form. To resolve this, we instead use an extension of McDiarmid?s inequality which can accommodate sequences which have bounded differences with high probability [3]. Theorem 2 (Convergence for sample-based gradient EM). Let ? be the contraction parameter in Theorem 1, and ? ? ?1/2 ? unif (n) = O(max{n M 3 (1 + Rmax )3 d max{1, log(?)}, (1 + Rmax )d/ n}). (5) If unif (n) ? (1 ? ?)a, then sample-based gradient EM satisfies t 1 unif ? ? i ? ??i ? ? t ?0 ? ?? 2 +  (n); 1?? ?i ? [M ] with probability at least 1 ? n?cd , where c is a positive constant. Remark 2. When data is observed in a streaming fashion, the gradient update can be modified into a stochastic gradient update, where the gradient is evaluated based on a single observation or a small batch. By the GS condition proved in Theorem 1, combined with Theorem 6 in [2], we immediately extend the guarantees of gradient EM into the guarantees for the stochastic gradient EM. 3.3 Initialization Appropriate initialization for EM is the key to getting good estimation within fewer restarts in practice. There have been a number of interesting initialization algorithms for estimating mixture models. It is pointed out in [9] that in practice, initializing the centers by uniformly drawing from the data is often more reasonable than drawing from a fixed distribution. Under this initialization strategy, we can bound the number of initializations required to find a ?good? initialization that falls in the contraction region in Theorem 1. The exact theorem statement and a discussion of random initialization can be found in Appendix D. More sophisticated strategy includes, an approximate solution to k-means on a projected low-dimensional space used in [1] and [10]. While it would be interesting to study different initialization schemes, that is part of future work. 4 Local Convergence of Population Gradient EM In this section we present the proof sketch for Theorem 1. The complete proofs in this section are deferred to Appendix B. To start with, we calculate the closed-form characterization of the gradient of q(?) as stated in the following lemma. 5 Lemma 1. Define q(?) = Q(?|?? ). The gradient of q(?) is ?q(?) = (diag(?) ? Id ) (?? ? ?). If we know the parameter ? in the gradient stability condition, then the convergence rate depends only on the condition number of the Hessian of q(?) and ?. Theorem 3 (Convergence rate for population gradient EM). If Q satisfies the GS condition with 2 , we have: parameter 0 < ? < ?min , denote dt := k?t ? ?? k, then with step size s = ?min +? max  t ?max ? ?min + 2? dt+1 ? d0 ?max + ?min The proof uses an approximation on gradient and standard techniques in analysis of gradient descent. Remark 3. It can be verified that the convergence rate is equivalent to that shown in [2] when applied to GMMs. The convergence slows down as the proportion imbalance ? = ?max /?min increases, which matches the observation in [19]. Now to verify the GS condition, we have the following theorem. p ? min{d, M }), and ?i ? Theorem 4 (GS condition for general GMM). If Rmin = p ?( p ? ? min{d, M } max(4 2[log(Rmin /4)]+ , 8 3), then B(??i , a), ?i ? [M ] where a ? Rmin 2 ? PM t ? ?? k?t ? ?? k, k??i Q(?|?t ) ? ??i q(?)k ? M i=1 k?i ? ?i k ?  M 
2 p 2 where ? = M 2 (2? + 4) (2Rmax + min{d, M }) exp ? Rmin min{d, M }/8 . 2 ?a Furthermore, k?Q(?|?t ) ? ?q(?)k ? ?k?t ? ?? k. Proof sketch of Theorem 4. W.l.o.g. we show the proof with the first cluster, consider the difference of the gradient corresponding to ?1 . ??1 Q(?t |?t ) ? ??1 q(?t ) =E(w1 (X; ?t ) ? w1 (X; ?? ))(X ? ?t1 ) For any given X, consider the function ? ? w1 (X; ?), we have ? ? w1 (X; ?)(1 ? w1 (X; ?))(X ? ?1 )T ? ?w1 (X; ?)w2 (X; ?)(X ? ?2 )T ? ? ? ?? w1 (X; ?) = ? ? .. ? ? . (6) (7) ?w1 (X; ?)wM (X; ?)(X ? ?M )T ? t ? M ? Let ?u = ?? + u(?t ? ?? ), ?u ? [0, 1], obviously ?u ? ?M i=1 B(?i , k?i ? ?i k) ? ?i=1 B(?i , a). By Taylor?s theorem, Z 1  u t t ? t E ? w (X; ? )du(X ? ? ) kE(w1 (X; ?1 ) ? w1 (X; ?1 ))(X ? ?1 )k = u 1 1 u=0 (8) X X ?U1 k?t1 ? ??1 k2 + Ui k?ti ? ??i k2 ? max {Ui } k?ti ? ??i k2 i6=1 i?[M ] i where U1 = sup kEw1 (X; ?u )(1 ? w1 (X; ?u ))(X ? ?t1 )(X ? ?u1 )T kop u?[0,1] Ui = sup kEw1 (X; ?u )wi (X; ?u )(X ? ?t1 )(X ? ?u2 )T kop u?[0,1] Bounding them with careful analysis on Gaussian distribution yields the result. The technical details are deferred to Appendix B. 5 Sample-based Convergence In this section we present the proof sketch for sample-based convergence of gradient EM. The main ingredient in proof of Theorem 2 is the result of the following theorem, which develops an uniform upper bound for the differences between sample-based gradient and population gradient on each cluster center. 6 Theorem 5 (Uniform bound for sample-based gradient EM). Denote A as the contraction region ? ?M i=1 B(?i , a). Under the condition of Theorem 1, with probability at least 1 ? exp (?cd log n), ?i ? [M ] sup G(i) (?) ? Gn(i) (?) < unif (n); ??A where unif(n) is as defined in Eq. (5). Remark 4. It is worth pointing out that, the first part of the bound on unif(n) in Eq. (5) comes from the Rademacher complexity, which is optimal in terms of the order of n and d. However the extra ? factor of d and log(n) comes from the altered McDiarmid?s inequality, tightening which will be left for future work. (i) Proof sketch of Theorem 5. Denote Zi = sup??A G(i) (?) ? Gn (?) . Recall giu (X) defined in Eq. (3), then it is not hard to see that Zi = supu?S d?1 giu (X). Consider a 21 -covering {u(1) , ? ? ? , u(K) } of the unit sphere S d?1 , where K is the covering number of an unit sphere in (j) d dimensions. We can show that Zi ? 2 maxj=1,??? ,K giu (X). As we state below in Lemma 2, we have for each u, with probability at least 1 ? exp (?cd log n), ? u ? giu (X) = O(max{R n (Fi ), (1 + Rmax )d/ n}). Plugging in the Rademacher complexity from Proposition 1, standard bounds on K, and applying union bound, we have ? ? ?1/2 ? Zi ? O(max{n M 3 (1 + Rmax )3 d max{1, log(?)}, (1 + Rmax )d/ n}) with probability at least 1 ? exp (2d ? cd log n) = 1 ? exp (?c0 d log n). Iteratively applying Theorem 5, we can bound the error in ? for the sample updates. The full proof is deferred to Appendix C. The key step is the following lemma, which bounds the giu (X) for any given u ? S d?1 . Without loss of generality we prove the result for i = 1. Lemma 2. Let u be a unit vector. Xi , i = 1, ? ? ? , n are i.i.d. samples from GMM(?, ?? , Id ). g1u (X) as defined in Eq. (3). Then with probability 1 ? exp(?cd log n) for some constant c > 0, ? u ? g1u (X) = O(max{R n (F1 ), (1 + Rmax )d/ n}). The quantity g1u (X) depends on the sample, the idea for proving Lemma 2 is to show it concentrates around its expectation when sample size is large. Note that when the function class has bounded differences (changing one data point changes the function by a bounded amount almost surely), as in the case in many risk minimization problems in supervised learning, the McDiarmid?s inequality can be used to achieve concentration. However the function class we define in Eq. (4) is not bounded almost everywhere, but with high probability, hence the classical result does not apply. Conditioning on the event where the difference is bounded, we use an extension of McDiarmid?s inequality by [3]. For convenience, we restate a weaker version of the theorem using our notation below. Theorem 6 ([3]). Consider independent random variable X = (X1 , ? ? ? , Xn ) in the product probQn ability space X = i=1 Xi , where Xi is the probability space for Xi . Also consider a function g : X ? R. If there exists a subset Y ? X , and a scalar c > 0, such that |g(x) ? g(y)| ? L, ?x, y ? Y, xj = yj , ?j 6= i.   2(?npL)2 Denote p = 1 ? P (X ? Y), then P (g(X) ? E[g(X)|X ? Y] ? ) ? p + exp ? nL2 + . It is worth pointing out that in Theorem 6, the concentration is shown in reference to the conditional expectation E[g(X)|X ? Y] when the data points are in the bounded difference set. So to fully achieve the type of bound given by McDiarmid?s inequality, we need to further bound the difference of the conditional expectation and the full expectation. Combining the two parts we will be able to show that, the empirical difference is upper bounded using the Rademacher complexity. Now it remains to derive the Rademacher complexity under the given function class. Note that when the function class is a contraction, or Lipschitz with respect to another function (usually of a simpler form), one can use the Ledoux-Talagrand contraction lemma [12] to reduce the Rademacher complexity of the original function class to the Rademacher complexity of the simpler function class. This is essential in getting the Rademacher complexities for complicated function classes. As 7 we mention in Section 3, our function class in Eq. (4) is unfortunately not Lipschitz due to the fact that it involves all cluster centers even for the gradient on one cluster. We get around this problem by introducing a vector valued function, and show that the functions in Eq. (4) are Lipschitz in terms of the vector-valued function. In other words, the absolute difference in the function when the parameter changes is upper bounded by the norm of the vector difference of the vector-valued function. Then we build upon the recent vector-contraction result from [15], and prove the following lemma under our setting. Lemma 3. Let X be nontrivial, symmetric and sub-gaussian. Then there exists a constant C < ?, depending only on the distribution of X, such that for any subset S of a separable Banach space and function hi : S ? R, fi : S ? Rk , i ? [n] satisfying ?s, s0 ? S, |hi (s)?hi (s0 )| ? Lkf (s)?f (s0 )k. If ik is an independent doubly indexed Rademacher sequence, we have, X X ? E sup i hi (s) ? E 2L sup ik fi (s)k , s?S s?S i i,k where fi (s)k is the k-th component of fi (s). Remark 5. In contrast to the original form in [15], we have a S as a subset of a separable Banach Space. The proof uses standard tools from measure theory, and is to be found in Appendix C. This equips us to prove Proposition 1. Proof sketch of Proposition 1. For any unit vector u, the Rademacher complexity of F1u is n Rn (F1u ) 1X i w1 (Xi ; ?)hXi ? ?1 , ui =EX E sup ??A n i=1 n n 1X 1X i w1 (Xi ; ?)hXi , ui + EX E sup i w1 (Xi ; ?)h?1 , ui ??A n i=1 ??A n i=1 {z } | {z } ? EX E sup | (D) (E) We bound the two terms separately. Define ?j (?) : R the k-th coordinate 2 [?j (?)]k = (9) Md ?R 2 M to be a vector valued function with k?1 k k?k k ? + hXj , ?k ? ?1 i + log 2 2  ?k ?1  ? It can be shown that 0 |w1 (Xj ; ?) ? w1 (Xj ; ? )| ? M k?j (?) ? ?j (?0 )k 4 (10) Now let ?1 (Xj ; ?) = w1 (Xj ; ?)hXj , ui. With Lipschitz property (10) and Lemma C.1, we have ? ? ? ? ? ? n n X M X X 2 M 1 E ? sup j wi (Xj ; ?)hXj , ui? ? E ? sup jk [?j (?)]k ? 4n ??A j=1 ??A n j=1 k=1 The right hand side can be bounded with tools regarding independent sum of sub-gaussian random variables. Similar techniques apply to the (E) term. Adding things up we get the final bound. 6 Experiments In this section we collect some numerical results. In all experiments we set the covariance matrix for each mixture component as identity matrix Id and define signal-to-noise ratio (SNR) as Rmin . Convergence Rate We first evaluate the convergence rate and compare with those given in Theorem 3 and Theorem 4. For this set of experiments, we use a mixture of 3 Gaussians in 2 dimensions. In both experiments Rmax /Rmin = 1.5. In different settings of ?, we apply gradient EM with varying SNR from 1 to 5. For each choice of SNR, we perform 10 independent trials with N = 12, 000 8 ? and the standard deviation are plotted versus iterations. In data points. The average of log k?t ? ?k Figure 1 (a) and (b) we plot balanced ? (? = 1) and unbalanced ? (? > 1) respectively. All settings indicate the linear convergence rate as shown in Theorem 3. As SNR grows, the parameter ? in GS condition decreases and thus yields faster convergence rate. Comparing left two panels in Figure 1, increasing imbalance of cluster weights ? slows down the local convergence rate as shown in Theorem 3. (a) (b) (c) (d) Figure 1: (a, b): The influence of SNR on optimization error in different settings. The figures represent the influence of SNR when the GMMs have different cluster centers and weights: (a) ? = (1/3, 1/3, 1/3). (b) ? = (0.6, 0.3, 0.1). (c) plots statistical error with different initializations arbitrarily close to the boundary of the contraction region. (d) shows the suboptimal stationary point when two centers are initialized from the midpoint of the respective true cluster centers. Contraction Region To show the tightness of the contraction region, we generate a mixture with ?? +?? ?? +?? M = 3, d = 2, and initialize the clusters as follows. We use ?02 = 2 2 3 ? , ?03 = 2 2 3 + , for shrinking , i.e. increasing a/Rmin and plot the error on the Y axis. Figure 1-(c) shows that gradient EM converges when initialized arbitrarily close to the boundary, thus confirming our near optimal contraction region. Figure 1-(d) shows that when  = 0, i.e. a = Rmin 2 , gradient EM can be trapped at a sub-optimal stationary point. 7 Concluding Remarks In this paper, we obtain local convergence rates and a near optimal contraction radius for population and sample-based gradient EM for multi-component GMMs with arbitrary mixing weights. For simplicity, we assume that the the mixing weights are known, and the covariance matrices are the same across components and known. For our sample-based analysis, we face new challenges where bears structural differences from the two-component, equal-weight setting, which are alleviated via the usage of non-traditional tools like a vector valued contraction argument and martingale concentrations bounds where bounded differences hold only with high probability. Acknowledgments PS was partially supported by NSF grant DMS 1713082. References [1] Pranjal Awasthi and Or Sheffet. Improved spectral-norm bounds for clustering. In APPROXRANDOM, pages 37?49. Springer, 2012. [2] Sivaraman Balakrishnan, Martin J. Wainwright, and Bin Yu. Statistical guarantees for the em algorithm: From population to sample-based analysis. Ann. Statist., 45(1):77?120, 02 2017. [3] Richard Combes. An extension of mcdiarmid?s inequality. arXiv preprint arXiv:1511.05240, 2015. [4] Denis Conniffe. Expected maximum log likelihood estimation. The Statistician, pages 317? 329, 1987. 9 [5] Sanjoy Dasgupta. Learning mixtures of gaussians. In Foundations of Computer Science, 1999. 40th Annual Symposium on, pages 634?644. IEEE, 1999. [6] Sanjoy Dasgupta and Leonard J Schulman. A two-round variant of em for gaussian mixtures. In Proceedings of the Sixteenth conference on Uncertainty in artificial intelligence, pages 152? 159. Morgan Kaufmann Publishers Inc., 2000. [7] Arthur P Dempster, Nan M Laird, and Donald B Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the royal statistical society. Series B (methodological), pages 1?38, 1977. [8] Zachary Friggstad, Mohsen Rezapour, and Mohammad R Salavatipour. Local search yields a ptas for k-means in doubling metrics. In Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium on, pages 365?374. IEEE, 2016. [9] Chi Jin, Yuchen Zhang, Sivaraman Balakrishnan, Martin J Wainwright, and Michael I Jordan. Local maxima in the likelihood of gaussian mixture models: Structural results and algorithmic consequences. In Advances in Neural Information Processing Systems, pages 4116?4124, 2016. [10] Amit Kumar and Ravindran Kannan. Clustering with spectral norm and the k-means algorithm. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, pages 299?308. IEEE, 2010. [11] Kenneth Lange. A gradient algorithm locally equivalent to the em algorithm. Journal of the Royal Statistical Society. Series B (Methodological), pages 425?437, 1995. [12] Michel Ledoux and Michel Talagrand. Probability in Banach Spaces: isoperimetry and processes. Springer Science & Business Media, 2013. [13] Yu Lu and Harrison H Zhou. Statistical and computational guarantees of lloyd?s algorithm and its variants. arXiv preprint arXiv:1612.02099, 2016. [14] Jir? Matou?sek. On approximate geometric k-clustering. Discrete & Computational Geometry, 24(1):61?84, 2000. [15] Andreas Maurer. A vector-contraction inequality for rademacher complexities. In International Conference on Algorithmic Learning Theory, pages 3?17. Springer, 2016. [16] Colin McDiarmid. On the method of bounded differences. 141(1):148?188, 1989. Surveys in combinatorics, [17] Dustin G Mixon, Soledad Villar, and Rachel Ward. Clustering subgaussian mixtures by semidefinite programming. arXiv preprint arXiv:1602.06612, 2016. [18] Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar. Foundations of machine learning. MIT press, 2012. [19] Iftekhar Naim and Daniel Gildea. Convergence of the em algorithm for gaussian mixtures with unbalanced mixing coefficients. arXiv preprint arXiv:1206.6427, 2012. [20] Richard A Redner and Homer F Walker. Mixture densities, maximum likelihood and the em algorithm. SIAM review, 26(2):195?239, 1984. [21] Santosh Vempala and Grant Wang. A spectral algorithm for learning mixture models. Journal of Computer and System Sciences, 68(4):841?860, 2004. [22] CF Jeff Wu. On the convergence properties of the em algorithm. The Annals of statistics, pages 95?103, 1983. [23] Ji Xu, Daniel J Hsu, and Arian Maleki. Global analysis of expectation maximization for mixtures of two gaussians. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, editors, Advances in Neural Information Processing Systems 29, pages 2676?2684. Curran Associates, Inc., 2016. 10 [24] Lei Xu and Michael I Jordan. On convergence properties of the em algorithm for gaussian mixtures. Neural computation, 8(1):129?151, 1996. [25] Bowei Yan and Purnamrita Sarkar. On robustness of kernel clustering. In Advances in Neural Information Processing Systems, pages 3090?3098, 2016. 11 

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Real Time Image Saliency for Black Box Classifiers Piotr Dabkowski pd437@cam.ac.uk University of Cambridge Yarin Gal yarin.gal@eng.cam.ac.uk University of Cambridge and Alan Turing Institute, London Abstract In this work we develop a fast saliency detection method that can be applied to any differentiable image classifier. We train a masking model to manipulate the scores of the classifier by masking salient parts of the input image. Our model generalises well to unseen images and requires a single forward pass to perform saliency detection, therefore suitable for use in real-time systems. We test our approach on CIFAR-10 and ImageNet datasets and show that the produced saliency maps are easily interpretable, sharp, and free of artifacts. We suggest a new metric for saliency and test our method on the ImageNet object localisation task. We achieve results outperforming other weakly supervised methods. 1 Introduction Current state of the art image classifiers rival human performance on image classification tasks, but often exhibit unexpected and unintuitive behaviour [6, 13]. For example, we can apply a small perturbation to the input image, unnoticeable to the human eye, to fool a classifier completely [13]. Another example of an unexpected behaviour is when a classifier fails to understand a given class despite having high accuracy. For example, if ?polar bear? is the only class in the dataset that contains snow, a classifier may be able to get a 100% accuracy on this class by simply detecting the presence of snow and ignoring the bear completely [6]. Therefore, even with perfect accuracy, we cannot be sure whether our model actually detects polar bears or just snow. One way to decouple the two would be to find snow-only or polar-bear-only images and evaluate the model?s performance on these images separately. An alternative is to use an image of a polar bear with snow from the dataset and apply a saliency detection method to test what the classifier is really looking at [6, 11]. Saliency detection methods show which parts of a given image are the most relevant to the model for a particular input class. Such saliency maps can be obtained for example by finding the smallest region whose removal causes the classification score to drop significantly. This is because we expect the removal of a patch which is not useful for the model not to affect the classification score much. Finding such a salient region can be done iteratively, but this usually requires hundreds of iterations and is therefore a time-consuming process. In this paper we lay the groundwork for a new class of fast and accurate model-based saliency detectors, giving high pixel accuracy and sharp saliency maps (an example is given in figure 1). We propose a fast, model agnostic, saliency detection method. Instead of iteratively obtaining saliency maps for each input image separately, we train a model to predict such a map for any input image in a single feed-forward pass. We show that this approach is not only orders-of-magnitude faster than iterative methods, but it also produces higher quality saliency masks and achieves better localisation results. We assess this with standard saliency benchmarks and introduce a new saliency measure. Our proposed model is able to produce real-time saliency maps, enabling new applications such as video-saliency which we comment on in our Future Research section. 2 Related work Since the rise of CNNs in 2012 [5] numerous methods of image saliency detection have been proposed. One of the earliest such methods is a gradient-based approach introduced in [11] which computes the gradient of the class with respect to the image and assumes that salient regions are at locations 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. (a) Input Image (b) Generated saliency map (c) Image multiplied by the mask (d) Image multiplied by inverted mask Figure 1: An example of explanations produced by our model. The top row shows the explanation for the "Egyptian cat" while the bottom row shows the explanation for the "Beagle". Note that produced explanations can precisely both highlight and remove the selected object from the image. with high gradient magnitude. Other similar backpropagation-based approaches have been proposed, for example Guided Backpropagation [12] or Excitation Backprop [16]. While the gradient-based methods are fast enough to be applied in real-time, they produce explanations of limited quality [16] and they are hard to improve and build upon. Zhou et al. [17] proposed an approach that iteratively removes patches of the input image (by setting them to the mean colour) such that the class score is preserved. After a sufficient number of iterations, we are left with salient parts of the original image. The maps produced by this method are easily interpretable, but unfortunately, the iterative process is very time consuming and not acceptable for real-time saliency detection. In another work, Cao et al. [1] introduced an optimisation method that aims to preserve only a fraction of network activations such that the class score is maximised. Again, after the iterative optimisation process, only activations that are relevant remain and their spatial location in the CNN feature map indicate salient image regions. Very recently (and in parallel to this work), another optimisation based method was proposed [2]. Similarly to Cao et al. [1], Fong and Vedaldi [2] also propose to use gradient descent to optimise for the salient region, but the optimisation is done only in the image space and the classifier model is treated as a black box. Essentially Fong and Vedaldi [2]?s method tries to remove as little from the image as possible, and at the same time to reduce the class score as much as possible. A removed region is then a minimally salient part of the image. This approach is model agnostic and the produced maps are easily interpretable because the optimisation is done in the image space and the model is treated as a black box. We next argue what conditions a good saliency model should satisfy, and propose a new metric for saliency. 3 Image Saliency and Introduced Evidence Image saliency is relatively hard to define and there is no single obvious metric that could measure the quality of the produced map. In simple terms, the saliency map is defined as a summarised explanation of where the classifier ?looks? to make its prediction. There are two slightly more formal definitions of saliency that we can use: ? Smallest sufficient region (SSR) ? smallest region of the image that alone allows a confident classification, 2 ? Smallest destroying region (SDR) ? smallest region of the image that when removed, prevents a confident classification. Similar concepts were suggested in [2]. An example of SSR and SDR is shown in figure 2. It can be seen that SSR is very small and has only one seal visible. Given this SSR, even a human would find it difficult to recognise the preserved image. Nevertheless, it contains some characteristic for ?seal? features such as parts of the face with whiskers, and the classifier is over 90% confident that this image should be labeled as a ?seal?. On the other hand, SDR has a much stronger and larger region and quite successfully removes all the evidence for seals from the image. In order to be as informative as possible, we would like to find a region that performs well as both SSR and SDR. Figure 2: From left to right: the input image; smallest sufficient region (SSR); smallest destroying region (SDR). Regions were found using the mask optimisation procedure from [2]. Both SDR and SSR remove some evidence from the image. There are few ways of removing evidence, for example by blurring the evidence, setting it to a constant colour, adding noise, or by completely cropping out the unwanted parts. Unfortunately, each one of these methods introduces new evidence that can be used by the classifier as a side effect. For example, if we remove a part of the image by setting it to the constant colour green then we may also unintentionally provide evidence for ?grass? which in turn may increase the probability of classes appearing often with grass (such as ?giraffe?). We discuss this problem and ways of minimising introduced evidence next. 3.1 Fighting the Introduced Evidence As mentioned in the previous section, by manipulating the image we always introduce some extra evidence. Here, let us focus on the case of applying a mask M to the image X to obtain the edited image E. In the simplest case we can simply multiply X and M element-wise: E=X M (1) This operation sets certain regions of the image to a constant ?0? colour. While setting a larger patch of the image to ?0? may sound rather harmless (perhaps following the assumption that the mean of all colors carries very little evidence), we may encounter problems when the mask M is not smooth. The mask M , in the worst case, can be used to introduce a large amount of additional evidence by generating adversarial artifacts (a similar observation was made in [2]). An example of such a mask is presented in figure 3. Adversarial artifacts generated by the mask are very small in magnitude and almost imperceivable for humans, but they are able to completely destroy the original prediction of the classifier. Such adversarial masks provide very poor saliency explanations and therefore should be avoided. Figure 3: The adversarial mask introduces very small perturbations, but can completely alter the classifier?s predictions. From left to right: an image which is correctly recognised by the classifier with a high confidence as a "tabby cat"; a generated adversarial mask; an original image after application of the mask that is no longer recognised as a "tabby cat". 3 There are a few ways to make the introduction of artifacts harder. For example, we may change the way we apply a mask to reduce the amount of unwanted evidence due to specifically-crafted masks: E = X M + A (1 M ) (2) where A is an alternative image. A can be chosen to be for example a highly blurred version of X. In such case mask M simply selectively adds blur to the image X and therefore it is much harder to generate high-frequency-high-evidence artifacts. Unfortunately, applying blur does not eliminate existing evidence very well, especially in the case of images with low spatial frequencies like a seashore or mountains. Another reasonable choice of A is a random constant colour combined with high-frequency noise. This makes the resulting image E more unpredictable at regions where M is low and therefore it is slightly harder to produce a reliable artifact. Even with all these measures, adversarial artifacts may still occur and therefore it is necessary to encourage smoothness of the mask M for example via a total variation (TV) penalty. We can also directly resize smaller masks to the required size as resizing can be seen as a smoothness mechanism. 3.2 A New Saliency Metric A standard metric to evaluate the quality of saliency maps is the localisation accuracy of the saliency map. However, it should be noted that saliency is not equivalent to localisation. For example, in order to recognise a dog we usually just need to see its head; legs and body are mostly irrelevant for the recognition process. Therefore, saliency map for a dog will usually only include its head while the localisation box always includes a whole dog with not-salient details like legs and tail. The saliency of the object highly overlaps with its localisation and therefore localisation accuracy still serves as a useful metric, but in order to better assess the quality and interpretability of the produced saliency maps, we introduce a new, highly tuned metric. According to the SSR objective, we require that the classifier is able to still recognise the object from the produced saliency map and that the preserved region is as small as possible. In order to make sure that the preserved region is free from adversarial artifacts, instead of masking we can crop the image. We propose to find the tightest rectangular crop that contains the entire salient region and to feed that rectangular region to the classifier to directly verify whether it is able to recognise the requested class. We define our saliency metric simply as: s(a, p) = log(? a) log(p) (3) with a ? = max(a, 0.05). Here a is the area of the rectangular crop as a fraction of the total image size and p is the probability of the requested class returned by the classifier based on the cropped region. The metric is almost a direct translation of the SSR. We threshold the area at 0.05 in order to prevent instabilities at low area fractions. Good saliency detectors will be able to significantly reduce the crop size without reducing the classification probability, and therefore a low value for the saliency metric is a characteristic of good saliency detectors. Interpreting this metric following information theory, this measure can be seen as the relative amount of information between an indicator variable with probability p and an indicator variable with probability a ? or the concentration of information in the cropped region. Because most image classifiers accept only images of a fixed size and the crop can have an arbitrary size, we resize the crop to the required size disregarding aspect ratio. This seems to work well in practice, but it should be noted that the proposed saliency metric works best with classifiers that are largely invariant to the scale and aspect ratio of the object. 3.3 The Saliency Objective Taking the previous conditions into consideration, we want to find a mask M that is smooth and performs well at both SSR and SDR; examples of such masks can be seen in figure 1. Therefore, more formally, given class c of interest, and an input image X, to find a saliency map M for class c, our objective function L is given by: L(M ) = 1 TV(M ) + 2 AV(M ) log(fc ( (X, M ))) + 3 fc ( (X, 1 M )) 4 (4) where fc is a softmax probability of the class c of the black box image classifier and TV(M ) is the total variation of the mask defined simply as: X X TV(M ) = (Mij Mij+1 )2 + (Mij Mi+1j )2 , (5) i,j i,j 4 AV(M ) is the average of the mask elements, taking value between 0 and 1, and i are regularisers. Finally, the function removes the evidence from the image as introduced in the previous section: (X, M ) = X M +A (1 M ). (6) In total, the objective function is composed of 4 terms. The first term enforces mask smoothness, the second term encourages that the region is small. The third term makes sure that the classifier is able to recognise the selected class from the preserved region. Finally, the last term ensures that the probability of the selected class, after the salient region is removed, is low (note that the inverted mask 1 M is applied). Setting 4 to a value smaller than 1 (e.g. 0.2) helps reduce this probability to very small values. 4 Masking Model The mask can be found iteratively for a given image-class pair by directly optimising the objective function from equation 4. In fact, this is the method used by [2] which was developed in parallel to this work, with the only difference that [2] only optimises the mask iteratively and for SDR (so they don?t include the third term of our objective function). Unfortunately, iteratively finding the mask is not only very slow, as normally more than 100 iterations are required, but it also causes the mask to greatly overfit to the image and a large TV penalty is needed to prevent adversarial artifacts from forming. Therefore, the produced masks are blurry, imprecise, and overfit to the specific image rather than capturing the general behaviour of the classifier (see figure 2). For the above reasons, we develop a trainable masking model that can produce the desired masks in a single forward pass without direct access to the image classifier after training. The masking model receives an image and a class selector as inputs and learns to produce masks that minimise our objective function (equation 4). In order to succeed at this task, the model must learn which parts of the input image are considered salient by the black box classifier. In theory, the model can still learn to develop adversarial masks that perform well on the objective function, but in practice it is not an easy task, because the model itself acts as some sort of a ?regulariser? determining which patterns are more likely and which are less. Figure 4: Architecture diagram of the masking model. In order to make our masks sharp and precise, we adopt a U-Net architecture [8] so that the masking model can use feature maps from multiple resolutions. The architecture diagram can be seen in figure 4. For the encoder part of the U-Net we use ResNet-50 [3] pre-trained on ImageNet [9]. It should be noted that our U-Net is just a model that is trained to predict the saliency map for the given black-box classifier. We use a pre-trained ResNet as a part of this model in order to speed up the training, however, as we show in our CIFAR-10 experiment in section 5.3 the masking model can also be trained completely from scratch. The ResNet-50 model contains feature maps of five different scales, where each subsequent scale block downsamples the input by a factor of two. We use the ResNet?s feature map from Scale 5 (which corresponds to downsampling by a factor of 32) and pass it through the feature filter. The purpose of the feature filter is to attenuate spatial locations which contents do not correspond to 5 the selected class. Therefore, the feature filter performs the initial localisation, while the following upsampling blocks fine-tune the produced masks. The output of the feature filter Y at spatial location i, j is given by: T Yij = Xij (Xij Cs ) (7) where Xij is the output of the Scale 5 block at spatial location i, j; Cs is the embedding of the selected class s and (?) is the sigmoid nonlinearity. Class embedding C can be learned as part of the overall objective. The upsampler blocks take the lower resolution feature map as input and upsample it by a factor of two using transposed convolution [15], afterwards they concatenate the upsampled map with the corresponding feature map from ResNet and follow that with three bottleneck blocks [3]. Finally, to the output of the last upsampler block (Upsampler Scale 2) we apply 1x1 convolution to produce a feature map with just two channels ? C0 , C1 . The mask Ms is obtained from: Ms = abs(C0 ) abs(C0 ) + abs(C1 ) (8) We use this nonstandard nonlinearity because sigmoid and tanh nonlinearities did not optimise properly and the extra degree of freedom from two channels greatly improved training. The mask Ms has resolution four times lower than the input image and has to be upsampled by a factor of four with bilinear resize to obtain the final mask M . The complexity of the model is comparable to that of ResNet-50 and it can process more than a hundred 224x224 images per second on a standard GPU (which is sufficient for real-time saliency detection). 4.1 Training process We train the masking model to directly minimise the objective function from equation 4. The weights of the pre-trained ResNet encoder (red blocks in figure 4) are kept fixed during the training. In order to make the training process work properly, we introduce few optimisations. First of all, in the naive training process, the ground truth label would always be supplied as a class selector. Unfortunately, under such setting, the model learns to completely ignore the class selector and simply always masks the dominant object in the image. The solution to this problem is to sometimes supply a class selector for a fake class and to apply only the area penalty term of the objective function. Under this setting, the model must pay attention to the class selector, as the only way it can reduce loss in case of a fake label is by setting the mask to zero. During training, we set the probability of the fake label occurrence to 30%. One can also greatly speed up the embedding training by ensuring T that the maximal value of (Xij Cs ) from equation 7 is high in case of a correct label and low in case of a fake label. Finally, let us consider again the evidence removal function (X, M ). In order to prevent the model from adapting to any single evidence removal scheme the alternative image A is randomly generated every time the function is called. In 50% of cases the image A is the blurred version of X (we use a Gaussian blur with = 10 to achieve a strong blur) and in the remainder of cases, A is set to a random colour image with the addition of a Gaussian noise. Such a random scheme greatly improves the quality of the produced masks as the model can no longer make strong assumptions about the final look of the image. 5 Experiments In the ImageNet saliency detection experiment we use three different black-box classifiers: AlexNet [5], GoogleNet [14] and ResNet-50 [3]. These models are treated as black boxes and for each one we train a separate masking model. The selected parameters of the objective function are 1 = 10, 3 , 3 = 5, 4 = 0.3. The first upsampling block has 768 output channels and with each 2 = 10 subsequent upsampling block we reduce the number of channels by a factor of two. We train each masking model as described in section 4.1 on 250,000 images from the ImageNet training set. During the training process, a very meaningful class embedding was learned and we include its visualisation in the Appendix. Example masks generated by the saliency models trained on three different black box image classifiers can be seen in figure 5, where the model is tasked to produce a saliency map for the ground truth 6 (a) Input Image (b) Model & AlexNet (c) Model & GoogleNet (d) Model & ResNet-50 (e) Grad [11] (f) Mask [2] Figure 5: Saliency maps generated by different methods for the ground truth class. The ground truth classes, starting from the first row are: Scottish terrier, chocolate syrup, standard schnauzer and sorrel. Columns b, c, d show the masks generated by our masking models, each trained on a different black box classifier (from left to right: AlexNet, GoogleNet, ResNet-50). Last two columns e, f show saliency maps for GoogleNet generated respectively by gradient [11] and the recently introduced iterative mask optimisation approach [2]. label. In figure 5 it can be clearly seen that the quality of masks generated by our models clearly outperforms alternative approaches. The masks produced by models trained on GoogleNet and ResNet are sharp and precise and would produce accurate object segmentations. The saliency model trained on AlexNet produces much stronger and slightly larger saliency regions, possibly because AlexNet is a less powerful model which needs more evidence for successful classification. Additional examples can be seen in the appendix A. 5.1 Weakly supervised object localisation As discussed in section 3.2 a standard method to evaluate produced saliency maps is by object localisation accuracy. It should be noted that our model was not provided any localisation data during training and was trained using only image-class label pairs (weakly supervised training). We adopt the localisation accuracy evaluation protocol from [1] and provide the ground truth label to the masking model. Afterwards, we threshold the produced saliency map at 0.5 and the tightest bounding box that contains the whole saliency map is set as the final localisation box. The localisation box has to have IOU greater than 0.5 with any of the ground truth bounding boxes in order to consider the localisation successful, otherwise, it is counted as an error. The calculated error rates for the three models are presented in table 1. The lowest localisation error of 36.7% was achieved by the saliency model trained on the ResNet-50 black box, this is a good achievement considering the fact that our method was not given any localisation training data and that a fully supervised approach employed by VGG [10] achieved only slightly lower error of 34.3%. The localisation error of the model trained on GoogleNet is very similar to the one trained on ResNet. This is not surprising because both models produce very similar saliency masks (see figure 5). The AlexNet trained model, on the other hand, has a considerably higher localisation error which is probably a result of AlexNet needing larger image contexts to make a successful prediction (and therefore producing saliency masks which are slightly less precise). We also compared our object localisation errors to errors achieved by other weakly supervised methods and existing saliency detection techniques. As a baseline we calculated the localisation error 7 Alexnet [5] GoogleNet [14] ResNet-50 [3] 39.8 36.9 36.7 Localisation Err (%) Table 1: Weakly supervised bounding box localisation error on ImageNet validation set for our masking models trained with different black box classifiers. of the centrally placed rectangle which spans half of the image area ? which we name "Center". The results are presented in table 2. It can be seen that our model outperforms other approaches, sometimes by a significant margin. It also performs significantly better than the baseline (centrally placed box) and iteratively optimised saliency masks. Because a big fraction of ImageNet images have a large, dominant object in the center, the localisation accuracy of the centrally placed box is relatively high and it managed to outperform two methods from the previous literature. Center Grad [11] Guid [12] CAM [18] Exc [16] Feed [1] Mask [2] This Work 46.3 41.7 42.0 48.1 39.0 38.7 43.1 36.9 Table 2: Localisation errors(%) on ImageNet validation set for popular weakly supervised methods. Error rates were taken from [2] which recalculated originally reported results using few different mask thresholding techniques and achieved slightly lower error rates. For a fair comparison, all the methods follow the same evaluation protocol of [1] and produce saliency maps for GoogleNet classifier [14]. 5.2 Evaluating the saliency metric To better assess the interpretability of the produced masks we calculate the saliency metric introduced in section 3.2 for selected saliency methods and present the results in the table 3. We include a few baseline approaches ? the "Central box" introduced in the previous section, and the "Max box" which simply corresponds to a box spanning the whole image. We also calculate the saliency metric for the ground truth bounding boxes supplied with the data, and in case the image contains more than one ground truth box the saliency metric is set as the average over all the boxes. Table 3 shows that our model achieves a considerably better saliency metric than other saliency approaches. It also significantly outperforms max box and center box baselines and is on par with ground truth boxes which supports the claim that the interpretability of the localisation boxes generated by our model is similar to that of the ground truth boxes. Localisation Err (%) Saliency Metric Ground truth boxes (baseline) Max box (baseline) Center box (baseline) 0.00 59.7 46.3 0.284 1.366 0.645 Grad [11] Exc [16] Masking model (this work) 41.7 39.0 36.9 0.451 0.415 0.318 Table 3: ImageNet localisation error and the saliency metric for GoogleNet. 5.3 Detecting saliency of CIFAR-10 To verify the performance of our method on a completely different dataset we implemented our saliency detection model for the CIFAR-10 dataset [4]. Because the architecture described in section 4 specifically targets high-resolution images and five downsampling blocks would be too much for 32x32 images, we modified the architecture slightly and replaced the ResNet encoder with just 3 downsampling blocks with 5 convolutional layers each. We also reduced the number of bottleneck blocks in each upsampling block from 3 to 1. Unlike before, with this experiment, we did not use a pre-trained masking model, but instead a randomly initialised one. We used a FitNet [7] trained to 92% validation accuracy as a black box classifier to train the masking model. All the training parameters were used following the ImageNet model. 8 Figure 6: Saliency maps generated by our model for images from CIFAR-10 validation set. The masking model was trained for 20 epochs. Saliency maps for sample images from the validation set are shown in figure 6. It can be seen that the produced maps are clearly interpretable and a human could easily recognise the original objects after masking. This confirms that the masking model works as expected even at low resolution and that FitNet model, used as a black box learned correct representations for the CIFAR-10 classes. More interestingly, this shows that the masking model does not need to rely on a pre-trained model which might inject its own biases into the generated masks. 6 Conclusion and Future Research In this work, we have presented a new, fast, and accurate saliency detection method that can be applied to any differentiable image classifier. Our model is able to produce 100 saliency masks per second, sufficient for real-time applications. We have shown that our method outperforms other weakly supervised techniques at the ImageNet localisation task. We have also developed a new saliency metric that can be used to assess the quality of explanations produced by saliency detectors. Under this new metric, the quality of explanations produced by our model outperforms other popular saliency detectors and is on par with ground truth bounding boxes. The model-based nature of our technique means that our work can be extended by improving the architecture of the masking network, or by changing the objective function to achieve any desired properties for the output mask. Future work includes modifying the approach to produce high quality, weakly supervised, image segmentations. Moreover, because our model can be run in real-time, it can be used for video saliency detection to instantly explain decisions made by black-box classifiers such as the ones used in autonomous vehicles. Lastly, our model might have biases of its own ? a fact which does not seem to influence the model performance in finding biases in other black boxes according to the various metrics we used. It would be interesting to study the biases embedded into our masking model itself, and see how these affect the generated saliency masks. 9 References [1] Chunshui Cao, Xianming Liu, Yi Yang, Yinan Yu, Jiang Wang, Zilei Wang, Yongzhen Huang, Liang Wang, Chang Huang, Wei Xu, Deva Ramanan, and Thomas S. Huang. Look and think twice: Capturing top-down visual attention with feedback convolutional neural networks. pages 2956?2964, 2015. doi: 10.1109/ICCV.2015.338. URL http://dx.doi.org/10.1109/ICCV.2015.338. [2] Ruth Fong and Andrea Vedaldi. Interpretable Explanations of Black Boxes by Meaningful Perturbation. arXiv preprint arXiv:1704.03296, 2017. [3] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. CoRR, abs/1512.03385, 2015. URL http://arxiv.org/abs/1512.03385. [4] Alex Krizhevsky. Learning Multiple Layers of Features from Tiny Images. Master?s thesis, 2009. URL http://www.cs.toronto.edu/~{}kriz/learning-features-2009-TR.pdf. 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Houdini: Fooling Deep Structured Visual and Speech Recognition Models with Adversarial Examples Moustapha Cisse Facebook AI Research moustaphacisse@fb.com Yossi Adi* Bar-Ilan University, Israel yossiadidrum@gmail.com Natalia Neverova* Facebook AI Research nneverova@fb.com Joseph Keshet Bar-Ilan University, Israel jkeshet@cs.biu.ac.il Abstract Generating adversarial examples is a critical step for evaluating and improving the robustness of learning machines. So far, most existing methods only work for classification and are not designed to alter the true performance measure of the problem at hand. We introduce a novel flexible approach named Houdini for generating adversarial examples specifically tailored for the final performance measure of the task considered, be it combinatorial and non-decomposable. We successfully apply Houdini to a range of applications such as speech recognition, pose estimation and semantic segmentation. In all cases, the attacks based on Houdini achieve higher success rate than those based on the traditional surrogates used to train the models while using a less perceptible adversarial perturbation. 1 Introduction Deep learning has redefined the landscape of machine intelligence [22] by enabling several breakthroughs in notoriously difficult problems such as image classification [20, 16], speech recognition [2], human pose estimation [35] and machine translation [4]. As the most successful models are permeating nearly all the segments of the technology industry from self-driving cars to automated dialog agents, it becomes critical to revisit the evaluation protocol of deep learning models and design new ways to assess their reliability beyond the traditional metrics. Evaluating the robustness of neural networks to adversarial examples is one step in that direction [32]. Adversarial examples are synthetic patterns carefully crafted by adding a peculiar noise to legitimate examples. They are indistinguishable from the legitimate examples by a human, yet they have demonstrated a strong ability to cause catastrophic failure of state of the art classification systems [12, 25, 21]. The existence of adversarial examples highlights a potential threat for machine learning systems at large [28] that can limit their adoption in security sensitive applications. It has triggered an active line of research concerned with understanding the phenomenon [10, 11], and making neural networks more robust [29, 7] . Adversarial examples are crucial for reliably evaluating and improving the robustness of the models [12]. Ideally, they must be generated to alter the task loss unique to the application considered directly. For instance, an adversarial example crafted to attack a speech recognition system should be designed to maximize the word error rate of the targetted system. The existing methods for generating adversarial examples exploit the gradient of a given differentiable loss function to guide the search in the neighborhood of legitimates examples [12, 25]. Unfortunately, the task loss of several structured prediction problems of interest is a combinatorial non-decomposable quantity that is not amenable * equal contribution original semantic segmentation framework adversarial attack compromised semantic segmentation framework Figure 1: We cause the network to generate a minion as segmentation for the adversarially perturbed version of the original image. Note that the original and the perturbed image are indistinguishable. to gradient-based methods for generating adversarial example. For example, the metric for evaluating human pose estimation is the (normalized) percentage of correct keypoints. Automatic speech recognition systems are assessed using their word (or phoneme) error rate. Similarly, the quality of a semantic segmentation is measured by the intersection over union (IOU) between the ground truth and the prediction. All these evalutation measures are non-differentiable. The solutions for this obstacle in supervised learning are of two kinds. The first route is to use a consistent differentiable surrogate loss function in place of the task loss [5]. That is a surrogate which is guaranteed to converge to the task loss asymptotically. The second option is to directly optimize the task loss by using approaches such as Direct Loss Minimization [14]. Both of these strategies have severe limitations. (1) The use of differentiable surrogates is satisfactory for classification because the relationship between such surrogates and the classification accuracy is well established [34]. The picture is different for the above-mentioned structured prediction tasks. Indeed, there is no known consistency guarantee for the surrogates traditionally used in these problems (e.g. the connectionist temporal classification loss for speech recognition) and designing a new surrogate is nontrivial and problem dependent. At best, one can only expect a high positive correlation between the proxy and the task loss. (2) The direct minimization approaches are more computationally involved because they require solving a computationally expensive loss augmented inference for each parameter update. Also, they are notoriously sensitive to the choice of the hyperparameters. Consequently, it is harder to generate adversarial examples for structured prediction problems as it requires significant domain expertise with little guarantee of success when surrogate does not tightly approximate the task loss. Results. In this work we introduce Houdini, the first approach for fooling any gradient-based learning machine by generating adversarial examples directly tailored for the task loss of interest be it combinatorial or non-differentiable. We show the tight relationship between Houdini and the task loss of the problem considered. We present the first successful attack on a deep Automatic Speech Recognition (ASR) system, namely a DeepSpeech-2 based architecture [1], by generating adversarial audio files not distinguishable from legitimate ones by a human (as validated by an ABX experiment). We also demonstrate the transferability of adversarial examples in speech recognition by fooling Google VoiceTM in a black box attack scenario: an adversarial example generated with our model and not distinguishable from the legitimate one by a human leads to an invalid transcription by the Google Voice application (see Figure 8). We also present the first successful untargeted and targetted attacks on a deep model for human pose estimation [26]. Similarly, we validate the feasibility of untargeted and targetted attacks on a semantic segmentation system [38] and show that we can make the system hallucinate an arbitrary segmentation of our choice for a given image. Figure 1 shows an experiment where we cause the network to hallucinate a minion. In all cases, our approach generates better quality adversarial examples than each of the different surrogates (expressly designed for the model considered) without additional computational overhead thanks to the analytical gradient of Houdini. 2 Related Work Adversarial examples. The empirical study of Szegedy et al. [32] first demonstrated that deep neural networks could achieve high accuracy on previously unseen examples while being vulnerable to small adversarial perturbations. This finding has recently aroused keen interest in the community [12, 28, 32, 33]. Several studies have subsequently analyzed the phenomenon [10, 31, 11] and various approaches have been proposed to improve the robustness of neural networks [29, 7]. More closely 2 related to our work are the different proposals aiming at generating better adversarial examples [12, 25]. Given an input (train or test) example (x, y), an adversarial example is a perturbed version of the original pattern x ? = x + ?x where ?x is small enough for x ? to be undistinguishable from x by a human, but causes the network to predict an incorrect target. Given the network g? (where ? is the set of parameters) and a p-norm, the adversarial example is formally defined as:
x ? = argmax ` g? (? x), y (1) x ?:k? x?xkp ? where  represents the strength of the adversary. Assuming the loss function `(?) is differentiable, Shaham et al. [31] propose to take the first order taylor expansion of x 7? `(g? (x), y) to compute ?x by solving the following simpler problem:
T x ? = argmax ?x `(g? (x), y) (? x ? x) (2) x ?:k? x?xkp ? When p = ?, then x ? = x + sign(?x `(g? (x), y)) which corresponds to the fast gradient sign method [12]. If instead p = 2, we obtain x ? = x + ?x `(g? (x), y) where ?x `(g? (x), y) is often normalized. Optionally, one can perform more iterations of these steps using a smaller norm. This more involved strategy has several variants [25]. These methods are concerned with generating adversarial examples assuming a differentiable loss function `(?). Therefore they are not directly applicable to the task losses of interest. However, they can be used in combination with our proposal which derives a consistent approximation of the task loss having an analytical gradient. Task Loss Minimization. Recently, several works have focused on directly minimizing the task loss. In particular, McAllester et al. [24] presented a theorem stating that a certain perceptron-like learning rule, involving feature vectors derived from loss-augmented inference, directly corresponds to the gradient of the task loss. While this algorithm performs well in practice, it is extremely sensitive to the choice of its hyper-parameter and needs two inference operations per training iteration. Do et al. [9] generalized the notion of the ramp loss from binary classification to structured prediction and proposed a tighter bound to the task loss than the structured hinge loss. The update rule of the structured ramp loss is similar to the update rule of the direct loss minimization algorithm, and similarly it needs two inference operations per training iteration. Keshet et al. [19] generalized the notion of the binary probit loss to the structured prediction case. The probit loss is a surrogate loss function naturally resulted in the PAC-Bayesian generalization theorems. it is defined as follows: `?probit (g? (x), y) = E?N (0,I) [`(y, g?+ (x))] (3) where  ? Rd is a d-dimensional isotropic Normal random vector. [18] stated finite sample generalization bounds for the structured probit loss and showed that it is strongly consistent. Strong consistency is a critical property of a surrogate since it guarantees the tight relationship to the task loss. For instance, an attacker of a given system can expect to deteriorate the task loss if she deteriorates the consistent surrogate of it. The gradient of the structured probit loss can be approximated by averaging over samples from the unit-variance isotropic normal distribution, where for each sample an inference with perturbed parameters is computed. Hundreds to thousands of inference operations are required per iteration to gain stability in the gradient computation. Hence the update rule is computationally prohibitive and limits the applicability of the structured probit loss despite its desirable properties. We propose a new loss named Houdini. It shares the desirable properties of the structured probit loss while not suffering from its limitations. Like the structured probit loss and unlike most surrogates used in structured prediction (e.g. structured hinge loss for SVMs), it is tightly related to the task loss. Therefore it allows to reliably generate adversarial examples for a given task loss of interest. Unlike the structured probit loss and like the smooth surrogates, it has an analytical gradient hence require only a single inference in its update rule. The next section presents the details of our proposal. 3 Houdini Let us consider a neural network g? parameterized by ? and the task loss of a given problem `(?). We assume `(y, y) = 0 for any target y. The score output by the network for an example (x, y) is g? (x, y) and the network?s decoder predicts the highest scoring target: y? = y? (x) = argmax g? (x, y). y?Y 3 (4) Using the terminology of section 2, finding an adversarial example fooling the model g? with respect to the task loss `(?) for a chosen p-norm and noise parameter  boils down to solving:
x ? = argmax ` y? (? x), y (5) x ?:k? x?xkp ? The task loss is often a combinatorial quantity which is hard to optimize, hence it is replaced with a ? ? (? differentiable surrogate loss, denoted `(y x), y). Different algorithms use different surrogate loss functions: structural SVM uses the structured hinge loss, Conditional random fields use the log loss, etc. We propose a surrogate named Houdini and defined as follows for a given example (x, y): h i `?H (?, x, y) = P??N (0,1) g? (x, y) ? g? (x, y?) < ? ? `(? y , y) (6) In words, Houdini is a product of two terms. The first term is a stochastic margin, that is the probability that the difference between the score of the actual target g? (x, y) and that of the predicted target g? (x, y?) is smaller than ? ? N (0, 1). It reflects the confidence of the model in its predictions. The second term is the task loss, which given two targets is independent of the model and corresponds to what we are ultimately interested in maximizing. Houdini is a lower bound of the task loss. Indeed denoting ?g(y, y?) = g? (x, y) ? g? (x, y?) as the difference between the scores assigned by the network to the ground truth and the prediction, we have P??N (0,1) (?g(y, y?) < ?) is smaller than 1. Hence when this probability goes to 1, or equivalently when the score assigned by the network to the target y? grows without a bound, Houdini converges to the task loss. This is a unique property not enjoyed by most surrogates used in the applications of interest in our work. It ensures that Houdini is a good proxy of the task loss for generating adversarial examples. We can now use Houdini in place of the task loss `(?) in the problem 5. Following 2, we resort to a first order approximation which requires the gradient of Houdini with respect to the input x. The latter is obtained following the chain rule:   ? `?H (?, x, y) ?g? (x, y) ?x `?H (?, x, y) = (7) ?g? (x, y) ?x To compute the RHS of the above quantity, we only need to compute the derivative of Houdini with respect to its input (the output of the network). The rest is obtained by backpropagation. The derivative of the loss with respect to the network?s output is: " # Z ? h h i i 2 1 ?g P??N (0,1) g? (x, y) ? g? (x, y?) < ? `(y, y?) = ?g ? e?v /2 dv `(y, y?) (8) 2? ?g(y,?y) ? Therefore, expanding the right hand side and denoting C = 1/ 2? we have: ? ?|?g(y,? y )|2 /2 ? `(y, y?), g = g? (x, y)   ??C ? e 2 ?g `?H (? y , y) = C ? e?|?g(y,?y)| /2 `(y, y?), (9) g = g? (x, y?) ? ?0, otherwise Equation 9 provides a simple analytical formula for computing the gradient of Houdini with respect to its input, hence an efficient way to obtain the gradient with respect to the input of the network x by backpropagation. The gradient can be used in combination with any gradient-based adversarial example generation procedure [12, 25] in two ways, depending on the form of attack considered. For an untargeted attack, we want to change the prediction of the network without preference on the final prediction. In that case, any alternative target y can be used (e.g. the second highest scorer as the target). For a targetted attack, we set the y to be the desired final prediction. Also note that, when the score of the predicted target is very close to that of the ground truth (or desired target), that is when ?g(y, y?) is small as we expect from the trained network we want to fool, we have 2 e?|?g(y,?y)| /2 ' 1. In the next sections, we show the effectiveness of the proposed attack scheme on human pose estimation, semantic segmentation and automatic speech recognition systems. 4 Human Pose Estimation We evaluate the effectiveness of Houdini loss in the context of adversarial attacks on neural models for human pose estimation. Compromising performance of such systems can be desirable for 4 (a) (b) (c) Figure 2: Convergence dynamics for pose estimation attacks: (a) perturbation perceptibility vs nb. iterations, (b) PCKh0.5 vs nb. iterations, (c) proportion of re-positioned joints vs perceptibility. manipulating surveillance cameras, altering the analysis of crime scenes, disrupting human-robot interaction or fooling biometrical authentication systems based on gate recognition. The pose estimation task is formulated as follows: given a single RGB image of a person, determine correct 2D positions of several pre-defined keypoints which typically correspond to skeletal joints. In practice, the performance is measured by the percentage of correctly detected keypoints (PCKh) (i.e. whose predicted locations are within a certain distance from the corresponding target positions) [3]: PN 1(kyi ? y?i k < ?h) PCKh? = i=1 , (10) N where y? and y are the predicted and the desired positions of a given joint respectively, h is the head size of the person (known at test time), ? is a threshold (set to 0.5), and N is the number of annotated keypoints. Pose estimation is a good example of a problem where we observe a discrepancy between the training objective and the final evaluation measure. Instead of directly minimizing the percentage of correctly detected keypoints, state-of-the-art methods rely upon a dense prediction of heatmaps, i.e. estimation of probabilities of every pixel corresponding to each of keypoint locations. These models can be trained with binary cross entropy [6], softmax [15] or MSE losses [26] applied to every pixel in the output space, separately for each plane corresponding to every keypoint. In our first experiment, we attack a state-of-the-art model for single person pose estimation based on Hourglass networks [26] and aim to minimize the value of PCKh0.5 metric given the minimal perturbation. For this task we choose y? as: y? = argmax g? (x, y?) (11) y?:kp?pk>?h ? where p is the pixel coordinate on the heatmap corresponding to the argmax value of vector y. We ?x perform the optimization iteratively till convergence with an update rule  ? k? where ?x are xk gradients with respect to the input and  = 0.1. We perform the evaluations on the validation subset of MPII dataset [3] consisting of 3000 images and defined as in P[26]. We evaluate the perceived degree of perturbation where perceptibility is expressed as ( n1 (x0i ? xi )2 )1/2 , where x and x0 are original and distorted images, and n is the number of pixels [32]. In addition, we report the structural similarity index (SSIM) [36] which is known to correlate well with the visual perception of image structure by a human. Figure 2 shows that Houdini only requires 100 iterations to maximally deteriorate the percentage of correct key-points from 89.4 down to 0.57 while MSE deteriorates the performance to only 24.12 after 100 iterations. This observation underlines the importance of the loss function used to generate adversarial examples in structured prediction problems. Also, for untargeted attacks optimized to convergence, the perturbation generated with Houdini is up to 50% less perceptible than the one obtained with MSE. In the second experiment, we perform a targetted attack in the form of pose transfer, i.e. we force the network to hallucinate an arbitrary pose (with success defined, as before, given the target metric PCKh0.5 ). The experimental setup is as follows: for a given pair of images (i, j) we force the network to output the ground truth pose of the picture i when the input is image j and vice versa. This task is more challenging and depends on the similarity between the original and target poses. Surprisingly, targetted attacks are still feasible even when the two ground truth poses are very different. Figure 3 shows an example where the model predicts the pose of a human body in horizontal position for an adversarially perturbed image depicting a standing person (and vice versa). A similar experiment with two persons in standing and sitting positions respectively is also shown in Figure 3. 5 perceptibility 0.145 perceptibility 0.211 perceptibility 0.016 perceptibility 0.210 Figure 3: Examples of successful targetted attacks on a pose estimation system. Despite the important difference between the images selected, it is possible to make the network predict the wrong pose by adding an imperceptible perturbation. The images are part of the MPI dataset. SSIM Method @mIoU/2 Perceptibility @mIoU lim @mIoU/2 @mIoUlim untargeted: NLL loss untargeted: Houdini loss 0.9989 0.9995 0.9950 0.9959 0.0037 0.0026 0.0117 0.0095 targetted: NLL loss targetted: Houdini loss 0.9972 0.9975 0.9935 0.9937 0.0074 0.0054 0.0389 0.0392 Table 1: Comparison of targetted and untargeted adversarial attacks on segmentation systems. mIoU/2 denotes 50% performance drop according to the target metric and mIoUlim corresponds to convergence or termination after 300 iterations. SSIM: the higher, the better, perceptibility: the lower, the better. Houdini based attacks are less perceptible. 5 Semantic segmentation Semantic segmentation uses another customized metric to evaluate performance, namely the mean Intersection over Union (mIoU) measure defined as averaging over classes the IoU = TP/(TP + FP + FN), where TP, FP and FN stand for true positive, false positive and false negative respectively, taken separately for each class. Compared to per-pixel accuracy, which appears to be overoptimistic on highly unbalanced datasets, and per-class accuracy, which under-penalizes false alarms for non-background classes, this metric favors accurate object localization with tighter masks (in instance segmentation) or bounding boxes (in detection). The models are trained with a per-pixel softmax or multi-class cross entropy losses depending on the task formulation, i.e. optimized for mean per-pixel or per-class accuracy instead of mIoU. Primary targets of adversarial attacks in this group of applications are self-driving cars and robots. Xie et al. [37] have previously explored adversarial attacks in the context of semantic segmentation. However, they exploited the same proxy used for training the network. We perform a series of experiments similar to the ones described in Sec. 4. That is, we show targetted and untargeted attacks on a semantic segmentation model. We use a pre-trained Dilation10 model for semantic segmentation [38] and evaluate the success of the attacks on the validation subset of Cityscapes dataset [8]. In the first experiment, we directly alter the target mIoU metric for a given test image in both targetted and untargeted attacks. As shown in Table 1, Houdini allows fooling the model at least as well as the training proxy (NLL) while using a less perceptible. Indeed, Houdini based adversarial perturbations generated to alter the performance of the model by 50% are about 30% less noticeable than the noise created with NLL. The second set of experiments consists of targetted attacks. That is, altering the input image to obtain an arbitrary target segmentation map as the network response. In Figure 4, we show an instance of such attack in a segmentation transfer setting, i.e. the target segmentation is the ground truth segmentation of a different image. It is clear that this type of attack is still feasible with a small 6 source image initial prediction adversarial adversarialprediction prediction perturbed noise noise source image initial prediction adversarial prediction perturbed noise target source Figure 4: Targetted attack on a semantic segmentation system: switching target segmentation between two images from Cityscapes dataset [8]. The last two columns are respectively zoomed-in parts of the perturbed image and the adversarial perturbation added to the original one. adversarial perturbation (even when after zooming in the picture). Figure 1 depicts a more challenging scenario where the target segmentation is an arbitrary map (e.g. a minion). Again, we can make the perturbed image network hallucinate the segmentation of our choice by adding a barely noticeable perturbation. 6 Speech Recognition We evaluate the effectiveness of Houdini concerning adversarial attacks on an Automatic Speech Recognition (ASR) system. Traditionally, ASR systems are composed of different components, (e.g. acoustic model, language model, pronunciation model, etc.) where each component is trained separately. Recently, ASR research has focused on deep learning based end-to-end models. These type of models get as input a speech segment and output a transcript with no additional post-processing. In this work, we use a deep neural network as our model with similar architecture to the one presented by [2]. The system is composed of two convolutional layers, followed by seven layers of Bidirectional LSTM [17] and one fully connected layer. We optimize the Connectionist Temporal Classification (CTC) loss function [13], which was specifically designed for ASR systems. The model gets as input raw spectrograms (extracted using a window size of 25ms, frame-size of 10ms and Hamming window), and outputs a transcript. A standard evaluating metric in speech recognition is the Word Error Rate (WER) or Character Error Rate (CER). These metrics were derived from the Levenshtein Distance [23], which is the number of substitutions, deletions, and insertions divided by the target length. The model achieves 12% Word Error Rate and 1.5% Character Error Rate on the Librispeech dataset [27], with no additional language modeling. In order to use Houdini for attacking an end-to-end ASR model, we need to get g? (x, y) and g? (x, y?), which are the scores for predicting y and y? respectively. Recall, in speech recognition, the target y is a sequence of characters. Hence, the score g? (x, y?) is the sum of all possible paths to output y?. Fortunately, we can use the forward-backward algorithm [30], which is at the core of the CTC, to get the score of a given label y. ABX Experiment We generated 100 audio samples of adversarial examples and performed an ABX test with about 100 humans. An ABX test is a standard way to assess the detectable differences between two choices of sensory stimuli. We present each human with two audio samples A and B. Each of these two samples is either the legitimate or an adversarial version of the same sound. These two samples are followed by a third sound X randomly selected to be either A or B. Next, the human must decide whether X is the same as A or B. For every audio sample, we executed the ABX test with at least nine (9) different persons. Overall, Only 53.73% of the adversarial examples could be distinguished from the original ones by the humans (the optimal ratio is 50%). Therefore the generated examples are not statistically significantly distinguishable by a human ear. Subsequently, we use such indistinguishable adversarial examples to test the robustness of ASR systems. Houdini vs Probit Loss Houdini and the Probit loss [19] are tightly related. We also initially experimented with Probit but decided not to consider it further because: (1) Houdini is computationally more efficient. It requires only one forward-backward pass to generate adversarial examples while Probit needs several times more passes as it must average many networks to reduce the variance 7  = 0.3 CTC Houdini  = 0.2  = 0.1  = 0.05 WER CER WER CER WER CER WER CER 68 96.1 9.3 12 51 85.4 6.9 9.2 29.8 66.5 4 6.5 20 46.5 2.5 4.5 Figure 5: CER and WER in (%) for adversarial examples generated by both CTC and Houdini. Frequency (Hz) 5000 Frequency (Hz) 5000 0 0 0 3.23 0 3.23 Time (s) Time (s) (a) a great saint saint francis zaviour (b) i great sinkt shink t frimsuss avir Figure 6: The model?s output for each of the spectrograms is located at the bottom of each spectrogram. The target transcription is: A Great Saint Saint Francis Xavier. of the gradients. (2) The "adversarial" examples generated with Probit are not adversarial in the sense that they are easily distinguishable from the original examples by a human. This is due to the noise (added to the parameters when computing the gradients with Probit) which seems to add white noise to the sound files. We calculated the character error rates (CER) and the percentage of examples that could be distinguished from original ones by a human (best is 50) for Houdini and Probit on the speech task. We used a perturbation of magnitude  = 0.05 and sampled 20 models for Probit (therefore 20x more computationally expensive than Houdini). In our results, while Probit and Houdini respectively achieve a CER of 5.97 and 4.50, adversarial examples generated with Probit are perfectly distinguishable by human (100%) in comparison to those generated with Houdini (53.73%). Untargeted Attacks In the first experiment, we compare network performance after attacking it with both Houdini and CTC. We generate two adversarial example, from each of the loss functions (CTC and Houdini), for every samples from the clean test set of Librispeech (2620 speech segments). We have experienced with a set of different distortion levels, using the `? norm and WER as `. For all adversarial examples, we use y? = "Forty Two", which is the "Answer to the Ultimate Question of Life, the Universe, and Everything." Results are summarized in 5. Notice that Houdini causes a bigger decrease regarding both CER and WER than CTC for all the distortions values we have tested. In particular, for small adversarial perturbation ( = 0.05) the word error rate (WER) caused by an attack with Houdini is 2.3x larger than the WER obtained with CTC. Similarly, the character error rate (CER) caused by an Houdini-based attack is 1.8x larger than a CTC-based one. Fig. 6 shows the original and adversarial spectrograms for a single speech segment. (a) shows a spectrogram of the original sound file and (b) shows the spectrogram of the adversarial one. They are visually undistinguishable. Targetted Attacks We push the model towards predicting a different transcription iteratively. In this case, the input to the model at iteration i is the adversarial example from iteration i ? 1. Corresponding transcription samples are shown in Table 7. We notice that while setting y? to be phonetically far from y, the model tends to predict wrong transcriptions but not necessarily similar to the selected target. However, when picking phonetically close ones, the model acts as expected and predict a transcription which is phonetically close to y?. Overall, targetted attacks seem to be much more challenging when dealing with speech recognition systems than when we consider artificial visual systems such as pose estimators or semantic segmentation systems. Black box Attacks Lastly, we experimented with a black box attack, that is attacking a system in which we do not have access to the models? gradients but only to its predictions. In Figure 8 we show few examples in which we use Google Voice application to predict the transcript for both original 8 Manual Transcription Adversarial Target Adversarial Prediction a great saint saint Francis Xavier a green thank saint frenzier a green thanked saint fredstus savia no thanks I am glad to give you such easy happiness notty am right to leave you soggy happiness no to ex i am right like aluse o yve have misser Figure 7: Examples of iteratively generated adversarial examples for targetted attacks. In all case the model predicts the exact target transcription of the original example. Targetted attacks are more difficult when the speech segments are phonetically very different. Groundtruth Transcription: The fact that a man can recite a poem does not show he remembers any previous occasion on which he has recited it or read it. G-Voice transcription of the original example: The fact that a man can decide a poem does not show he remembers any previous occasion on which he has work cited or read it. G-Voice transcription of the adversarial example: The fact that I can rest I?m just not sure that you heard there is any previous occasion I am at he has your side it or read it. Groundtruth Transcription: Her bearing was graceful and animated she led her son by the hand and before her walked two maids with wax lights and silver candlesticks. G-Voice transcription of the original example: The bearing was graceful an animated she let her son by the hand and before he walks two maids with wax lights and silver candlesticks. G-Voice transcription of the adversarial example: Mary was grateful then admitted she let her son before the walks to Mays would like slice furnace filter count six. Figure 8: Transcriptions from Google Voice application for original and adversarial speech segments. and adversarial audio files. The original audio and their adversarial versions generated with our DeepSpeech-2 based model are not distinguishable by human according to our ABX test. We play each audio clip in front of an Android based mobile phone and report the transcription produced by the application. As can be seen, while Google Voice could get almost all the transcriptions correct for legitimate examples, it largely fails to produce good transcriptions for the adversarial examples. As with images [32], adversarial examples for speech recognition also transfer between models. 7 Conclusion We have introduced a novel approach to generate adversarial examples tailored for the performance measure unique to the task of interest. We have applied Houdini to challenging structured prediction problems such as pose estimation, semantic segmentation and speech recognition. In each case, Houdini allows fooling state of the art learning systems with imperceptible perturbation, hence extending the use of adversarial examples beyond the task of image classification. What the eyes see and the ears hear, the mind believes. (Harry Houdini) Acknowledgments The authors thank Alexandre Lebrun, Pauline Luc and Camille Couprie for valuable help with code and experiments. We also thank Antoine Bordes, Laurens van der Maaten, Nicolas Usunier, Christian Wolf, Herve Jegou, Yann Ollivier, Neil Zeghidour and Lior Wolf for their insightful comments on the early draft of this paper. 9 References [1] D. Amodei, R. Anubhai, E. Battenberg, C. Case, J. Casper, B. Catanzaro, J. Chen, M. Chrzanowski, A. Coates, G. Diamos, et al. Deep speech 2: End-to-end speech recognition in english and mandarin. arXiv preprint arXiv:1512.02595, 2015. [2] D. Amodei, R. Anubhai, E. Battenberg, C. Case, J. Casper, B. Catanzaro, J. Chen, M. Chrzanowski, A. Coates, G. Diamos, et al. 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Efficient and Flexible Inference for Stochastic Systems Stefan Bauer? Department of Computer Science ETH Zurich bauers@inf.ethz.ch Nico S. Gorbach? Department of Computer Science ETH Zurich ngorbach@inf.ethz.ch ?orde ? Miladinovi?c Department of Computer Science ETH Zurich djordjem@inf.ethz.ch Joachim M. Buhmann Department of Computer Science ETH Zurich jbuhmann@inf.ethz.ch Abstract Many real world dynamical systems are described by stochastic differential equations. Thus parameter inference is a challenging and important problem in many disciplines. We provide a grid free and flexible algorithm offering parameter and state inference for stochastic systems and compare our approch based on variational approximations to state of the art methods showing significant advantages both in runtime and accuracy. 1 Introduction A dynamical system is represented by a set of K stochastic differential equations (SDE?s) with model parameters ? that describe the evolution of K states X(t) = [x1 (t), x2 (t), . . . , xK (t)]T such that: dX(t) = f (X(t), ?)dt + ?dWt , (1) where Wt is a Wiener process. A sequence of observations, y(t) is usually contaminated by some measurement error which we assume to be normally distributed with zero mean and variance for each of the K states, i.e. E ? N (0, D), with Dik = ?k2 ?ik . Thus for N distinct time points the overall system may be summarized as Y = AX + E, where X = [x(t1 ), . . . , x(tN )] = [x1 , . . . , xK ]T Y = [y(t1 ), . . . , y(tN )] = [y1 , . . . , yK ]T , where xk = [xk (t1 ), . . . , xk (tN )]T is the k?th state sequence and yk = [yk (t1 ), . . . , yk (tN )]T are the observations. Given the observations Y and the description of the dynamical system (1), the aim is to estimate both state variables X and parameters ?. Related Work. Classic approaches for solving the inverse problem i.e. estimating the parameters given some noisy observations of the process, include the Kalman Filter or its improvements [e.g. Evensen, 2003, Torn?e et al., 2005] and MCMC based approaches [e.g. Lyons et al., 2012]. However, ? The first two authors contributed equally to this work. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. MCMC based methods do not scale well since the number of particles required for a given accuracy grows exponentially with the dimensionality of the inference problem [Snyder et al., 2008], which is why approximations to the inference problem became increasingly more popular in recent years. Archambeau et al. [2008] proposed a variational formulation for parameter and state inference of stochastic diffuion processes using a linear dynamic approximation: In an iterated two-step approach the mean and covariance of the approximate process (forward propagation) and in the second step the time evolution of the Lagrange multipliers, which ensure the consistency constraints for mean and variance (backward propagation), are calculated in order to obtain a smooth estimate of the states. Both forward and backward smoothing require the repeated solving of ODEs. In order to obtain a good accuracy a fine time grid is additionally needed, which makes the approach computational expensive and infeasible for larger systems [Vrettas et al., 2015]. For parameter estimation the smoothing algorithm is used in the inner loop of a conjugate gradient algorithm to obtain an estimate of the optimal approximation process (given a fixed set of parameters) while in the outer loop a gradient step is taken to improve the current estimate of the parameters. An extension of Archambeau et al. [2008] using local polynomial approximations and mean-field approximations was proposed in Vrettas et al. [2015]. Mean-field approximations remove the need of Lagrange multipliers and thus of the backward propagation while the polynomial approximations remove the need of solving ODEs iteratively in the forward propagation step which makes the smoothing algorithm and thus the inner loop for parameter estimation feasible, even for large systems while achieving a comparable accuracy [Vrettas et al., 2015]. Our contributions. While established methods often assume full observability of the stochastic system for parameter estimation, we solve the more difficult problem of inferring parameters in systems which include unobserved variables by combining state and parameter estimation in one step. Despite the fact that we compare our approach to other methods which solve a simpler problem, we offer improved accuracy in parameter estimation at a fraction of the computational cost. 2 Random Ordinary Differential Equations Compared to stochastic differential equations, random ordinary differential equations (RODEs) have been less popular even though both frameworks are highly connected. RODEs are pathwise ordinary differential equations that contain a stochastic process in their vector field functions. In Kloeden and Jentzen [2007] RODEs have been studied to derive better numerical integration schemes for SDEs, which e.g. allows for stronger pathwise results compared to the L2 results given in Ito stochastic calculus. Moreover, RODEs sometimes have an advantage over SDEs by allowing more realistic noise for some applications e.g. correlated noise or noise with limited variance. Let (?, F, P) be a complete probability space, (?t )t?[0,T ] be a Rm -valued stochastic process with continuous sample paths and f : Rm ? Rd ? Rd a continuous function. Then dx(t) = f (x(t), ?t (?)) dt (2) is a scalar RODE, that is, an ODE dx(t) = F? (t, x) := f (x(t), ?(t)), (3) dt for all ? ? ?. Following Kloeden and Jentzen [2007], we likewise assume that f is arbitrary smooth i.e. f ? C ? and thus locally Lipschitz in x such that the initial value problem (3) has a unique solution, which we assume to exist on the finite time interval [0, T ]. A simple example for a RODE is Example 1 (RODE). dx(t) = ?x + sin(Wt (?)), dt (4) where Wt is a Wiener process. Taylor-like schemes for directly solving RODEs (2) were derived e.g. in Gr?ne and Kloeden [2001], Jentzen and Kloeden [2009]. One approach for solving the RODE (2) is to use sampling to obtain many ODE?s (3) which can then be solved pathwise using deterministic calculus. However, this pathwise solution of RODEs implies that a massive amount of deterministic ODEs have to be solved efficiently. A study with a high performance focus was conducted in 2 Riesinger et al. [2016], where parallelized pathwise inference for RODEs was implemented using GPU?s. While in principle classic numerical schemes for deterministic systems e.g. Runge-Kutta can be used for each path, they will usually converge with a lower order since the vector field is not smooth enough in time [Asai et al., 2013]. Since the driving stochastic process ?t has at most H?lder continuous sample paths, the sample paths of the solution t ? x(t) are continuously differentiable but the derivatives of the solution sample paths are at most H?lder continuous in time. This is caused by the fact that F? (t, x) of the ODE (3) is usually only continuous, but not differentiable in t, no matter how smooth the function f is in its variables. RODEs offer the opportunity to use deterministic calculus (pathwise), yet being highly connected with an SDE since any RODE with a Wiener process can be written as SDE Jentzen and Kloeden [2011]. To illustrate the point, the example 1 above can be re-written as an SDE by: Example 2 (SDE transformed RODE).       Xt ?Xt + sin(Yt ) 0 dWt . d = + Yt 0 1 (5) It likewise holds that SDEs can be transformed into RODEs. This transformation was first described in Sussmann [1978] and Doss [1977] and generalized to all finite dimensional stochastic differential equations by Imkeller and Schmalfuss [2001]. RODEs can thus be used to find pathwise solutions for SDEs but SDEs can likewise be used to find better solution for RODEs Asai and Kloeden [2013]. Due to space limitations and to circumvent the introduction of a large mathematical framework, we only show the transformation for additive SDE?s following [Jentzen and Kloeden, 2011, chapter 2]. Proposition 1. Any finite dimensional SDE can be transformed into an RODE and the other way round: dxt = f (xt )dt + dWt ?? dz(t) = f (zt + Ot ) + Ot , dt (6) where z(t) := xt ? Ot and Ot is the Ornstein-Uhlenbeck stochastic stationary process satisfying the linear SDE dOt = ?Ot dt + dWt (7) Typically a stationary Ornstein-Uhlenbeck process is used to replace the white noise of the SDE in its transformation to an RODE. By continuity and the Fundamental Theorem of Calculus it then follows that z(t) is pathwise differentiable. While we only showed the transformation for additive SDE?s, it generally holds true that any RODE with a Wiener process can be transformed into an SDE and any finite dimensional SDE with regular coefficients can be transformed into an RODE. This includes nonlinear drifts and diffusions and is true for univariate and multivariate processes [Han and Kloeden, 2017]. There are cases for which this does not hold e.g. a RODE which includes fractional Brownian motion as the driving noise. While the presented method is thus even more general since RODE?s can be solved, we limit ourselves to the problem of solving additive SDE?s by transforming them into a RODE. Since the solution of a RODE is continuously differentiable in time (but not further differentiable in time), classic numerical methods for ODEs rarely do achieve their traditional order and thus efficiency [Kloeden and Jentzen, 2007]. In the following we describe a scalable variational formulation to infer states and parameters of stochastic differential equations by providing an ensemble learning type algorithm for inferring the parameters of the corresponding random ordinary differential equation. 3 Variational Gradient Matching Gradient matching with Gaussian processes was originally motivated in Calderhead et al. [2008] and offers a computationally efficient shortcut for parameter inference in deterministic systems. While the original formulation was based on sampling, Gorbach et al. [2017] proposed a variational formulation offering significant runtime and accuracy improvements. Gradient matching assumes that the covariance kernel C?k (with hyper-parameters ?k ) of a Gaussian process prior on state variables is once differentiable to obtain a conditional distribution over state 3 Figure 1: Noise. The left plot shows three typical Wiener processes generated with mean zero and the corresponding Ornstein-Uhlenbeck (OU) process having the same Wiener process in its diffusion (right). The scale on the y-axis shows the mean-reverting behaviour of the OU process (compared to the Wiener process). derivatives using the closure property under differentiation of Gaussian processes: ? | X, ?) = p(X Y N (x? k | mk , Ak ), (8) k where the mean and covariance is given by: mk := 0 C?k C?1 ?k xk , 0 Ak := C00?k ? 0 C?k C?1 ?k C?k , C00?k denotes the auto-covariance for each state-derivative with C0?k and 0 C?k denoting the crosscovariances between the state and its derivative. The posterior distribution over state-variables is p(X | Y, ?, ?) = Y N (?k (yk ), ?k ) , (9) k where ?k (yk ) := C?k (C?k + ?k2 I)?1 yk and ?k := ?k2 C?k (C?k + ?k2 I)?1 . Inserting the GP based prior in the right hand side of a differential equation and assuming additive, normally distributed noise with state-specific error variance ?k one obtains a distribution of state derivatives ? | X, ?, ?) = p(X Y N (x? k | fk (X, ?), ?k I) . (10) k which is combined with the smoothed distribution obtained from the data fit (9) in a product of experts approach: ? | X, ?, ?, ?) ? p(X ? | X, ?)p(X ? | X, ?, ?). p(X After analytically integrating out the latent state-derivatives Y
p(? | X, ?, ?) ? p(?) N fk (X, ?) | mk , ??1 k ) . (11) k where ??1 k := Ak + ?k I one aims to determine the maximum a posteriori estimate (MAP) of the parameters Z ? ? : = arg max ln p(? | X, ?, ?)p(X | Y, ?)dX, (12) ? Since the integral in (12) is in most cases analytically intractable (even for small systems due to the non-linearities and couplings induced by the drift function), a lower bound is established through the 4 introduction of an auxiliary distribution Q: Z ln p(? | X, ?, ?)p(X | Y, ?)dX R Z Q(X)dX (a) = ? Q(X)dX ln R p(? | X, ?, ?)p(X | Y, ?)dX Z (b) Q(X) ? ? Q(X) ln dX p(? | X, ?, ?)p(X | Y, ?) = H(Q) + EQ ln p(? | X, ?, ?) + EQ ln p(X | Y, ?) =: LQ (?) (13) R where H(Q) is the entropy. In (a) the auxiliary distribution Q(X), Q(X)dX = 1 is introduced and in (b) is using Jensens?s inequality. The lower bound holds with equality whenever p(? | X, ?, ?)p(X | Y, ?) (c) Q? (X) : = R = p(X | Y, ?, ?, ?), p(? | X, ?, ?)p(X | Y, ?)dX where in (c) Bayes rule is used. Unfortunately Q? is analytically intractable because its normalization given by the integral in the denominator is in most cases analytically intractable due to the strong couplings induced by the nonlinear drift function f in (1). Using mean-field approximations   Y Q := Q : Q(X, ?) = q(? | ?) q(xu | ? u ) , (14) u where ? and ? u are the variational parameters. Assuming that the drift in (1) is linear in the parameters ? and that states only appear as monomial factors in arbitrary large products of states the true conditionals p(? | X, Y, ?) and p(xu | ?, X?u , Y, ?) are Gaussian distributed, where X?u denotes all states excluding state xu (i.e. X?u := {x ? X | x 6= xu }) and thus q(? | ?) and q(xu | ? u ) are designed to be Gaussian. b This Q Qposterior distribution over states is then approximated as p(X|Y, ?, ?, ?, ?) ? Q(X) = b?kt and the log transformed distribution over the ODE parameters given the observations as k tq ln p(?|Y, ?, ?, ?) ? LQ? (?). Algorithm 1 Ensemble based parameter estimation for SDEs 1: Transform the SDE 1 into a RODE 2 2: Simulate a maximum number Nmax of OU-processes and insert them in 2 to obtain Nmax ODEs 3: For each ODE obtain approximate solutions using variational gradient matching [Gorbach et al., 2017] b to obtain an estimate of the parameters for the RODE 2 4: Combine the solutions ? 5: Transform the solutions of the RODE 2 back into solutions of the SDE 1. Gorbach et al. [2017] then use an EM-type approach illustrated in figure 2 iteratively optimizing parameters and the variational lower bound LQ? (?). The variational parameters can be derived analytically and the algorithm scales linearly in the number states of the differential equation and is thus ideally suited to infer the solutions of the massive number of pathwise ODEs required for the pathwise solution of the RODE formulation of the SDE. Since solution paths of the RODE are only once differentiable, gradient matching (which only makes this assumption w.r.t. solution paths) is ideally suited for estimating the parameters. Our approach is summarized in algorithm 1. However, the application of variational gradient matching [Gorbach et al., 2017] for the pathwise solution of the RODE is not straightforward since e.g. in the case for scalar stochastic differential equations one has to solve dz(t) = f? (zt + Ot ) + Ot , (15) dt for a sampled trajectory Ot of an Ornstein-Uhlenbeck process rather than the classic ODE formulation dz(t) dt = f (zt ). We account for the increased uncertainty by assuming an additional state specific Gaussian noise factor ? i.e. assuming f (x + Ot ) + Ot + ? for a sampled trajectory Ot in the gradient matching formulation (10). 5 74 5 Experiments 75 6 lower Discussion tight variational bounds that are analytically tractable provided that the ODE is such th state variables appear in quadratic form in equation 6. ODE?s such as the Lotka-Volterra s 76 ODE The contribution ofwhereas this paper is tosystems integratesuch out the latent state variables instead 80 full-fill such requirements other as the Fitz-High Nagumo sytemo 77 lower in previous work. over stateprovided variablesthat is not trac ??bounds ? (i?1) ?the integration 78 tight variational thatSince are analytically tractable theanalytically ODE is such ? ? 81 D Q (X) p(?, X | Y, ?, ?) KL 78 appear tight variational lower bounds that are analytically that the OD 79 state variables in quadratic form in equation 6. ODE?s tractable such as provided the Lotka-Volterr 79 state variables appear in quadratic form in equation ??. ODE?s such as the Lotk 80 full-fill 82 LQsuch (t) (?)ODE requirements whereas other systems such as the Fitz-High Nagumo syte 80 full-fill such ODE requirements whereas other systems such as the Fitz-High Nag ?? ? ?(i?1) ? tight variational lower bounds that are analytically tractable ?kernel ??) ?Y, ?p(?, ?X ? provided that the ODE is such that t GP 81 D83KLlogQ p(?,81 (X) | Y, ?, X |D ?, ?)dX (i?1) ? ? (X) p(?, X | Y, ?, ?) KL Q 78 79 Flexibility and Efficiency Algorithm 1 offers 78 parameters appear in quadratic form in equation 6. ODE?s such as the Lotka-Volterra syste a flexible framework for inference in stochas- 79 state variables state-derivative (i+1) 82 L84Qsuch state ?(?) ODE requirements noisesuch as the Fitz-High Nagumo sytem do n ? 82 LQobservations (?) whereas other systems b 80 full-fill tic dynamical systems e.g. if the parameters ? ??78 tight ? ? (i?1) bounds that are analytically tractable provided that ? variational ?? lower ? ? ? Q p(?, (X) X ?)dX | p(?, Y, ?, X??79 |83p(?, Y,log ?, X ?) | Y,appear ?, ?)dX are known they can be set to the true values in 81 D83KL log state variables in quadratic in equation 6.? ODE?s such 2as ?th ? k | Fkform max log N Y (Y, ?, ? I) N such Xk as| Y ? I 85 References k not analytically tractable 80 full-fill such ODE requirements whereas other systems thek ,Fitz-Hi (i) each iteration, and algorithm 1 then just corre- 82 L84Q ??(?) ? ? Y ??kbounds ?Girolami ? 78 84 tight variational lower that are analytically tractable provided that the ? References 86 B. Calderhead, M. and N. Lawrence, ?Accelerating bayesian inference over no (i?1) latent state ? ? ??) ? ? sponds to a smoothing algorithm. Compared to 83 log p(?, X |E-??step 81 DKL Q (X) p(?, X | Y, ??, Y, ?)dX 79 ?, state variableswith appear invariables quadratic form in equation 6. suchSystems, as the Lv 87 differential equations gaussian processes,? Neural Information Processing ? kODE?s = max H(Q) + E log N Y | F (Y, ?, ? I) 85 B. Calderhead, M. Girolami and N. Lawrence, ?Accelerating bayesian infere the smoothing algorithm in Archambeau et al. Q k k full-fill such(?) ODE?requirements whereas other systems such as the Fitz-High N 88 no.80429-443, 82 LQ2008. (i?1) 85 References 86 differential with gaussian ?? ? ? (i?1) equations k ? processes,? Neural Information Processin [2008] it does not require the computational ex- 84 ?? ?2008. ?S. 78Dondelinger, tight variational lower bounds are analytically tractable that the ad O ? provided 87 no. 429-443, M-??step 81 83 D Q p(?, X |that Y, ? 89 F. M. Filippone, Rogers and?, D.?) Husmeier, ?Ode parameter inference using KL logGirolami p(?, (X) X |and Y, ?, ?)dX 86 B. Calderhead, M. N. Lawrence, ?Accelerating bayesian inference over ?in ? ? pensive forward and backward propagation us? k | such 79gradient state variables appear in quadratic form equation 6.pp.ODE?s as the Lot 90 matching with with gaussian processes,? AISTATS, vol. 31,N 216?228, 2013. Y F (Y, ?, ? I k k ? ? 87 differential equations gaussian processes,? Neural Information Processing Systems 88 F. Dondelinger, M. Filippone, S. Rogers and D. Husmeier, ?Ode parameter infere (i) 82 LQ ?such (?)ODE requirements? 80 full-fill whereas such as the Fitz-High Na ?other ? systems ing an ODE solver. If the parameters are not 85 References ? + D Q(?) 84 ?? gradient KL 89 2008. matching with gaussian processes,? AISTATS, vol. 31, pp. 216?228, 2 88 no.E-??step 429-443, ? ??analytically tractable for? a ?? ? ? k | Fk (Y, ?, ?k I) 83 logQ(i?1) p(?,(X) X |?Y, ?, ?)dX ?p(?, known then algorithm 1 offers a grid free infer- 86 B. Calderhead, restricted of ?Accelerating ODE's k bayesian ?kN Y 81 D X |family Y, ?, ?) M. and N. Lawrence, inference over nonlin KLGirolami Y 89 F. Dondelinger, M. Filippone, S. Rogers and Neural D. Husmeier, ?Ode parameter equations with gaussian processes,? Information Processinginference Systems, using vol. 2 ence procedure for estimating the parameters. 87 90differential 85? (i)References 82 L (?) gradient matching with gaussian processes,? AISTATS, vol. 31, pp. 216?228, 2013. 84 ? M-??step 2008. Q no. 429-443, Opposite to Vrettas et al. [2011] which consider 88 61 ? We can establish touching lower bounds since we can solve the in 86 B. X Calderhead, M. Girolami and N. Lawrence, ?Accelerating bayesian 83 log p(?, | S. Y,Rogers ?, ?)dX unobserved state variables in the case of smooth- 89 F. Dondelinger, M. 87Filippone, and D. with Husmeier, ?Ode parameterNeural inference using adapti differential equations gaussian processes,? Information Pro (ODE parameters) ? 8562 References gradient matching gaussian processes,? 429-443, 2008. AISTATS, vol. 31, pp. 216?228, 2013. ing but assume the system to be fully observed if 90 ? (i)88 withno. 84 ? 86 B. Calderhead, M. Girolami and N. Lawrence, ?Accelerating bayesian infe parameters are estimated, the outlined approach Figure 2: Illustration F. Dondelinger, M. climbing" Filippone, S. Rogers and D.Neural Husmeier, ?Ode parameter of theequations "hill algo87 89 differential with gaussian processes,? Information Process offers an efficient inference framework for the rithm in Gorbach matching with difference gaussian processes,? AISTATS, vol. 31, pp. 216? 88 90 no. 429-443, 2008.. The et gradient al. [2017] 85 References much more complicated problem of inferring between the lower bound L (?) andandthe log in(?) bM. 89 Calderhead, F. Dondelinger, Filippone, S.N. Rogers and D. ?Accelerating Husmeier, ?Odebayesian parameter infe Q 86 B. M. Girolami Lawrence, infer the parameters while not all states are observed 90by gradient matching with with gaussian gaussian processes,? vol. 31, pp. Processin 216?228 87 differential equations processes,?AISTATS, Neural Information tegral is given the Kullback-Leibler divergence and still scales linearly in the states if pathwise no. 429-443, 2008. (red line). 88 inference of the RODE is done in parallel. (t) (t) (t) (i+1) (i) (i?1) 89 F. Dondelinger, M. Filippone, S. Rogers and D. Husmeier, ?Ode parameter infer 90 gradient matching with gaussian processes,? AISTATS, vol. 31, pp. 216?228, 2 The conceptual difference between the approach of Vrettas et al. [2015] and Gorbach et al. [2017] is illustrated in figure 3. Figure 3: Conceptual Difference. The red line represents an artificial function which has to be approximated. Our approach (right) is grid free and based on the minimization of the differences of the slopes. That is why convergence is vertical with each iteration step corresponding to a dashed line (thickness of the line indicating the convergence direction). Vrettas et al. [2015] approximate 4 the true process by a linearized dynamic process which is discretized (left) and improved by iterated forward and backward smoothing. 4 Experiments 4 4 We compare our approach on two established benchmark models for stochastic systems especially 4 used for weather forecasts. Vrettas et al. [2011] provide an extensive comparison of the approach of Archambeau et al. [2008] and its improvements compared to classic Kalman filtering as well as more advanced and state of the art inference schemes like 4D-Var [Le Dimet and Talagrand, 1986]. We use the reported results there as a comparison measure. 4 4 The drift function for the Lorenz96 system consists of equations of the form: fk (x(t), ?) = (xk+1 ? xk?2 )xk?1 ? xk + ? where ? is a scalar forcing parameter, x?1 = xK?1 , x0 = xK and xK+1 = x1 (with K being the number of states in the stochastic system (1)). The Lorenz96 system can be seen as a minimalistic weather model [Lorenz and Emanuel, 1998]. 6 4 4 The three dimensional Lorenz attractor is described by the parameter vector ? = (?, ?, ?) and the following time evolution: " # ?(x2 (t) ? x1 (t)) 1 dX(t) = ?x1 (t) ? x2 (t) ? x1 (t)x3 (t) dt + ? 2 dWt x1 (t)x2 (t) ? ?x3 (t) The runtime for state estimation using the approach of Vrettas et al. [2011] and our method is indicated in table 1. While parameter and state estimation are combined in one step in our approach, parameter estimation using the approach of Vrettas et al. [2011] would imply the iterative use of the smoothing algorithm and thus a multiple factor of the runtime indicated in table 1. While we solve a much more difficult problem by inferring parameters and states at the same time our runtime is only a fraction of the runtime awarded for a single run of the inner loop for parameter estimation in Vrettas et al. [2011]. Method VGPA_MF Our approach L63/D=3 31s 2.4s L96/D=40 6503s 14s L96/D=1000 17345s 383s Table 1: Runtime for one run of the smoothing algorithm of the approach of Vrettas et al. [2015] vs the runtime of our approach in parallel implementation (using 51 OU sample paths). While parameter estimation is done simultaneously in our approach, Vrettas et al. [2015] use the smoothing algorithm iteratively for state estimation in an inner loop such that the runtime for parameter estimations is multiple times higher than the indicated runtime for just one run of the smoothing algorithm. We use our method to infer the states and drift parameters for the Lorenz attractor where the dimension y is unobserved. The estimated state trajectories are shown in figure 4. Estimated Trajectory z z Simulated Trajectory x y x y Figure 4: Lorenz attractor. The Lorenz attractor trajectories are shown on the right -hand side for inferred solutions using an SDE solver, while the left-hand side plot shows the inferred trajectory using our method. Our method was able to accurately resolve the typical ?butterfly? pattern despite not observing the drift parameters as well as not observing the dimension y. Only the dimensions x and z were observed. The estimated trajectories for one sample path are also shown in the time domain in section 5.2 of the supplementary material. Our approach offers an appealing shortcut to the inference problem for stochastic dynamical systems and is robust to the noise in the diffusion term. Figure 5 shows the dependence of the inferred parameters on the variance in the diffusion term of the stochastic differential equation. Increasing the time interval of the observed process e.g. from 10 to 60 secs leads to a converging behaviour to the true parameters (figure 6). This is in contrast to the reported results of Archambeau et al. [2008], reported in Vrettas et al. [2011, Figure 29] and shows the asymptotic time consistency of our approach. Figure 5 shows, that in the near noiseless scenario we approximately identify sigma correctly. Estimating the ? term in Figure 6 is more difficult than the other two parameters in the drift 7 est 11 est 30 est 2.8 2.6 10 29 2.4 9 28 2.2 8 2 27 7 1.8 26 6 1.6 5 30 18 6 25 1 30 18 6 1.4 1 30 18 diffusion diffusion 6 1 diffusion Figure 5: Lorenz attractor. Boxplots indicate the median of the inferred parameters over 51 generated OU sample paths. Using a low variance for the diffusion term in simulating one random sample path from the SDE, our approach infers approximately the correct parameters and does not completely deteriorate if the variance is increased by a factor of 30. est 11 est 28.5 est 2.7 28 10 2.6 27.5 2.5 9 27 2.4 8 26.5 7 2.3 26 6 10 20 30 40 50 60 25.5 2.2 10 20 final time 30 40 50 60 2.1 10 20 final time 30 40 50 60 final time Figure 6: Lorenz attractor. Increasing the time interval of the observed process leads to a convergence towards the true parameters opposed to the results in [Vrettas et al., 2011, Figure 29]. function of the Lorenz attractor system, since the variance of the diffusion and the observation noise unfortunately lead to an identifiability problem for the parameter sigma, which is why longer time periods in Figure 6 do not improve the estimation accuracy for ?. est 10 est 9 8 9 7 6 8 5 4 7 3 6 2 1 1 5 10 20 0 diffusion fully obs. 2/3 observed 1/2 observed 1/3 observed Figure 7: Lorenz96. Left hand side shows the accuracy of the parameter estimation with increasing diffusion variance (right to left) for a 40 dimensional system, while the plot on the right hand side shows the accuracy with decreasing number of observations. Red dots show the results of the approach of Archambeau et al. [2008] when available as reported in Vrettas et al. [2011]. The correct parameter has the value 8 and our approach performs significantly better, while having a lower runtime and is furthermore able to include unobserved variables (right) For the Lorenz96 system our parameter estimation approach is likewise robust to the variance in the diffusion term (figure 7). It furthermore outperforms the approach of Archambeau et al. [2008] in the cases where results were reported in Vrettas et al. [2011]. The performance level is equal when, for our approach, we assume that only one third of the variables are unobserved. 8 The estimated trajectories for one sample path of the Lorenz96 system are shown in section 5.3 of the supplementary material. 5 Discussion Parameter inference in stochastic systems is a challenging but important problem in many disciplines. Current approaches are based on exploration in the parameter space which is computationally expensive and infeasible for larger systems. Using a gradient matching formulation and adapting it to the inference of random ordinary differential equations, our proposal is a flexible framework which allows to use deterministic calculus for inference in stochastic systems. While our approach tackles a much more difficult problem by combining state and parameter estimation in one step, it offers improved accuracy and is orders of magnitude faster compared to current state of the art methods based on variational inference. Acknowledgements This research was partially supported by the Max Planck ETH Center for Learning Systems and the SystemsX.ch project SignalX. 9 References C?dric Archambeau, Manfred Opper, Yuan Shen, Dan Cornford, and John S Shawe-taylor. Variational inference for diffusion processes. Neural Information Processing Systems (NIPS), 2008. Yusuke Asai and Peter E Kloeden. Numerical schemes for random odes via stochastic differential equations. Commun. Appl. Anal, 17(3):521?528, 2013. Yusuke Asai, Eva Herrmann, and Peter E Kloeden. Stable integration of stiff random ordinary differential equations. Stochastic Analysis and Applications, 31(2):293?313, 2013. Ben Calderhead, Mark Girolami and Neil D. Lawrence. Accelerating bayesian inference over nonliner differential equations with gaussian processes. Neural Information Processing Systems (NIPS), 2008. Halim Doss. Liens entre ?quations diff?rentielles stochastiques et ordinaires. In Annales de l?IHP Probabilit?s et statistiques, volume 13, pages 99?125, 1977. Geir Evensen. The ensemble kalman filter: Theoretical formulation and practical implementation. Ocean dynamics, 53(4):343?367, 2003. Nico S Gorbach, Stefan Bauer, and Joachim M Buhmann. Scalable variational inference for dynamical systems. arXiv preprint arXiv:1895944, 2017. Lars Gr?ne and PE Kloeden. Pathwise approximation of random ordinary differential equations. BIT Numerical Mathematics, 41(4):711?721, 2001. Xiaoying Han and Peter E Kloeden. Random ordinary differential equations and their numerical solution, 2017. Peter Imkeller and Bj?rn Schmalfuss. The conjugacy of stochastic and random differential equations and the existence of global attractors. Journal of Dynamics and Differential Equations, 13(2): 215?249, 2001. Arnulf Jentzen and Peter E Kloeden. Pathwise taylor schemes for random ordinary differential equations. BIT Numerical Mathematics, 49(1):113?140, 2009. Arnulf Jentzen and Peter E Kloeden. Taylor approximations for stochastic partial differential equations. SIAM, 2011. Peter E Kloeden and Arnulf Jentzen. Pathwise convergent higher order numerical schemes for random ordinary differential equations. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, volume 463, pages 2929?2944. The Royal Society, 2007. Fran?ois-Xavier Le Dimet and Olivier Talagrand. Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus A: Dynamic Meteorology and Oceanography, 38(2):97?110, 1986. Edward N Lorenz and Kerry A Emanuel. Optimal sites for supplementary weather observations: Simulation with a small model. Journal of the Atmospheric Sciences, 55(3):399?414, 1998. Simon Lyons, Amos J Storkey, and Simo S?rkk?. The coloured noise expansion and parameter estimation of diffusion processes. Neural Information Processing Systems (NIPS), 2012. Christoph Riesinger, Tobias Neckel, and Florian Rupp. Solving random ordinary differential equations on gpu clusters using multiple levels of parallelism. SIAM Journal on Scientific Computing, 38(4): C372?C402, 2016. Chris Snyder, Thomas Bengtsson, Peter Bickel, and Jeff Anderson. Obstacles to high-dimensional particle filtering. Monthly Weather Review, 136(12):4629?4640, 2008. H?ctor J Sussmann. On the gap between deterministic and stochastic ordinary differential equations. The Annals of Probability, pages 19?41, 1978. 10 Christoffer W Torn?e, Rune V Overgaard, Henrik Agers?, Henrik A Nielsen, Henrik Madsen, and R implementation, application, E Niclas Jonsson. Stochastic differential equations in nonmem : and comparison with ordinary differential equations. Pharmaceutical research, 22(8):1247?1258, 2005. Michail D Vrettas, Dan Cornford, and Manfred Opper. Estimating parameters in stochastic systems: A variational bayesian approach. 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When Cyclic Coordinate Descent Outperforms Randomized Coordinate Descent Mert G?rb?zbalaban?, Asuman Ozdaglar?, Pablo A. Parrilo?, N. Denizcan Vanli? ? Rutgers University, mg1366@rutgers.edu ? Massachusetts Institute of Technology, {asuman,parrilo,denizcan}@mit.edu Abstract The coordinate descent (CD) method is a classical optimization algorithm that has seen a revival of interest because of its competitive performance in machine learning applications. A number of recent papers provided convergence rate estimates for their deterministic (cyclic) and randomized variants that differ in the selection of update coordinates. These estimates suggest randomized coordinate descent (RCD) performs better than cyclic coordinate descent (CCD), although numerical experiments do not provide clear justification for this comparison. In this paper, we provide examples and more generally problem classes for which CCD (or CD with any deterministic order) is faster than RCD in terms of asymptotic worst-case convergence. Furthermore, we provide lower and upper bounds on the amount of improvement on the rate of CCD relative to RCD, which depends on the deterministic order used. We also provide a characterization of the best deterministic order (that leads to the maximum improvement in convergence rate) in terms of the combinatorial properties of the Hessian matrix of the objective function. 1 Introduction We consider solving smooth convex optimization problems using the coordinate descent (CD) method. The CD method is an iterative algorithm that performs (approximate) global minimizations with respect to a single coordinate (or several coordinates in the case of block CD) in a sequential manner. More specifically, at each iteration k, an index ik 2 {1, 2, . . . , n} is selected and the decision vector is updated to approximately minimize the objective function in the ik -th coordinate [3, 4]. The CD method can be deterministic or randomized depending on the choice of the update coordinates. If the coordinate indices ik are chosen in a cyclic manner from the set {1, 2, . . . , n}, then the method is called the cyclic coordinate descent (CCD) method. When ik is sampled uniformly from the set {1, 2, . . . , n}, the resulting method is called the randomized coordinate descent (RCD) method.1 The CD method has a long history in optimization and its convergence has been studied extensively in 80s and 90s (cf. [5, 12, 13, 18]). It has seen a resurgence of recent interest because of its applicability and superior empirical performance in machine learning and large-scale data analysis [7, 8]. Several recent influential papers established non-asymptotic convergence rate estimates under various assumptions. Among these are Nesterov [15], which provided the first global non-asymptotic convergence rates of RCD for convex and smooth problems (see also [11, 21, 22] for problems with non-smooth terms), and Beck and Tetruashvili [1], which provided rate estimates for block coordinate gradient descent method that yields rate results for CCD with exact minimization for quadratic problems. Tighter rate estimates (with respect to [1]) for CCD are then presented in [23]. These rate estimates suggest that CCD can be slower than RCD (precisely O(n2 ) times slower for quadratic 1 Note that there are other coordinate selection rules as well (such as the Gauss-Southwell rule [17]). However, in this paper, we focus on cyclic and randomized rules. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. problems, where n is the dimension of the problem), which is puzzling in view of the faster empirical performance of CCD over RCD for various problems (e.g., see numerical examples in [1, 24]). This gap was investigated in [24], which provided a quadratic problem that attains this performance gap. In this paper, we investigate performance comparison of deterministic and randomized coordinate descent and provide examples and more generally problem classes for which CCD (or CD with any deterministic order) is faster than RCD in terms of asymptotic worst-case convergence. Furthermore, we provide lower and upper bounds on the amount of improvement on the rate of deterministic CD relative to RCD. The amount of improvement depends on the deterministic order used. We also provide a characterization of the best deterministic order (that leads to the maximum improvement in convergence rate) in terms of the combinatorial properties of the Hessian matrix of the objective function. In order to clarify the rate comparison between CCD and RCD, we focus on quadratic optimization problems. In particular, we consider the problem2 min x2Rn 1 T x Ax, 2 (1) where A is a positive definite matrix. We consider two problem classes: i) A is a 2-cyclic matrix, whose formal definition is given in Definition 4.4, but an equivalent and insightful definition is the bipartiteness of the graph induced by the matrix A D, where D is the diagonal part of A; ii) A is an M-matrix, i.e., the off-diagonal entries of A are nonpositive. These matrices arise in a large number of applications such as in inference in attractive Gaussian-Markov random fields [14] and in minimization of quadratic forms of graph Laplacians (for which A = D W , where W Pdenotes the weighted adjacency matrix of the graph and D is the diagonal matrix given by Di,i = j Wi,j ), for example in spectral partitioning [6] and semisupervised learning [2]. We build on the seminal work of Young [27] and Varga [25] on the analysis of Gauss-Seidel method for solving linear systems of equations (with matrices satisfying certain properties) and provide a novel analysis that allows us to compare the asymptotic worst-case convergence rate of CCD and RCD for the aforementioned class of problems and establish the faster performance of CCD with any deterministic order. Outline: In the next section, we formally introduce the CCD and RCD methods. In Section 3, we present the notion of asymptotic convergence rate to compare the CCD and RCD methods and provide a motivating example for which CCD converges faster than RCD. In Section 4, we present classes of problems for which the asymptotic convergence rate of CCD is faster than that of RCD. We provide numerical experiments in Section 5 and concluding remarks in Section 6. Notation: For a matrix H, we let Hi denote its ith row and Hi,j denote its entry at the ith row and jth column. For a vector x, we let xi denote its ith entry. Throughout the paper, we reserve superscripts for iteration counters of iterative algorithms and use x? to denote the optimal solution of problem (1). For a vector x, kxk denotes its Euclidean norm and for a matrix H, ||H|| denotes its operator norm. For matrices, and ? are entry-wise operators. The matrices I and 0 denote the identity matrix and the zero matrix respectively and their dimensions can be understood from the context. 2 Coordinate Descent Method Starting from an initial point x0 2 Rn , the coordinate descent (CD) method, at each iteration k, picks a coordinate of x, say ik , and updates the decision vector by performing exact minimization along the ik th coordinate, which for problem (1) yields xk+1 = xk 1 Aik ,ik Ai k x k e i k , k = 0, 1, 2, . . . , (2) where eik is the unit vector, whose ik th entry is 1 and the rest of its entries are 0. Note that this is a special case of the coordinate gradient projection method (see [1]), which at each iteration updates a single coordinate, say coordinate ik , along the gradient component direction (with the particular step size of Ai 1,i ). The coordinate index ik can be selected according to a deterministic or randomized k k rule: 2 For ease of presentation, we consider minimization of 12 xT Ax, yet our results directly extend for problems of the type 12 xT Ax bT x for any b 6= 0. 2 ? When ik is chosen using the cyclic rule with order 1, . . . , n (i.e., ik = k (mod n) + 1), the resulting algorithm is called the cyclic coordinate descent (CCD) method. In order to write the CCD iterations in a matrix form, we introduce the following decomposition A=D L LT , where D is the diagonal part of A and L is the strictly lower triangular part of A. Then, over each epoch ` 0 (where an epoch is defined to be consecutive n iterations), the CCD iterations given in (2) can be written as (`+1)n xCCD = C x`n CCD , where C = (D L) 1 LT . (3) Note that the epoch in (3) is equivalent to one iteration of the Gauss-Seidel (GS) method applied to the first-order optimality condition of (1), i.e., applied to the linear system Ax = 0 [26]. ? When ik is chosen at random among {1, . . . , n} with probabilities {p1 , . . . , pn } independently at each iteration k, the resulting algorithm is called the randomized coordinate descent (RCD) method3 . Given the kth iterate generated by the RCD algorithm, i.e., xkRCD , we have ? ? k Ek xk+1 SD 1 A xkRCD , (4) RCD | xRCD = I where S = diag(p1 , . . . , pn ) contains the coordinate sampling probabilities and the conditional expectation Ek is taken over the random variable ik given xkRCD . Using the nested property of the expectations, the RCD iterations in expectation over each epoch ` 0 satisfy (`+1)n ExRCD 3 = R Ex`n RCD with R := I SD 1 A n . (5) Comparison of the Convergence Rates of CCD and RCD Methods In the following subsection, we define our basis of comparison for rates of CCD and RCD methods. To measure the performance of these methods, we use the notion of the average worst-case asymptotic rate that has been studied extensively in the literature for characterizing the rate of iterative algorithms [25]. In Section 3.2, we construct an example, for which the rate of CCD is more than twice the rate of RCD. This raises the question whether the best known convergence rates of CCD in the literature are tight or whether there exist a class of problems for which CCD provably attains better convergence rates than the best known rates for RCD, a question which we will answer in Section 4. 3.1 Asymptotic Rate of Converge for Iterative Algorithms Consider an iterative algorithm with update rule x(`+1)n = Cx`n (e.g., the CCD algorithm). The reduction in the distance to the optimal solution of the iterates generated by this algorithm after ` epochs is given by x`n x? C ` (x0 x? ) = . (6) 0 ? ||x x || ||x0 x? || Note that the right hand side of (6) can be as large as C ` , hence in the worst-case, the average decay of distance at each epoch of this algorithm is C ` 1/` 1/` . Over any finite epochs ` 1, we 1/` have C ?(C) and C ! ?(C) as ` ! 1 by Gelfand?s formula. Thus, we define the asymptotic worst-case convergence rate of an iterative algorithm (with iteration matrix C) as follows ! x`n x? 1 Rate(CCD) := lim sup log = log (?(C)) . (7) `!1 x0 2Rn ` ||x0 x? || ` ` We emphasize that this notion has been used extensively for studying the performance of iterative methods such as GS and Jacobi methods [5, 18, 25, 27]. Note that according to our definition in (7), larger rate means faster algorithm and we will use these terms interchangably in throughout the paper. 3 sd 3 Analogously, for a randomized algorithm with expected update rule Ex(`+1)n = R Ex`n (e.g., the RCD algorithm), we consider the asymptotic convergence of the expected iterate error E(x`n ) x? and define the asymptotic worst-case convergence rate as ! E(x`n ) x? 1 Rate(RCD) := lim sup log = log (?(R)) , (8) `!1 x0 2Rn ` ||x0 x? || Note that in (8), we use the distance of the expected iterates Ex`n x? as our convergence criterion. One can also use the expected distance (or the squared distance) of the iterates E x`n x? as the convergence criterion, which is a stronger convergence criterion than the one in (8). This follows since E x`n x? Ex`n x? by Jensen?s inequality and any convergence rate on `n ? E x x immediately implies at least the same convergence rate on Ex`n x? as well. Since we consider the reciprocal case, i.e., obtain a convergence rate on Ex`n x? and show that it is slower than that of CCD, our results naturally imply that the convergence rate on E x`n x? is also slower than that of CCD. 3.2 A Motivating Example In this section, we provide an example for which the (asymptotic worst-case convergence) rate of CCD is better than the one of RCD and building on this example, in Section 4, we construct a class of problems for which CCD attains a better rate than RCD. For some positive integer n 1, consider the 2n ? 2n symmetric matrix ? 1 0 0n?n A = I L LT , where L = 2 n?n , (9) n 1n?n 0n?n and 1n?n is the n ? n matrix with all entries equal to 1 and 0n?n is the n ? n zero matrix. Noting that A has a special structure (A is equal to the sum of the identity matrix and the rank-two matrix L LT ), it is easy to check that 1 1/n and 1 + 1/n are eigenvalues of A with the corresponding T T 11?n ] . The remaining 2n 2 eigenvalues of A are eigenvectors [11?n 11?n ] and [11?n equal to 1. The iteration matrix of the CCD algorithm when applied to the problem in (1) with the matrix (9) can be found as ? 1 0 1 C = n?n n12 n?n . 0n?n n3 1n?n The eigenvalues of C are all zero except the eigenvalue of 1/n2 with the corresponding eigenvector [n11?n , 11?n ]T . Therefore, ?(C) = 1/n2 and Rate(CCD) = log(?(C)) = 2 log n. On the other hand, the spectral radius of the expected iteration matrix of RCD can be found as ? ?n 1 min (A) ?(R) = 1 1 , min (A) = n n which yields Rate(RCD) = log(?(R)) ? log n. Thus, we conclude Rate(CCD) Rate(RCD) 2, for all n 1. That is, CCD is at least twice as fast as RCD in terms of the the asymptotic rate. This motivates us to investigate if there exists a more general class of problems for which the asymptotic worst-case rate of CCD is larger than that of RCD. The answer to this question turns out to be positive as we describe in the following section. 4 When Deterministic Orders Outperform Randomized Sampling In this section, we present special classes of problems (of the form (1)) for which the asymptotical worst-case rate of CCD is larger than that of RCD. We begin our discussion by highlighting the main assumption we will use in this section. Assumption 4.1. A is a symmetric positive definite matrix whose smallest eigenvalue is ? and the diagonal entries of A are 1. 4 If A is a positive semidefinite matrix, then our results will still hold, where ? is the smallest non-zero eigenvalue of A and x? is the projection of x0 onto the null space of A. Moreover, given any positive definite matrix A with diagonals D 6= I, the diagonal entries of the preconditioned matrix D 1/2 AD 1/2 are 1. Therefore, Assumption 4.1 is mild. The relationship between the smallest eigenvalue of the original matrix and the preconditioned matrix are as follows. Let > 0 and Lmax denote the smallest eigenvalue and the largest diagonal entry of the original matrix, respectively. Then, the smallest eigenvalue of the preconditioned matrix satisfies ? /Lmax . Remark 4.2. For the RCD algorithm, the coordinate index ik 2 {1, . . . , n} (at iteration k) can be chosen using different probability distributions {p1 , . . . , pn }. Two common choices of distributions A are pi = n1 , for all i 2 {1, . . . , n} and pi = PN i,iA [15]. Since by Assumption 4.1, the diagonal J=1 j,j entries of A are 1, both of these distributions reduces to pi = n1 , for all i 2 {1, . . . , n}. Therefore, in the rest of the paper, we consider the RCD algorithm with uniform and independent coordinate selection at each iteration. In the following lemma, we characterize the spectral radius of the RCD method. This worst-case rate has been presented in many works in the literature for strongly convex optimization problems [15, 26]. The proof is deferred to Appendix. Lemma 4.3. Suppose Assumption 4.1 holds. Then, the spectral radius of the expected iteration matrix R of the RCD algorithm (defined in (5)) is given by ? ? ?n ?(R) = 1 . (10) n 4.1 Convergence Rate of CCD for 2-Cyclic Matrices In this section, we introduce the class of 2-cyclic matrices and show that the asymptotic worst-case rate of CCD is more than two times faster than that of RCD. Definition 4.4 (2-Cyclic Matrix). A matrix H is 2-cyclic if there exists a permutation matrix P such that ? 0 B1 T P HP = D + , (11) B2 0 where the diagonal null submatrices are square and D is a diagonal matrix. This definition can be interpreted as follows. Let H be a 2-cyclic matrix, i.e., H satisfies (11). Then, the graph induced by the matrix H D is bipartite. The definition in (11) is first introduced in [27], where it had an alternative name: Property A. A generalization of this property is later introduced by Varga to the class of p-cyclic matrices [25] where p 2 can be arbitrary. We next introduce the following definition that will be useful in Theorem 4.12 and explicitly identify the class of matrices that satisfy this definition in Lemma 4.6. Definition 4.5 (Consistently Ordered Matrix). For a matrix H, let H = HD HL HU be its decomposition such that HD is a diagonal matrix, HL (and HU ) is a strictly lower (and upper) triangular matrix. If the eigenvalues of the matrix ?HL + ?HU HD are independent of ? for any 2 R and ? 6= 0, then H is said to be consistently ordered. Lemma 4.6. [27, Theorem 4.5] A matrix H is 2-cyclic if and only if there exists a permutation matrix P such that P HP T is consistently ordered. In the next theorem, we characterize the convergence rate of CCD algorithm applied to a 2-cyclic matrix. Since ?(R) 1 ? by Lemma 4.3, the following theorem indicates that the spectral radius of the CCD iteration matrix is smaller than ?2 (R). Theorem 4.7. Suppose Assumption 4.1 holds and A is a consistently ordered 2-cyclic matrix. Then, the spectral radius of the CCD algorithm is given by ?(C) = (1 2 ?) . Remark 4.8. Note that our motivating example in (9) is an example of a consistently ordered 2-cyclic matrix, for which Theorem 4.7 is applicable. In particular, for the example in (9), we can apply Theorem 4.7 with ? = 1 1/n leading to ?(C) = 1/n2 , which exactly coincides with our previous computations of ?(C) in Section 3.2. We also give an example in Appendix F, for which CCD is twice faster than RCD for any arbitrary initialization with probability one. 5 The following corollary states that the asymptotic worst-case rate of CCD is more than twice larger than that of RCD for quadratic problems whose Hessian is a 2-cyclic matrix. This corollary directly follows by Theorem 4.7 and definitions (7)-(8). Corollary 4.9. Suppose Assumption 4.1 holds and A is a consistently ordered 2-cyclic matrix. Then, the asymptotic worst-case rates of CCD and RCD satisfy Rate(CCD) = 2?n , Rate(RCD) where ?n := log(1 ?) . n log 1 n? (12) In the following remark, we highlight several properties of the constant ?n . Remark 4.10. ?n is a monotonically increasing function of n over the interval [1, 1), where ?1 = 1 ?) and limn!1 ?n = log(1 > 1. Furthermore, lim?!0+ ?n = 1. ? We emphasize that the CCD method applied to 1 is equivalent to the Gauss-Seidel (GS) algorithm applied to the linear system Ax = 0 and when A is a 2-cyclic matrix, the GS algorithm is twice as fast as the Jacobi algorithm [25, 27]. Hence, when A is a 2-cyclic matrix and ? is sufficiently small, the RCD method is approximately as fast as the Jacobi algorithm. 4.2 Convergence Rate of CCD for Irreducible M-Matrices In this section, we first define the class of M-matrices and then present the convergence rate of the CCD algorithm applied to quadratic problems whose Hessian is an M-matrix. Definition 4.11 (M-matrix). A real matrix A with Ai,j ? 0 for all i 6= j is an M-matrix if A has the decomposition A = sI B such that B 0 and s ?(B). We emphasize that M-matrices arise in a variety of applications such as in belief propagation over Gaussian graphical models [14] and in distributed control of positive systems [20]. Furthermore, graph Laplacians are M-matrices, therefore solving linear systems with M-matrices (or equivalently solving (1) for an M-matrix A) arise in a variety of applications for analyzing random walks over graphs as well as distributed optimization and consensus problems over graphs (cf. [10] for a survey). For quadratic problems, the Hessian is an M-matrix if and only if the gradient descent mapping is an isotone operator [5, 22] and in Gaussian graphical models, M-matrices are often referred as attractive models [14]. In the following theorem, we provide lower and upper bounds on the spectral radius of the iteration matrix of CCD for quadratic problems, whose Hessian matrix is an irreducible M-matrix. In particular, we show that the spectral radius of the iteration matrix of CCD is strictly smaller than that of RCD for irreducible M-matrices. Theorem 4.12. Suppose Assumption 4.1 holds, A is an irreducible M-matrix and n 2. Then, the iteration matrix of the CCD algorithm C = (I L) 1 LT satisfies the following inequality (1 ?)2 ? ?(C) ? 1 ? , 1+? (13) where the inequality on the left holds with equality if and only if A is a consistently ordered matrix. An immediate consequence of Theorem 4.12 is that for quadratic problems whose Hessian is an irreducible M-matrix, the best cyclic order that should be used in CCD can be characterized as follows. Remark 4.13. The standard CCD method follows the standard cyclic order (1, 2, . . . , n) as described in Section 2. However, we can construct a CCD method that follows an alternative deterministic order by considering a permutation ? of {1, 2, . . . , n}, and choosing the coordinates according to the order (?(1), ?(2), . . . , ?(n)) instead. For any given order ?, (1) can be reformulated as follows min x? 2Rn 1 T x A? x ? , 2 ? where A? := P? AP?T and x? = P? x, where P? is the corresponding permutation matrix of ?. Supposing that Assumption 4.1 holds, the corresponding CCD iterations for this problem can be written as follows x(`+1)n = C? x`n ? ? , where C? = (I 6 L? ) 1 LT? and L? = P? LP? . If A is an irreducible M-matrix and satisfies Assumptions 4.1, then so does A? . Consequently, Theorem 4.12 yields the same upper and lower bounds (in (13)) on ?(C? ) as well, i.e., the spectral radius of the iteration matrix of CCD with any cyclic order ? satisfies (1 ?)2 ? ?(C? ) ? 1 ? , 1+? (14) where the inequality on the left holds with equality if and only if A? is a consistently ordered matrix. Therefore, if a consistent order ? ? exists, then the CCD method with the consistent order ? ? attains the smallest spectral radius (or equivalently, the fastest asymptotic worst-case convergence rate) among the CCD methods with any cyclic order. Remark 4.14. The irreducibility of A is essential to derive the lower bound in (13) of Theorem 4.12. However, the upper bound in (13) holds even when A is a reducible matrix. We next compare the spectral radii bounds for CCD (given in Theorem 4.12) and RCD (given in Lemma 4.3). Since ? > 0, the right-hand side of (13) can be relaxed to (1 ?)2 ? ?(C) < 1 ?. A direct consequence of this inequality is the following corollary, which states that the asymptotic worst-case rate of CCD is strictly better than that of RCD at least by a factor that is strictly greater than 1. Corollary 4.15. Suppose Assumption 4.1 holds, A is an irreducible M-matrix and n 2. Then, the asymptotic worst-case rates of CCD and RCD satisfy 1 < ?n < Rate(CCD) ? 2?n , Rate(RCD) where ?n := log(1 ?) , n log 1 n? (15) and the inequality on the right holds with equality if and only if A is a consistently ordered matrix. In the following corollary, we highlight that as the smallest eigenvalue of A goes to zero, the asymptotic worst-case rate of the CCD algorithm becomes twice the asymptotic worst-case rate of the RCD algorithm. Corollary 4.16. Suppose Assumption 4.1 holds, A is an irreducible M-matrix and n 2. Then, we have Rate(CCD) lim = 2. ?!0+ Rate(RCD) 5 Numerical Experiments In this section, we compare the performance of CCD and RCD through numerical examples. First, we consider the quadratic optimization problem in (1), where A is an n ? n matrix defined as follows ? 1 0 0 T A = I L L , where L = (16) n n 1 0 , ? n 2 2 and 1 n2 ? n2 is the n2 ? n2 matrix with all entries equal to 1. Here, it can be easily checked that A is a consistently ordered, 2-cyclic matrix. By Theorem 4.7 and Corolloary 4.9, the asymptotic worst-case convergence rate of CCD on this example is 2?n = 2 log(1 ?) log(0.5) ? 2.77 ? = 1 n log 1 n 50 log 1 200 (17) times faster than that of RCD. This is illustrated in Figure 1 (left), where the distance to the optimal solution is plotted in a logarithmic scale over epochs. Note that even if our results our asymptotic, we see the same difference in performances on the early epochs (for small `). On the other hand, when the matrix A is not consistently ordered, according to Theorem 4.12, CCD is still faster but the difference in the convergence rates decreases with respect to the consistent ordering case. To illustrate this, we need to generate an inconsistent ordering of the matrix A. For this goal, we generate a permutation matrix P and replace A with AP := P AP T in the optimization problem (1) (This is equivalent to solving the system AP x = 0) so that AP is not consistently ordered (We generate P randomly and compute AP ). Figure 1 (right) shows that for this inconsistent ordering CCD is still faster compared to RCD, but not as fast (the slope of the decay of error line in blue marker is less steep) predicted by our theory. 7 Consistent Ordering, Worst-Case Initialization Inconsistent Ordering, Worst-Case Initialization ?4 ?6 x? || ?2 ?4 ?6 ?8 log ||x` ?2 x? || 0 log ||x` 0 ?8 ?10 ?10 CCD RCD Expected RCD ?12 ?14 1 2 3 4 CCD RCD Expected RCD ?12 5 6 7 Number of Epochs ` 8 9 ?14 10 1 2 3 4 5 6 7 Number of Epochs ` 8 9 10 Figure 1: Distance to the optimal solution of the iterates of CCD and RCD for the cyclic matrix in (16) (left figure) and a randomly permuted version of the same matrix (right figure) where the y-axis is on a logarithmic scale. The left (right) figure corresponds to the consistent (inconsistent) ordering for the same quadratic optimization problem. M-Matrix, Worst-Case Initialization M-Matrix, Random Initialization ?4 ?4 ? ?2 ||x` x? || ||x0 x? || ?2 ? 0 ||x` x? || ||x0 x? || 0 ? ? log ?6 log ?6 ?8 ?8 ?10 ?12 ?10 CCD RCD Expected RCD 0 20 40 60 Number of Epochs ` 80 ?12 100 CCD RCD Expected RCD 0 20 40 60 Number of Epochs ` 80 100 Figure 2: Distance to the optimal solution of the iterates of CCD and RCD for the M-matrix matrix in (18) for the worst-case initialization (left figure) and a random initialization (right figure). We next consider the case, where A is an irreducible positive definite M-matrix. In particular, we consider the matrix A = (1 + )I 1n?n , (18) 1 where 1n?n is the n ? n matrix with all entries equal to 1 as before and = n+5 . We set n = 100 and plot the performance of CCD and RCD methods for the quadratic problem defined by this matrix. In Figure 2, we compare the convergence rate of CCD and RCD for an initial point that corresponds to a worst-case (left figure) and for a random choice of an initial point (right figure). We conclude that the asymptotic rate of CCD is faster than that of RCD demonstrating our results in Theorem 4.12 and Corolloary 4.15. 6 Conclusion In this paper, we compare the CCD and RCD methods for quadratic problems, whose Hessian is a 2-cyclic matrix or an M-matrix. We show by a novel analysis that for these classes of quadratic problems, CCD is always faster than RCD in terms of the asymptotic worst-case rate. We also give a characterization of the best cyclic order to use in the CCD algorithm for these classes of problems and show that with the best cyclic order, CCD enjoys more than a twice faster asymptotic worst-case rate with respect to RCD. We also provide numerical experiments that show the tightness of our results. References [1] A. Beck and L. Tetruashvili. On the convergence of block coordinate descent type methods. SIAM Journal on Optimization, 23(4):2037?2060, 2013. [2] M. Belkin, P. Niyogi, and V. Sindhwani. Manifold regularization: A geometric framework for learning from labeled and unlabeled examples. Journal of Machine Learning Research, 7:2399?2434, 2006. 8 [3] D. P. Bertsekas. Nonlinear programming. Athena Scientific, 1999. [4] D. P. Bertsekas. Convex Optimization Algorithms. Athena Scientific, 2015. [5] D. P. Bertsekas and J. N. Tsitsiklis. Parallel and Distributed Computation: Numerical Methods. PrenticeHall, Inc., 1989. [6] F. R. K. Chung. Spectral Graph Theory. American Mathematical Society, 1997. [7] J. Friedman, T. Hastie, H. H?fling, and R. Tibshirani. Pathwise coordinate optimization. The Annals of Applied Statistics, 1(2):302?332, 2007. [8] J. Friedman, T. Hastie, and R. Tibshirani. Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software, 33(1):1?22, 2010. [9] J. F. C. Kingman. A convexity property of positive matrices. The Quarterly Journal of Mathematics, 12(1):283?284, 1961. [10] S. J. Kirkland and M. Neumann. Group inverses of M-matrices and their applications. CRC Press, 2012. [11] Z. Lu and L. Xiao. On the complexity analysis of randomized block-coordinate descent methods. Mathematical Programming, 152(1):615?642, 2015. [12] Z.-Q. Luo and P. Tseng. On the convergence of the coordinate descent method for convex differentiable minimization. Journal of Optimization Theory and Applications, 72(1):7?35, 1992. [13] Z.-Q. Luo and P. Tseng. Error bounds and convergence analysis of feasible descent methods: a general approach. Annals of Operations Research, 46(1):157?178, 1993. [14] D. M. Malioutov, J. K. Johnson, and A. S. Willsky. Walk-sums and belief propagation in gaussian graphical models. Journal of Machine Learning Research, 7:2031?2064, 2006. [15] Y. Nesterov. Efficiency of coordinate descent methods on huge-scale optimization problems. SIAM Journal on Optimization, 22(2):341?362, 2012. [16] Roger D. Nussbaum. Convexity and log convexity for the spectral radius. Linear Algebra and its Applications, 73(Supplement C):59 ? 122, 1986. [17] J. Nutini, M. Schmidt, I. H. Laradji, M. Friedlander, and H. Koepke. Coordinate descent converges faster with the gauss-southwell rule than random selection. In Proceedings of the 32nd International Conference on International Conference on Machine Learning, pages 1632?1641, 2015. [18] J. Ortega and W. Rheinboldt. Iterative Solution of Nonlinear Equations in Several Variables. Society for Industrial and Applied Mathematics, 2000. [19] R. J. Plemmons. M-matrix characterizations.I?nonsingular m-matrices. Linear Algebra and its Applications, 18(2):175 ? 188, 1977. [20] A. Rantzer. Distributed control of positive systems. ArXiv:1203.0047, 2014. [21] P. Richt?rik and M. Tak??c. Parallel coordinate descent methods for big data optimization. Mathematical Programming, 156(1):433?484, 2016. [22] A. Saha and A. Tewari. On the nonasymptotic convergence of cyclic coordinate descent methods. SIAM Journal on Optimization, 23(1):576?601, 2013. [23] R. Sun and M. Hong. Improved iteration complexity bounds of cyclic block coordinate descent for convex problems. In Advances in Neural Information Processing Systems 28, pages 1306?1314. 2015. [24] R. Sun and Y. Ye. Worst-case Complexity of Cyclic Coordinate Descent: O(n2 ) Gap with Randomized Version. ArXiv:1604.07130, 2016. [25] R. S. Varga. Matrix iterative analysis. Springer Science & Business Media, 2009. [26] S. J. Wright. Coordinate descent algorithms. Mathematical Programming, 151(1):3?34, 2015. [27] D. M. Young. Iterative solution of large linear systems. Academic Press, 1971. 9 

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Active Learning from Peers Keerthiram Murugesan Jaime Carbonell School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 {kmuruges,jgc}@cs.cmu.edu Abstract This paper addresses the challenge of learning from peers in an online multitask setting. Instead of always requesting a label from a human oracle, the proposed method first determines if the learner for each task can acquire that label with sufficient confidence from its peers either as a task-similarity weighted sum, or from the single most similar task. If so, it saves the oracle query for later use in more difficult cases, and if not it queries the human oracle. The paper develops the new algorithm to exhibit this behavior and proves a theoretical mistake bound for the method compared to the best linear predictor in hindsight. Experiments over three multitask learning benchmark datasets show clearly superior performance over baselines such as assuming task independence, learning only from the oracle and not learning from peer tasks. 1 Introduction Multitask learning leverages the relationship between the tasks to transfer relevant knowledge from information-rich tasks to information-poor ones. Most existing work in multitask learning focuses on how to take advantage of these task relationships, either by sharing data directly [1] or learning model parameters via cross-task regularization techniques [2, 3, 4]. This paper focuses on a specific multitask setting where tasks are allowed to interact by requesting labels from other tasks for difficult cases. In a broad sense, there are two settings to learn multiple related tasks together: 1) batch learning, in which an entire training set is available to the learner 2) online learning, in which the learner sees the data sequentially. In recent years, online multitask learning has attracted increasing attention [5, 6, 7, 8, 9, 10]. The online multitask setting starts with a learner at each round t, receiving an example (along with a task identifier) and predicts the output label. One may also consider learning multiple tasks simultaneously by receiving K examples for K tasks at each round t. Subsequently, the learner receives the true label and updates the model(s) as necessary. This sequence is repeated over the entire data, simulating a data stream. In this setting, the assumption is that the true label is readily available for the task learner, which is impractical in many applications. Recent works in active learning for sequential problems have addressed this concern by allowing the learner to make a decision on whether to ask the oracle to provide the true label for the current example and incur a cost or to skip this example. Most approach in active learning for sequential problems use a measure such a confidence of the learner in the current example [11, 12, 13, 14, 15]. In online multitask learning, one can utilize the task relationship to further reduce the total number of labels requested from the oracle. This paper presents a novel active learning for the sequential decision problems using peers or related tasks. The key idea is that when the learner is not confident on the current example, the learner is allowed to query its peers, which usually has a low cost, before requesting a true label from the oracle and incur a high cost. Our approach follows a perceptron-based update rule in which the model for a given task is updated only when the prediction for that task is 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. in error. The goal of an online learner in this setting is to minimize errors attempting to reach the performance of the full hindsight learner and at the same time, reduce the total number of queries issued to the oracle. There are many useful application areas for online multitask learning with selective sampling, including optimizing financial trading, email prioritization and filtering, personalized news, crowd source-based annotation, spam filtering and spoken dialog system, etc. Consider the latter, where several automated agents/bots servicing several clients. Each agent is specialized or trained to answer questions from customers on a specific subject such as automated payment, troubleshooting, adding or cancelling services, etc. In such setting, when one of the automated agents cannot answer a customer?s question, it may request the assistance of another automated agent that is an expert in the subject related to that question. For example, an automated agent for customer retention may request some help from an automated agent for new services to offer new deals for the customer. When both the agents could not answer the customer?s question, the system may then direct the call to a live agent. This may reduce the number of service calls directed to live agents and the cost associated with such requests. Similarly in spam filtering, where some spam is universal to all users (e.g. financial scams), some messages might be useful to certain affinity groups, but spam to most others (e.g. announcements of meditation classes or other special interest activities), and some may depend on evolving user interests. In spam filtering each user is a task, and shared interests and dis-interests formulate the inter-task relationship matrix. If we can learn the task relationship matrix as well as improving models from specific decisions from peers on difficult examples, we can perform mass customization of spam filtering, borrowing from spam/not-spam feedback from users with similar preferences. The primary contribution of this paper is precisely active learning for multiple related tasks and its use in estimating per-task model parameters in an online setting. 1.1 Related Work While there is considerable literature in online multitask learning, many crucial aspects remain largely unexplored. Most existing work in online multitask learning focuses on how to take advantage of task relationships. To achieve this, Lugosi et. al [7] imposed a hard constraint on the K simultaneous actions taken by the learner in the expert setting, Agarwal et. al [16] used matrix regularization, and Dekel et. al [6] proposed a global loss function, as an absolute norm, to tie together the loss values of the individual tasks. In all these works, their proposed algorithms assume that the true labels are available for each instance. Selective sampling-based learners in online setting, on the other hand, decides whether to ask the human oracle for labeling of difficult instances [11, 12, 13, 14, 15]. It can be easily extended to online multitask learning setting by applying selective sampling for each individual task separately. Saha et. al [9] formulated the learning of task relationship matrix as a Bregman-divergence minimization problem w.r.t. positive definite matrices and used this task-relationship matrix to naively select the instances for labelling from the human oracle. Several recent works in online multitask learning recommended updating all the task learners on each round t [10, 9, 8]. When a task learner makes a mistake on an example, all the tasks? model parameters are updated to account for the new examples. This significantly increases the computational complexity at each round, especially when the number of tasks is large [17]. Our proposed method avoids this issue by updating only the learner of the current example and utilize the knowledge from peers only when the current learner requested them. This work is motivated by the recent interests in active learning from multiple (strong or weak) teachers [18, 19, 12, 20, 21, 22]. Instead of single all-known oracle, these papers assume multiple oracles (or teachers) each with a different area of expertise. At round t, some of the teachers are experts in the current instance but the others may not be confident in their predicted labels. Such learning setting is very common in crowd-sourcing platform where multiple annotators are used to label an instance. Our learning setting is different from their approaches where, instead of learning from multiple oracles, we learn from our peers (or related tasks) without any associated high cost. Finally, our proposed method is closely related to learning with rejection option [23, 24] where the learner may choose not to predict label for an instance. To reject an instance, they use a measure of 2 1. Receive an example x(t) for the task k 2. If the task k is not confident in the prediction for this example, ask the peers or related tasks whether they can give a confident label to this example. 3. If the peers are not confident enough, ask the oracle for the true label y (t) . Figure 1: Proposed learning approach from peers. confidence to identify difficult instances. We use a similar approach to identify when to query peers and when to query the human oracle for true label. 2 Problem Setup Suppose we are given K tasks where the k th task is associated with Nk training examples. For brevity, we consider a binary classification problem for each task, but the methods generalize to multi-class settings and are also applicable to regression tasks. We denote by [N ] consecutive integers  (i) (i) Nk (i) ranging from 1 to N . Let (xk , yk ) i=1 be data for task k where xk ? Rd is the ith instance (i) from the k th task and yk is its corresponding true label. When the notation is clear from the context, we drop the index k and write ((x(i) , k), y (i) ). Let {wk? }k?[K] be any set of arbitrary vectors where wk? ? Rd . The hinge losses on the example


(t)? (t)? ? (x(t) , k), y (t) are given by `kk = 1 ? y (t) hx(t) , wk? i + and `km = 1 ? y (t) hx(t) , wm i +, (t) (t) respectively, where (z)+ = max(0, z). Similarly, we define hinge losses `kk and `km for the linear (t) predictors {wk }k?[K] learned at round t. Let Z (t) be a Bernoulli random variable to indicate whether the learner requested a true label for the example x(t) . Let M (t) be a binary variable to indicate whether the learner made a mistake on the example x(t) . We expected hingelosses  use the following  P P (t) (t) (t)? (t) (t) (t)? ? ? for our theoretical analysis: Lkk = E Z `kk and Lkm = E Z `km . tM tM We start with our proposed active learning from peers algorithm based on selective sampling for online multitask problems and study the mistake bound for the algorithm in Section 3. We report our experimental results and analysis in Section 4. Additionally, we extend our learning algorithm to learning multiple task in parallel in the supplementary. 3 Learning from Peers We consider multitask perceptron for our online learning algorithm. On each round t, we receive an example (x(t) , k) from task k 1 . Each perceptron learner for the task k maintains a model represented (t?1) by wk learned from examples received until round t ? 1. Task k predicts a label for the received (t?1) example x(t) using hk (x(t) ) = hwk , x(t) i 2 . As in the previous works [11, 12, 23], we use (t) |hk (x )| to measure the confidence of the k th task learner on this example. When the confidence is higher, the learner doesn?t require the need to request the true label y (t) from the oracle. Built on this idea, [11] proposed a selective sampling algorithm using the margin |hk (x(t) )| to decide whether to query the oracle or not. Intuitively, if |hk (x(t) )| is small, then the k th task learner is not confident in the prediction of x(t) and vice versa. They consider a Bernoulli random variable P (t) for the event |hk (x(t) )| ? b1 with probability b1 +|hbk1(x(t) )| for some predefined constant b1 ? 0. If 1 We will consider a different online learning setting later in the supplementary section where we simultaneously receive K examples at each round, one for each task k (t?1) (t?1) 2 We also use the notation p?kk = hwk , x(t) i and p?km = hwm , x(t) i 3 P (t) = 1 (confidence is low), then the k th learner requests the oracle for the true label. Similarly when P (t) = 0 (confidence is high), the learner skips the request to the oracle. This considerably saves a lot of label requests from the oracle. When dealing with multiple tasks, one may use similar idea and apply selective sampling for each task individually [25]. Unfortunately, such approach doesn?t take into account the inherent relationship between the tasks. In this paper, we consider a novel active learning (or selective sampling) for online multitask learning to address the concerns discussed above. Our proposed learning approach can be summarized in Figure 1. Unlike in the previous work [8, 9, 10], we update only the current task parameter wk when we made a mistake at round t, instead of updating all the task model parameters wm , ?m ? [K], m 6= k. Our proposed method avoids this issue by updating only the learner of the current example and share the knowledge from peers only when the assistance is needed. In addition, the task relationship is taken into account, to measure whether the peers are confident in predicting this example. This approach provides a compromise between learning them independently and learning them by updating all the learners when a specific learner makes a mistake. As in traditional selective sampling algorithm [11], we consider a Bernoulli random variable P (t) for the event |hk (x(t) )| ? b1 with probability b1 +|hbk1(x(t) )| . In addition, we consider a second Bernoulli random variable Q(t) for the event |hm (x(t) )| ? b2 with probability b2 . P (t?1) b2 + m?[K],m6=k ?km |hm (x(t) )| The idea is that when the weighted sum of the confidence of the peers on the current example is high, then we use the predicted label y?(t) from the peers for the perceptron update instead of requesting a true label y (t) from the oracle. In our experiment in Section 4, we consider the confidence of most related task instead of the weighted sum to reduce the computational complexity at each round. We set Z (t) = P (t) Q(t) and set M (t) = 1 if we made a mistake at round t i.e., (y (t) 6= y?(t) ) (only when the label is revealed/queried). The pseudo-code is in Algorithm 1. Line 14 is executed when we request a label from the oracle or when peers are confident on the label for the current example. Note the two terms in (M (t) Z (t) y (t) + Z? (t) y?(t) ) are mutually exclusive (when P (t) = 1). Line (15-16) computes the relationship between tasks ?km based on the recent work by [10]. It maintains a distribution over peers w.r.t the current task. The value of ? is updated at each round using the cross-task error `km . In addition, we use the ? to get the confidence of the most-related task rather than the weighted sum of the confidence of the peers to get the predicted label from the peers (see Section 4 for more details). When we are learning with many tasks [17], it provides a faster computation without significantly compromising the performance of the learning algorithm. One may use different notion of task relationship based on the application at hand. Now, we give the bound on the expected number of mistakes. 
T Theorem 1. let Sk = (x(t) , k), y (t) t=1 be a sequence of T examples given to Algorithm 1 where x(t) ? Rd , y (t) ? {?1, +1} and X = maxt kx(t) k. Let P (t) be a Bernoulli random variable for the event |hk (x(t) )| ? b1 with probability b1 +|hbk1(x(t) )| and let Q(t) be a Bernoulli random variable for the event |hm (x(t) )| ? b2 with probability b2 . b2 +maxm?[K] |hm (x(t) )| m6=k Let Z (t) = P (t) Q(t) and M (t) = I(y (t) 6= y?(t) ). If the Algorithm 1 is run with b1 > 0 and b2 > 0 (b2 ? b1 ), then ?t ? 1 and ? > 0 we have X  
b2 (2b1 + X 2 )2 (t) ? 2 E M ? kwk? k2 + max kwm k ? 8b1 ? m?[K],m6=k t?[T ]  X 2
 ? ? km + 1+ Lkk + max L 2b1 m?[K],m6=k Then, the expected number of label requests to the oracle by the algorithm is X t b1 b2 b1 + |hk (x(t) )| b2 + maxm?[K] |hm (x(t) )| m6=k 4 Algorithm 1: Active Learning from Peers 1 2 3 4 5 6 Input : b1 > 0, b2 > 0 s.t., b2 ? b1 , ? > 0, Number of rounds T (0) Initialize wm = 0 ?m ? [K], ? (0) . for t = 1 . . . T do Receive (x(t) , k) (t) (t?1) Compute p?kk = hx(t) , wk i (t) (t) Predict y? = sign(? pkk ) b1 Draw a Bernoulli random variable P (t) with probability (t) b1 +|p ?kk | if P (t) = 1 then (t) (t?1) Compute p?km = hx(t) , wm i ?m 6= k, m ? [K] P (t?1) (t) (t) Compute p? = m6=k,m?[K] ?km p?km and y?(t) = sign(? p(t) ) 7 8 9 Draw a Bernoulli random variable Q(t) with probability 10 b2 b2 +|p ?(t) | end Set Z (t) = P (t) Q(t) & Z? (t) = P (t) (1 ? Q(t) ) Query true label y (t) if Z (t) = 1 and set M (t) = 1 if y?(t) 6= y (t) (t) (t?1) Update wk = wk + (M (t) Z (t) y (t) + Z? (t) y?(t) )x(t) Update ? : 11 12 13 14 15 16 (t?1) (t) ?km ?km e? = P 17 m0 ?[K] m0 6=k Z (t) `(t) km ? ?km0 e? (t?1) Z (t) `(t) ? km0 m ? [K], m 6= k (1) end The proof is given in Appendix A. It follows q from Theorem 1 in [11] and Theorem 1 in [10] and X2 2 ? kwk k 2 , 2 2 ? Lkk where b1 = X2 1 + kw4? ? 2 ? . Theorem 1 states that the quality k kX ? kk and the maximum of {L ? km }m?[K],m6=k . In other words, the worstof the bound depends on both L setting b2 = b1 + + case regret will be lower if the k th true hypothesis wk? can predict the labels for training examples in both the k th task itself as well as those in all the other related tasks in high confidence. In addition, we consider a related problem setting in which all the K tasks receive an example simultaneously. We give the learning algorithm and mistake bound for this setting in Appendix B. 4 Experiments We evaluate the performance of our algorithm in the online setting. All reported results in this section are averaged over 10 random runs on permutations of the training data. We set the value of b1 = 1 for all the experiments and the value of b2 is tuned from 20 different values. Unless otherwise specified, all model parameters are chosen via 5-fold cross validation. 4.1 Benchmark Datasets We use three datasets for our experiments. Details are given below: Landmine Detection3 consists of 19 tasks collected from different landmine fields. Each task is a binary classification problem: landmines (+) or clutter (?) and each example consists of 9 features extracted from radar images with four moment-based features, three correlation-based features, one energy ratio feature and a spatial variance feature. Landmine data is collected from two different terrains: tasks 1-10 are from highly foliated regions and tasks 11-19 are from desert regions, therefore tasks naturally form two clusters. Any hypothesis learned from a task should be able to utilize the information available from other tasks belonging to the same cluster. 3 http://www.ee.duke.edu/~lcarin/LandmineData.zip 5 Spam Detection4 We use the dataset obtained from ECML PAKDD 2006 Discovery challenge for the spam detection task. We used the task B challenge dataset which consists of labeled training data from the inboxes of 15 users. We consider each user as a single task and the goal is to build a personalized spam filter for each user. Each task is a binary classification problem: spam (+) or non-spam (?) and each example consists of approximately 150K features representing term frequency of the word occurrences. Since some spam is universal to all users (e.g. financial scams), some messages might be useful to certain affinity groups, but spam to most others. Such adaptive behavior of user?s interests and dis-interests can be modeled efficiently by utilizing the data from other users to learn per-user model parameters. Sentiment Analysis5 We evaluated our algorithm on product reviews from Amazon on a dataset containing reviews from 24 domains. We consider each domain as a binary classification task. Reviews with rating > 3 were labeled positive (+), those with rating < 3 were labeled negative (?), reviews with rating = 3 are discarded as the sentiments were ambiguous and hard to predict. Similar to the previous dataset, each example consists of approximately 350K features representing term frequency of the word occurrences. We choose 3040 examples (160 training examples per task) for landmine, 1500 emails for spam (100 emails per user inbox) and 2400 reviews for sentiment (100 reviews per domain) for our experiments. We use the rest of the examples for test set. On average, each task in landmine, spam, sentiment has 509, 400 and 2000 examples respectively. Note that we intentionally kept the size of the training data small to drive the need for learning from other tasks, which diminishes as the training sets per task become large. 4.2 Results To evaluate the performance of our proposed approach, we compare our proposed methods to 2 standard baselines. The first baseline selects the examples to query randomly (Random) and the second baseline chooses the examples via selective sampling independently for each task (Independent) [11]. We compare these baselines against two versions of our proposed algorithm 1 with different confidence measures for predictions from peer tasks: PEERsum where the confidence p?(t) at round t is computed by the weighted sum of the confidence of each task as shown originally in Algorithm 1 and (t) PEERone where the confidence p?(t) is set to the confidence of the most related task k (? pk ), sampled (t) from the probability distribution ?km , m ? [K], m 6= k. The intuition is that, for multitask learning with many tasks [17], PEERone provides a faster computation without significantly compromising the performance of the learning algorithm. The task weights ? are computed based on the relationship between the tasks. As mentioned earlier, the ? values can be easily replaced by other functions based on the application at hand 6 . In addition to PEERsum and PEERone, we evaluated a method that queries the peer with the highest confidence, instead of the most related task as in PEERone, to provide the label. Since this method uses only local information for the task with highest confidence, it is not necessarily the best peer in hindsight, and the results are worse than or comparable (in some cases) to the Independent baseline. Hence, we do not report its results in our experiment. Figure 2 shows the performance of the models during training. We measure the average rate of mistakes (cumulative measure), the number of label requests to the oracle and the number of peer query requests to evaluate the performance during the training time. From Figure 2 (top and middle), we can see that our proposed methods (PEERsum and PEERone) outperform both the baselines. Among the proposed methods, PEERsum outperforms PEERone as it uses the confidence from all the tasks (weighted by task relationship) to measure the final confidence. We notice that during the earlier part of the learning, all the methods issue more query to the oracle. After a few initial set of label requests, peer requests (dotted lines) steadily take over in our proposed methods. We can see three noticeable phases in our learning algorithm: initial label requests to the oracle, label requests to peers, and as task confidence grows, learning with less dependency on other tasks. 4 http://ecmlpkdd2006.org/challenge.html http://www.cs.jhu.edu/~mdredze/datasets/sentiment 6 Our algorithm and theorem can be easily generalized to other types of functions on ? 5 6 0.5 Random Independent PEERsum PEERone 0.6 0.4 0.2 0 0 500 1000 1500 2000 2500 0.32 0.4 0.35 0.3 0.25 0.2 3000 Average rate of mistakes 0.8 0.34 0.45 Average rate of mistakes Average rate of mistakes 1 0 500 Number of samples 1000 800 600 400 200 0 500 1000 1500 2000 2500 3000 800 600 400 200 0 0 500 1000 1500 0 Random Independent PEERsum PEERone 0.18 0 500 1000 1500 Number of samples 2000 2500 Average rate of mistakes 0.32 Average rate of mistakes 0.34 0.24 0.3 0.28 0.26 0.24 0.22 0.2 2500 0 500 0 500 1000 1500 Number of samples 1000 1500 2000 2500 2000 2500 Number of samples 0.32 0.26 2000 500 0.34 0.28 1500 1000 Number of samples 0.3 1000 1500 0.32 0.2 500 2000 0.34 0.22 0 Number of samples 1000 Number of samples Average rate of mistakes 0.22 2500 Number of Label/Peer Requests 1200 Number of Label/Peer Requests Number of Label/Peer Requests Random (Label Request) Independent (Label Request) PEERsum (Label Request) PEERone (Label Request) PEERsum (Peer Request) PEERone (Peer Request) 1400 0.24 0.2 1500 1200 1600 0.26 Number of samples 1800 0 1000 0.3 0.28 2000 2500 0.3 0.28 0.26 0.24 0.22 0.2 0 500 1000 1500 Number of samples Figure 2: Average rate of mistakes vs. Number of examples calculated for compared models on the three datasets (top). Average number of label and peer requests on the three datasets (middle). Average rate of (training) mistakes vs. Number of examples with the query budget of (10%, 20%, 30%) of the total number of examples T on sentiment (bottom). These plots are generated during the training. In order to efficiently evaluate the proposed methods, we restrict the total number of label requests issued to the oracle during training, that is we give all the methods the same query budget: (10%, 20%, 30%) of the total number of examples T on sentiment dataset. After the desired number of label requests to the oracle reached the said budget limit, the baseline methods predicts label for the new examples based on the earlier assistance from the oracle. On the other hand, our proposed methods continue to reduce the average mistake rate by requesting labels from peers. This shows the power of learning from peers when human expert assistance is expensive, scarce or unavailable. Table 1 summarizes the performance of all the above algorithms on the test set for the three datasets. In addition to the average accuracy ACC scores, we report the average total number of queries or label requests to the oracle (#Queries) and the CPU time taken (seconds) for learning from T examples (Time). From the table, it is evident that PEER* outperforms all the baselines in terms of both ACC and #Queries. In case of landmine and sentiment, we get a significant improvement in the test set accuracy while reducing the total number of label requests to the oracle. As in the training set results before, PEERsum performs slightly better than PEERone. Our methods perform slightly better than Independent in spam, we can see from Figure 2 (middle) for spam dataset, the number of peer queries are lower compared to that of the other datasets. The results justify our claim that relying on assistance from peers in addition to human intervention leads to improved performance. Moreover, our algorithm consumes less or comparable CPU time than the baselines which take into account inter-task relationships and peer requests. Note that PEERone takes a little more training time than PEERsum. This is due to our implementation that takes more time in (MATLAB?s) inbuilt sampler to draw the most related task. One may improve the sampling procedure to get better run time. However, the time spent on selecting the most related tasks is small compared to the other operations when dealing with many tasks. Figure 3 (left) shows the average test set accuracy computed for 20 different values of b2 for PEER* methods in sentiment. We set b1 = 1. Each point in the plot corresponds to ACC (y-axis) and #Queries (x-axis) computed for a specific value of b2 . We find the algorithm performs well for 7 Table 1: Average test accuracy on three datasets: means and standard errors over 10 random shuffles. Models Random Independent PEERsum PEERone Landmine Detection ACC #Queries Time (s) 0.8905 1519.4 0.38 (0.007) (31.9) 0.9040 1802.8 0.29 (0.016) (35.5) 0.9403 265.6 0.38 (0.001) (18.7) 0.9377 303 1.01 (0.003) (17) Spam Detection ACC #Queries Time (s) 0.8117 753.4 8 (0.021) (29.1) 0.8309 1186.6 7.9 (0.022) (18.3) 0.8497 1108.8 8 (0.007) (32.1) 0.8344 1084.2 8.3 (0.018) (24.2) jewelry camera baby Sentiment Analysis ACC #Queries Time (s) 0.7443 1221.8 35.6 (0.028) (22.78) 0.7522 2137.6 35.6 (0.015) (19.1) 0.8141 1494.4 36 (0.001) (68.59) 0.8120 1554.6 36.3 (0.01) (92.2) 0 20 0 spo rts 0 20 0 0 ok s 0.9 0 0 he kitc 0 20 ines az n ag 0.81 m PEERsum PEERone hea 0 gourmet 0 20 lth 0 grocery 0 0.79 20 0 0 music 0 nics 20 elec 0 men stru 0 ts 0.76 tro 0 ativein tom au 0.77 20 be 0 au ty office 20 0.78 Average Test Accuracy 20 dvd 0 0.8 0.8 apparel Average Test set Accuracy bo o vide 20 0.82 0.7 0.6 0.5 SHAMPO PEERsum PEERone 0.4 ys to 0 20 0.75 1400 20 1500 1600 1700 1800 1900 2000 2100 2200 com puter 0 0 0 cell_phones Number of Label Requests 20 outdoor so are ftw 0.3 1 2 3 4 ? Figure 3: Average test set ACC calculated for different values of b2 (left). A visualization of the peer query requests among the tasks in sentiment learned by PEERone (middle) and comparison of proposed methods against SHAMPO in parallel setting. We report the average test set accuracy (right). b2 > b1 and the small values of b2 . When we increase the value of b2 to ?, our algorithm reduces to the baseline (Independent), as all request are directed to the oracle instead of the peers. Figure 3 (middle) shows the snapshot of the total number of peer requests between the tasks in sentiment at the end of the training of PEERone. Each edge says that there was one peer query request from a task/domain to another related task/domain (based on the task relationship matrix ? ). The edges with similar colors show the total number of peer requests from a task. It is evident from the figure that all the tasks are collaborative in terms of learning from each other. Figure 3 (right) compares the PEER* implementation of Algorithm 2 in Appendix B against SHAMPO in terms of test set accuracy for sentiment dataset (See Supplementary material for more details on the Algorithm). The algorithm learns multiple tasks in parallel, where at most ? out of K label requests to the oracle are allowed at each round. While SHAMPO ignores the other tasks, our PEER* allows peer query to related tasks and thereby improves the overall performance. As we can see from the figure, when ? is set to small values, PEER* performs significantly better than SHAMPO. 5 Conclusion We proposed a novel online multitask learning algorithm that learns to perform each task jointly with learning inter-task relationships. The primary intuition we leveraged in this paper is that task performance can be improved both by querying external oracles and by querying peer tasks. The former incurs a cost or at least a query-budget bound, but the latter requires no human attention. Hence, our hypothesis was that with bounded queries to the human expert, additionally querying peers should improve task performance. Querying peers requires estimating the relation among tasks. The key idea is based on smoothing the loss function of each task w.r.t. a probabilistic distribution over all tasks, and adaptively refining such distribution over time. In addition to closedform updating rules, we provided a theoretical bound on the expected number of mistakes. The effectiveness of our algorithm is empirically verified over three benchmark datasets where in all cases task accuracy improves both for PEERsum (sum of peer recommendations weighted by task similarity) and PEERone (peer recommendation from the most highly related task) over baselines such as assuming task independence. 8 References [1] Koby Crammer and Yishay Mansour. Learning multiple tasks using shared hypotheses. In Advances in Neural Information Processing Systems, pages 1475?1483, 2012. 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Experimental Design for Learning Causal Graphs with Latent Variables Murat Kocaoglu? Department of Electrical and Computer Engineering The University of Texas at Austin, USA mkocaoglu@utexas.edu Karthikeyan Shanmugam? IBM Research NY, USA karthikeyan.shanmugam2@ibm.com Elias Bareinboim Department of Computer Science and Statistics Purdue University, USA eb@purdue.edu Abstract We consider the problem of learning causal structures with latent variables using interventions. Our objective is not only to learn the causal graph between the observed variables, but to locate unobserved variables that could confound the relationship between observables. Our approach is stage-wise: We first learn the observable graph, i.e., the induced graph between observable variables. Next we learn the existence and location of the latent variables given the observable graph. We propose an efficient randomized algorithm that can learn the observable graph using O(d log2 n) interventions where d is the degree of the graph. We further propose an efficient deterministic variant which uses O(log n + l) interventions, where l is the longest directed path in the graph. Next, we propose an algorithm that uses only O(d2 log n) interventions that can learn the latents between both nonadjacent and adjacent variables. While a naive baseline approach would require O(n2 ) interventions, our combined algorithm can learn the causal graph with latents using O(d log2 n + d2 log (n)) interventions. 1 Introduction Causality shapes how we view, understand, and react to the world around us. It is arguably a key ingredient in building intelligent systems that are autonomous and can act efficiently in complex environments. Not surprisingly, the task of automating the learning of cause-and-effect relationships have attracted great interest in the artificial intelligence and machine learning communities. This effort has led to a general theoretical and algorithmic understanding of the assumptions under which causeand-effect relationships can be inferred from data. These results have started to percolate through the applied fields ranging from genetics to medicine, from psychology to economics [5, 26, 33, 25]. The endeavour of algorithmically learning causal relations may have started from the independent discovery of the IC [35] and PC algorithms [33], which almost identically, and contrary to previously held beliefs, showed the feasibility of recovering these relations from purely observational, nonexperimental data. A plethora of methods followed this breakthrough, and now we understand, at least in principle, the limits of what can be inferred from purely observational data, including (not exhaustively) [31, 14, 21, 27, 19, 13]. There are a number of assumptions that have been considered about the data-generating model when attempting to unveil the causal structure. One of the most ? Equal contribution. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. popular assumptions is that the data-generating model is causally sufficient, which means that no latent (unmeasured) variable affects more than one observed variable. In practice, this is a very stringent condition since the existence of latents affecting more than one observed variable, and generating what is called confounding bias, is one of the main concerns of empirical scientists. The problem of causation is deemed challenging in most of the empirical fields because scientists recognize that not all the variables influencing the observed phenomenon can be measured. The general question that arises is then how much of the observed behavior of the system is truly causal, or whether it is due to some external, unobserved forces [26, 5]. To account for the latent variables in the context of structural learning, the IC* [35] and FCI [33] algorithms were introduced, which showed the possibility of recovering causal structures even when latent variables may be confounding the observed behavior 2 . One of the main challenges faced by these algorithms is that although some ancestral relations as well as certain causal edges can be learned [36, 7], many observationally equivalent architectures cannot be distinguished. Despite the practical challenges when collecting the data (e.g., finite samples, selection bias, missing data), we now have a complete characterization of what structures are recoverable from observational data based on conditional independence constraints [33, 2, 37]. Inferences will be constrained within an equivalence class. Initial works leveraged ideas of experimental design and the availability of interventional data to move from the equivalence class to a specific graph, but almost exclusively considering causally sufficient systems [9, 15, 11, 12, 30, 18]. For causally insufficient systems, there is a growing interest in identifying experimental quantities and structures based on partially observed interventional data [4, 32, 29, 28, 24, 16, 8, 34, 22], but without the goal of designing the optimal set of interventions. Perhaps the most relevant paper to our setup is [23]. Authors identify the experiments needed to learn the causal graph under latents, given the output of FCI algorithm. However, they are not interested in minimizing the number of experiments. In this paper, we propose the first efficient non-parametric algorithm for learning a causal graph with latent variables. It is known that log(n) interventions are necessary (across all graphs) and sufficient to learn a causal graph without latent variables [12], and we show, perhaps surprisingly, that there exists an algorithm that can learn any causal graph with latent variables which requires poly(log n) interventions when the observable graph is sparse. More specifically, our contributions are as follow: ? We introduce a deterministic 3 algorithm that can learn any causal graph and the existence and location of the latent variables using O(d log(n) + l) interventions, where d is the largest node degree and l is the longest directed path of the causal graph. ? We design a randomized algorithm that can learn the observable graph and all the latent variables using O(d log2 (n) + d2 log(n)) interventions with high probability, where d is the largest node degree. The first algorithm is useful in practical settings where the longest directed path is not very deep, e.g., O(log(n)). This includes bipartite, time-series, and relational type of domains where the underlying causal topology is somewhat sparse. As an example application, consider the problem of inferring the causal effect of a set of genes on a set of phenotypes, that could be cast as learning a bipartite causal system. For the more general setting, we introduce a randomized algorithm that with high probability is capable of unveiling the true causal structure. Background We assume for simplicity that all the random variables are discrete. We use the language of Structural Causal Models (SCM) [26, pp. 204-207]. Formally, an SCM M is a 4-tuple hU , V, F, P (u)i, where U is a set of exogenous (unobserved, latent) variables, V is a set of endogenous (measured) variables. We partition the set of exogenous variables into two disjoint sets: Exogenous variables with one observable child, denoted by E, exogenous variables with two observable children, denoted by L. F = {fi } is a collection of functions such that each endogenous variable Vi 2 V is determined by a function fi 2 F : Each fi is a mapping from the respective domain of the exogenous variables associated with Vi and a set of observable variables associated with Vi , called P Ai , into Vi . The 2 Hereafter, latent variable refers to any unmeasured variable that affects more than one observed variable. We assume access to an oracle that outputs a size-O(d2 log (n)) independent set cover for the non-edges of a given graph. This oracle can be implemented using another randomized algorithm as we explain in Section 5. 3 2 set of exogenous variables associated with Vi can be divided into two classes, the one with a single observable child, denoted by Ei 2 E, and those with two observable children, denoted by Li ? L. Hence fi maps from the domain of Ei [ P Ai [ Li to Vi . The entire set F forms a mapping from U to V. The uncertainty is encoded through a product probability distribution over the exogenous variables P (E, L). For simplicity we refer to L as the set of latents, and E as the set of exogenous variables. Within the structural semantics, performing an action S = s is represented through the do-operator, do(S = s), which encodes the operation of replacing the original equation of S by the constant s and induces a submodel MS (also for when S is not a singleton). We denote the post-interventional distribution by PS (?). For a detailed discussion on the properties of structural models, we refer readers to [5, 23, 24, Ch. 7]. Define D` = (V [ L, E` ) to be the causal graph with latents. We define the observable graph to be the induced subgraph on V which is D = (V, E). In practice, we use an independent random variable Wi taking values uniformly at random in the state space of Vi , to implement an intervention do(Vi ). A conditional independence statement, e.g., X is independent from Y given Z ? V with respect to causal model MS , in shown by (X ?? Y |Z)MS , or (X ?? Y |Z)S when the causal model is clear from the context. These conditional independencies are with respect to the post-interventional joint probability distribution PS (?). In this paper, we assume that an oracle to conditional independence (CI) tests is available. The mutilated or post-interventional causal graph, denoted D` [S] = (V [ L, E` [S]), is identical to D` except that all the incoming edges incident on any vertex in the interventional set S is absent, i.e., E` [S] = E` {(Y, V ) : V 2 S, (Y, V ) 2 E` }. We define the transitive closure, denoted Dtc , of an observable causal DAG D as follows: If there is a directed path from Vi to Vj in D, there is a directed edge from Vi to Vj in Dtc . Essentially, a directed edge in Dtc represents an ancestral relation in D. For any DAG D = (V, E), a set of nodes S ? V d-separates two nodes a and b if and only if S blocks all paths between a and b. ?Blocking? is a graphical criterion associated with d-separation 4 . A probability distribution is said to be faithful (or stable) to a graph, if and only if every conditional independence statement can be read off from the graph using d-separation, see [26, Ch. 2] for a review. We assume that faithfulness holds in the observational and post-interventional distributions following [12]. Results and outline of the paper The skeleton of the proposed learning algorithms can be split into 3 steps, namely: (a) (b) (c) ; ! Transitive Closure ! Observable graph ! Observable graph with Latent variables Each step requires different tools and graph theoretic concepts: (a) We use a pairwise independence test under interventions that reveals the ancestral relations. This is combined in an efficient manner with separating systems to discover the transitive closure of D in O(log n) interventions. (b) We rely on the transitive reduction of directed acyclic graphs that can be efficiently computed only from their transitive closure. A key property we observe is that the transitive reduction reveals a subset of the true edges. For our randomized algorithm, we use a sequence of transitive reductions computed from transitive closures (obtained using step (a)) of different post-interventional graphs. (c) Given the observable graph, it is possible to discover latents between non-adjacent nodes using CI tests under suitable interventions. We use an edge-clique cover on the complement graph to optimize the number of experiments. For latents between adjacent nodes, we use a relatively unknown test called the do-see test, i.e., leveraging the equivalence between observing and intervening on the node. We implement it using induced matching cover of the observable graph. The modularity of our approach allows us to solve subproblems: given the ancestral graph, we can use (b) to discover the observable graph D. If D is known, we can learn the latents with (c). Some pictorial illustrations of the main results in the technical sections are found in the full version [20]. 2 Identifying the Observable Graph: A simple baseline We discuss a natural and a simple deterministic baseline algorithm that finds the observable graph with experiments when confounders are present. To our knowledge, a provably complete algorithm 4 For convenience, detailed definitions of blocking and non-blocking paths are provided in the full version [20]. 3 that recovers the observable graph under this setting and is superior than this simple baseline in the worst case is not known. We start from the following observation. Suppose X ! Y where X, Y are observable variables and let L be a latent variable such that L ! X, L ! Y . Consider the post interventional graph D` [{X}] where we intervene on X. It is easy to see that, X and Y are dependent in the post interventional graph too because of the direct causal relationship. However, if X is not a parent of Y , then in the post interventional graph D` [{X}] even with or without the latent L between X and Y , X is independent of Y since X is intervened on. It is possible to recreate this condition between any target variable Y and any one of its direct parents X when many other observable variables are involved. Simply, we consider the post-interventional graph where we intervene on all observable variables but Y . In D` [V {Y }], Y and X are dependent if and only if X ! Y is a directed edge in the observable graph D, because every variable except X becomes independent of all other variables in the post interventional graph. Therefore, one needs n interventions, each of size n 1 to find out the parent set of every node. We basically show in the next two sections that when the graph D has constant degree, it is enough to do O(log2 (n)) interventions representing the first provably exponential improvement. 3 Learning Ancestral Relations In this section, we show that separating systems can be used to construct sequences of pairwise CI tests to discover the transitive closure of the observable causal graph, i.e., the graph that captures all ancestral relations. The following lemma relates post-interventional statistical dependencies with the ancestral relations in the graph with latents. Lemma 1. [Pairwise Conditional Independence Test] Consider a causal graph with latents D` . Consider an intervention on the set S ? V of observable variables. Then, under the post-interventional faithfulness assumption, for any pair Xi 2 S, Xj 2 V\S, (Xi 6?? Xj )D` [S] if and only if Xi is an ancestor of Xj in the post-interventional observable graph D[S]. Lemma 1 constitutes, for any ordered pair of variables (Xi , Xj ) in the observable graph D, a test for whether Xi is an ancestor of Xj or not. Note that a single test is not sufficient to discover the ancestral relation between a pair (Xi , Xj ), e.g., if Xi ! Xk ! Xj and Xi , Xk 2 S, Xj 2 / S, the ancestral relation will not be discovered. This issue can be resolved by using a sequence of interventions guided by a separating system, and later finding the transitive closure of the learned graph. Separating systems were first defined by [17], and has been subsequently used in the context of experimental design [10]. A separating system on a ground set S is a collection of subsets of S, S = {S1 , S2 . . .} such that for every pair (i, j), there is a set that contains only one, i.e., 9k such that i 2 Sk , j 2 / Sk or j 2 Sk , i 2 / Sk . We require a stronger notion which is captured by a strongly separating system. Definition 1. An (m, n) strongly separating system is a family of subsets {S1 , S2 . . . Sm } of the ground set [n] such that for any two pairs of nodes i and j, there is a set S in the family such that i 2 S, j 2 / S and also another set S 0 such that i 2 / S0, j 2 S0. Similar to separating systems, one can construct strongly separating systems using O(log(n)) subsets: Lemma 2. An (m, n) strong separating system exists on a ground set [n] where m ? 2dlog ne. We propose Algorithm 1 to discover the ancestral relations between the observable variables. It uses the subsets of a strongly separating system on the ground set of all observable variables as intervention sets, to assure that the ancestral relation between every ordered pair of observable variables is tested. The following theorem shows the number of experiments and the soundness of Algorithm 1. Theorem 1. Algorithm 1 requires only 2dlog ne interventions and conditional independence tests on samples obtained from each post-interventional distribution and outputs the transitive closure Dtc . 4 Learning the Observable Graph We introduce a deterministic and a randomized algorithm for learning the observable causal graph D from ancestral relations. D encodes every direct causal connection between the observable nodes. 4 Algorithm 1 LearnAncestralRelations- Given access to a conditional independence testing oracle (CI oracle), query access to samples from any post-interventional causal model derived out of M (with causal graph D` ), outputs all ancestral relationships between observable variables, i.e., Dtc 1: function L EARNA NCESTRAL R ELATIONS(M) 2: E = ;. 3: Consider a strongly sep. system of size ? 2 log n on the ground set V - {S1 , S2 ..S2dlog ne }. 4: for i in [1 : 2dlog ne] do 5: Intervene on the set Si of nodes. 6: for X 2 Si , Y 2 / Si , Y 2 V do 7: Use samples from MSi and use the CI-oracle to test the following. 8: if (X 6?? Y )D` [S] then 9: E E [ (X, Y ). 10: end if 11: end for 12: end for 13: return The transitive closure of the graph (V, E) 14: end function 4.1 A Deterministic Algorithm Based on Section 3, assume that we are given the transitive closure of the observable graph. We show in Lemma 3 that, when the intervention set contains all parents of Xi , the only variables dependent with Xi in the post-interventional observable graph are the parents of Xi in the observable graph. Lemma 3. For variable Xi , consider an intervention on S where P ai ? S. Then {Xj 2 S : (Xi 6?? Xj )D[S] } = P ai . Let the longest directed path of Dtc be r. Consider the partial order <Dtc implied by Dtc on the vertex set V. Define {Ti : i 2 [r + 1]} as the unique partitioning of vertices of Dtc where Ti <Dtc Tj , 8i < j and each node in Ti is a set of mutually incomparable elements. In other words, 1 Ti are the set of nodes at layer i of the transitive closure graph Dtc . Define Ti = [ik=1 Tk . We have the following observation: P ai ? Ti . This paves the way for Algorithm 2 that leverages Lemma 3. Algorithm 2 LearnObservableGraph/Deterministic - Given the ancestral graph, access to a conditional independence testing oracle (CI oracle) and outputs the graph induced on observable nodes. 1: function L EARN O BSERVABLE G RAPH /D ETERIMINISTIC(M) 2: E = ;. 3: for i in {r + 1, r, r 1, . . . , 2} do 4: Intervene on the set Ti of nodes. 5: Use samples from MTi and use the CI-oracle to test the following. 6: for X in Ti do 7: if (X 6?? Y )D` [Ti ] then 8: E E [ (X, Y ). 9: end if 10: end for 11: end for 12: return Observable graph 13: end function The correctness of Algorithm 2 follows from Lemma 3, which is stated explicitly in the sequel. Theorem 2. Let r be the length of the longest directed path in the causal graph D` . Algorithm 2 requires only r interventions and conditional independence tests on samples obtained from each one of the post-interventional distributions and outputs the observable graph D. 4.2 A Randomized Algorithm We propose a randomized algorithm that repeatedly uses the ancestor graph learning algorithm from Section 3 to learn the observable graph 5 . A key structure that we use is the transitive reduction: 5 Note that this algorithm does not require learning the ancestral graph first. 5 V1 V1 V2 V2 V3 V3 V4 V4 Observable Graph D (a) V1 V1 Transitive reduction of D V2 V2 V3 V3 V4 V4 Post-interventional graph D[{V2}] After intervention on V2 (b) (c) Transitive reduction of D[{V2}] (d) Figure 1: Illustration of Lemma 5 - (a) An example of an observable graph D without latents (b): Transitive reduction of D. The highlighted red edge (V1 , V3 ) has not been revealed under the operation of transitive reduction. c) Intervention on node V2 and its post interventional graph D[{V2 }] d) Since all parents of V3 above V1 in the partial order have been intervened on, by Lemma 5, the edge (V1 , V3 ) is revealed in the transitive reduction of D[{V2 }]. Definition 2 (Transitive Reduction). Given a directed acyclic graph D = (V, E), let its transitive closure be Dtc . Then Tr(D) = (V, Er ) is a directed acyclic graph with minimum number of edges such that its transitive closure is identical to Dtc . Lemma 4. [1] Tr(D) is known to be unique if D is acyclic. Further, the set of directed edges of Tr(D) is a subset of the directed edges of D, i.e., Er ? E. Computing Tr(D) from D takes the same time as transitive closure of a DAG D, which takes time poly(n). We note that Tr(D) = Tr(Dtc ). Now, we provide an algorithm that outputs an observable graph based on samples from the post-interventional distribution after a sequence of interventions. Let us assume an ordering ? on the observable vertices V that satisfies the partial order relationships in the observable causal graph D. The key insight behind the algorithm is given by the following Lemma. Lemma 5. Consider an intervention on a set S ? V of nodes in the observable causal graph D. Consider the post-interventional observable causal graph D[S]. Suppose for a specific observable node Vi , Vi 2 S c . Let Y be a direct parent of Vi in D such that all the direct parents of Vi above Y in the partial order6 ?(?) is in S, i.e., {X : ?(X) > ?(Y ), (X, V ) 2 D} ? S. Then, Tr(D[S]) will contain the directed edge (Y, Vi ) and it can be computed from Tr((D[S])tc ) We illustrated Lemma 5 through an example in Figure 1. The red edge in Figure 1(a) is not revealed in the transitive reduction. The edge is revealed when computing the transitive reduction of the post-interventional graph D[{V2 }]. This is possible because all parents of V3 above V1 in the partial order (in this case node V2 ) have been intervened on. Lemma 5 motivates Algorithm 3. The basic idea is to intervene in randomly, then compute the transitive closure of the post-interventional graph using the algorithm in the previous section, compute the transitive reduction, and then accumulate all the edges found in the transitive reduction at every stage. We will show in Theorem 3 that with high probability, the observable graph can be recovered. Theorem 3. Let dmax be greater than the maximum in-degree in the observable graph D. Algorithm 3 requires at most 8cdmax (log n)2 interventions and CI tests on samples obtained from post-interventional distributions, and outputs the observable graph with probability at least 1 nc1 2 . Remark. The above algorithm takes as input a parameter dmax that needs to be estimated. One practical option is to gradually increase dmax and run Algorithm 3. 6 The nodes above with respect to the partial order of a graph are those that are closer to the source nodes. 6 Algorithm 3 LearnObservable- Given access to a conditional independence testing oracle (CI oracle), a parameter dmax outputs induced subgraph between observable variables, i.e. D 1: function L EARN O BSERVABLE /R ANDOMIZED(M, dmax ) 2: E = ;. 3: for i in [1 : c ? 4 ? dmax log n] do 4: S = ;. 5: for V 2 V do 6: S S [ V randomly with probability 1 1/dmax . 7: end for ? S = LearnAncestralRelations(M). Let D ? = (V, E). ? 8: D ? ? S )) according to the algorithm in [1]. 9: Compute the transitive reduction of D(Tr( D ? 10: Add the edges of the transitive reduction to the set E if not already there, i.e. E E [ E. 11: end for 12: return The directed graph (V, E). 13: end function 5 Learning Latents from the Observable Graph The final stage of our framework is learning the existence and location of latent variables given the observable graph. We divide this problem into two steps ? first, we devise an algorithm that can learn the latent variables between any two variables that are non-adjacent in the observable graph; later, we design an algorithm that learns the latent variables between every pair of adjacent variables. 5.1 Baseline Algorithm for Detecting Latents between Non-edges Consider two variables X and Y such that X L ! Y and where L is a latent variable. Clearly, to distinguish it from the case where X and Y are disconnected and have no latents, one needs check if X 6?? Y or not. This is a conditional independence test. For any non edge (X, Y ) in the observable graph D, when the observable graph D is known, to check for latents between them, when other variables and possible confounders are around, one has to simply intervene on the rest of the n 2 variables and do a independence test between X and Y in the post interventional graph. This requires a distinct intervention for every pair of variables. If the observable graph has maximum degree d = o(n), this requires ?(n2 ) interventions. We will reduce this to O(d2 log n) interventions which is an exponential improvement for constant degree graphs. 5.2 Latents between Non-adjacent Nodes We start by noting the following fact about causal systems with latent variables: Theorem 4. Consider two non-adjacent nodes Xi , Xj . Let S be the union of the parents of Xi , Xj , S = P ai [ P aj . Consider an intervention on S. Then we have (Xi 6?? Xj )MS if and only if there exists a latent variable Li,j such that Xj Li,j ! Xi . The statement holds under an intervention S such that P ai [ P aj ? S, Xi , Xj 2 / S. The above theorem motivates the following approach: For a set of nodes which forms an independent set, an intervention on the union of parents of the nodes of the independent set allows us to learn the latents between any two nodes in the independent set. We leverage this observation using the following lemma on the number of such independent sets needed to cover all non-edges. Lemma 6. Consider a directed acyclic graph D = (V, E) with degree (out-degree+in-degree) d. Then there exists a randomized algorithm that returns a family of m = O(4e2 (d + 1)2 log(n)) independent sets I = {I1 , I2 , . . . , Im } that cover all non-edges of D: 8i, j such that (Xi , Xj ) 2 /E and (Xj , Xi ) 2 / E, 9k 2 [m] such that Xi 2 Ik and Xj 2 Ik , with probability at least 1 n12 . Note that this is a randomized construction and we are not aware of any deterministic construction. Our deterministic causal learning algorithm requires oracle access to such a famiy of independent sets, whereas our randomized algorithm can directly use this randomized construction. Now, we use this observation to construct a procedure to identify latents between non-edges (see Algorithm 4). The following theorem about its performance follows from Lemma 6 and Theorem 4. 7 Algorithm 4 LearnLatentNonEdge- Given access to a CI oracle, observable graph D with max degree d (in-degree+out-degree), outputs all latents between non-edges 1: function L EARN L ATENT N ON E DGE(M, dmax ) 2: L = ;. 3: Apply the randomized algorithm in Lemma 6 to find a family of independent sets I = {I1 , I2 , . . . , Im } that cover all non-edges in D such that m ? 4e2 (d + 1)2 log(n). 4: for j 2 [1 : m] do 5: Intervene on the parent set of the nodes in Ij . 6: for every pair of nodes X, Y in Ij do 7: if (X 6?? Y )D` [Ij ] then 8: L L [ {X, Y }. 9: end if 10: end for 11: end for 12: return The set of non-edges L. 13: end function U M L L Y X T Z X Y M Z G2: do(PaY) is needed G1: do(PaX) is needed Figure 2: Left: A graph where intervention on the parents of X is needed for do-see test to succeed. Right: A graph where intervention on the parents of Y is needed for do-see test to succeed. Theorem 5. Algorithm 4 outputs a list of non-edges L that have latent variables between them, given the observable graph D, with probability at least 1 n12 . The algorithm requires 4e2 (d + 1)2 log(n) interventions where d is the max-degree (in-degree+out-degree) of the observable graph. 5.3 Latents between Adjacent Nodes We construct an algorithm that can learn latent variables between the variables adjacent in the observable graph. Note that the approach of CIT testing in the post-interventional graph is not helpful. Consider the variables X ! Y . To see the effect of the latent path, one needs to cut the direct edge from X to Y . This requires intervening on Y . However, such an intervention disconnects Y from its latent parent. Thus we resort to a different approach compared to the previous stages and exploit a different characterization of causal Bayesian networks called a ?do-see? test. A do-see test can be described as follows: Consider again a graph where X ! Y . If there are no latents, we have P(Y |X) = P(Y |do(X)). Assume that there is a latent variable Z which causes both X and Y , then excepting the pathological cases7 , P(Y |X) 6= P(Y |do(X)). Figure 2 illustrates the challenges associated with a do-see test in bigger graphs with latents. Graphs G1 and G2 are examples where parents of both nodes involved in the test need to be included in the intervention set for the Do-see test to work. In G1, suppose we condition on X, as required by the ?see? test. This opens up a non-blocking path X U T M Y . Since X ! Y is not the only d-connecting path, it is not necessarily true that P(Y |X) = P(Y |do(X)). Now suppose we perform the do-see test under the intervention do(Z). Then the aforementioned path is closed since X is not a descendant of T in the post interventional graph. Hence we have P(Y |X, do(Z)) = P(Y |do(X, Z)). Similarly G2 shows that intervening on the parent set of Y is also necessary. We have the following theorem, which shows that we can perform the do-see test between X, Y under do(P aX , P aY ): 7 These cases are fully identified in the full version [20]. 8 Theorem 6. [Interventional Do-see test] Consider a causal graph D on the set of observable variables V = {Vi }i2[n] and latent variables L = {Li }i2[m] with edge set E. If (Vi , Vj ) 2 E, then Pr(Vj |Vi = vi , do(P ai = pai , P aj = paj )) = Pr(Vj |do(Vi = vi , P ai = pai , P aj = paj )), iff @k such that (Lk , Vi ) 2 E and (Lk , Vj ) 2 E, where P ai is the set of parents of Vi in V . Quantities on both sides are invariant irrespective of additional interventions elsewhere. Next we need a subgraph structure to perform multiple do-see tests at once in order to efficiently discover the latents between the adjacent nodes. Performing the test for every edge would take O(n) even in graphs with constant degree. We use strong edge coloring of sparse graphs. Definition 3. A strong edge coloring of an undirected graph with k colors is a map : E ! [k] such that every color class is an induced matching. Equivalently, it is an edge coloring such that any two nodes adjacent to distinct edges with the same color are non-adjacent. Graphs of maximum degree d can be strongly edge-colored with at most 2d2 colors. Lemma 7. [6] A graph of maximum degree d can be strongly edge-colored with at most 2d2 colors. A simple greedy algorithm that colors edges in sequence achieves this. Now observe that a color class of the edges forms an induced matching. We show that due to this, the ?do? part (RHS of Theorem 6) of all the do-see tests in a color class can be performed with a single intervention while the ?see? part (RHS of Theorem 6) can be again performed with another intervention. We argue that we need exactly two different interventions per color class. The following theorem uses this property to prove correctness of Algorithm 5. Algorithm 5 LearnLatentEdge- Observable graph D with max degree d (in-degree+out-degree), outputs all latents between edges 1: function L EARN L ATENT E DGE(M, d) 2: L = ;. 3: Apply the greedy algorithm in Lemma 7 to color the edges of D with k ? 2d2 colors. 4: for j 2 [1 : k] do 5: Let Aj be the nodes involved with the edges that form color class j. Let Pj be the union of parents of all nodes in Aj except the nodes in Aj . 6: Let the set of tail nodes of all edges be Tj . 7: Following loop requires the intervention on the set Tj [ Pj , i.e. do({Tj , Pj }). 8: for Every directed edge (Vt , Vh ) in color class j do 9: Calculate S(Vt , Vh ) = P (Vh |do(Tj , Pj )) using post interventional samples. 10: end for 11: Following loop requires the intervention on the set Pj . 12: for Every directed edge (Vt , Vh ) in color class j do 13: Calculate S 0 (Vt , Vh ) = P (Vh |Vt , do(Pj )) using post interventional samples. 14: if S 0 (Vt , Vh ) 6= S(Vt , Vh ) then 15: L L [ (Vt , Vh ) 16: end if 17: end for 18: end for 19: return The set of edges L that have latents between them. 20: end function Theorem 7. Algorithm 5 requires at most 4d2 interventions and outputs all latents between the edges in the observable graph. 6 Conclusions Learning cause-and-effect relations is one of the fundamental challenges in science. We studied the problem of learning causal models with latent variables using experimental data. Specifically, we introduced two efficient algorithms capable of learning direct causal relations (instead of ancestral relations) and finding the existence and location of potential latent variables. 9 References [1] Alfred V. Aho, Michael R Garey, and Jeffrey D. Ullman. The transitive reduction of a directed graph. SIAM Journal on Computing, 1(2):131?137, 1972. [2] Ayesha R. Ali, Thomas S. Richardson, Peter L. Spirtes, and Jiji Zhang. Towards characterizing markov equivalence classes for directed acyclic graphs with latent variables. In Proc. of the Uncertainty in Artificial Intelligence, 2005. [3] Noga Alon. Covering graphs by the minimum number of equivalence relations. Combinatorica, 6(3):201?206, 1986. [4] E. Bareinboim and J. Pearl. Causal inference by surrogate experiments: z-identifiability. 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Learning to Model the Tail Yu-Xiong Wang Deva Ramanan Martial Hebert Robotics Institute, Carnegie Mellon University {yuxiongw,dramanan, hebert}@cs.cmu.edu Abstract We describe an approach to learning from long-tailed, imbalanced datasets that are prevalent in real-world settings. Here, the challenge is to learn accurate ?fewshot? models for classes in the tail of the class distribution, for which little data is available. We cast this problem as transfer learning, where knowledge from the data-rich classes in the head of the distribution is transferred to the data-poor classes in the tail. Our key insights are as follows. First, we propose to transfer meta-knowledge about learning-to-learn from the head classes. This knowledge is encoded with a meta-network that operates on the space of model parameters, that is trained to predict many-shot model parameters from few-shot model parameters. Second, we transfer this meta-knowledge in a progressive manner, from classes in the head to the ?body?, and from the ?body? to the tail. That is, we transfer knowledge in a gradual fashion, regularizing meta-networks for few-shot regression with those trained with more training data. This allows our final network to capture a notion of model dynamics, that predicts how model parameters are likely to change as more training data is gradually added. We demonstrate results on image classification datasets (SUN, Places, and ImageNet) tuned for the long-tailed setting, that significantly outperform common heuristics, such as data resampling or reweighting. 1 Motivation Deep convolutional neural networks (CNNs) have revolutionized the landscape of visual recognition, through the ability to learn ?big models? with hundreds of millions of parameters [1, 2, 3, 4]. Such models are typically learned with artificially balanced datasets [5, 6, 7], in which objects of different classes have approximately evenly distributed, very large number of human-annotated images. In real-world applications, however, visual phenomena follow a long-tailed distribution as shown in Fig. 1, in which the number of training examples per class varies significantly from hundreds or thousands for head classes to as few as one for tail classes [8, 9, 10]. Long-tail: Minimizing the skewed distribution by collecting more tail examples is a notoriously difficult task when constructing datasets [11, 6, 12, 10]. Even those datasets that are balanced along one dimension still tend to be imbalanced in others [13]; e.g., balanced scene datasets still contain long-tail sets of objects [14] or scene subclasses [8]. This intrinsic long-tail property poses a multitude of open challenges for recognition in the wild [15], since the models will be largely dominated by those few head classes while degraded for many other tail classes. Rebalancing training data [16, 17] is the most widespread state-of-the-art solution, but this is heuristic and suboptimal ? it merely generates redundant data through over-sampling or loses critical information through under-sampling. Head-to-tail knowledge transfer: An attractive alternative is to transfer knowledge from data-rich head classes to data-poor tail classes. While transfer learning from a source to target task is a well studied problem [18, 19], by far the most common approach is fine-tuning a model pre-trained on the source task [20]. In the long-tailed setting, this fails to provide any noticeable improvement since 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. 1200 # Occurrences Knowledge Transfer Living Room 1000 ?" ?# 800 ?% 600 Head 400 ?& 200 0 Long tail 0 50 100 150 200 250 Class index 300 350 ? ?? 400 (a) Long-tail distribution on the SUN-397 dataset. ?" Library ? ?& ?? (b) Knowledge transfer from head to tail classes. Figure 1: Head-to-tail knowledge transfer in model space for long-tail recognition. Fig. 1a shows the number of examples by scene class on SUN-397 [14], a representative dataset that follows an intrinsic long-tailed distribution. In Fig. 1b, from the data-rich head classes (e.g., living rooms), we introduce a meta-learner F to learn the model dynamics ? a series of transformations (denoted as solid lines) that represents how few k-shot models ?k start from ?1 and gradually evolve to the underlying many-shot models ?? trained from large sets of samples. The model parameters ? are visualized as points in the ?dual? model (parameter) space. We leverage the model dynamics as prior knowledge to facilitate recognizing tail classes (e.g., libraries) by hallucinating their model evolution trajectories (denoted as dashed lines). pre-training on the head is quite similar to training on the unbalanced long-tailed dataset (which is dominated by the head) [10]. Transferring meta-knowledge: Inspired by the recent work on meta-learning [21, 22, 23, 24, 25, 26], we instead transfer meta-level knowledge about learning to learn from the head classes. Specifically, we make use of the approach of [21], which describes a method for learning from small datasets (the ?few-shot? learning problem) through estimating a generic model transformation. To do so, [21] learns a meta-level network that operates on the space of model parameters, which is specifically trained to regress many-shot model parameters (trained on large datasets) from few-shot model parameters (trained on small datasets). Our meta-level regressor, which we call MetaModelNet, is trained on classes from the head of the distribution and then applied to those from the tail. As an illustrative example in Fig. 1, consider learning a scene classifier on a long-tailed dataset with many living-rooms but few outside libraries. We learn both many-shot and few-shot living-room models (by subsampling the training data as needed), and train a regressor that maps between the two. We can then apply the regressor on few-shot models of libraries learned from the tail. Progressive transfer: The above description suggests that we need to split up a long-tailed training set into a distinct set of source classes (the head) and target classes (the tail). This is most naturally done by thresholding the number of training examples per class. But what is the correct threshold? A high threshold might result in a meta-network that simply acts as an identity function, returning the input set of model parameters. This certainly would not be useful to apply on few-shot models. Similarly, a low threshold may not be useful when regressing from many-shot models. Instead, we propose a ?continuous? strategy that builds multiple regressors across a (logarithmic) range of thresholds (e.g., 1-shot, 2-shot, 4-shot regressors, etc.), corresponding to different head-tail splits. Importantly, these regressors can be efficiently implemented with a single, chained MetaModelNet that is naturally regularized with residual connections, such that the 2-shot regressor need only predict model parameters that are fed into the 4-shot regressor, and so on (until the many-shot regressor that defaults to the identity). By doing so, MetaModelNet encodes a trajectory over the space of model parameters that captures their evolution with increasing sample sizes, as shown in Fig. 1b. Interestingly, such a network is naturally trained in a progressive manner from the head towards the tail, effectively capturing the gradual dynamics of transferring meta-knowledge from data-rich to data-poor regimes. Model dynamics: It is natural to ask what kind of dynamics are learned by MetaModelNet ? how can one consistently predict how model parameters will change with more training data? We posit that the network learns to capture implicit data augmentation ? for example, given a 1-shot model trained with a single image, the network may learn to implicitly add rotations of that single image. But rather 2 than explicitly creating data, MetaModelNet predicts their impact on the learned model parameters. Interestingly, past work tends to apply the same augmentation strategies across all input classes. But perhaps different classes should be augmented in different ways ? e.g., churches maybe viewed from consistent viewpoints and should not be augmented with out-of-plane rotations. MetaModelNet learns class-specific transformations that are smooth across the space of models ? e.g., classes with similar model parameters tend to transform in similar ways (see Fig. 1b and Fig. 4 for more details). Our contributions are three-fold. (1) We analyze the dynamics of how model parameters evolve when given access to more training examples. (2) We show that a single meta-network, based on deep residual learning, can learn to accurately predict such dynamics. (3) We train such a meta-network on long-tailed datasets by through a recursive approach that gradually transfers meta-knowledge learned from the head to the tail, significantly improving long-tail recognition on a broad range of tasks. 2 Related Work A widespread yet suboptimal strategy is to resample and rebalance training data in the presence of the long tail, either by sampling examples from the rare classes more frequently [16, 17], or reducing the number of examples from the common classes [27]. The former generates redundancy and quickly runs into the problem of over-fitting to the rare classes, whereas the latter loses critical information contained within the large-sample sets. An alternative practice is to introduce additional weights for different classes, which, however, makes optimization of the models very difficult in the large-scale recognition scenarios [28]. Our underlying assumption that model parameters across different classes share similar dynamics is somewhat common in meta-learning [21, 22, 25]. While [22, 25] consider the dynamics during stochastic gradient descent (SGD) optimization, we address the dynamics as more training data is gradually made available. In particular, the model regression network from [21] empirically shows a generic nonlinear transformation from small-sample to large-sample models for different types of feature spaces and classifier models. We extend [21] for long-tail recognition by introducing a single network that can model transformations across different samples sizes. To train such a network, we introduce recursive algorithms for head-to-tail transfer learning and architectural modifications based on deep residual networks (that ensure that transformations of large-sample models default to the identity). Our approach is broadly related to different meta-learning concepts such as learning-to-learn, transfer learning, and multi-task learning [29, 30, 18, 31, 32]. Such approaches tend to learn shared structures from a set of relevant tasks and generalize to novel tasks. Specifically, our approach is inspired by early work on parameter prediction that modifies the weights of one network using another [33, 34, 35, 36, 37, 38, 39, 26, 40]. Such techniques have also been recently explored in the context of regressing classifier weights from training sample [41, 42, 43]. From an optimization perspective, our approach is related to work on learning to optimize, which replaces hand-designed update rules (e.g., SGD) with a learned update rule [22, 24, 25]. The most related formulation is that of one/few-shot learning [44, 45, 46, 47, 32, 48, 21, 49, 36, 50, 23, 51, 52, 53, 54, 55, 56]. Past work has explored strategies of using the common knowledge captured among a set of one-shot learning tasks during meta-training for a novel one-shot learning problem [52, 25, 36, 53]. These techniques, however, are typically developed for a fixed set of few-shot tasks, in which each class has the same, fixed number of training samples. They appear difficult to generalize to novel tasks with a wide range of sample sizes, the hallmark of long-tail recognition. 3 Head-to-Tail Meta-Knowledge Transfer Given a long-tail recognition task of interest and a base recognition model such as a deep CNN, our goal is to transfer knowledge from the data-rich head to the data-poor tail classes. As shown in Fig. 1, knowledge is represented as trajectories in model space that capture the evolution of parameters with more and more training examples. We train a meta-learner (MetaModelNet) to learn such model dynamics from head classes, and then ?hallucinate? the evolution of parameters for the tail classes. To simplify exposition, we first describe the approach for a fixed split of our training dataset into a head and tail. We then generalize the approach to multiple splits. 3 2? Shot ? BN ?) Res0 Res1 ? Leaky ReLU ?& (? = 2& ) Res? Weight ? Res? ?? BN Leaky ReLU 1Shot ? 2Shot ? 2& Shot ? 2% Shot ? (a) Learning a sample-size dependent transformation. Weight (b) Structure of residual blocks. Figure 2: MetaModelNet architecture for learning model dynamics. We instantiate MetaModelNet as a deep residual network with residual blocks i = 0, 1, . . . , N in Fig. 2a, which accepts few-shot model parameters ? (trained on small datasets across a logarithmic range of sample sizes k, k = 2i ) as (multiple) inputs and regresses them to many-shot model parameters ?? (trained on large datasets) as output. The skip connections ensure the identity regularization. Fi denotes the meta-learner that transforms (regresses) k-shot ? to ?? . Fig. 2b shows the structure of the residual blocks. Note that the meta-learners Fi for different k are derived from this single, chained meta-network, with nested circles (subnetworks) corresponding to Fi . 3.1 Fixed-size model transformations Let us write Ht for the ?head? training set of (x, y) data-label pairs constructed by assembling those classes for which there exist more than t training examples. We will use Ht to learn a meta-network thats maps few-shot model parameters to many-shot parameters, and then apply this network on few-shot models from the tail classes. To do so, we closely follow the model regression framework from [21], but introduce notation that will be useful later. Let us write a base learner as g(x; ?) as a feedforward function g(?) that processes an input sample x given parameters ?. We first learn a set of ?optimal? model parameters ?? by tuning g on Ht with a standard loss function. We also learn few-shot models by randomly sampling a smaller fixed number of examples per class from Ht . We then train a meta-network F(?) to map or regress the few-shot parameters to ?? . Parameters: In principle, F(?) applies to model parameters from multiple CNN layers. Directly regressing parameters from all layers is, however, difficult to do because of the larger number of parameters. For example, recent similar methods for meta-learning tend to restrict themselves to smaller toy networks [22, 25]. For now, we focus on parameters from the last fully-connected layer for a single class ? e.g., ? ? R4096 for an AlexNet architecture. This allows us to learn regressors that are shared across classes (as in [21]), and so can be applied to any individual test class. This is particularly helpful in the long-tailed setting, where the number of classes in the tail tends to outnumber the head. Later we will show that (nonlinear) fine-tuning of the ?entire network? during head-to-tail transfer can further improve performance. Loss function: The meta-network F(?) is itself parameterized with weights w. The objective function for each class is: X ??kShot(Ht ) n ||F(?; w) ? ?? ||2 + ? X 
o loss g x; F(?; w) , y . (1) (x,y)?Ht The final loss is averaged over all the head classes and minimized with respect to w. Here, kShot(Ht ) is the set of few-shot models learned by subsampling k examples per class from Ht , and loss refers to the performance loss used to train the base network (e.g., cross-entropy). ? > 0 is the regularization parameter used to control the trade-off between the two terms. [21] found that the performance loss was useful to learn regressors that maintained high accuracy on the base task. This formulation can be viewed as an extension to those in [21, 25]. With only the performance loss, Eqn. (1) reduces to the loss function in [25]. When the performance loss is evaluated on the subsampled set, Eqn. (1) reduces to the loss function in [21]. 4 Training: What should be the value of k, for the k-shot models being trained? One might be tempted to set k = t, but this implies that there will be some head classes near the cutoff that have only t training examples, implying ? and ?? will be identical. To ensure that a meaningful mapping is learned, we set k = t/2. In other terms, we intentionally learn very-few-shot models to ensure that target model parameters are sufficiently more general. 3.2 Recursive residual transformations We wish to apply the above module on all possible head-tail splits of a long-tailed training set. To do so, we extend the above approach in three crucial ways: ? (Sample-size dependency) Generate a sequence of different meta-learners Fi each tuned for a specific k, where k = k(i) is an increasing function of i (that will be specified shortly). Though a straightforward extension, prior work on model regression [21] learns a single fixed meta-learner for all the k-shot regression tasks. ? (Identity regularization) Ensure that the meta-learner defaults to the identity function for large i: Fi ? I as i ? ?.   ? (Compositionality) Compose meta-learners out of each other: ?i < j, Fi (?) = Fj Fij (?) where Fij is the regressor that maps between k(i)-shot and k(j)-shot models. Here we dropped the explicit dependence of F(?) on w for notational simplicity. These observations emphasize the importance of (1) the identity regularization and (2) sample-size dependent regressors for long-tailed model transfer. We operationalize these extensions with a recursive residual network:   Fi (?) = Fi+1 ? + f (?; wi ) , (2) where f denotes a residual block parameterized by wi and visualized in Fig. 2b. Inspired by [57, 21], f consists of batch normalization (BN) and leaky ReLU as pre-activation, followed by fully-connected weights. By construction, each residual block transforms an input k(i)-shot model to a k(i + 1)-shot model. The final MetaModelNet can be efficiently implemented through a chained network of N + 1 residual blocks, as shown in Fig. 2a. By feeding in a few-shot model at a particular block, we can derive any meta-learner Fi from the central underlying chain. 3.3 Training Given the network structure defined above, we now describe an efficient method for training based on two insights. (1) The recursive definition of MetaModelNet suggests a recursive strategy for training. We begin with the last block and train it with the largest threshold (e.g., those few classes in the head with many examples). The associated k-shot regressor should be easy to learn because it is similar to an identity mapping. Given the learned parameters for the last block, we then train the next-to-last block, and so on. (2) Inspired by the general observation that recognition performance improves on a logarithmic scale as the number of training samples increases [8, 9, 58], we discretize blocks accordingly, to be tuned for 1-shot, 2-shot, 4-shot, ... recognition. In terms of notation, we write the recursive training procedure as follows. We iterate over blocks i from N to 0, and for each i: ? Using Eqn.(1), train parameters of the residual block wi on the head split Ht with k-shot model regression, where k = 2i and t = 2k = 2i+1 . The above ?back-to-front? training procedure works because whenever block i is trained, all subsequent blocks (i + 1, . . . , N ) have already been trained. In practice, rather than holding all subsequent blocks fixed, it is natural to fine-tune them while training block i. One approach might be fine-tuning them on the current k = 2i -shot regression task being considered at iteration i. But because MetaModelNet will be applied across a wide range of k, we fine-tune blocks in a multi-task manner across the current viable range of k = (2i , 2i+1 , . . . , 2N ) at each iteration i. 5 3.4 Implementation details We learn the CNN models on the long-tailed recognition datasets in different scenarios: (1) using a CNN pre-trained on ILSVRC 2012 [1, 59, 60] as off-the-shelf feature; (2) fine-tuning the pre-trained CNN; and (3) training a CNN from scratch. We use ResNet152 [4] for its state-of-the-art performance and use ResNet50 [4] and AlexNet [1] for their easy computation. When training the residual block i, we use the corresponding threshold t and obtain Ct head classes. We generate the Ct -way many-shot classifiers on Ht . For few-shot models, we learn Ct -way k-shot classifiers on random subsets of Ht . Through random sampling, we generate S model mini-batches and each model mini-batch consists of Ct weight vector pairs. In addition, to minimize the loss function (1), we randomly sample 256 image-label pairs as a data mini-batch from Ht . We then use Caffe [59] to train our MetaModelNet on the generated model and data mini-batches based on standard SGD. ? is cross-validated. We use 0.01 as the negative slope for leaky ReLU. Computation is naturally divided into two stages: (1) training a collection of few/many-shot models and (2) learning MetaModelNet from those models. (2) is equivalent to progressively learning a nonlinear regressor. (1) can be made efficient because it is naturally parallelizable across models, and moreover, many models make use of only small training sets. 4 Experimental Evaluation In this section, we explore the use of our MetaModelNet on long-tail recognition tasks. We begin with extensive evaluation of our approach on scene classification of the SUN-397 dataset [14], and address the meta-network variations and different design choices. We then visualize and empirically analyze the learned model dynamics. Finally, we evaluate on the challenging large-scale, scene-centric Places [7] and object-centric ImageNet datasets [5] and show the generality of our approach. 4.1 Evaluation and analysis on SUN-397 Dataset and task: We start our evaluation by fine-tuning a pre-trained CNN on SUN-397, a mediumscale, long-tailed dataset with 397 classes and 100?2,361 images per class [14]. To better analyze trends due to skewed distributions, we carve out a more extreme version of the dataset. Following the experimental setup in [61, 62, 63], we first randomly split the dataset into train, validation, and test parts using 50%, 10%, and 40% of the data, respectively. The distribution of classes is uniform across all the three parts. We then randomly discard 49 images per class for the train part, leading to a long-tailed training set with 1?1,132 images per class (median 47). Similarly, we generate a small long-tailed validation set with 1?227 images per class (median 10), which we use for learning hyper-parameters. We also randomly sample 40 images per class for the test part, leading to a balanced test set. We report 397-way multi-class classification accuracy averaged over all classes. 4.1.1 Comparison with state-of-the-art approaches We first focus on fine-tuning the classifier module while freezing the representation module of a pre-trained ResNet152 CNN model [4, 63] for its state-of-the-art performance. Using MetaModelNet, we learn the model dynamics of the classifier module, i.e., how the classifier weight vectors change during fine-tuning. Following the design choices in Section 3.2, our MetaModelNet consists of 7 residual blocks. For few-shot models, we generate S = 1000 1-shot, S = 500 2-shot, and S = 200 4-shot till 64-shot models from the head classes for learning MetaModelNet. At test time, given the weight vectors of all the classes learned through fine-tuning, we feed them as inputs to the different residual blocks according to their training sample size of the corresponding class. We then ?hallucinate? the dynamics of these weight vectors and use the outputs of MetaModelNet to modify the parameters of the final recognition model as in [21]. Baselines: In addition to the ?plain? baseline that fine-tunes on the target data following the standard practice, we compare against three state-of-the-art baselines that are widely used to address the imbalanced distributions. (1) Over-sampling [16, 17], which uses the balanced sampling via label shuffling as in [16, 17]. (2) Under-sampling [27], which reduces the number of samples per class to 47 at most (the median value). (3) Cost-sensitive [28], which introduces additional weights in the loss function for each class with inverse class frequency. For a fair comparison, fine-tuning is performed 6 Method Acc (%) Plain [4] 48.03 Over-Sampling [16, 17] 52.61 Down-Sampling [27] 51.72 Cost-Sensitive [28] 52.37 MetaModelNet (Ours) 57.34 Table 1: Performance comparison between our MetaModelNet and state-of-the-art approaches for long-tailed scene classification when fine-tuning the pre-trained ILSVRC ResNet152 on the SUN-397 dataset. We focus on learning the model dynamics of the classifier module while freezing the CNN representation module. By benefiting from the learned generic model dynamics from head classes, ours significantly outperforms all the baselines for the long-tail recognition. 1200 MetaModelNet (Ours) Over-Sampling 60 1000 40 800 20 600 0 400 -20 200 -40 0 50 100 150 200 Class index 250 300 350 # Occurrences Relative accuracy gain (%) 80 0 400 Figure 3: Detailed per class performance comparison between our MetaModelNet and the state-ofthe-art over-sampling approach for long-tailed scene classification on the SUN-397 dataset. X-axis: class index. Y-axis (Left): per class classification accuracy improvement relative to the plain baseline. Y-axis (Right): number of training examples. Ours significantly improves for the few-shot tail classes. for around 60 epochs using SGD with an initial learning rate of 0.01, which is reduced by a factor of 10 around every 30 epochs. All the other hyper-parameters are the same for all approaches. Table 1 summarizes the performance comparison averaged over all classes and Fig. 3 details the per class comparison. Table 1 shows that our MetaModelNet provides a promising way of encoding the shared structure across classes in model space. It outperforms existing approaches for long-tail recognition by a large margin. Fig. 3 shows that our approach significantly improves accuracy in the tail. 4.1.2 Ablation analysis We now evaluate variations of our approach and provide ablation analysis. Similar as in Section 4.1.1, we use ResNet152 in the first two sets of experiments and only fine-tune the classifier module. In the last set of experiments, we use ResNet50 [4] for easy computation and fine-tune through the entire network. Tables 2 and 3 summarize the results. Sample-size dependent transformation and identity regularization: We compare to [21], which learns a single transformation for a variety of sample sizes and k-shot models, and importantly, learns a network without identity regularization. For a fair comparison, we consider a variant of MetaModelNet trained on a fixed head and tail split, selected by cross-validation. Table 2 shows that training for a fixed sample size and identity regularization provide a noticeable performance boost (2%). Recursive class splitting: Adding multiple head-tail splits through recursion further improves accuracy by a small but noticeable amount (0.5% as shown in Table 2). We posit that progressive knowledge transfer outperforms the traditional approach because ordering classes by frequency is a natural form of curriculum learning. Joint feature fine-tuning and model dynamics learning: We also explore (nonlinear) fine-tuning of the ?entire network? during head-to-tail transfer by jointly learning the classifier dynamics and the feature representation using ResNet50. We explore two approaches: (1) We first fine-tune the whole 7 Method Acc (%) Model Regression [21] 54.68 MetaModelNet+Fix Split (Ours) 56.86 MetaModelNet+ Recur Split (Ours) 57.34 Table 2: Ablation analysis of variations of our MetaModelNet. In a fixed head-tail split, ours outperforms [21], showing the merit of learning a sample-size dependent transformation. By recursively partitioning the entire classes into different head-tail splits, our performance is further improved. Scenario Method Acc (%) Pre-Trained Features Plain [4] MetaModelNet (Ours) 46.90 54.99 Plain [4] 49.40 Fine-Tuned Features (FT) Fix FT + MetaModelNet (Ours) Recur FT + MetaModelNet (Ours) 58.53 58.74 Table 3: Ablation analysis of joint feature fine-tuning and model dynamics learning on a ResNet50 base network. Though results with pre-trained features underperform those with a deeper base network (ResNet152, the default in our experiments), fine-tuning such features significantly improves results, even outperforming the deeper base network. By progressively fine-tuning the representation during the recursive training of MetaModelNet, performance significantly improves from 54.99% (changing only the classifier weights) to 58.74% (changing the entire CNN). CNN on the entire long-tailed training dataset, and then learn the classifier dynamics using the fixed, fine-tuned representation. (2) During the recursive head-tail splitting, we fine-tune the entire CNN on the current head classes in Ht (while learning the many-shot parameters ?? ), and then learn classifier dynamics using the fine-tuned features. Table 3 shows that progressively learning classifier dynamics while fine-tuning features performs the best. 4.2 Understanding model dynamics Because model dynamics are highly nonlinear, a theoretical proof is rather challenging and outside the scope of this work. Here we provide some empirical analysis of model dynamics. When analyzing the ?dual model (parameter) space?, in which models parameters ? can be viewed as points, Fig. 4 shows that our MetaModelNet learns an approximately-smooth, nonlinear warping of this space that transforms (few-shot) input points to (many-shot) output points. For example, iceberg and mountain scene classes are more similar to each other than to bedrooms. This implies that few-shot iceberg and mountain scene models lie near each other in parameter space, and moreover, they transform in similar ways (when compared to bedrooms). This single meta-network hence encodes class-specific model transformations. We posit that the transformation may capture some form of (class-specific) data-augmentation. Finally, we find that some properties of the learned transformations are quite class-agnostic and apply in generality. Many-shot model parameters tend to have larger magnitudes and norms than few-shot ones (e.g., on SUN-397, the average norm of 1-shot models is 0.53; after transformations through MetaModelNet, the average norm of the output models becomes 1.36). This is consistent with the common empirical observation that classifier weights tend to grow with the amount of training data, showing that they become more confident about their prediction. 4.3 Generalization to other tasks and datasets We now focus on the more challenging, large-scale scene-centric Places [7] and object-centric ImageNet [5] datasets. While we mainly addressed the model dynamics when fine-tuning a pretrained CNN in the previous experiments, here we train AlexNet models [1] from scratch on the target tasks. Table 4 shows the generality of our approach and shows that MetaModelNets facilitate the recognition of other long-tailed datasets with significantly different visual concepts and distributions. Scene classification on the Places dataset: Places-205 [7] is a large-scale dataset which contains 2,448,873 training images approximately evenly distributed across 205 classes. To generate its long-tailed version and better analyze trends due to skewed distributions, we distribute it according to the distribution of SUN and carve out a more extreme version (p2 , or 2? the slope in log-log plot) out of the Places training portion, leading to a long-tailed training set with 5?9,900 images per class (median 73). We use the provided validation portion as our test set with 100 images per class. Object classification on the ImageNet dataset: The ILSVRC 2012 classification dataset [5] contains 1,000 classes with 1.2 million training images (approximately balanced between the classes) and 50K validation images. There are 200 classes used for object detection which are defined as 8 0.6 Mountain Snowy 0.4 Mountain Iceberg 40 20 0.2 Bedroom -0.2 -0.4 -0.4 0 Hotel Room 0 -20 Living Room -0.2 0 0.2 0.4 -40 -40 0.6 (a) PCA visualization. -20 0 20 40 60 (b) t-SNE visualization. Figure 4: Visualizing model dynamics. Recall that ? is a fixed-dimensional vector of model parameters ? e.g., ? ? R2048 when considering parameters from the last layer of ResNet. We visualize models as points in this ?dual? space. Specifically, we examine the evolution of parameters predicted by MetaModelNet with dimensionality reduction ? PCA (Fig. 4a) or t-SNE [64] (Fig. 4b). 1-shot models (purple) to many-shot models (red) are plotted in a rainbow order. These visualizations show that MetaModelNet learns an approximately-smooth, nonlinear warping of this space that transforms (few-shot) input points to (many-shot) output points. PCA suggests that many-shot models tend to have larger norms, while t-SNE (which nonlinearly maps nearby points to stay close) suggests that similar semantic classes tend to be close and transform in similar ways, e.g., the blue rectangle encompasses ?room? classes while the red rectangle encompasses ?wintry outdoor? classes. Dataset Method Acc (%) Plain [1] 23.53 Places-205 [7] MetaModelNet (Ours) 30.71 ILSVRC-2012 [5] Plain [1] MetaModelNet (Ours) 68.85 73.46 Table 4: Performance comparisons on long-tailed, large-scale scene-centric Places [7] and objectcentric ImageNet [5] datasets. Our MetaModelNets facilitate the long-tail recognition with significantly diverse visual concepts and distributions. higher-level classes of the original 1,000. Taking the ILSVRC 2012 classification dataset and merging the 1,000 classes into the 200 higher-level classes, we obtain a natural long-tailed distribution. 5 Conclusions In this work we proposed a conceptually simple but powerful approach to address the problem of long-tail recognition through knowledge transfer from the head to the tail of the class distribution. Our key insight is to represent the model dynamics through meta-learning, i.e., how a recognition model transforms and evolves during the learning process when gradually encountering more training examples. To do so, we introduce a meta-network that learns to progressively transfer meta-knowledge from the head to the tail classes. We present several state-of-the-art results on benchmark datasets (SUN, Places, and Imagenet) tuned for the long-tailed setting, that significantly outperform common heuristics, such as data resampling or reweighting. Acknowledgments. We thank Liangyan Gui and Olga Russakovsky for valuable and insightful discussions. This work was supported in part by ONR MURI N000141612007 and U.S. Army Research Laboratory (ARL) under the Collaborative Technology Alliance Program, Cooperative Agreement W911NF-10-2-0016. DR was supported in part by the National Science Foundation (NSF) under grant number IIS-1618903, Google, and Facebook. 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Stochastic Mirror Descent in Variationally Coherent Optimization Problems Zhengyuan Zhou Stanford University zyzhou@stanford.edu Nicholas Bambos Stanford University bambos@stanford.edu Panayotis Mertikopoulos Univ. Grenoble Alpes, CNRS, Inria, LIG panayotis.mertikopoulos@imag.fr Stephen Boyd Stanford University boyd@stanford.edu Peter Glynn Stanford University glynn@stanford.edu Abstract In this paper, we examine a class of non-convex stochastic optimization problems which we call variationally coherent, and which properly includes pseudo-/quasiconvex and star-convex optimization problems. To solve such problems, we focus on the widely used stochastic mirror descent (SMD) family of algorithms (which contains stochastic gradient descent as a special case), and we show that the last iterate of SMD converges to the problem?s solution set with probability 1. This result contributes to the landscape of non-convex stochastic optimization by clarifying that neither pseudo-/quasi-convexity nor star-convexity is essential for (almost sure) global convergence; rather, variational coherence, a much weaker requirement, suffices. Characterization of convergence rates for the subclass of strongly variationally coherent optimization problems as well as simulation results are also presented. 1 Introduction The stochastic mirror descent (SMD) method and its variants[1, 7, 8] is arguably one of the most widely used family of algorithms in stochastic optimization ? convex and non-convex alike. Starting with the orginal work of [16], the convergence of SMD has been studied extensively in the context of convex programming (both stochastic and deterministic), saddle-point problems, and monotone variational inequalities. Some of the most important contributions in this domain are due to Nemirovski et al. [15], Nesterov [18] and Xiao [23], who provided tight convergence bounds for the ergodic average of SMD in stochastic/online convex programs, variational inequalities, and saddle-point problems. These results were further boosted by recent work on extra-gradient variants of the algorithm [11, 17], and the ergodic relaxation of [8] where the independence assumption on the gradient samples is relaxed and is replaced by a mixing distribution that converges in probability to a well-defined limit. However, all these works focus exclusively on the algorithm?s ergodic average (also known as timeaverage), a mode of convergence which is strictly weaker than the convergence of the algorithm?s last iterate. In addition, most of the analysis focuses on establishing convergence "in expectation" and then leveraging sophisticated martingale concentration inequalities to derive "large deviations" results that hold true with high probability. Last (but certainly not least), the convexity of the objective plays a crucial role: thanks to the monotonicity of the gradient, it is possible to exploit regret-like bounds and transform them to explicit convergence rates.1 1 For the role of variational monotonicity in the context of convex programming, see also [22]. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. By contrast, the gradient operator of the non-convex programs studied in this paper does not satisfy any reasonable monotonicity property (such as quasi-/pseudo-monotonicity, monotonicity-plus, or any of the standard variants encountered in the theory of variational inequalities [9]. Furthermore, given that there is no inherent averaging in the algorithm?s last iterate, it is not possible to employ a regret-based analysis such as the one yielding convergence in convex programs. Instead, to establish convergence, we use the stochastic approximation method of Bena?m and Hirsch [2, 3] to compare the evolution of the SMD iterates to the flow of a mean, underlying dynamical system.2 By a judicious application of martingale limit theory, we then exploit variational coherence to show that the last iterate of SMD converges with probability 1, recovering in the process a large part of the convergence analysis of the works mentioned above. Our Contributions. We consider a class of non-convex optimization problems, which we call variationally coherent and which strictly includes convex, pseudo/quasi-convex and star-convex optimization problems. For this class of optimization problems, we show that the last iterate of SMD with probability 1 to a global minimum under i.i.d. gradient samples. To the best of our knowledge, this strong convergence guarantee (almost sure of the last iterate of SMD) is not known even for stochastic convex problems. As such, this results contributes to the landscape of non-convex stochastic optimization by making clear that neither pseudo-/quasi-convexity nor star-convexity is essential for global convergence; rather, variational coherence, a much weaker requirement, suffices. Our analysis leverages the Lyapunov properties of the Fenchel coupling [14], a primal-dual divergence measure that quantifies the distance between primal (decision) variables and dual (gradient) variables, and which serves as an energy function to establish recurrence of SMD (Theorem 3.4). Building on this recurrence, we consider an ordinary differential equation (ODE) approximation of the SMD scheme and, drawing on various results from the theory of stochastic approximation and variational analysis, we connect the solution of this ODE to the last iterate of SMD. In so doing, we establish the algorithm?s convergence with probability 1 from any initial condition (Thereom 4.4) and, to complete the circle, we also provide a convergence rate estimate for the subclass of strongly variationally coherent optimization problems. Importantly, although the ODE approximation of discrete-time Robbins?Monro algorithms has been widely studied in control and stochastic optimization [10, 13], converting the convergence guarantees of the ODE solution back to the discrete-time process is a fairly subtle affair that must be done on an case-by-case basis. Further, even if such conversion goes through, the results typically have the nature of convergence-in-distribution: almost sure convergence is much harder to obtain [5]. 2 Setup and Preliminaries Let X be a convex compact subset of a d-dimensional real space V with norm k?k. Throughout this paper, we focus on the stochastic optimization problem minimize g(x), subject to x ? X , (Opt) where the objective function g : X ? R is of the form g(x) = E[G(x; ?)] (2.1) for some random function G : X ? ? ? R defined on an underlying (complete) probability space (?, F, P). We make the following assumptions regarding (Opt): Assumption 1. G(x, ?) is continuously differentiable in x for almost all ? ? ?. Assumption 2. ?G(x; ?) has bounded second moments and is Lipschitz continuous in the mean: 2 E[k?G(x; ?)k? ] < ? for all x ? X and E[?G(x; ?)] is Lipschitz on X .3 Assumption 1 is a token regularity assumption which can be relaxed to account for nonsmooth objectives by using subgradient devices (as opposed to gradients). However, this would make 2 For related approaches based on the theory of dynamical systems, see [21] and [12]. In the above, gradients are treated as elements of the dual space V ? of V and kvk? = sup{hv, xi : kxk ? 1} denotes the dual norm of v ? V ? . We also note that ?G(x; ?) refers to the gradient of G(x; ?) with respect to x; since ? need not have a differential structure, there is no danger of confusion. 3 2 the presentation significantly more cumbersome, so we stick with smooth objectives throughout. Assumption 2 is also standard in the stochastic optimization literature: it holds trivially if ?G is uniformly Lipschitz (another commonly used condition) and, by the dominated convergence theorem, it further implies that g is smooth and ?g(x) = ? E[G(x; ?)] = E[?G(x; ?)] is Lipschitz continuous. As a result, the solution set X ? = arg min g (2.2) of (Opt) is closed and nonempty (by the compactness of X and the continuity of g). Remark 2.1. An important special case of (Opt) is when G(x; ?) = g(x) + h?, xi for some V ? -valued 2 random vector ? such that E[?] = 0 and E[k?k? ] < ?. This gives ?G(x; ?) = ?g(x) + ?, so (Opt) can also be seen as a model for deterministic optimization problems with noisy gradient observations. 2.1 Variational Coherence With all this at hand, we now define the class of variationally coherent optimization problems: Definition 2.1. We say that (Opt) is variationally coherent if h?g(x), x ? x? i ? 0 for all x ? X , x? ? X ? , (VC) ? with equality if and only if x ? X . Remark 2.2. (VC) can be interpreted in two ways. First, as stated, it is a non-random condition for g, so it applies equally well to deterministic optimization problems (with or without noisy gradient observations). Alternatively, by the dominated convergence theorem, (VC) can be written as: E[h?G(x; ?), x ? x? i] ? 0. (2.3) In this form, it can be interpreted as saying that G is variationally coherent ?on average?, without any individual realization thereof satisfying (VC). Remark 2.3. Importantly, (VC) does not have to be stated in terms of the solution set of (Opt). Indeed, assume that C is a nonempty subset of X such that h?g(x), x ? pi ? 0 for all x ? X , p ? C, (2.4) with equality if and only if x ? C. Then, as the next lemma (see appendix) indicates, C = arg min g: Lemma 2.2. Suppose that (2.4) holds for some nonempty subset C of X . Then C is closed, convex, and it consists precisely of the global minimizers of g. Corollary 2.3. If (Opt) is variationally coherent, arg min g is convex and compact. Remark 2.4. All the results given in this paper also carry through for ?-variationally coherent optimization problems, a further generalization of variational coherence. More precisely, we say that (Opt) is ?-variationally coherent if there exists a (component-wise) positive vector ? ? Rd such that d X i=1 ?i ?g (xi ? x?i ) ? 0 ?xi for all x ? X , x? ? X ? , (2.5) with equality if and only if x ? X ? . For simplicity, our analysis will be carried out in the ?vanilla" variational coherence framework, but one should keep in mind that the results to following also hold for ?-coherent problems. 2.2 Examples of Variational Coherence Example 2.1 (Convex programs). If g is convex, ?g is a monotone operator [19], i.e. h?g(x) ? ?g(x0 ), x ? x0 i ? 0 for all x, x0 ? X . (2.6) By the first-order optimality conditions for g, we have hg(x? ), x ? x? i ? 0 for all x ? X . Hence, by monotonicity, we get h?g(x), x ? x? i ? h?g(x? ), x ? x? i ? 0 ? for all x ? X , x? ? X ? . ? ? (2.7) ? By convexity, it follows that h?g(x), x ? x i < 0 whenever x ? X and x ? X \ X , so equality holds in (2.7) if and only if x ? X ? . 3 Example 2.2 (Pseudo/Quasi-convex programs). The previous example shows that variational coherence is a weaker and more general notion than convexity and/or operator monotonicity. In fact, as we show below, the class of variationally coherent problems also contains all pseudo-convex programs, i.e. when h?g(x), x0 ? xi ? 0 =? g(x0 ) ? g(x), (PC) for all x, x0 ? X . In this case, we have: Proposition 2.4. If g is pseudo-convex, (Opt) is variationally coherent. Proof. Take x? ? X ? and x ? X \ X ? , and assume ad absurdum that h?g(x), x ? x? i ? 0. By (PC), this implies that g(x? ) ? g(x), contradicting the choice of x and x? . We thus conclude that h?g(x), x ? x? i > 0 for all x? ? X ? , x ? X \ X ? ; since h?g(x), x ? x? i ? 0 if x ? X ? , our claim follows by continuity.  We recall that every convex function is pseudo-convex, and every pseudo-convex function is quasiconvex (i.e. its sublevel sets are convex). Both inclusions are proper, but the latter is fairly thin: Proposition 2.5. Suppose that g is quasi-convex and non-degenerate, i.e. hg(x), zi = 6 0 for all nonzero z ? TC(x), x ? X \ X ? , (2.8) where TC(x) is the tangent cone vertexed at x. Then, g is pseudo-convex (and variationally coherent). Proof. This follows from the following characterization of quasi-convex functions [6]: g is quasiconvex if and only if g(x0 ) ? g(x) implies that h?g(x), x0 ? xi ? 0. By contraposition, this yields the strict part of (PC), i.e. g(x0 ) > g(x) whenever h?g(x), x0 ? xi > 0. To complete the proof, if h?g(x), x0 ? xi = 0 and x ? X ? , (PC) is satisfied trivially; otherwise, if h?g(x), x0 ? xi = 0 but x ? X \ X ? , (2.8) implies that x0 ? x = 0, so g(x0 ) = g(x) and (PC) is satisfied as an equality.  The non-degeneracy condition (2.8) is satisfied by every quasi-convex function after an arbitrarily small perturbation leaving its minimum set unchanged. By this token, Propositions 2.4 and 2.5 imply that essentially all quasi-convex programs are also variationally coherent. Example 2.3 (Star-convex programs). If g is star-convex, then h?g(x), x ? x? i ? g(x) ? g(x? ) for all x ? X , x? ? X ? . This is easily seen to be a special case of variational coherence because h?g(x), x ? x? i ? g(x) ? g(x? ) ? 0, with the last inequality strict unless x ? X ? . Note that star-convex functions contain convex functions as a subclass (but not necessarily pseudo/quasi-convex functions). Example 2.4 (Beyond quasi-/star-convexity). A simple example of a function that is variationally coherent without being quasi-convex or star-convex is given by: g(x) = 2 d X ? 1 + xi , x ? [0, 1]d . (2.9) i=1 When d ? 2, it is easy to see g is not quasi-convex: ? ? for instance, taking d = 2, x = (0, 1) and x0 = (1, 0) yields g(x/2 + x0 /2) = 2 6 > 2 2 = max{g(x), g(x0 )}, so g is not quasiconvex. It is also instantly clear this function is not star-convex even when d = 1 (in which case it is a concave function). On the other hand, to estabilish (VC), simply note that X ? = {0} and ? Pd h?g(x), x ? 0i = i=1 xi / 1 + xi > 0 for all x ? [0, 1]d \{0}. For a more elaborate example of a variationally coherent problem that is not quasi-convex, see Figure 2. 2.3 Stochastic Mirror Descent To solve (Opt), we focus on the widely used family of algorithms known as stochastic mirror descent (SMD), formally given in Algorithm 1.4 Heuristically, the main idea of the method is as follows: At each iteration, the algorithm takes as input an independent and identically distributed (i.i.d.) sample 4 Mirror descent dates back to the original work of Nemirovski and Yudin [16]. More recent treatments include [1, 8, 15, 18, 20] and many others; the specific variant of SMD that we are considering here is most closely related to Nesterov?s ?dual averaging? scheme [18]. 4 Y0 Y = V? ??1 ?G(X0 ; ?1 ) ??2 ?G(X1 ; ?2 ) Y1 Y 2 Q Q Q Q X2 X ?V X0 X1 Figure 1: Schematic representation of stochastic mirror descent (Algorithm 1). of the gradient of G at the algorithm?s current state. Subsequently, the method takes a step along this stochastic gradient in the dual space Y ? V ? of V (where gradients live), the result is ?mirrored? back to the problem?s feasible region X to obtain a new solution candidate, and the process repeats. In pseudocode form, we have: Algorithm 1 Stochastic mirror descent (SMD) Require: Initial score variable Y0 1: n ? 0 2: repeat 3: Xn = Q(Yn ) 4: Yn+1 = Yn ? ?n+1 ?G(Xn , ?n+1 ) 5: n?n+1 6: until end 7: return solution candidate Xn In the above representation, the key elements of SMD (see also Fig. 1) are: 1. The ?mirror map? Q : Y ? X that outputs a solution candidate Xn ? X as a function of the auxiliary score variable Yn ? Y. In more detail, the algorithm?s mirror map Q is defined as Q(y) = arg max{hy, xi ? h(x)}, (2.10) x?X where h(x) is a strongly convex function that plays the role of a regularizer. Different choices of the regularizer h yields different specific algorithm. Due to space limitation, we mention in 2 passing two well-known examples: When h(x) = 12 kxk2 (i.e. Euclidean regularizer), mirror Pd descent becomes gradient descent. When h(x) = i=1 xi log xi (i.e. entropic regularizer), mirror descent becomes exponential gradient (aka exponential weights). 2. The step-size sequence ?n > 0, chosen to satisfy the ?`2 ? `1 ? summability condition: ? X ?n2 < ?, n=1 ? X ?n = ?. (2.11) n=1 3. A sequence of i.i.d. gradient samples ?G(x; ?n+1 ).5 3 Recurrence of SMD In this section, we characterize an interesting recurrence phenomenon that will be useful later for establishing global convergence. Intuitively speaking, for a variationally coherent program of the 5 The specific indexing convention for ?n has been chosen so that Yn and Xn are both adapted to the natural filtration Fn of ?n . 5 general form(Opt), any neighborhood of X ? will almost surely be visited by iterates Xn infinitely often. Note that this already implies that at least a subsequence of iterates converges to global minima almost surely. To that end, we first define an important divergence measure between a primal variable x and a dual variable y, called Fenchel coupling, that plays an indispensable role of an energy function. Definition 3.1. Let h : X ? R be a regularizer with respect to k ? k that is K-strongly convex. 1. The convex conjugate function h? : Rn ? R of h is defined as: h? (y) = max{hx, yi ? h(x)}. x?X 2. The mirror map Q : Rn ? X associated with the regularizer h is defined as: Q(y) = arg max{hx, yi ? h(x)}. x?X 3. The Fenchel coupling F : X ? Rn ? R is defined as: F (x, y) = h(x) ? hx, yi + h? (y). Note that the naming of Fenchel coupling is natural as it consists of all the terms in the well-known Fenchel?s inequality: h(x) + h? (y) ? hx, yi. The Fenchel?s inequality says that Fenchel coupling is always non-negative. As indicated by part 1 of the following lemma, a stronger result can be obtained. We state the two key properties Fenchel coupling next. Lemma 3.2. Let h : X ? R be a K-strongly convex regularizer on X . Then: 1. F (x, y) ? 21 KkQ(y) ? xk2 , ?x ? X , ?y ? Rn . 2. F (x, y?) ? F (x, y) + h? y ? y, Q(y) ? xi + 1 y? 2K k? yk2? , ?x ? X , ?? y , y ? Rn . We assume that we are working with mirror maps that are regular in the following weak sense:6 Assumption 3. The mirror map Q is regular: if Q(yn ) ? x, then F (x, yn ) ? 0. Definition 3.3. Given a point x ? X , a set S ? X and a norm k ? k. 1. Define the point-to-set normed distance and Fenchel coupling distance respectively as: dist(x, S) , inf s?S kx ? sk and F (S, y) = inf s?S F (s, y). 2. Given ? > 0, define B(S, ?) , {x ? X | dist(x, S) < ?}. ? 3. Given ? > 0, define B(S, ?) , {Q(y) | F (S, y) < ?}. We then have the following recurrence result for a variationally coherent optimization problem Opt. Theorem 3.4. Under Assumptions 1?3, for any ? > 0, ? > 0 and any Xn , the (random) iterates Xn ? ? , ?) infinitely often almost surely. generated in Algorithm 1 enter both B(X ? , ?) and B(X 4 Global Convergence Results 4.1 Deterministic Convergence When a perfect gradient ?g(x) is available (in Line 4 of Algorithm 1), SMD recovers its deterministic counterpart: mirror descent (Algorithm 2). We first characterize global convergence in this case. 6 Mirror maps induced by many common regularizers are regular, including the Euclidean regularizer and the entropic regularizer. 6 ??????????? ?? ?????????? ?????? ??????? ??? ??? ??? -??? -??? -??? -??? ??? ??? ??? Figure 2: Convergence of stochastic mirror descent for the mean objective g(r, ?) = (2 + cos ?/2 + cos(4?))r2 (5/3 ? r) expressed in polar coordinates over the unit ball (r ? 1). In the left subfigure, we have plotted the graph of g; the plot to the right superimposes a typical SMD trajectory over the contours of g. Algorithm 2 Mirror descent (MD) Require: Initial score variable y0 1: n ? 0 2: repeat 3: xn = Q(yn ) 4: xn+1 = xn ? ?n+1 ?g(xn ) 5: n?n+1 6: until end 7: return solution candidate xn Theorem 4.1. Consider an optimization problem Opt that is variationally coherent. Let xn be the iterates generated by MD. Under Assumption 3, limt?? dist(xn , X ? ) = 0, for any y0 . Remark 4.1. Here we do not require ? g(x) to be Lipschitz continuous. If ? g(x) is indeed (locally) Lipschitz continuous, then Theorem 4.1 follows directly from Theorem 4.4. Otherwise, Theorem 4.1 requires a different argument, briefly outlined as follows. Theorem 3.4 implies that (in the special case of perfect gradient), iterates xn generated from MD enter B(X ? , ?) infinitely often. Now, by exploiting the properties of Fenchel coupling on a finer-grained level (compared to only using it to establish recurrence), we can establish that for any ?-neighborhood B(X ? , ?), after a certain number of iterations, once the iterate xn enters B(X ? , ?), it will never exit. Convergence therefore follows. 4.2 Stochastic Almost Sure Convergence We begin with minimal mathematical preliminaries [4] needed that will be needed. Definition 4.2. A semiflow ? on a metric space (M, d) is a continuous map ? : R+ ? M ? M : (t, x) ? ?t (x), such that the semi-group properties hold: ?0 = identity, ?t+s = ?t ? ?s for all (t, s) ? R+ ? R+ . Definition 4.3. Let ? be a semiflow on the metric space (M, d). A continuous function s : R+ ? M is an asymptotic pseudotrajectory (APT) for ? if for every T > 0, the following holds: lim sup d(s(t + h), ?h (s(t))) = 0. t?? 0?h?T (4.1) We are now ready to state the convergence result. See Figure 2 for a simulation example. Theorem 4.4. Consider an optimization problem Opt that is variationally coherent. Let Xn be the iterates generated by SMD (Algorithm 1). Under Assumptions 1?3, if ? g(x) is locally Lipschitz continuous on X , then dist(xn , X ? ) ? 0 almost surely as t ? ?, irrespective of Y0 . 7 Remark 4.2. The proof is rather involved and contains several ideas. To enhance the intuition and understanding, we outline the main steps here, each of which will be proved in detail in the appendix. To simplify the notation, we assume there is a unique optimal (i.e. X ? is a singleton set). The proof is identical in the multiple minima case, provide we replace x? by X ? and use the point-to-set distance. 1. We consider the following ODE approximation of SMD: y? = v(x), x = Q(y), where v(x) = ? ? g(x). We verify that the ODE admits a unique solution for y(t) for any initial condition. Consequently, this solution induces a semiflow7 , which we denote ?t (y): it is the state at time t given it starts at y initially. Note that we have used y as the initial point (as opposed to y 0 ) to indicate that the semiflow representing the solution trajectory should be viewed as a function of the initial point y. 2. We now relate the iterates generated by SMD to the above ODE?s solution. Connect linearly Pk?1 the SMD iterates Y1 , Y2 , . . . , Yk , . . . at times 0, ?1 , ?1 + ?2 , . . . , i=0 ?i , . . . respectively to form a continuous, piecewise affine (random) curve Y (t). We then show that Y (t) is almost surely an asymptotic pseudotrajectory of the semi-flow ? induced by the above ODE. 3. Having characterized the relation between the SMD trajectory (affine interpolation of the discrete SMD iterates) and the ODE trajectory (the semi-flow), we now turn to studying the latter (the semiflow given by the ODE trajectory). A desirable property of ?t (y) is that the distance F (x? , ?t (y)) between the optimal solution x? and the dual variable ?t (y) (as measured by Fenchel coupling) can never increase as a function of t. We refer to this as the monotonicity property of Fenchel coupling under the ODE trajectory, to be contrasted to the discrete-time dynamics, where such monotonicity is absent (even when perfect information on the gradient is available). More formally, we show that ?y, ?0 ? s ? t, F (x? , ?s (y)) ? F (x? , ?t (y)). (4.2) 4. Continuing on the previous point, not only the distance F (x? , ?t (y)) can never increase as t increases, but also, provided that ?t (y) is not too close to x? , F (x? , ?t (y)) will decrease no slower than linearly. This suggests that either ?t (y) is already close to x? (and hence x(t) = Q(?t (y)) is close to x? ), or their distance will be decreased by a meaningful amount in (at least) the ensuing short time-frame. We formalize this discussion as follows: ? ? ?? > 0, ?y, ?s > 0, F (x? , ?s (y)) ? max{ , F (x? , y) ? }. (4.3) 2 2 5. Now consider an arbitrary fixed horizon T . If at time t, F (x? , ?0 (Y (t))) is small, then by the monotonicity property in Claim 3, F (x? , ?h (Y (t))) will remain small on the entire interval h ? [0, T ]. Since Y (t) is an asymptotic pseudotrajectory of ? (Claim 2), Y (t + h) and ?h (Y (t)) should be very close for h ? [0, T ], at least for t large enough. This means that F (x? , Y (t + h)) should also be small on the entire interval h ? [0, T ]. This can be made precise as follows: ??, T > 0, ?? (?, T ) > 0 such that ?t ? ?, ?h ? [0, T ]: ? F (x? , Y (t + h)) < F (x? , ?h (Y (t))) + , a.s.. (4.4) 2 6. Finally, we are ready to put the above pieces together. Claim 5 gives us a way to control the amount by which the two Fenchel coupling functions differ on the interval [0, T ]. Claim 3 and Claim 4 together allow us to extend such control over successive intervals [T, 2T ), [2T, 3T ), . . . , thereby establishing that, at least for t large enough, if F (x? , Y (t)) is small, then F (x? , Y (t + h)) will remains small ?h > 0. As it turns out, this means that ? ? , ?), it will (almost surely) be forever trapped after long enough time, if xn ever visits B(x ? ? , 2?)). Since Theorem 3.4 ensures that inside the neighborhood twice that size (i.e. B(x ? ? , ?) infinitively often (almost surely), the hypothesis is guaranteed to be true. xn visits B(x Consequently, this leads to the following claim: ?? > 0, ??0 (a positive integer), such that: F (x? , Y (?0 + h)) < ?, ?h ? [0, ?), a.s.. 7 (4.5) A crucial point to note is that since C may not be invertible, there may not exist a unique solution for x(t). 8 ??????????? ?? ?????????? ?????? ??????? ???????? ????? ? ? ? ? ? ? ? ? ? ? ? ????? ????? ? ???? ??????? ? ??????? ??????? ????? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ????????? ? ?? ?????????? ?????????? ? ?? ???????????? ?? ?????????????? ???? ??????????????? ?????? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? ? ? ? ? ? ? ????? ? ??? ????? ? ?? ???? ?????? ????? ???????? ?? ?? ? ? ??????????? ? ? ? ? ? ? ? ?? ?? ? ?? ???? ?? ? ?? ? ? ? ? ? ? ? ???? ? ? ? ?? ?? ? ? ??-? ? ? ?? ????????? ?? ??? Figure 3: SMD run on the objective function of Fig. 2 with ?n ? n?1/2 and Gaussian random noise with standard deviation about 150% the mean value of the gradient. Due to the lack of convexity, the algorithm?s last iterate converges much faster than its ergodic average. To conclude, Equation (4.5) implies that F (x? , Yn ) ? 0, a.s. as t ? ?, where the SMD iterates Yn are values at integer time points of the affine trajectory Y (? ). Per Statement 1 in Lemma 3.2, this gives kQ(Yn ) ? x? k ? 0, a.s. as t ? ?, thereby establishing that Xn = Q(Yn ) ? x? , a.s.. 4.3 Convergence Rate Analysis At the level of generality at which (VC) has been stated, it is unlikely that any convergence rate can be obtained, because unlike in the convex case, one has no handle on measuring the progress of mirror descent updates (recall that in (VC), only non-negativity is guaranteed for the inner product). Consequently, we focus here on the class ? of strongly coherent problems (a generalization of strongly convex problems) and derive a O(1/ T ) convergence rate in terms of the squared distance to a solution of (Opt). Definition 4.5. We say that g is c-strongly variationally coherent (or c-strongly coherent for short) if, for some x? ? X , we have: h?g(x), x ? x? i ? c 2 kx ? x? k 2 for all x ? X . Theorem 4.6. If (Opt) is c-strongly coherent, then k? xT ? x? k2 ? x ?T = PT n=0 ?n xn P , T n=0 ?n PT ? 2 B )+ 2K 2 F (x ,y0P n=0 ?n , T c n=0 ?n (4.6) where K is the strong convexity coefficient of h and B = maxx?X k ? g(x)k2? . The proof of Theorem 4.6 is given in the supplement. We mention a few implications of Theorem 4.6. ? T ) (note First, in a strongly coherent optimization problem, if ?n = ?1n , then k? xT ? x? k2 = O( log T that here `2 ? `1 summability is not required for global convergence). By appropriately choosing ? the step-size sequence, one can further shave off the log T term above and obtain an O(1/ T ) convergence rate. This rate matches existing rates when applying gradient descent to strongly convex functions, although strongly variational coherence is a strict superset of strong convexity. Finally, note that even though we have characterized the rates in the mirror descent (i.e. perfect gradient case), ? one can easily obtain a mean O(1/ T ) rate in the stochastic case by using a similar argument. This discussion is omitted due to space limitation. We end the section (and the paper) with an interesting observation from the simulation shown in Figure 3. The rate characterized in Theorem 4.6 is with respect to the ergodic average of the mirror descent iterates, while global convergence results established in Theorem 4.1 and Theorem 4.4 are both last iterate convergence. Figure 3 then provides a convergence speed comparison on the function given in Figure 2. It is apparent that the last iterate of SMD (more specifically, stochastic gradient descent) converges much faster than the ergodic average in this non-convex objective. 9 5 Acknowledgments Zhengyuan Zhou is supported by Stanford Graduate Fellowship and would like to thank Yinyu Ye and Jose Blanchet for constructive discussions and feedback. Panayotis Mertikopoulos gratefully acknowledges financial support from the Huawei Innovation Research Program ULTRON and the ANR JCJC project ORACLESS (grant no. ANR?16?CE33?0004?01). References [1] A. B ECK AND M. T EBOULLE, Mirror descent and nonlinear projected subgradient methods for convex optimization, Operations Research Letters, 31 (2003), pp. 167?175. [2] M. B ENA?M, Dynamics of stochastic approximation algorithms, in S?minaire de Probabilit?s XXXIII, J. Az?ma, M. ?mery, M. Ledoux, and M. Yor, eds., vol. 1709 of Lecture Notes in Mathematics, Springer Berlin Heidelberg, 1999, pp. 1?68. [3] M. B ENA?M AND M. W. H IRSCH, Asymptotic pseudotrajectories and chain recurrent flows, with applications, Journal of Dynamics and Differential Equations, 8 (1996), pp. 141?176. [4] M. B ENA ? M AND M. W. H IRSCH, Asymptotic pseudotrajectories and chain recurrent flows, with applications, Journal of Dynamics and Differential Equations, 8 (1996), pp. 141?176. [5] V. S. B ORKAR, Stochastic Approximation: A Dynamical Systems Viewpoint, Cambridge University Press and Hindustan Book Agency, 2008. [6] S. B OYD AND L. VANDENBERGHE, Convex Optimization, Berichte ?ber verteilte messysteme, Cambridge University Press, 2004. [7] N. C ESA -B IANCHI , P. G AILLARD , G. L UGOSI , AND G. S TOLTZ, Mirror descent meets fixed share (and feels no regret), in Advances in Neural Information Processing Systems, 989-997, ed., vol. 25, 2012. [8] J. C. D UCHI , A. AGARWAL , M. J OHANSSON , AND M. I. J ORDAN, Ergodic mirror descent, SIAM Journal on Optimization, 22 (2012), pp. 1549?1578. [9] F. FACCHINEI AND J.-S. 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Bayesian Backpropagation Over 1-0 Functions Rather Than Weights David H. Wolpert The Santa Fe Institute 1660 Old Pecos Trail Santa Fe, NM 87501 Abstract The conventional Bayesian justification of backprop is that it finds the MAP weight vector. As this paper shows, to find the MAP i-o function instead one must add a correction tenn to backprop. That tenn biases one towards i-o functions with small description lengths, and in particular favors (some kinds of) feature-selection, pruning, and weight-sharing. 1 INTRODUCTION In the conventional Bayesian view ofbackpropagation (BP) (Buntine and Weigend, 1991; Nowlan and Hinton,1994; MacKay,I992; Wolpert, 1993), one starts with the "likelihood" conditional distribution P(training set = t I weight vector w) and the "prior" distribution P(w). As an example, in regression one might have a "Gaussian likelihood", P(t I w) oc: exp[-x2(w, t)] == I1i exp [-(net(w, tx(i? - ty(i) )2/2c?] for some constant CJ. (tx(i) and ty(i) are the successive input and output values in the training set respectively, and net(w, .) is the function, induced by w, taking input neuron values to output neuron values.) As another example, the "weight decay" (Gaussian) prior is P(w) oc: eXp(-a(w2? for some constant a. Bayes' theorem tells us that P(w I t) oc P(t I w) P(w). Accordingly, the most probable weight given the data - the "maximum a posteriori" (MAP) w - is the mode over w of P(t I w) P(w), which equals the mode over w of the "cost function" L(w, t) == In[P(t I w)] + In[P(w)]. So for example with the Gaussian likelihood and weight decay prior, the most probable w given the data is the w minimizing X2(w, t) + aw2. Accordingly BP with weight decay can be viewed as a scheme for trying to find the function from input neuron values to output neuron values (i-o function) induced by the MAP w. 200 Bayesian Backpropagation over 1-0 Functions Rather Than Weights One peculiar aspect of this justification of weight-decay BP is the fact that rather than the i-o function induced by the most probable weight vector. in practice one would usually prefer to know the most probable i-o function. (In few situations would one care more about a weight vector than about what that weight vector parameterizes.) Unfortunately. the difference between these two i-o functions can be large; in general it is not true that "the most probable output corresponds to the most probable parameter (Denker and LeCun. 1991). U This paper shows that to fmd the MAP i-o function rather than the MAP w one adds a "correction term to conventional BP. That term biases one towards i-o functions with small description lengths. and in particular favors feature-selection. pruning and weight-sharing. In this that term constitutes a theoretical justification for those techniques. U Although cast in terms of neural nets. this paper?s analysis applies to any case where convention is to use the MAP value of a parameter encoding Z to estimate the value of Z. 2 BACKPROP OVER 1-0 FUNCTIONS Assume the nee s architecture is fixed. and that weight vectors w live in a Euclidean vector space W of dimension IWI. Let X be the set of vectors x which can be loaded on the input neurons. and 0 the set of vectors 0 which can be read off the output neurons. Assume that the number of elements in X (lXI) is finite. This is always the case in the real world. where measuring devices have finite accuracy. and where the computers used to emulate neural nets are finite state machines. For similar reasons 0 is also finite in practice. However for now assume that 0 is very large and "fine-grained". and approximate it as a Euclidean vector space of dimension 101. (This assumption usually holds with neural nets. where output values are treated as real-valued vectors.) This assumption will be relaxed later. Indicate the set of functions taking X to 0 by cl>. (net(w?.) is an element of cl>.) Any cI? E cl> is an (lXI x 101)-dimensional Euclidean vector. Accordingly. densities over W are related to densities over cl> by the usual rules for transforming densities between IWI-dimensional and (IXI x IOI)-dimensional Euclidean vector spaces. There are three cases to consider: 1) IWI < IXIIOL In general. as one varies over all w's the corresponding i-o functions net(w, .) map out a sub-manifold of cl> having lower dimension than cl>. 2) IWI > IXIIOL There are an infinite number of w's corresponding to each cI?. 3) IWI = IXIIOI. This is the easiest case to analyze in detail. Accordingly I will deal with it first, deferring discussion of cases one and two until later. With some abuse of notation, let capital letters indicate random variables and lower case letters indicate values of random variables. So for example w is a value of the weight vector random variable W. Use 'p' to indicate probability densities. So for example P<l>IT<cI? I t) is the density of the i-o function random variable cl>, conditioned on the training set random variable T, and evaluated at the values cl> = cI? and T = t. In general, any i-o function not expressible as net(w, .) for some w has zero probability. For the other i-o functions, with S(.) being the multivariable Dirac delta function, p<l>(net(w?.? = jdw' Pw(w') S(net(w', .) - net(w, When the mapping cl> (1) =net(W, .) is one-to-one, we can evaluate equation (1) to get p4>lT<net(w, .) I t) = Pwrr(w I t) / J<I>.W<w), where J<I> w(w) is the Jacobian of the W ~ cl> mapping: ? .?. (2) 201 202 Wolpert J<I>,W(w) == I det[ ()<l>i / dWj lew) I = I det[ d net(w, ')i / dWj ] I. (3) "net(w, .)t means the i'th component of the i-o function net(w, .). "net(w, x)" means the vector 0 mapped by net(w, .) from the input x, and "net(w, X)k" is the k'th component of o. So the "i" in "net(w, .)( refers to a pair of values {x, k}. Each matrix value a~ dWj is the partial derivative of net(w, x>t with respect to some weight, for some x and k. J<I>,w(w) can be rewritten as detla [gij(W)], where gij(w) == ~ [(d<!>t / dWi) (d~ / dWj)] is the metric of the W ~ ct? mapping. This form of J~,w(w) is usually more difficult to evaluate though. / * Unfortunately, cI? = net(w, .) is not one-to-one; where J<I>,w(w) 0 the mapping is locally one-to-one, but there are global symmetries which ensure that more than one w corresponds to each cI?. (Such symmetries arise from things like permuting the hidden neurons or changing the sign of all weights leading into a hidden neuron - see (Fefferman, 1993) and references therein.) To circumvent this difficulty we must make a pair of assumptions. To begin, restrict attention to W inj , those values w of the variable W for which the Jacobian is non-zero. This ensures local injectivity of the map between W and ct?. Given a particular w E W inj' let k be the number of w' E W inj such that net(w, .) = net(w', .). (Since net(w,.) = net(w, .), k ~ 1.) Such a set ofk vectors form an equivalence class, {w}. The first assumption is that for all w E W inj the size of (w) (i.e., k) is the same. This will be the case if we exclude degenerate w (e.g.? w's with all first layer weights set to 0). The second assumption is that for all w' and w in the same equivalence class, PWID (w I d) = PWID (w' I d). This is usually the case. (For example, start with w' and relabel hidden neurons to get a new WE (w'). If we assume the Gaussian likelihood and prior, then since neither differs for the two w's the weight-posterior is also the same for the two w's.) Given these assumptions, p<l>IT(net(w, .) I t) = k pWlnw I t) / J<I>,w(w). So rather than minimize the usual cost function, L(w, t), to find the MAP ct? BP should minimize L'(w. t) == L(w, t) + In[ J~W<w)]. The In[ J~w(w)] term constitutes a correction term to conventional BP. , , One should not confuse the correction term with the other quantities in the neural net literature which involve partial derivative matrices. As an example, one way to characterize the "quality" of a local peak w' of a cost function involves the Hessian of that cost function (Buntine and Weigend, 1991). The correction term doesn't directly concern the validity of such a Hessian-based quality measure. However it does concern the validity of some implementations of such a measure. In particular. the correction term changes the location of the peak w'. It also suggests that a peak's quality be measured by the Hessian of L'(w', t) with respect to cI?, rather than by the Hessian of L(w', t) with respect to w. (As an aside on the subject of Hessians, note that some workers incorrectly use Hessians when they attempt to evaluate quantities like output-variances. See (Wolpert, 1994).) If we stipulate that the pcI>lncI? I t) one encounters in the real world is independent of how one chooses to parameterize ct?, then the probability density of our parameter must depend on how it gets mapped to ct?. This is the basis of the correction term. As this suggests, the correction term won't arise if we use non-pcI>lncl? I t)-based estimators, like maximum-likelihood estimators. (This is a basic difference between such estimators and MAP estimators with a uniform prior.) The correction term is also irrelevant if it we use an MAP estimate but J~,w(w) is independent of w (as when net (w?.) depends linearly on w). And for nonlinear net(w, .), the correction term has no effect for some non-MAP-based ways to apply Bayesianism to neural nets, like guessing the posterior average ct? (Neal, 1993): Bayesian Backpropagation over 1-0 Functions Rather Than Weights E(ct? It) == Idct> Pcl>lnct> It) ct> = Idw PWIT(w I t) net(w, .), (4) so one can calculate E(ct? I t) by working in W, without any concern for a correction tenn. (Loosely speaking, the Jacobian associated with changing integration variables cancels the Jacobian associated with changing the argument of the probability density. A formal derivation - applicable even when IWI-:/: IXI x 101 - is in the appendix of (Wolpert, 1994).) One might think that since it's independent of 1, the correction term can be absorbed into Pw(w). Ironically, it is precisely because quantities like E(ct? I t) aren't affected by the correction tenn that this is impossible: Absorb the correction term into the prior, giving a new prior P*w(w) == d x Pw(w) x J<I),w(w) (asterisks refers to new densities, and d is a normalization constant). Then p*<I)IT(net(w, .) I t) = pwrr(w I t). So by redefining what we call the prior we can justify use of conventional uncorrected BP; the (new) MAP ct> corresponds to the w minimizing L(w, t). However such a redefinition changes E(ct? I t) (amongst other things): Idct> P*<I)IT(ct> I t) ct> = Idw P*Wlnw I t) net(w, .) -:/: Idw Pwrr(w I t) net(w, .) = Idct> Pcl>lnct> I t) ct>. So one can either modify BP (by adding in the correction term) and leave E(ct? I t) alone, or leave BP alone but change E(ct? I t); one can not leave both unchanged. Moreover, some procedures involve both prior-based modes and prior-based integrals, and therefore are affected by the correction tenn no matter how Pw(w) is redefined. For example, in the evidence procedure (Wolpert, 1993; MacKay, 1992) one fixes the value of a hyperparameter r (e.g., ex from the introduction) to the value 1 maximizing Pr IT(11 t). Next one find the value s' maximizing PSIT,r (s' I t, 1) for some variable S. Finally, one guesses the <I> associated with s'. Now it's hard to see why one should use this procedure with S = W (as is conventional) rather than with S = ct?. But with S ct? rather than W, one must factor in the correction term when calculating PSIT,r (s I t, 1), and therefore the guessed ct> is different from when S = W. If one tries to avoid this change in the guessed ct> by absorbing the correction tenn into the prior PWIr(w Iy), then Pn nY I t) - which is given by an integral involving that prior - changes. This in turn changes 1, and therefore the guessed ct> again is different. So presuming one is more directly interested in ct? rather than W, one can't avoid having the correction term affect the evidence procedure. It should be noted that calculating the correction tenn can be laborious in large nets. One should bear in mind the determinant-evaluation tricks mentioned in (Buntine and Weigend, 1991), as well as others like the identity In[ J<I),w(w) ] = Tr(ln[ ~ / dwj ]) == Tr(ln*[ d$i / dwj n, where In*(.) is In(.) evaluated to several orders. = 3 EFFECTS OF THE CORRECTION TERM To illustrate the effects of the correction term, consider a perceptron with a single output neuron, N input neurons and a unary input space: 0 = tanh(w . x), and x always consist of a single one and N - 1 zeroes. For this scenario d<l>i / dwj is an N x N diagonal matrix, and In[ J<I),w(w)] = -2 ~~=l In[ COSh(Wk)]. Assume the Gaussian prior and likelihood of the introduction, and for simplicity take 2cr2 = 1. Both L(w, t) and L'(w, t) are sums of terms each of which only concerns one weight and the corresponding input neuron. Accordingly, it suffices to consider just the i' th weight and the corresponding input neuron. Let xCi) be the input vector which has its 1 in neuron i. Let ojCi) be the output of the j'th of the pairs in the training set with input x(i), and mi the number of such pairs. With ex = 0 203 204 Wolpert (no weight decay), L(w, t) = X2 (1, w), which is minimized by W'i = tanh-l [ l:j~l oj(i) / mil. If we instead try to minimize X2(t, w) + Jw.MW) though, then for low enough mi (e.g., mi = 1), we find that there is no minimum. The correction term pushes waway from 0, and for low enough mi the likelihood isn't strong enough to counteract this push. . -o o -, -, -, o o Figures 1 through 3: Train using unmodified BP on training set 1, and feed input x into the resultant net. The horizontal axis gives the output you get If t and x were still used but training had been with modified BP, the output would have been the value on the vertical axis. In succession, the three figures have a = .6, .4, .4, and m = 1,4, 1. _ .._..... ......................................- I o ?1 - I - I -, - _... . . . _ .... ..... .... ............... . ... - I - I - ao o o Figure 3. Figure 4: The horizontal axis is IWil. The top curve depicts the weight decay regularizer, aw?, and the bottom curve shows that regularizer modified by the correction term. a = .2. When weight-decay is used though, modified BP finds a solution, just like unmodified BP does. Since the correction term "pushes out" w, and since tanh(.) grows with its argument, a <I> found by modified BP has larger (in magnitude) values of 0 than does the corresponding <I> found by unmodified BP. In addition, unlike unmodified BP, modified BP has multiple extrema over certain regimes. All of this is illustrated in figures (1) through (3), which graph the value of 0 resulting from using modified BP with a particular training set t and input value x vs. the value of 0 resulting from using unmodified BP with t and x. Figure (4) depicts the wi-dependences of the weight decay term and of the weight-decay term plus the correction term. (When there's no data, BP searches for minima of those curves.) Now consider multi-layer nets, possibly with non-unary X. Denote a vector of the compo- Bayesian Backpropagation over 1-0 Functions Rather Than Weights nents of w which lead from the input layer into hidden neuron K by w[K)' Let x? be the input vector consisting of all O's. Then a tanh(W[K) . x') I aWj = 0 for any j, w, and K, and for any w, there is a row of ~I j which is all zeroes. This in tum means that Jw,w(w) = for any w, which means that W inj is empty, and PWIT(' I t) is independent of the data t. (Intuitively, this problem arises since the 0 corresponding to x? can't vary with w, and therefore the dimension of ct> is less than IWI) So we must forbid such an all-zeroes x'. The easiest way to do this is to require that one input neuron always be on, i.e., introduce a bias unit. An alternative is to redefine ct> to be the functions from the set {X - (0, 0, ... , O)} to 0 rather than from the set X to O. Another alternative, appropriate when the original X is the set of all input neuron vectors consisting of O's and 1's, is to instead have input neuron values E {z 0, I}. (In general z -1 though; due to the symmetries of the tanh, for many architectures z = -1 means that two rows of a<l>i I j are identical up to an overall sign, which means that Jw,w(w) = 0.) This is the solution implicitly assumed from now on. aw o * * aw Jw,w(w) will be small - and therefore Pw(net(w, .? will be large - whenever one can make large changes to w without affecting, = net(w, .) much. In other words, pw(net(w, will be large whenever we don't need to specify w very accurately. So the correction factor favors those w which can be expressed with few bits. In other words, the correction factor enforces a sort of automatic MDL (Rissanen, 1986; Nowlan and Hinton, 1994). .? More generally, for any multi-layer architecture there are many "singular weights" w sin ~ W inj such that Jw.w(wsin) is not just small but equals zero exactly. Pw(w) must compensate for these singularities, or the peaks of PCI)rr<, I t) won't depend on t. So we need to have pw(w) ~ 0 as w ~ wsin' Sometimes this happens automatically. For example often Wsin includes infinite-valued w's, since tanh'(oo) = O. Because Pw(oo) = 0 for the weightdecay prior, that prior compensates for the infmite-w singularities in the correction term. For other w sin there is no such automatic compensation, and we have to explicitly modify pwCw) to avoid singularities. In doing so though it seems reasonable to maintain a "bias" towards the wsin, that Pw(w) goes to zero slowly enough so that the values pw(net(w, .? are "enhanced" for w near wsin' Although a full characterization of such enhanced w is not in hand, it's easy to see that they include certain kinds of pruned nets (Hassibi and Stork, 1992), weight-shared nets (Nowlan and Hinton, 1994), and feature-selected nets. To see that (some kinds of) pruned nets have singular weights, let w* be a weight vector with a zero-valued weight coming out of hidden neuron K. By (1) Pw(net(w*, .? = Jdw' Pw(w,) S(net(w', .) - net(w*, Since we can vary the value of each weight w*i leading into neuron K without affecting net(w*, .), the integral diverges. So w* is singUlar; removing a hidden neuron results in an enhanced probability. This constitutes an a priori argument in favor of trying to remove hidden neurons during training. .?. This argument does not apply to weights leading into a hidden neuron; Jw,w(w) treats weights in different layers differently. This fact suggests that however pw(w) compensates for the singularities in Jw.w(w), weights in different layers should be treated differently by Pw(w). This is in accord with the advice given in (MacKay, 1992). To see that some kinds of weight-shared nets have singular weights, let w' be a weight vector such that for any two hidden neurons K and K' the weight from input neuron i to K equals the weight from i to K', for all input neurons i. In other words, w is such that all hid- 20S 206 Wolpert den neurons compute identical functions of x. (For some architectures we'll actually only need a single pair of hidden neurons to be identical.) Usually for such a situation there is a pair of columns of the matrix ~ / awj which are exactly proportional to one another. (For example, in a 3-2-1 architecture, with X = {z, I} 3 , IWI = IXI x 101 = 8, and there are four such pairs of columns.) This means that JfI>,W(w') = 0; w' has an enhanced probability, and we have an a priori argument in favor of trying to equate hidden neurons during training. The argument that feature-selected nets have singular weights is architecture-dependent, and there might be reasonable architectures for which it fails. To illustrate the argument, consider the 3-2-1 architecture. Let xl(k) and x2(k) with k = {I, 2,3) designate three distinct pairs of input vectors. For each k have xl (k) and x2(k) be identical for all input neurons except neuron A, for which they differ. (Note there are four pairs of input vectors with this property, one for each of the four possible patterns over input neurons B and C.) Let w' be a weight vector such that both weights leaving A equal zero. For this situation net(w', xl(k? = net(w', x2(k? for all k. In addition a net(w, xl(k? / awj = net(w, x2(k? / awj for all weights Wj except the two which lead out of A. So k = 1 gives us a pair of rows of the matrix a~ awj which are identical in all but two entries (one row for Xl (k) and one for x2(k?. We get another such pair of rows, differing from each other in the exact same two entries, for k = 2, and yet another pair for k = 3. So there is a linear combination of these six rows which is all zeroes. This means that JfI>, w(w') = O. This constitutes an a priori argument in favor of trying to remove input neurons during training. a / Since it doesn't favor any Pw(w), the analysis of this paper doesn't favor any pfl>( <1?. However when combined with empirical knowledge it suggests certain pfl>(cj). For example, there are functions g(w) which empirically are known to be good choices for pfl>(net(w, .? (e.g., g(w) oc:exp[awl]). There are usually problems with such choices of Pfl>(cj) though. For example, these g(w) usually make more sense as a prior over W than as a prior over <1>, which would imply pfl>(net(w, .? = g(w) / J<I>W(w). Moreover it's empirically true that , enhanced w should be favored over other w, as advised by the correction term. So it makes sense to choose a compromise between g(w) and g(w) / J<I>W(w). An example is pfl>(cj) oc: , g(w) / [A} + tanh(~ x JfI>,w(w?] for two hyperparameters A} > 0 and ~ > O. 4 BEYOND THE CASE OF BACKPROP WITH IWI = IXIIOI When 0 does not approximate a Euclidean vector space, elements of <1> have probabilities rather than probability densities, and P(cj) It) = jdw PWl'r(w I t) S(net(w, .), cj), (0(., .) being a Kronecker delta function). Moreover, if 0 is a Euclidean vector space but WI > IXI 101, then again one must evaluate a difficult integral; <1> = net(W, .) is not one-to-one so one must use equation (1) rather than (2). Fortunately these two situations are relatively rare. The final case to consider is IWI < IXIIOI (see section two). Let Sew) be the surface in <1> which is the image (under net(W, .? ofW. For all <I> PfI>(cj) is either zero (when cj) Ii': S(W? or infinite (when cj) E S(W?. So as conventionally defined, "MAP cj)" is not meaningful. One way to deal with this case is to embed the net in a larger net, where that larger net's output is relatively insensitive to the values of the newly added weights. An alternative that is applicable when IWI / 101 is an integer is to reduce X by removing "uninteresting" x's. A third alternative is to consider surface densities over Sew), Ps(W)(cj), instead of vol- Bayesian Backpropagation over 1-0 Functions Rather Than Weights ume densities over <%?. P?l>(e!?). Such surface densities are given by equation (2). if one uses the metric form of J?l>,w(w). (Buntine has emphasized that the Jacobian form is not even defined for IWI < IXIIOI. since ()cj)i / aWj is not square then (personal communication).) As an aside, note that restricting P?l>(e!?) to Sew) is an example of the common theoretical assumption that "target functions" come from a pre-chosen "concept class". In practice such an assumption is usually ludicrous - whenever it is made there is an implicit hope that it constitutes a valid approximation to a more reasonable P?l>(e!?). When decision theory is incorporated into Bayesian analysis. only rarely does it advise us to evaluate an MAP quantity (Le.. use BP). Instead Bayesian decision theory usually advises us to evaluate quantities like E(<%? I t) (Wolpert. 1994). Just as it does for the use of MAP estimators. the analysis of this paper has implications for the use of such E(<%? I t) estimators. In particular. one way to evaluate E(<%?I t) = jdw PwIT(w I t) net(w?.) is to expand net(w ?.) to low order and then approximate PWlnw I t) as a sum of Gaussians (Buntine and Weigend. 1991). Equation (4) suggests that instead we write E(<%? I t) as jde!? P?l>lne!? I t) e!? and approximate P?l>IT(e!? I t) as a sum of Gaussians. Since fewer approximations are used (no low order expansion of net(w ?. this might be more accurate. ?, Acknowledgements Thanks to David Rosen and Wray Buntine for stimulating discussion. and to TXN and the SF! for funding. This paper is a condensed version of (Wolpert 1994). References Buntine. W.? Weigend. A. (1991). Bayesian back-propagation. Complex Systems. S.p. 603. Denker. J., LeCun, Y. (1991). Transforming neural-net output levels to probability distributions. In Neural Information Processing Systems 3, R. Lippman et al. (Eds). Fefferman, C. (1993). Reconstructing a neural net from its output. Sarnoff Research Center TR 93-01. Hassibi. B., and Stork, D. (1992). Second order derivatives for network pruning: optimal brain surgeon. Ricoh Tech Report CRC-TR-9214. MacKay, D. (1992). Bayesian Interpolation, and A Practical Framework for Backpropagation Networks. Neural Computation. 4. pp. 415 and 448. Neal, R. (1993). Bayesian learning via stochastic dynamics. In Neural Information Processing Systems 5, S. Hanson et al. (Eds). Morgan Kaufmann. Nowlan, S., and Hinton. G. (1994). Simplifying Neural Networks by Soft Weight-Sharing. In Theories of Induction: Proceedings of the SFIICNLS Workshop on Formal Approaches to Supervised Learning, D. Wolpert (Ed.). Addison-Wesley, to appear. Rissanen, J. (1986). Stochastic complexity and modeling. Ann. Stat .? 14, p. 1080. Wolpert, D. (1993). On the use of evidence in neural networks. In Neural Information Processing Systems 5, S. Hanson et aI. (Eds). Morgan-Kauffman. Wolpert, D. (1994). Bayesian back-propagation over i-o functions rather than weights. SF! tech. report. ftp'ablefrom archive.cis.ohio-state.edu, as pub/neuroprose/wolpert.nips.93.Z. 207 

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On Separability of Loss Functions, and Revisiting Discriminative Vs Generative Models Adarsh Prasad Machine Learning Dept. CMU adarshp@andrew.cmu.edu Alexandru Niculescu-Mizil NEC Laboratories America Princeton, NJ, USA alex@nec-labs.com Pradeep Ravikumar Machine Learning Dept. CMU pradeepr@cs.cmu.edu Abstract We revisit the classical analysis of generative vs discriminative models for general exponential families, and high-dimensional settings. Towards this, we develop novel technical machinery, including a notion of separability of general loss functions, which allow us to provide a general framework to obtain `1 convergence rates for general M -estimators. We use this machinery to analyze `1 and `2 convergence rates of generative and discriminative models, and provide insights into their nuanced behaviors in high-dimensions. Our results are also applicable to differential parameter estimation, where the quantity of interest is the difference between generative model parameters. 1 Introduction Consider the classical conditional generative model setting, where we have a binary random response Y 2 {0, 1}, and a random covariate vector X 2 Rp , such that X|(Y = i) ? P?i for i 2 {0, 1}. Assuming that we know P (Y ) and {P?i }1i=0 , we can use the Bayes rule to predict the response Y given covariates X. This is said to be the generative model approach to classification. Alternatively, consider the conditional distribution P (Y |X) as specified by the Bayes rule, also called the discriminative model corresponding to the generative model specified above. Learning this conditional model directly is said to be the discriminative model approach to classification. In a classical paper [8], the authors provided theoretical justification for the common wisdom regarding generative and discriminative models: when the generative model assumptions hold, the generative model estimators initially converge faster as a function of the number of samples, but have the same asymptotic error rate as discriminative models. And when the generative model assumptions do not hold, the discriminative model estimators eventually overtake the generative model estimators. Their analysis however was for the specific generative-discriminative model pair of Naive Bayes, and logistic regression models, and moreover, was not under a high-dimensional sampling regime, when the number of samples could even be smaller than the number of parameters. In this paper, we aim to extend their analysis to these more general settings. Doing so however required some novel technical and conceptual developments. To motivate the machinery we develop, consider why the Naive Bayes model estimator might initially converge faster. The Naive Bayes model makes the conditional independence assumption that P (X|Y ) = Qp s=1 P (Xs |Y ), so that the parameters of each of the conditional distributions P (Xs |Y ) for s 2 {1, . . . , p} could be estimated independently. The corresponding log-likelihood loss function is thus fully ?separable? into multiple components. The logistic regression log-likelihood on the other hand is seemingly much less ?separable?, and in particular, it does not split into multiple components each of which can be estimated independently. In general, we do not expect the loss functions underlying statistical estimators to be fully separable into multiple components, so that we need a more flexible notion of separability, where different losses could be shown to be separable to differing degrees. In a very related note, though it might seem unrelated at first, the analysis of `1 convergence rates of 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. statistical estimators considerably lags that of say `2 rates (see for instance, the unified framework of [7], which is suited to `2 rates but is highly sub-optimal for `1 rates). In part, the analysis of `1 rates is harder because it implicitly requires analysis at the level of individual coordinates of the parameter vector. While this is thus harder than an `2 error analysis, intuitively this would be much easier if the loss function were to split into independent components involving individual coordinates. While general loss functions might not be so ?fully separable?, they might perhaps satisfy a softer notion of separability motivated above. In a contribution that would be of independent interest, we develop precisely such a softer notion of separability for general loss functions. We then use this notion of separability to derive `1 convergence rates for general M -estimators. Given this machinery, we are then able to contrast generative and discriminative models. We focus on the case where the generative models are specified by exponential family distributions, so that the corresponding discriminative models are logistic regression models with the generative model sufficient statistics as feature functions. To compare the convergence rates of the two models, we focus on the difference of the two generative model parameters, since this difference is also precisely the model parameter for the discriminative model counterpart of the generative model, via an application of the Bayes rule. Moreover, as Li et al. [3] and others show, the `2 convergence rates of the difference of the two parameters is what drives the classification error rates of both generative as well as discriminative model classifiers. Incidentally, such a difference of generative model parameters has also attracted interest outside the context of classification, where it is called differential parameter learning [1, 14, 6]. We thus analyze the `1 as well as `2 rates for both the generative and discriminative models, focusing on this parameter difference. As we show, unlike the case of Naive Bayes and logistic regression in low-dimensions as studied in [8], this general highdimensional setting is more nuanced, and in particular depends on the separability of the generative models. As we show, under some conditions on the models, generative and discriminative models not only have potentially different `1 rates, but also differing ?burn in? periods in terms of the minimum number of samples required in order for the convergence rates to hold. The choice of a generative vs discriminative model, namely that with a better sample complexity, thus depends on their corresponding separabilities. As a minor note, we also show how generative model M -estimators are not directly suitable in high-dimensions, and provide a simple methodological fix in order to obtain better `2 rates. We instantiate our results with two running examples of isotropic and non-isotropic Gaussian generative models, and also corroborate our theory with instructive simulations. 2 Background and Setup. We consider the problem of differential parameter estimation under the following generative model. Let Y 2 {0, 1} denote a binary response variable, and let X = (X1 , . . . , Xp ) 2 Rp be the covariates. For simplicity, we assume P[Y = 1] = P[Y = 0] = 12 . We assume that conditioned on the response variable, the covariates belong to an exponential family, X|Y ? P?Y? (?), where: P?Y? (X|Y ) = h(X) exp(h?Y? , (X)i A(?Y? )). (1) Here, is the vector of the true canonical parameters, A(?) is the log-partition function and (X) (0) is the sufficient statistic. We assume access to two sets of samples X0n = {xi }ni=1 ? P?0? and ?Y? (1) X1n = {xi }ni=1 ? P?1? . Given these samples, as noted in the introduction, we are particularly ? interested in estimating the differential parameter ?diff := ?1? ?0? , since this is also the model parameter corresponding to the discriminative model, as we show below. In high dimensional ? ? sampling settings, we additionally assume that ?diff is at most s-sparse, i.e. ||?diff ||0 ? s. We will be using the following two exponential family generative models as running examples: isotropic and non-isotropic multivariate Gaussian models. Isotropic Gaussians (IG) Let X = (X1 , . . . , Xp ) ? N (?, Ip ) be an isotropic gaussian random variable; it?s density can be written as: ? ? 1 1 T P? (x) = p exp (x ?) (x ?) . (2) 2 (2?)p Gaussian MRF (GMRF). Let X = (X1 , . . . , Xp ) denote a zero-mean gaussian random vector; 1 it?s density is fully-parametrized as by the inverse covariance or concentration matrix ? = (?) 0 2 and can be written as: P? (x) = r 1 ? (2?)p det (?) 1 ? exp ? ? 1 T x ?x . 2 (3) Let d? = maxj2[p] ?(:,j) 0 is the maximum number non-zeros in a row of ?. Let ?? ? = Pp (?? ) 1 1 , where |||M |||1 is the `1 /`1 operator norm given by |||M |||1 = max k=1 |Mjk |. j=1,2,...,p Generative Model Estimation. Here, we proceed by estimating the two parameters {?i? }1i=0 individually. Letting ?b1 and ?b0 be the corresponding estimators, we can then estimate the difference of the parameters as ?bdiff = ?b1 ?b0 . The most popular class of estimators for the individual parameters is based on Maximum likelihood Estimation (MLE), where we maximize the likelihood of the given data. For isotropic gaussians, the negative log-likelihood function can be written as: ?T ? ?T ? b, (4) 2 P n where ? b = n1 i=1 xi . In the case of GGMs the negative log-likelihood function can be written as: DD EE b LnGGM (?) = ?, ? log(det(?)), (5) P P b = 1 n xi xT is the sample covariance matrix and hhU, V ii = where ? i i=1 i,j Uij Vij denotes n the trace inner product on the space of symmetric matrices. In high-dimensional sampling regimes (n << p), regularized MLEs, for instance with `1 -regularization under the assumption of sparse model parameters, have been widely used [11, 10, 2]. Discriminative Model Estimation. Using Bayes rule, we have that: P[X|Y = 1]P[Y = 1] P[Y = 1|X] = P[X|Y = 0]P[Y = 0] + P[X|Y = 1]P[Y = 1] 1 = (6) ? 1 + exp ( (h?1 ?0? , (x)i + c? )) where c? = A(?0? ) A(?1? ). The conditional distribution is simply a logistic regression model, with the generative model sufficient statistics as the features, and with optimal parameters being precisely ? the difference ?diff := ?1? ?0? of the generative model parameters. The corresponding negative log-likelihood function can be written as n 1X Llogistic (?, c) = ( yi (h?, (xi )i + c) + (h?, (xi )i + c)) (7) n i=1 LnIG (?) = where (t) = log(1 + exp(t)). In high dimensional sampling regimes, under the assumption that the model parameters are sparse, we would use the `1 -penalized version ?bdiff of the MLE (7) to estimate ? ?diff . Outline. We proceed by studying the more general problem of `1 error for parameter estimation for any loss function Ln (?). Specifically, consider the general M -estimation problem, where we are given n i.i.d samples Z1n = {z1 , z2 , . . . , zn }, zi 2 Z from some distribution P, and we are interested in estimating some parameter ?? of the distribution P. Let ` : Rp ? Z 7! R be a twice differentiable and convex function which assigns a loss `(?; z) to any parameter ? 2 Rp , for a given ? observation z. Also assume that the loss is Fisher consistent so that ?? 2 argmin? L(?) where def ? L(?) = Ez?P [`(?; z)] is the population loss. We are then interested in analyzing the M -estimators b ?? that minimize Pn the empirical loss i.e. ? 2 argmin? Ln (?), or regularized versions thereof, where Ln (?) = n1 i=1 L(?; Zi ). We introduce a notion of the separability of a loss function, and show how more separable losses require fewer samples to establish convergence for ?b ?? . We then instantiate our separability 1 results from this general setting for both generative and discriminative models. We calculate the number of samples required for generative and discriminative approaches to estimate the differential ? parameter ?diff , for consistent convergence rates with respect to `1 and `2 norm. We also discuss the consequences of these results for high dimensional classification for Gaussian Generative models. 3 3 Separability Let R( ; ?? ) = rLn (?? + ) rLn (?? ) r2 Ln (?? ) be the error in the first order approximation of the gradient at ?? . Let B1 (r) = {?| ||?||1 ? r} be an `1 ball of radius r. We begin by analyzing the low dimensional case, and then extend it to high dimensions. 3.1 Low Dimensional Sampling Regimes In low dimensional sampling regimes, we assume that the number of samples n p. In this setting, we make the standard assumption that the empirical loss function Ln (?) is strongly convex. Let ?b = argmin? Ln (?) denote the unique minimizer of the empirical loss function. We begin by defining a notion of separability for any such empirical loss function Ln . Definition 1. Ln is (?, , ) locally separable around ?? if the remainder term R( ; ?? ) satisfies: ||R( ; ?? )||1 ? 1 ? || ||1 8 2 B1 ( ) This definition might seem a bit abstract, but for some general intuition, indicates the region where it is separable, ? indicates the conditioning of the loss, while it is that quantifies the degree of separability: the larger it is, the more separable the loss function. Next, we provide some additional intuition on how a loss function?s separability is connected to (?, , ). Using the mean-value theorem, we can write ||R( , ?? )||1 = r2 Ln (?? + t ) r2 Ln (?? ) for some t 2 (0, 1). This can 1 be further simplified as ||R( , ?? )||1 ? r2 Ln (?? + t ) r2 Ln (?? ) 1 || ||1 . Hence, ? and 1/ measure the smoothness of Hessian (w.r.t. the `1 /`1 matrix norm) in the neighborhood of ?? , with ? being the smoothness exponent, and 1/ being the smoothness constant. Note that the Hessian of the loss function r2 Ln (?) is a random matrix and can vary from being a diagonal matrix for a fully-separable loss function to a dense matrix for a heavily-coupled loss function. Moreover, from standard concentration arguments, the `1 /`1 matrix norm for a diagonal ("separable") subgaussian random matrix has at most logarithmic dimension dependence1 , but for a dense ("non-separable") random matrix, the `1 /`1 matrix norm could possibly scale linearly in the dimension. Thus, the scaling of `1 /`1 matrix norm gives us an indication how ?separable? the matrix is. This intuition is captured by (?, , ), which we further elaborate in future sections by explicitly deriving (?, , ) for different loss functions and use them to derive `2 and `1 convergence rates. Theorem 1. Let Ln be a strongly convex loss function which is (?, , ) locally separable function ? 1 1 ? 1 ? 1} around ?? . Then, if ||rLn (?? )||1 ? min{ 2? , 2? where ? = r2 Ln (?? ) 1 1 . ?b ?? 1 ? 2? ||rLn (?? )||1 Proof. (Proof Sketch). The proof begins by constructing a suitable continuous function F , for which b = ?b ?? is the unique fixed point. Next, we show that F (B1 (r)) ? B1 (r) for r = 2? ||rLn (?? )||1 . Since F is continuous and `1 -ball is convex and compact, the contraction property coupled with Brouwer?s fixed point theorem [9], shows that there exists some fixed point of F , such that || ||1 ? 2? ||rLn (?? )||1 . By uniqueness of the fixed point, we then establish our result. See Figure 1 for a geometric description and Section A for more details 3.2 High Dimensional Sampling Regimes In high dimensional sampling regimes (n << p), estimation of model parameters is typically an under-determined problem. It is thus necessary to impose additional assumptions on the true model parameter ?? . We will focus on the popular assumption of sparsity, which entails that the number of non-zero coefficients of ?? is small, so that ||?? ||0 ? s. For this setting, we will be focusing in particular on `1 -regularized empirical loss minimization: 1 Follows from the concentration of subgaussian maxima [12] 4 F b F(b) = b F (B1 (2? ||rLn (?? )||1 )) B1 (2? ||rLn (?? )||1 ) Figure 1: Under the conditions of Theorem 1, F ( ) = r2 Ln (?? ) 1 (R( ; ?? ) + rLn (?? )) is contractive over B1 (2? ||rLn (?? )||1 ) and has b = ?b ?? as its unique fixed point. Using these two observations, we can conclude that b ? 2? ||rLn (?? )|| . 1 1 ?b n = argmin Ln (?) + ? n (8) ||?||1 Let S = {i | ?i? 6= 0} be the support set of the true parameter and M(S) = {v|vS c = 0} be the corresponding subspace. Note that under a high-dimensional sampling regime, we can no longer assume that the empirical loss Ln (?) is strongly convex. Accordingly, we make the following set of assumptions: ? Assumption 1 (A1): Positive Definite Restricted Hessian. r2SS Ln (?? ) % min I ? Assumption 2 (A2): Irrepresentability. There exists some 2 (0, 1] such that r2S c S Ln (?? ) r2SS Ln (?? ) 1 1 ?1 ? Assumption 3 (A3). Unique Minimizer. When restricted to the true support, the solution to the `1 penalized loss minimization problem is unique, which we denote by: ?? = argmin {Ln (?) + n ||?|| } . (9) n 1 ?2M(S) Assumptions 1 and 2 are common in high dimensional analysis. We verify that Assumption 3 holds for different loss functions individually. We refer the reader to [13, 5, 11, 10] for further details on these assumptions. For this high dimensional sampling regime, we also modify our separability notion to a restricted separability, which entails that the remainder term be separable only over the model subspace M(S). Definition 2. Ln is (?, , ) restricted locally separable around ?? over the subspace M(S) if the remainder term R( ; ?? ) satisfies: ||R( ; ?? )||1 ? 1 ? || ||1 8 2 B1 ( ) \ M(S) We present our main deterministic result in high dimensions. Theorem 2. Let Ln be a (?, , ) locally separable function around ?? . If ( that, ? 8 n ||rLn (?? )||1 . ? ||rLn (?? )||1 + n ? min n 2? , 1 2? ? ? 1 1 ? 1 o Then we have that support(?b n ) ? support(?? ) and where ? = r2SS Ln (?? ) 1 ?b n ?? 1 ? 2? (||rLn (?? )||1 + 1 5 n) n , rLn (? ? )) are such Proof. (Proof Sketch). The proof invokes the primal-dual witness argument [13] which when combined with Assumption 1-3, gives ?b n 2 M(S) and that ?b n is the unique solution of the restricted problem. The rest of the proof proceeds similar to Theorem 1, by constructing a suitable function F : R|S| 7! R|S| for which b = ?b n ?? is the unique fixed point, and showing that F is contractive over B1 (r; ?? ) for r = 2? (||rLn (?? )||1 + n ).See Section B for more details. Discussion. Theorems 1 and 2 provide a general recipe to estimate the number of samples required by any loss `(?, z) to establish `1 convergence. The first step is to calculate the separability constants (?, , ) for the corresponding empirical loss function Ln . Next, since the loss ` is Fisher consistent, ? ? ) = 0, the upper bound on ||rLn (?? )|| can be shown to hold by analyzing the so that rL(? 1 concentration of rLn (?? ) around its mean. We emphasize that we do not impose any restrictions on the values of (?, , ). In particular, these can scale with the number of samples n; our results hold so long as the number of samples n satisfy the conditions of the theorem. As a rule of thumb, the smaller that either or get for any given loss `, the larger the required number of samples. 4 `1 -rates for Generative and Discriminative Model Estimation In this section we study the `1 rates for differential parameter estimation for the discriminative and generative approaches. We do so by calculating the separability of discriminative and generative loss functions, and then instantiate our previously derived results. 4.1 Discriminative Estimation As discussed before, the discriminative approach uses `1 -regularized logistic regression with the sufficient statistic as features to estimate the differential parameter. In addition to A1-A3, we Pn 2 assume column normalization of the sufficient statistics, i.e. i=1 ([ (xi )]j ) ? n. Let n = maxi || (x)i ||1 , ?n = maxi ||( (x)i )S ||2 . Firstly, we characterize the separability of the logistic loss. ? ? Lemma 1. The logistic regression negative log-likelihood LnLogistic from (7) is 2, s n1? 2 , 1 ren stricted local separable around ?? . Combining Lemma 1 with Theorem 2, we get the following corollary. Corollary 3. (Logistic Regression) Consider the model in (1), then there exist q universal positive constants C1 , C2 and C3 such that for n differential estimate ?bdiff , satisfies 4.2 support(?bdiff ) ? C 1 ?2 s 2 ? support(?diff ) 2 4 n ?n and Generative Estimation log p and ?bdiff ? ?diff n = C2 1 ? C3 log p n , r the discriminative log p . n We characterize the separability of Generative Exponential Families. The negative log-likelihood function can be written as: Ln (?) = A(?) h?, n i , P n where n = n1 i=1 (xi ). In this setting, the remainder term is independent of the data and can be written as R( ) = rA(?? + ) rA(?? ) r2 A(?? ) and rLn (?? ) = E[ (x)] n1 (xi ). Hence, ||rLn (?? )||1 is a measure of how well the sufficient statistics concentrate around their mean. Next, we show the separability of our running examples Isotropic Gaussians and Gaussian Graphical Models. Lemma 2. The isotropic Gaussian negative log-likelihood LnIG from (4) is (?, 1, 1) locally separable around ?? . ? ? Lemma 3. The Gaussian MRF negative log-likelihood LnGGM from (5) is 2, 3d? 2?3 , 3d? 1??? ? ? ?? restricted locally separable around ?? . 6 Comparing Lemmas 1, 2 and 3, we see that the separability of the discriminative model loss depends weakly on the feature functions. On the other hand, the separability for the generative model loss depends critically on the underlying sufficient statistics. This has consequences for their differing sample complexities for differential parameter estimation, as we show next. Corollary 4. (Isotropic Gaussians) Consider the model in (2). Then there exist universal constants C1 , C2 , C3 such that if the number of samples scale as n C1 log p, then with probability atleast 1 1/pC2 , the generative estimate of the differential parameter ?bdiff satisfies r log p ? b ?diff ?diff ? C3 . n 1 Comparing Corollary 3 and Corollary 4, we see that for isotropic gaussians, both the discriminative and generative approach achieve the same `1 convergence rates, but at different sample complexities. Specifically, the sample complexity for the generative method depends only logarithmically on the dimension p, and is independent of the differential sparsity s, while the sample complexity of the discriminative method depends on the differential sparsity s. Therefore in this case, the generative method is strictly better than its discriminative counterpart, assuming that the generative model assumptions hold. Corollaryp5. (Gaussian MRF) Consider the model in (3), and suppose that the scaled covari? are subgaussian with parameter 2 . Then there exist universal positive conates Xk / ?kk stants C2 , C3 , C4 such that if the number of samples for the two generative models scale as ni C2 ?2i ?6(?? ) 1 d2?? log p, for i 2 {0, 1}, then with probability at least 1 1/pC3 , the geni i bdiff = ? b1 ? b0 , satisfies erative estimate of the differential parameter, ? r log p ? bdiff ?diff ? ? C4 , n 1 bi ) ? support(?? ) for i 2 {0, 1}. and support(? i Comparing Corollary 3 and Corollary 5, we see that for Gaussian Graphical Models, both the discriminative and generative approach achieve the same `1 convergence rates, but at different sample complexities. Specifically, the sample complexity for the generative method depends only on row-wise sparsity of the individual models d2?? , and is independent of sparsity s of the differential i ? parameter ?diff . In contrast, the sample complexity of the discriminative method depends only on the sparsity of the differential parameter, and is independent of the structural complexities of the individual model parameters. This suggests that in high dimensions, even when the generative model assumptions hold, generative methods might perform poorly if the underlying model is highly non-separable (e.g. d = ?(p)), which is in contrast to the conventional wisdom in low dimensions. Related Work. Note that results similar to Corollaries 3 and 5 have been previously reported in [11, 5] separately. Under the same set of assumptions as ours, Li et al. [5] provide a unified analysis for support recovery and `1 -bounds for `1 -regularized M-estimators. While they obtain the same rates as ours, their required sample complexities are much higher, since they do not exploit the separability of the underlying loss function. As one example, in the case of GMRFs, their results require the number of samples to scale as n > k 2 log p, where k is the total number of edges in the graph, which is sub-optimal, and in particular does not match the GMRF-specific analysis of [11]. On the other hand, our unified analysis is tighter, and in particular, does match the results of [11]. 5 `2 -rates for Generative and Discriminative Model Estimation In this section we study the `2 rates for differential parameter estimation for the discriminative and generative approaches. 5.1 Discriminative Approach The bounds for the discriminative approach are relatively straightforward. Corollary 3 gives bounds b ? support(?? ). Since the true model parameter is on the `1 error and establishes that support(?) p s-sparse, ||?? ||0 ? s, the `2 error can be simply bounded as s k?b ?? k1 . 7 5.2 Generative Approach In the previous section, we saw that the generative approach is able to exploit the inherent separability of the underlying model, and thus is able to get `1 rates for differential parameter estimation at a much lower sample complexity. Unfortunately, it does not q have support consistency. Hence a na?ve p generative estimator will have an `2 error scaling with p log n , which in high dimensions, would ? make it unappealing. However, one can exploit the sparsity of ?diff and get better rates of convergence in `2 -norm by simply soft-thresholding the generative estimate. Moreover, soft-thresholding also leads to support consistency. Definition 3. We denote the soft-thresholding operator ST ST n (?) = argmin w 1 ||w 2 n (?), defined as: 2 ?||2 + n ||w||1 . Lemma 4. Suppose ? = ?? + ? for some s-sparse ?? . Then there exists a universal constant C1 such that for n 2 ||?||1 , p ||ST n (?) ?? ||2 ? C1 s ||?||1 and ||ST n (?) ?? ||1 ? C1 s ||?||1 (10) Note that this is a completely deterministic result and has no sample complexity requirement. Motivated by this, we introduce a thresholded generative estimator that has two stages: (a) compute ?bdiff using the generative model estimates, and (b) soft-threshold the generative estimate with n = ? c ?bdiff ?diff . An elementary application of Lemma 4 can then be shown to provide `2 error 1 ? bounds for ?bdiff given its `1 error bounds, and that the true parameter ?diff is s-sparse. We instantiate these `2 -bounds via corollaries for our running examples of Isotropic Gaussians, and Gaussian MRFs. Lemma 5. (Isotropic Gaussians) Consider the model in (2). Then there exist universal constants C1 , C2 , C3 such that if the number of samples scale as n C1 log p, then with probability ? ?atleast 1 1/pC2 , the soft-thresholded generative estimate of the differential parameter ST n ?bdiff , with q the soft-thresholding parameter set as n = c logn p for some constant c, satisfies: r ? ? s log p ? b ST n ?diff ?diff ? C3 . n 2 Lemma 6. MRF) Consider the model in Equation 3, and suppose that the covarip (Gaussian ? are subgaussian with parameter 2 . Then there exist universal positive conates Xk / ?kk stants C2 , C3 , C4 such that if the number of samples for the two generative models scale as ni C2 ?2i ?6(?? ) 1 d2?? log p, for i 2 {0, 1}, for i 2 {0, 1}, then with probability at least 1 1/pC3 , i i ? ? bdiff , with the softthe soft-thresholded generative estimate of the differential parameter, ST n ? q thresholding parameter set as n = c logn p for some constant c, satisfies: r ? ? s log p ? bdiff ST n ? ?diff ? C4 . n 2 Comparing Lemmas 5 and 6 to Section 5.1, we can see that the additional soft-thresholding step allows the generative methods to achieve the same `2 -error rates as the discriminative methods, but at different sample complexities. The sample complexities of the generative estimates depend on the separabilities of the individual models, and is independent of the differential sparsity s, where as the sample complexity of the discriminative estimate depends only on the differential sparsity s. 6 Experiments: High Dimensional Classification In this section, we corroborate our theoretical results on `2 -error rates for generative and discriminative model estimators, via their consequences for high dimensional classification. We focus on the case of isotropic Gaussian generative models X|Y ? N (?Y , Ip ), where ?0 , ?1 2 Rp are unknown 8 0-1 Error for s=4,p=512,d=1 0.5 0.4 0-1 Error 0.3 0.25 0.35 0.3 0.25 0.2 50 100 150 200 250 300 350 400 0.15 0.3 0.2 0 50 100 150 200 n (a) 0.35 0.25 0.2 0 Gen-Thresh Logistic 0.45 0.4 0-1 Error 0-1 Error 0.4 0-1 Error for s=64,p=512,d=1 0.5 Gen-Thresh Logistic 0.45 0.35 0.15 0-1 Error for s=16,p=512,d=1 0.5 Gen-Thresh Logistic 0.45 250 300 350 400 0.15 0 50 100 n (b) s = 4, p = 512 150 200 250 300 350 400 n (c) s = 16, p = 512 Figure 2: Effect of sparsity s on excess 0 s = 64, p = 512 1 error. and ?1 ?0 is s-sparse. Here, we are interested in a classifier C : Rp 7! {0, 1} that achieves low classification error EX,Y [1 {C(X) 6= Y }]. Under this setting, it can be shown that the Bayes classifier, that achieves the lowest possible classification error, is given by the linear discriminant ?T ? ?T ? classifier C ? (x) = 1 xT w? + b? > 0 , where w? = (?1 ?0 ) and b? = 0 0 2 1 1 . Thus, the coefficient w? of the linear discriminant is precisely the differential parameter, which can be estimated via both generative and discriminative approaches as detailed in the previous section. Moreover, the classification error can also be related to the `2 error of thenestimates. Under o some mild assumptions, T b b Li et al. [3] showed that for any linear classifier C(x) = 1 x w b + b > 0 , the excess classification error can be bounded as: ? b ? C1 ||w E(C) b 2 w? ||2 + bb b? 2 2 ? , for some constant C1 > 0, and where E(C) = EX,Y [1 {C(X) 6= Y }] EX,Y [1 {C ? (X) 6= Y }] is the excess 0-1 error. In other words, the excess classification error is bounded by a constant times the `2 error of the differential parameter estimate. Methods. In this setting, as discussed in previous sections, the discriminative model is simply a logistic regression model with linear features (6), so that the discriminative estimate of the differential parameter w b as well as the constant bias term bb can be simply obtained via `1 -regularized logistic regression. For the generative estimate, we use our two stage estimator from Section 5, which proceeds by estimating ? b0 , ? b1 using the empirical means, and then estimating the differential parameter by soft-thresholding q the difference of the generative model parameter estimates w bT = ST n (b ?1 ? b0 ) where ?bT = n 1 2 = C1 log p n for some constant C1 . The corresponding estimate for b? is given by hw bT , ? b1 + ? b0 i. Experimental Setup.? For our experimental? setup, we consider isotropic Gaussian models with 1s 1s means ?0 = 1p p1s , ?1 = 1p + p1s , and vary the sparsity level s. For both methods, 0p s 0p s p we set the regularization parameter 2 as n = log(p)/n. We report the excess classification error for the two approaches, averaged over 20 trials, in Figure 2. Results. As can be seen from Figure 2, our two-staged thresholded generative estimator is always better than the discriminative estimator, across different sparsity levels s. Moreover, the sample complexity or ?burn-in? period of the discriminative classifier strongly depends on the sparsity level, which makes it unsuitable when the true parameter is not highly sparse. For our two-staged generative estimator, we see that the sparsity s has no effect on the ?burn-in? period of the classifier. These observations validate our theoretical results from Section 5. 2 See Appendix J for cross-validated plots. 9 Acknowledgements A.P. and P.R. acknowledge the support of ARO via W911NF-12-1-0390 and NSF via IIS-1149803, IIS-1447574, DMS-1264033, and NIH via R01 GM117594-01 as part of the Joint DMS/NIGMS Initiative to Support Research at the Interface of the Biological and Mathematical Sciences. References [1] Alberto de la Fuente. From ?differential expression?to ?differential networking??identification of dysfunctional regulatory networks in diseases. Trends in genetics, 26(7):326?333, 2010. [2] Christophe Giraud. Introduction to high-dimensional statistics, volume 138. CRC Press, 2014. [3] Tianyang Li, Adarsh Prasad, and Pradeep K Ravikumar. Fast classification rates for high-dimensional gaussian generative models. In Advances in Neural Information Processing Systems, pages 1054?1062, 2015. [4] Tianyang Li, Xinyang Yi, Constantine Carmanis, and Pradeep Ravikumar. Minimax gaussian classification & clustering. In Artificial Intelligence and Statistics, pages 1?9, 2017. [5] Yen-Huan Li, Jonathan Scarlett, Pradeep Ravikumar, and Volkan Cevher. Sparsistency of 1-regularized m-estimators. In AISTATS, 2015. [6] Song Liu, John A Quinn, Michael U Gutmann, Taiji Suzuki, and Masashi Sugiyama. Direct learning of sparse changes in markov networks by density ratio estimation. Neural computation, 26(6):1169?1197, 2014. [7] Sahand Negahban, Bin Yu, Martin J Wainwright, and Pradeep K Ravikumar. A unified framework for highdimensional analysis of m-estimators with decomposable regularizers. In Advances in Neural Information Processing Systems, pages 1348?1356, 2009. [8] Andrew Y Ng and Michael I Jordan. On discriminative vs. generative classifiers: A comparison of logistic regression and naive bayes. Advances in neural information processing systems, 2:841?848, 2002. [9] James M Ortega and Werner C Rheinboldt. Iterative solution of nonlinear equations in several variables. SIAM, 2000. [10] Pradeep Ravikumar, Martin J Wainwright, John D Lafferty, et al. High-dimensional ising model selection using `1-regularized logistic regression. The Annals of Statistics, 38(3):1287?1319, 2010. [11] Pradeep Ravikumar, Martin J Wainwright, Garvesh Raskutti, Bin Yu, et al. High-dimensional covariance estimation by minimizing `1 -penalized log-determinant divergence. Electronic Journal of Statistics, 5: 935?980, 2011. [12] JM Wainwright. High-dimensional statistics: A non-asymptotic viewpoint. preparation. University of California, Berkeley, 2015. [13] Martin J Wainwright. Sharp thresholds for high-dimensional and noisy sparsity recovery using-constrained quadratic programming (lasso). IEEE transactions on information theory, 55(5):2183?2202, 2009. [14] Sihai Dave Zhao, T Tony Cai, and Hongzhe Li. Direct estimation of differential networks. Biometrika, page asu009, 2014. 10 

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Maxing and Ranking with Few Assumptions Moein Falahatgar Yi Hao Alon Orlitsky Venkatadheeraj Pichapati Vaishakh Ravindrakumar University of California, San Deigo {moein,yih179,alon,dheerajpv7,vaishakhr}@ucsd.edu Abstract 1 1.1 PAC maximum selection (maxing) and ranking of n elements via random pairwise comparisons have diverse applications and have been studied under many models and assumptions. With just one simple natural assumption: strong stochastic transitivity, we show that maxing can be performed with linearly many comparisons yet ranking requires quadratically many. With no assumptions at all, we show that for the Borda-score metric, maximum selection can be performed with linearly many comparisons and ranking can be performed with O(n log n) comparisons. Introduction Motivation Maximum selection (maxing) and sorting using pairwise comparisons are among the most practical and fundamental algorithmic problems in computer science. As is well-known, maxing requires n ? 1 comparisons, while sorting takes ?(n log n) comparisons. The probabilistic version of this problem, where comparison outcomes are random, is of significant theoretical interest as well, and it too arises in many applications and diverse disciplines. In sports, pairwise games with random outcomes are used to determine the best, or the order, of teams or players. Similarly Trueskill [1] matches video gamers to create their ranking. It is also used for a variety of online applications such as to learn consumer preferences with the popular A/B tests, in recommender systems [2], for ranking documents from user clickthrough data [3, 4], and more. The popular crowd sourcing website GIFGIF [5] shows how pairwise comparisons can help associate emotions with many animated GIF images. Visitors are presented with two images and asked to select the one that better corresponds to a given emotion. For these reasons, and because of its intrinsic theoretical interest, the problem received a fair amount of attention. 1.2 Terminology and previous results One of the first studies in the area, [6] assumed n totally-ordered elements, where each comparison errs with the same, known, probability ? < 12 . It presented a maxing algorithm that uses O( ?n2 log 1 ) comparisons to output the maximum with probability ? 1 ? , and a ranking algorithm that uses O( ?n2 log n ) comparisons to output the ranking with probability ? 1 ? . These results have been and continue to be of great interest. Yet this model has two shortcomings. It assumes that there is only one random comparison probability, ?, and that its value is known. In practice, comparisons have different, and arbitrary, probabilities, and they are not known in advance. To address more realistic scenarios, researchers considered more general probabilistic models. Consider a set of n elements, without loss of generality [n] = {1, 2, . . . , n}. A probabilistic model, or model for short, is an assignment of a preference probability pi,j ? [0, 1] for every i ? j ? [n], reflecting the probability that i is preferred when compared with j. We assume that repeated comparisons are independent and that there are no ?draws?, hence pj,i = 1 ? pi,j . def 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. If pi,j ? 12 , we say that i is preferable to j and write i ? j. Element i is maximal in a model if i ? j for all j ? i. And a permutation `1 , . . . ,`n is a ranking if `i ? `j for all i ? j. Observe that the first element of any ranking is always maximal. For example, for n = 3, p1,2 = 1?2, p1,3 = 1?3, and p2,3 = 2?3, we have 1 ? 2, 2 ? 1, 3 ? 1, and 2 ? 3. Hence 2 is the unique maximum, and 2,3,1 is the unique ranking. We seek algorithms that without knowing the underlying model, use pairwise comparisons to find a maximal element and a ranking. Two concerns spring to mind. First, there may be two elements i, j with pi,j arbitrarily close to half, requiring arbitrarily many comparisons just to determine which is preferable to the other. This concern has a common remedy, that we also adopt. The PAC paradigm, e.g. [7, 8], that requires the algorithm?s output to be only Probably Approximately Correct. Let p?i,j = pi,j ? 12 be the centered preference probability. Note that p?i,j ? 0 iff i is preferable to j. If p?i,j ? ?? we say that i is ?-preferable to j. For 0 < ? < 1?2, an element i ? [n] is ?-maximum if it is ?-preferable to all other elements, namely, p?i,j ? ?? ?j ? i. Given ? > 0, 12 ? > 0, a PAC maxing algorithm must output an ?-maxima with probability ? 1 ? , henceforth abbreviated with high probability (WHP). Similarly, a permutation `1 , . . . ,`n of {1, . . . ,n} is an ?-ranking if `i is ?-preferable to `j for all i ? j, and a PAC ranking algorithm must output an ?-ranking WHP. Note that in this paper, we consider ? 12 , the more practical regime. For larger values of , one can use our algorithms with = 12 . def The second concern is that not all models have a ranking, or even a maximal element. For example, for p1,2 = p2,3 = p3,1 = 1, or the more opaque yet interesting non-transitive coins [9], each element is preferable to the cyclically next, hence there is no maximal element and no ranking. A standard approach, that again we too will adopt, to address this concern is to consider structured models. The simplest may be parametric models, of which one of the more common is Placket Luce (PL) [10, 11], where each element i is associated with an unknown positive number ai and i pi,j = aia+a . [12] derived a PAC maxing algorithm that uses O( ?n2 log ?n ) comparisons and a PAC j ranking algorithm that uses O( ?n2 log n log ?n ) comparisons for any PL model. Related results for the Mallows model under a non-PAC paradigm were derived by [13]. But significantly more general, and more realistic, non-parametric, models may also have maxima and rankings. A model is strongly stochastically transitive (SST), if i ? j and j ? k imply pi,k ? max(pi,j , pj,k ). By simple induction, every SST model has a maximum element and a ranking. And one additional property, that is perhaps more difficult to justify, has proved helpful in constructing maxing and sorting PAC algorithms. A tournament satisfies the stochastic triangle inequality if i ? j and j ? k imply that p?i,k ? p?i,j + p?j,k . In Section 4 we show that if a model has a ranking, then an ?-ranking can be found WHP via 2 O( n?2 log n ) comparisons. For all models that satisfy both SST and triangle inequality, [7] derived a PAC maxing algorithm that uses O( ?n2 log ?n ) comparisons. [14] eliminated the log n? factor and showed that O? ?n2 log 1 ? comparisons suffice and are optimal, and constructed a nearly-optimal log n) PAC ranking algorithm that uses O( n log n(log ) comparisons for all ? n1 , off by a factor ?2 of O((log log n)3 ) from optimum. Lower-bounds follow from an analogy to [15, 6]. Observe that since the PL model satisfies both SST and triangle inequality, these results also improve the corresponding PL results. 3 Finally, we consider models that are not SST, or perhaps don?t have maximal elements, rankings, or even their ?-equivalents. In all these cases, one can apply a weaker order relation. The Borda def score s(i) = n1 ?j pi,j is the probability that i is preferable to another, randomly selected, element. Element i is Borda maximal if s(i) = maxj s(j), and ?-Borda maximal if s(i) ? maxj s(j) ? ?. A PAC Borda-maxing algorithm outputs an ?-Borda maximal element WHP (with probability ? 1 ? ). Similarly, a Borda ranking is a permutation i1 , . . . ,in such that for all 1 ? j ? n ? 1, s(ij ) ? s(ij+1 ). An ?-Borda ranking is a permutation where for all 1 ? j ? k ? n, s(ij ) ? s(ik ) ? ?. A PAC Borda-ranking algorithm outputs an ?-Borda ranking WHP. Recall that Borda scores apply to all models. As noted in [16, 17, 8, 18] considering elements with nearly identical Borda scores shows that exact Borda-maxing and ranking requires arbitrarily many comparisons. [8] derived a PAC Borda ranking, and therefore also maxing, algorithms that use 2 n O( n?2 ) comparisons. [19] derived a O( n log log( n )) PAC Borda ranking algorithm for restricted ?2 setting. However note that several simple models, including p1,2 = p2,3 = p3,1 = 1 do not belong to this model. 2 [20, 21, 22] considered deterministic adversarial versions of this problem that has applications in [23]. Finally, we note that all our algorithms are adaptive, where each comparison is chosen based on the outcome of previous comparisons. Non-adaptive algorithms were discussed in [24, 25, 26, 27]. 2 Results and Outline Our goal is to find the minimal assumptions that enable efficient algorithms for these problems. In particular, we would like to see if we can eliminate the somewhat less-natural triangle inequality. With two algorithmic problems: maxing and ranking, and one property?SST and one special metric? Borda scores, the puzzle consists of four main questions. 1) With just SST (and no triangle inequality) are there: a) PAC maxing algorithms with O(n) comparisons? b) PAC ranking algorithms with near O(n log n) comparisons? 2) With no assumptions at all, but for the Borda-score metric, are there: a) PAC Borda-maxing algorithms with O(n) comparisons? b) PAC Borda-ranking algorithms with near O(n log n) comparisons? We essentially resolve all four questions. 1a) Yes. In Section 3, Theorem 6, we use SST alone to derive a O? ?n2 log 1 ? comparisons PAC maxing algorithm. Note that this is the same complexity as with triangle inequality, and it matches the lower bound. 1b) No. In Section 4, Theorem 7, we show that there are SST models where any PAC ranking algorithm with ? ? 1?4 requires ?(n2 ) comparisons. This is significantly higher than the roughly O(n log n) comparisons needed with triangle inequality, and is close to the O(n2 log n) comparisons required without any assumptions. 2a) Yes. In Section 5, Theorem 8, we derive a PAC Borda maxing algorithm that without any model assumptions requires O? ?n2 log 1 ? comparisons which is order optimal. 2b) Yes. In Section 5, Theorem 9, we derive a PAC Borda ranking algorithm that without any model assumptions requires O? ?n2 log n ? comparisons. Beyond the theoretical results sections, in Section 6, we provide experiments on simulated data. In Section 7, we discuss the results. 3 3.1 Maxing S EQ -E LIMINATE Our main building block is a simple, though sub-optimal, algorithm S EQ -E LIMINATE that sequentially eliminates one element from input set to find an ?-maximum under SST. S EQ -E LIMINATE uses O? ?n2 log n ? comparisons and w.p.? 1? , finds an ?-maximum. Even for simpler models [15] we know that an algorithm needs ?? ?n2 log 1 ? comparisons to find an ?-maximum w.p.? 1 ? . Hence the number of comparisons used by S EQ -E LIMINATE is optimal up to a constant factor when ? n1 but can be log n times the lower bound for = 12 . By SST, any element that is ?-preferable to absolute maximum element of S is an ?-maximum of S. Therefore if we can reduce S to a subset S ? of size O( logn n ) that contains an absolute maximum of S using O? ?n2 log 1 ? comparisons, we can then use S EQ -E LIMINATE to find an ?-maximum of S ? and the number of comparisons is optimal up to constants. We provide one such reduction in subsection 3.2. Sequential elimination techniques have been used before [13] to find an absolute maximum. In such approaches, a running element is maintained, and is compared and replaced with a competing element in S if the latter is found to be better with confidence ? 1 ? ?n. Note that if the running and competing elements are close to each other, this technique can take an arbitrarily long time to declare the winner. But since we are interested in finding only an ?-maximum, S EQ -E LIMINATE circumvents this issue. We later show that S EQ -E LIMINATE needs to update the running element r with the competing element c if p?c,r ? ? and retain r if p?c,r ? 0. If 0 < p?c,r < ?, replacing or 3 retaining r doesn?t affect the performance of S EQ -E LIMINATE significantly. Thus, in other words we?ve reduced the problem to testing whether p?c,r ? 0 or p?c,r ? ?. Assuming that testing problem always returns the right answer, since S EQ -E LIMINATE never replaces the running element with a worse element, either the output is the absolute maximum b? or b? is never the running element. If b? is eliminated against running element r then p?b? ,r ? ? and hence r is an ?-maximum and since the running element only gets better, the output is an ?-maximum. We first present a testing procedure C OMPARE that we use to update the running element in S EQ E LIMINATE. 3.1.1 C OMPARE C OMPARE(i, j, ?l , ?u , ) takes two elements i and j, and two biases ?u > ?l , and with confidence ? 1 ? , determines whether p?i,j is ? ?l or ? ?u . For this, C OMPARE compares the two elements 2?(?u ? ?l )2 log(2? ) times. Let p?i,j be the fraction ?i,j def of times i beats j, and let p? = p?i,j ? 12 . If p??i,j < (?l + ?u )?2, C OMPARE declares p?i,j ? ?l (returns 1), and otherwise it declares p?i,j ? ?u (returns 2). Due to lack of space, we present the algorithm C OMPARE in Appendix A.1 along with certain improvements for better performance in practice . In the Lemma below, we bound the number of comparisons used by C OMPARE and prove its correctness. Proof is in A.2. 2 2 Lemma 1. For ?u > ?l , C OMPARE(i, j, ?l , ?u , ) uses ? (?u ?? comparisons and if p?i,j ? ?l , 2 log l) then w.p.? 1 ? , it returns 1, else if p?i,j ? ?u , w.p.? 1 ? , it returns 2. Now we present S EQ -E LIMINATE that uses the testing subroutine C OMPARE and finds an ?maximum. 3.1.2 S EQ -E LIMINATE Algorithm S EQ -E LIMINATE takes a variable set S, selects a random running element r ? S and repeatedly uses C OMPARE(c, r, 0, ?, ?n) to compare r to a random competing element c ? S ? r. If C OMPARE returns 1 i.e., deems p?c,r ? 0, it retains r as the running element and eliminates c from S, but if C OMPARE returns 2 i.e., deems p?c,r ? ?, it eliminates r from S and updates c as the new running element. Algorithm 1 S EQ -E LIMINATE 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: inputs Set S, bias ?, confidence n ? ?S? r ? a random c ? S, S = S ? {r} while S ? ? do Pick a random c ? S, S = S ? {c}. if C OMPARE(c, r, 0, ?, n ) = 2 then r?c end if end while return r We now bound the number of comparisons used by S EQ -E LIMINATE(S, ?, ) and prove its correctness. Proof is in A.3. Theorem 2. S EQ -E LIMINATE(S, ?, ) uses O? ?S? log ?S? ? comparisons, and w.p.? 1 ? outputs an ?2 ?-maximum. 4 3.2 Reduction Recall that, for ? n1 , S EQ -E LIMINATE is order-wise optimal. For ? n1 , here we present a reduction procedure that uses O? ?n2 log 1 ? comparisons and w.p.? 1 ? , outputs a subset S ? of size ? O( n log n) and an element a such that either a is a 2??3-maximum or S ? contains an absolute maximum of S. Combining the reduction with S EQ -E LIMINATE results in an order-wise optimal algorithm. We form the reduced subset S ? by pruning S. We compare each element e ? S with an anchor element a, test whether p?e,a ? 0 or p?e,a ? 2??3 using C OMPARE, and? retain all elements e for which C OMPARE returns the second hypothesis.? For S ? to be of size O( n log n) we?d like to pick an anchor element that is among the top O( n log n) elements. But this can be computationally ? hard and we show that it suffices to pick an anchor that is not ??3-preferable to at most O( n log n) elements in S. An element a is called an (?, n? )-good anchor if a is not ?-preferable to at most n? elements, i.e., ?{e ? e ? S and p?e,a > ?}? ? n? . We now present the subroutine P ICK -A NCHOR that finds a good anchor element. 3.2.1 Picking Anchor Element P ICK -A NCHOR(S, n? , ?, ) uses O? nn? ?2 log 1 log nn? ? comparisons and w.p.? 1 ? , outputs an (?, n? )-good anchor element. P ICK -A NCHOR first picks randomly a set Q of nn? log 2 elements from S without replacement. This ensures that w.p.? 1 ? , Q contains at least one of the top n? elements. We then use S EQ -E LIMINATE to find an ?-maximum of Q. Let the absolute maximum element of Q be denoted as q ? . Now an ?-maximum of Q is ?-preferable to q ? . Further, if Q contains an element in the top n? elements, there exists n ? n? elements worse than q ? in S. Thus by SST, the ?-maximum of Q is also ?-preferable to these n ? n? elements and hence the output of P ICK -A NCHOR is an (?, n? )-good anchor element. P ICK -A NCHOR is shown in appendix A.4 We now bound the number of comparisons used by P ICK -A NCHOR and prove its correctness. Proof is in A.5. Lemma 3. P ICK -A NCHOR(S, n? , ?, ) uses O? nn? ?2 log 1 log nn? ? comparisons and w.p.? 1 ? , outputs an (?, n? )-good anchor element. Remark 4. Note that P ICK -A NCHOR(S, cn, ?, ) uses Oc ? ?12 ?log 1 ? ? comparisons where the constant depends only on c but not on n. Hence it is advantageous to use this method to pick nearmaximum element when n is large. 2 We now present P RUNE that takes an anchor element as input and prunes the set S using the anchor. 3.2.2 Pruning Given an (?l , n? )-good anchor element a, w.p.? 1 ? ?2, P RUNE(S, a, n? , ?l , ?u , ) outputs a subset S ? of size ? 2n? . Further, any element e that is at least ?u -better than a i.e., p?e,a ? ?u is in S ? w.p.? 1 ? ?2. P RUNE prunes S in multiple rounds. In each round t, for every element e in S, P RUNE tests whether p?e,a ? ?l or p?e,a ? ?u using C OMPARE(e, a, ?l , ?u , ?2t+1 ) and eliminates e if the first hypothesis i.e., p?e,a ? ?l is returned. By Lemma 1, an element e that is ?u better than a i.e., p?e,a ? ?u passes the tth round of pruning w.p.? 1 ? ?2t+1 . Thus by union bound, the probability that such an element t+1 is not present in the pruned set is ? ?? ? ?2. t=1 ?2 Now for element e that is not ?l -better than a i.e., p?e,a ? ?l , by Lemma 1, the first hypothesis is returned w.p.? 1 ? ?4. Hence w.h.p., the number of such elements (not ?l -better elements) is reduced by a factor of after each round. Since a is an (?l , n? )-good anchor element, there are at most n? elements atleast ?l -better than a. Thus the number of elements left in the pruned set after round t is at most n? + n t . Thus P RUNE succeeds eventually in reducing the size to ? 2n? (in ? log1? nn? rounds). 5 Algorithm 2 P RUNE 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: inputs Set S, element a, size n? , lower bias ?l , upper bias ?u , confidence . t?1 S1 ? S while ?St ? > 2n? and t < log2 n do Initialize: Qt ? ? for e in St do if C OMPARE(e, a, ?l , ?u , ?2t+1 ) = 1 then Qt ? Qt ?{e} end if end for St+1 ? St ? Qt t?t+1 end while return St . We now bound the number of comparisons used by P RUNE and prove its correctness. Proof is in A.6. ? Lemma 5. If n? ? 6n log n, ? n1 and a is an (?l , n? )-good anchor element, then w.p.? 1 ? 2 , n 1 P RUNE(S, a, n? , ?l , ?u , ) uses O? (?u ?? ? comparisons and outputs a set of size less than 2 log l) 2n? . Further if a is not an ?u -maximum of S then w.p.? 1 ? 2 , the output set contains an absolute maximum element of S. 3.3 Full Algorithm We now present the main algorithm, O PT-M AXIMIZE that w.p.? 1? , uses O? ?n2 log 1 ? comparisons and outputs an ?-maximum. For ? n1 , S EQ -E LIMINATE uses O( ?n2 log 1 ) comparisons and hence we directly use S EQ -E LIMINATE. Below we assume > n1 . ? Here O PT-M AXIMIZE first finds an (??3, 6n log n)-good ?anchor element a using ? P ICK -A NCHOR(S, 6n log n, ??3, 4 ). Then using P RUNE(S, a, 6n log n, ??3, 2??3, 4 ) with ? a, O PT-M AXIMIZE prunes S to a subset S ? of size ? 2 6n log n such that if a is not a 2??3 maximum i.e. p?b? ,a > 2??3, S ? contains the absolute maximum b? w.p.? 1 ? ?2. O PT-M AXIMIZE then checks if a is a 2??3 maximum by using C OMPARE(e, a, 2??3, ?, ?(4n)) for every element e ? S ? . If C OMPARE returns first hypothesis for every e ? S ? then O PT-M AXIMIZE outputs a or else O PT-M AXIMIZE outputs S EQ -E LIMINATE(S ? , ?, 4 ). Note that only one of these three cases is possible: (1) a is a 2??3-maximum, (2) a is not an ?maximum and (3) a is an ?-maximum but not a 2??3-maximum. In case (1), since a is a 2??3maximum, by Lemma 1, w.p.? 1 ? ?4, C OMPARE returns the first hypothesis for every e ? S ? and O PT-M AXIMIZE outputs a. In both cases (2) and (3), as stated above, w.p.? 1 ? ?2, S ? contains the absolute maximum b? . Now in case (2) since a is not an ?-maximum, by Lemma 1, w.p.? 1 ? ?(4n), C OMPARE(b? , a, 2??3, ?, ?(4n)) returns the second hypothesis. Thus O PT-M AXIMIZE outputs S EQ -E LIMINATE(S ? , ?, ?4), which w.p.? 1 ? ?4, returns an ?-maximum of S ? (recall that an ?-maximum of S ? is an ?-maximum of S if S ? contains b? ). Finally in case (3), O PT-M AXIMIZE either outputs a or S EQ -E LIMINATE(S ? , ?, ?4) and either output is an ?-maximum w.p.? 1 ? . In the below Theorem, we bound comparisons used by O PT-M AXIMIZE and prove its correctness. Proof is in A.7. Theorem 6. W.p.? 1 ? , O PT-M AXIMIZE(S, ?, ) uses O( ?n2 log 1 ) comparisons and outputs an ?-maximum. 4 Ranking Recall that [14] considered a model with both SST and stochastic triangle inequality and derived log n)3 an ?-ranking with O? n log n(log ? comparisons for = n1 . By constrast, we consider a more ?2 6 Algorithm 3 O PT-M AXIMIZE 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: inputs Set S, bias ?, confidence . if ? n1 then return S EQ -E LIMINATE(S, ?, ) end if ? a ? P ICK -A NCHOR(S, 6n log n, ??3, 4 ) ? S ? ? P RUNE(S, a, 6n log n, ??3, 2??3, 4 ) for element e in S ? do if C OMPARE(e, a, 2? , ?, 4n ) = 2 then 3 return S EQ -E LIMINATE(S ? , ?, 4 ) end if end for return a general model without stochastic triangle inequality and show that even a 1?4-ranking with just SST takes ?(n2 ) comparisons for ? 18 . To establish the lower bound, we reduce the problem of finding 1?4-ranking to finding a coin with bias 1 among n(n?1) ? 1 other fair coins. For this, we consider the following model with n elements 2 {a1 , a2 , ..., an }: p?a1 ,an = 12 , p?ai ,aj = ?(0 < ? < 1?n10 ), when i < j and (i, j) ? (1, n). Note that this model satisfies SST but not stochastic triangle inequality. Also note that any ranking where a1 precedes an is an 1?4-ranking and thus the algorithm only needs to order a1 and an correctly. Now the output of a comparison between any two elements other than a1 and an is essentially a fair coin toss (since ? is very small). Thus if we output a ranking without querying comparison between a1 and an , then the ranking is correct w.p.? 12 since a1 and an must necessarily be ordered correctly. Now if an algorithm uses only n2 ?20 comparisons then the probability that the algorithm queried at least one comparison between a1 and an is less than 12 and hence cannot achieve a confidence of 78 . Proof sketch in B.1. Theorem 7. There exists a model that satisfies SST for which any algorithm requires ?(n2 ) comparisons to find a 1?4-ranking with probability ? 7?8. We also present a trivial ?-ranking algorithm in Appendix B.2 that for any stochastic model with 2 ranking (Weak Stochastic Transitivity), uses O( n?2 log n ) comparisons and outputs an ?-ranking w.p.? 1 ? . 5 Borda Scores We show that for general models, using O( ?n2 log 1 ) comparisons w.p.? 1? , we can find an ?-Borda maximum and using O( ?n2 log n ) comparisons w.p.? 1 ? , we can find an ?-Borda ranking. Recall that Borda score s(e) of an element e is the probability that e is preferable to an element picked randomly from S i.e., s(e) = n1 ?f ?S p?e,f . We first make a connection between Borda scores of elements and the traditional multi armed bandit setting. In the Bernoulli multi armed setting, every arm a is associated with a parameter q(a) and pulling that arm results in a reward B(q(a)), a Bernoulli random variable with parameter q(a). Observe that we can simulate our pairwise comparisons setting as a traditional bandit arms setting by comparing an element with a random element where in our setting, for every element e, the associated parameter is s(e). Thus PAC optimal algorithms derived under traditional bandit setting work for PAC Borda score setting too. [28] and several others derived a PAC maximum arm selection algorithms that use O( ?n2 log 1 ) comparisons and find an arm with parameter at most ? less than the highest. This implies an ?-Borda maxing algorithm with the same complexity. Proof follows from reduction to Bernoulli multi-armed bandit setting. Theorem 8. There exists an algorithm that uses O( ?n2 log 1 ) comparisons and w.p.? 1 ? , outputs an ?-Borda maximum. 7 For ?-Borda ranking, we note that if we compare an element e with ?22 log 2n random elements, w.p. ? 1 ? ?n, the fraction of times e wins approximates the Borda score of e to an additive error of ? . Ranking based on these approximate scores results in an ?-Borda ranking. We present B ORDA 2 R ANKING in C.1 that uses 2n log 2n comparisons and w.p.? 1 ? outputs an ?-Borda ranking. Proof ?2 in C.1. Theorem 9. B ORDA -R ANKING(S, ?, ) uses 2n log 2n comparisons and w.p.? 1 ? outputs an ?2 ?-Borda ranking. 6 Experiments In this section we validate the performance of our algorithms using simulated data. Since we essentially derived a negative result for ?-ranking, we consider only our ?-maxing algorithms - S EQ E LIMINATE and O PT-M AXIMIZE for experiments. All results are averaged over 100 runs. 16 105 14 OPT-MAXIMIZE SEQ-ELIMINATE 12 12 Sample Complexity Sample Complexity 14 10 8 6 4 OPT-MAXIMIZE SEQ-ELIMINATE 10 8 6 4 2 2 0 0 106 200 400 600 800 1000 1200 1400 1600 1800 0 0 2000 Number of elements 5000 10000 15000 Number of elements (a) small values of n (b) large values of n Figure 1: Comparison of S EQ -E LIMINATE and O PT-M AXIMIZE Similar to [14, 7], we consider the stochastic model pi,j = 0.6 ?i < j. We use maxing algorithms to find 0.05-maximum with error probability = 0.1. Note that i = 1 is the unique 0.05-maximum under this model. In Figure 1, we compare the performance of S EQ -E LIMINATE and O PT-M AXIMIZE over different ranges of n. Figures 1(a), 1(b) show that for small n i.e., n ? 1300 S EQ -E LIMINATE performs well and for large n i.e., n ? 1300, O PT-M AXIMIZE performs well. Since we are using 1 = 0.1, the experiment suggests that for ? n1?3 , O PT-M AXIMIZE uses fewer comparisons as com1 pared to S EQ -E LIMINATE. Hence it would be beneficial to use S EQ -E LIMINATE for ? n1?3 and O PT-M AXIMIZE for higher values of . In further experiments, we use = 0.1 and n < 1000 so we use S EQ -E LIMINATE for better performance. We compare S EQ -E LIMINATE with BTM-PAC [7], KNOCKOUT [14], MallowsMPI [13], and AR [16] . KNOCKOUT and BTM-PAC are PAC maxing algorithms for models with SST and stochastic triangle inequality requirements. AR finds an element with maximum Borda score. Mallows finds the absolute best element under Weak Stochastic Transitivity. We again consider the model: pi,j = 0.6 ?i < j and try to find a 0.05-maximum with error probability = 0.1. Note that this model satisfies both SST and stochastic triangle inequality and under this model all these algorithms can find an ?-maximum. From Figure 2(a), we can see that BTM-PAC performs worse for even small values of n and from Figure 2(b), we can see that AR performs worse for higher values of n. One possible reason is that BTM-PAC is tailored for reducing regret in the bandit setting and in the case of AR, Borda scores of elements become approximately the same with increasing number of elements, leading to more comparisons. For this reason, we drop BTM-PAC and AR for further experiments. We also tried PLPAC [12] but it fails to achieve required accuracy of 1 ? since it is designed primarily for Plackett-Luce. For example, we considered the previous setting pi,j = 0.6 ?i < j with n = 100 and tried to find a 0.09-maximum with = 0.1. Even though PLPAC used almost same number of comparisons (57237) as S EQ -E LIMINATE (56683), PLPAC failed to find 0.09-maxima 20 out of 100 runs whereas S EQ -E LIMINATE found the maximum in all 100 runs. In figure 3, we compare algorithms S EQ -E LIMINATE, KNOCKOUT [14] and MallowsMPI [13] for models that do not satisfy stochastic triangle inequality. In Figure 3(a), we consider the stochastic model p1,j = 12 + q? ?j ? n?2, p1,j = 1 ?j > n?2 and pi,j = 12 + q? ?1 < i < j where q? ? 0.05 and we pick n = 10. Observe that this model satisfies SST but not stochastic triangle inequality. Here 8 10 6 5 10 SEQ-ELIMINATE KNOCKOUT MallowsMPI AR BTM-PAC Sample Complexity Sample Complexity 10 104 9 SEQ-ELIMINATE KNOCKOUT MallowsMPI AR 108 107 106 105 103 7 10 104 15 50 100 Number of elements 200 500 Number of elements (a) small values of n (b) large valuesof n Figure 2: Comparison of Maxing Algorithms with Stochastic Triangle Inequality again, we try to find a 0.05-maximum with = 0.1. Note that any i ? n?2 is a 0.05 maximum. From Figure 3(a), we can see that MallowsMPI uses more comparisons as q? decreases since MallowsMPI is not a PAC algorithm and tries to find the absolute maximum. Even though KNOCKOUT performs better than MallowsMPI, it fails to output a 0.05 maximum with probability 0.12 for q? = 0.001 and 0.26 for q? = 0.0001. Thus KNOCKOUT can fail when the model doesn?t satisfy stochastic triangle inequality. We give an explanation for this behavior in Appendix D. By constrast, even for q? = 0.0001, S EQ -E LIMINATE outputted a 0.05 maximum in all runs and outputted the abosulte maximum in 76% of trials. We can also see that S EQ -E LIMINATE uses much fewer comparisons compared to the other two algorithms. In Figure 3(b), we compare S EQ -E LIMINATE and MallowsMPI on the Mallows model, a model which doesn?t satisfy stochastic triangle inequality. Mallows model can be specified with one parameter . We consider n = 10 elements and find a 0.05-maximum with error probablility = 0.05. From Figure 3(b) we can see that the performance of MallowsMPI gets worse as approaches 1, since comparison probabilities get close to 12 whereas S EQ -E LIMINATE is not affected. 1012 107 SEQ-ELIMINATE 1010 KNOCKOUT Sample Complexity Sample Complexity SEQ-ELIMINATE MallowsMPI 108 10 6 104 106 MallowsMPI 105 104 103 0.04 0.02 0.01 0.001 102 0.0001 (a) No Triangle Inequality 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) Mallows Model Figure 3: Comparison of S EQ -E LIMINATE and M ALLOWS MPI over Mallows Model One more experiment is presented in Appendix E. 7 Conclusion We extended the study of PAC maxing and ranking to general models which satisfy SST but not stochastic triangle inequality. For PAC maxing, we derived an algorithm with linear complexity. For PAC ranking, we showed a negative result that any algorithm needs ?(n2 ) comparisons. We thus showed that removal of stochastic triangle inequality constraint does not affect PAC maxing but affects PAC ranking. We also ran experiments over simulated data and showed that our PAC maximum selection algorithms are better than other maximum selection algorithms. For unconstrained models, we derived algorithms for PAC Borda maxing and PAC Borda ranking by making connections with traditional multi-armed bandit setting. 9 Acknowledgments We thank 1619448. NSF for supporting this work through grants CIF-1564355 and CIF- References [1] Ralf Herbrich, Tom Minka, and Thore Graepel. Trueskill: a bayesian skill rating system. In Proceedings of the 19th International Conference on Neural Information Processing Systems, pages 569?576. MIT Press, 2006. 1.1 [2] Jialei Wang, Nathan Srebro, and James Evans. Active collaborative permutation learning. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 502?511. ACM, 2014. 1.1 [3] Filip Radlinski and Thorsten Joachims. Active exploration for learning rankings from clickthrough data. 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On clustering network-valued data Soumendu Sundar Mukherjee Department of Statistics University of California, Berkeley Berkeley, California 94720, USA soumendu@berkeley.edu Purnamrita Sarkar Department of Statistics and Data Sciences University of Texas, Austin Austin, Texas 78712, USA purna.sarkar@austin.utexas.edu Lizhen Lin Department of Applied and Computational Mathematics and Statistics Univeristy of Notre Dame Notre Dame, Indiana 46556, USA lizhen.lin@nd.edu Abstract Community detection, which focuses on clustering nodes or detecting communities in (mostly) a single network, is a problem of considerable practical interest and has received a great deal of attention in the research community. While being able to cluster within a network is important, there are emerging needs to be able to cluster multiple networks. This is largely motivated by the routine collection of network data that are generated from potentially different populations. These networks may or may not have node correspondence. When node correspondence is present, we cluster networks by summarizing a network by its graphon estimate, whereas when node correspondence is not present, we propose a novel solution for clustering such networks by associating a computationally feasible feature vector to each network based on trace of powers of the adjacency matrix. We illustrate our methods using both simulated and real data sets, and theoretical justifications are provided in terms of consistency. 1 Introduction A network, which is used to model interactions or communications among a set of agents or nodes, is arguably among one of the most common and important representations for modern complex data. Networks are ubiquitous in many scientific fields, ranging from computer networks, brain networks and biological networks, to social networks, co-authorship networks and many more. Over the past few decades, great advancement has been made in developing models and methodologies for inference of networks. There are a range of rigorous models for networks, starting from the relatively simple Erd?s-R?nyi model [12], stochastic blockmodels and their extensions [15, 17, 6], to infinite dimensional graphons [28, 13]. These models are often used for community detection, i.e. clustering the nodes in a network. Various community detection algorithms or methods have been proposed, including modularity-based methods [21], spectral methods [25], likelihood-based methods [8, 11, 7, 4], and optimization-based approaches like those based on semidefinite programming [5], etc. The majority of the work in the community detection literature including the above mentioned focus on finding communities among the nodes in a single network. While this is still a very important problem with many open questions, there is an emerging need to be able to detect clusters among multiple network-valued objects, where a network itself is a fundamental unit 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. of data. This is largely motivated by the routine collection of populations or subpopulations of network-valued data objects. Technological advancement and the explosion of complex data in many domains has made this a somewhat common practice. There has been some notable work on graph kernels in the Computer Science literature [27, 26]. In these works the goal is to efficiently compute different types of kernel similarity matrices or their approximations between networks. In contrast, we ask the following statistical questions. Can we cluster networks consistently from a mixture of graphons, when 1) there is node correspondence and 2) when there isn?t. The first situation arises when one has a dynamic network over time, or multiple instantiations of a network over time. If one thinks of them as random samples from a mixture of graphons, then can we cluster them? Note that this is not answered by methods which featurize graphs using different statistics. Our work proposes a simple and general framework for the first question - viewing the data as coming from a mixture model on graphons. This is achieved by first obtaining a graphon estimate of each of the networks, constructing a distance matrix based on the graphon estimates, and then performing spectral clustering on the resulting distance matrix. We call this algorithm Network Clustering based on Graphon Estimates (NCGE). The second situation arises when one is interested in global properties of a network. This setting is closer to that of graph kernels. Say we have co-authorship networks from Computer Science and High Energy Physics. Are these different types of networks? There has been a lot of empirical and algorithmic work on featurizing networks or computing kernels between networks. But most of these features require expensive computation. We propose a simple feature based on traces of powers of the adjacency matrix for this purpose which is very cheap to compute as it involves only matrix multiplication. We cluster these features and call this method Network Clustering based on Log Moments (NCLM). We provide some theoretical guarantees for our algorithms in terms of consistency, in addition to extensive simulations and real data examples. The simulation results show that our algorithms clearly outperform the naive yet popular method of clustering (vectorized) adjacency matrices in various settings. We also show that in absence of node correspondence, our algorithm is consistently better and faster than methods which featurize networks with different global statistics and graphlet kernels. We also show our performance on a variety of real world networks, like separating out co-authorship networks form different domains and ego networks. The rest of the paper is organized as follows. In Section 2 we briefly describe graphonestimation methods and other related work. Next, in Section 3 we formally describe our setup and introduce our algorithms. Section 4.1 contains some theory for these algorithms. In Section 5 we provide simulations and real data examples. We conclude with a discussion in Section 6. 2 Related work The focus of this paper is on 1) clustering networks which have node correspondence based on estimating the underlying graphon and 2) clustering networks without node correspondence based on global properties of the networks. We present the related work in two parts: first we cite two such methods of obtaining graphon estimates, which we will use in our first algorithm. Second, we present existing results that summarize a network using different statistics and compare those to obtain a measure of similarity. A prominent estimator of graphons is the so called Universal Singular Value Thresholding (USVT) estimator proposed by [9]. The main idea behind USVT is to essentially approximate the rank of the population matrix by thresholding the singular values of the observed matrix at an universal threshold, and then compute an approximation of the population using the top singular values and vectors. A recent work [29] proposes a novel, statistically consistent and computationally efficient approach for estimating the link probability matrix by neighborhood smoothing. Typically for large networks USVT is a lot more scalable than the neighborhood-smoothing approach. There are several other methods for graphon estimations, e.g., by fitting a stochastic blockmodel [24]. These methods can also be used in our algorithm. 2 In [10], a graph-based method for change-point detection is proposed, where an independent sequence of observations are considered. These are generated i.i.d. under the null hypothesis, whereas under the alternative, after a change point, the underlying distribution changes. The goal is to find this change point. The observations can be high-dimensional vectors or even networks, with the latter bearing some resemblance with our first framework. Essentially the authors of [10] present a statistically consistent clustering of the observations into ?past? and ?future?. We remark here that our graphon-based clustering algorithm suggests an alternative method for change point detection, namely by looking at the second eigenvector of the distance matrix between estimated graphons. Another related work is due to [14] which aims to extend the classical large sample theory to model network-valued objects. For comparing global properties of networks, there has been many interesting works which featurize networks based on global features [3]. In the Computer Science literature, graph kernels have gained much attention [27, 26]. In these works the goal is to efficiently compute different types of kernel similarity matrices or their approximations between networks. 3 A framework for clustering networks Let G be a binary random network or graph with n nodes. Denote by A its adjacency matrix, which is an n by n symmetric matrix with binary entries. That is, Aij = Aji ? {0, 1}, 1 ? i < j ? n, where Aij = 1 if there is an observed edge between nodes i and j, and Aij = 0 otherwise. All the diagonal elements of A are structured to be zero (i.e. Aii = 0). We assume the following random Bernoulli model with: Aij | Pij ? Bernoulli(Pij ), i < j, (1) where Pij = P (Aij = 1) is the link probability between nodes i and j. We denote the link probability matrix as P = ((Pij )). The edge probabilities are often modeled using the so-called graphons. A graphon f is a nonnegative bounded, measurable symmetric function f : [0, 1]2 ? [0, 1]. Given such an f , one can use the model Pij = f (?i , ?j ), (2) where ?i , ?j are i.i.d. uniform random variables on (0, 1). In fact, any (infinite) exchangeable network arises in this way (by Aldous-Hoover representation [2, 16]). Intuitively speaking, one wishes to model a discrete network G using some continuous object f . Our current work focuses on the problem of clustering networks. Unlike in a traditional setup, where one observes a single network (with potentially growing number of nodes) and the goal is often to cluster the nodes, here we observe multiple networks and are interested in clustering these networks viewed as fundamental data units. 3.1 Node correspondence present A simple and natural model for this is what we call the graphon mixture model for obvious reasons: there are only K (fixed) underlying graphons f1 , . . . , fK giving rise to link probability matrices ?1 , . . . , ?K and we observe T networks sampled i.i.d. from the mixture model Pmix (A) = K X qi P?i (A), (3) i=1 Q Auv where the qi ?s are the mixing proportions and PP (A) = u<v Puv (1 ? Puv )1?Auv is the probability of observing the adjacency matrix A when the link probability matrix is given by P . Consider n nodes, and T independent networks Ai , i ? [T ], which define edges between these n nodes. We propose the algorithm the following simple and general algorithm (Algorithm 3.1) for clustering them: 3 Algorithm 1 Network Clustering based on Graphon Estimates (NCGE) 1: Graphon estimation. Given A1 , . . . , AT , estimate their corresponding link probability matrices P1 , . . . , PT using any one of the ?blackbox? algorithms such as USVT ([9]), the neighborhood smoothing approach by [29] etc. Call these estimates P?1 , . . . , P?T . ? with D ? ij = 2: Forming a distance matrix. Compute the T by T distance matrix D ? ? ? kPi ? Pj kF , where k?kF is the Frobenius norm. D is considered an estimate of D = ((Dij )) where Dij = kPi ? Pj kF . ? 3: Clustering. Apply the spectral clustering algorithm to the distance matrix D. We will from now on denote the above algorithm with the different graphon estimation (?blackbox?) approaches as follows: the algorithm with USVT a blackbox will be denoted by CL-USVT and the one with the neighborhood smoothing method will be denoted by CL-NBS. We will compare these two algorithms with the CL-NAIVE method which does not estimate the underlying graphon, but uses the vectorized binary string representation of the adjacency matrices, and clusters those (in the spirit of [10]). 3.2 Node correspondence absent We will use certain graph statistics to construct a feature vector. The basic statistics we choose are the trace of powers of the adjacency matrix, suitably normalized and we call them graph moments: mk (A) = trace(A/n)k . (4) These statistics are related to various path/subgraph counts. For example, m2 (A) is the normalized count of the total number of edges, m3 (A) is the normalized triangle count of A. Higher order moments are actually counts of closed walks (or directed circuits). The reason we use graph moments instead of subgraph counts is that the latter are quite difficult to compute and present day algorithms work only for subgraphs up to size 5. On the contrary, graph moments are easy to compute as they only involve matrix multiplication. While it may seem that this is essentially the same as comparing the eigenspectrum, it is not clear how many eigenvalues one should use. Even if one could estimate the number of large eigenvalues using an USVT type estimator, the length is different for different networks. The trace takes into account the relative magnitudes of ?i naturally. In fact, we tried (see Section 5) using the top few eigenvalues as the sole features; but the results were not as satisfactory as using mk . We now present our second algorithm (Algorithm 2) Network Clustering with Log Moments (NCLM). In step 1, for some positive integer J ? 2, we compute gJ (A) := (log m2 (A), . . . , log mJ (A)) ? RJ . Our feature map here is g(A) = gJ (A). For step 2, ? ij = kgi ? gj k. we use the Euclidean norm, i.e. D Algorithm 2 Network Clustering based on Log Moments (NCLM) 1: Moment calculation. For a network Ai , i ? [T ] and a positive integer J, compute the feature vector gJ (A) := (log m1 (A), log m2 (A), . . . , log mJ (A)) (see Eq 4). 2: Forming a distance matrix. d(A1 , A2 ) := d(gJ (A1 ), gJ (A2 )). ? 3: Clustering. Apply the spectral clustering algorithm to the distance matrix D. Note: The rationale behind taking a logarithm of the graph moments is that, if we have two graphs with the same degree density but different sizes, then the degree density will not play any role in the the distance (which is necessary because the degree density will subdue any other difference otherwise). The parameter J counts, in some sense, the effective number of eigenvalues we are using. 4 4 Theory We will only mention our main results and discuss some of the consequences here. All the proofs and further details can be found in the supplementary article [1]. 4.1 Results on NCGE ? ij as estimating Dij = kPi ? Pj kF . We can think of D Theorem 4.1. Suppose D = ((Dij )) has rank K. Let V (resp. V? ) be the T ? K matrix whose columns correspond to the leading K eigenvectors (corresponding to the K largest-in? Let ? = ?(K, n, T ) be the K-th smallest eigenvalue magnitude eigenvalues) of D (resp. D). ? such that value of D in magnitude. Then there exists an orthogonal matrix O X ? ? V k2F ? 64T kV? O kP?i ? Pi k2F . 2 ? i Corollary 4.2. Assume for some absolute constants ?, ? > 0 the following holds for each i = 1, . . . , T : kP?i ? Pi k2F ? Ci n?? (log n)? , (5) n2 either in expectation P or with high probability (? 1 ? i,n ). Then in expectation or with high probability (? 1 ? i i,n ) we have that ? ? V k2F ? kV? O 64CT T 2 n2?? (log n)? . ?2 (6) where CT = maxi?i?T Ci . (If there are K (fixed, not growing with T ) underlying graphons, the constant CT does not depend on T .) Table 1 reports values of ?, ? for various graphon estimation procedures (under assumptions on the underlying graphons, that are described in the supplementary article [1]). Table 1: Values of ?, ? for various graphon estimation procedures. Procedure ? ? USVT 1/3 0 NBS 1/2 1/2 Minimax rate 1 1 While it is hard to obtain an explicit bound on ? in general, let us consider a simple equal weight mixture of two graphons to illustrate the relationship between ? and separation between graphons. Let the distance between the population graphons be dn We have D = ZDZ T , where the 2 ? 2 population matrix be D has D(i, j) = D(j, i) = dn. Here ZDZ T = dn(ET ? ZZ T ), where ET is the T ? T matrix of all ones, and the ith row of the binary matrix Z has a single one at position l if network Ai is sampled from ?l . The eigenvalues of this matrix are ?T nd/2 and ?T nd/2. Thus in this case ? = T nd/2. As a result (6) becomes ?? ? ? ? V k2F ? 256CT n (log n) . kV? O (7) 2 d Let us look at a more specific case of blockmodels with the same number (= m) of clusters of equal sizes (= n/m) to gain some insight into d. Let C be a n ? m binary matrix of memberships such that Cib = 1 if node i within a blockmodel comes from cluster b. Consider two blockmodels ?1 = CB1 C T with B1 = (p ? q)Im + qEm and ?2 = CB2 C T with B2 = (p0 ? q 0 )Im + q 0 Em , where Im is the identity matrix of order k (here the only difference between the models come from link formation probabilities within/between blocks, the blocks remaining the same). In this case   k?1 ? ?2 k2F 1 1 0 2 d2 = = (p ? p ) + 1 ? (q ? q 0 )2 . n2 m m 5 The bound (6) can be turned into a bound on the proportion of ?misclustered? networks, defined appropriately. There are several ways to define misclustered nodes in the context of community detection in stochastic blockmodels that are easy to analyze with spectral clustering (see, e.g., [25, 18]). These definitions work in our context too. For example, if we use Definition 4 of [25] and denote by M the set of misclustered networks, then from the proof of their Theorem 1, we have ? ? V k2F , |M| ? 8mT kV? O where mT = maxj=1,...,K (Z T Z)jj is the maximum number of networks coming from any of the graphons. 4.2 Results on NCLM We first establish concentration of trace(Ak ). The proof uses Talagrand?s concentration inequality, which requires additional results on Lipschitz continuity and convexity. This is obtained via decomposing A 7? trace(Ak ) into a linear combination of convex-Lipschitz functions. Theorem 4.3 (Concentration of moments). Let A be the adjacency matrix of an inhomogen neous random graph with link-probability matrix P . Then for any k. Let ?k (A) := k? m (A). 2 k Then ? P(|?k (A) ? E?k (A)| > t) ? 4 exp(?(t ? 4 2)2 /16). As a consequence of this, we can show that gJ (A) concentrates around g?J (A) := (log Em2 (A), . . . , log EmJ (A)). Theorem 4.4 (Concentration of gJ (A)). Let EA = ?S, where ? ? (0, 1), mini,j Sij = ?(1), P and i,j Sij = n2 . Then k? gJ (A)k = ?(J 3/2 log(1/?)) and for any 0 < ? < 1 satisfying ?J log(1/?) = ?(1), we have P(kgJ (A) ? g?J (A)k ? ?J 3/2 log(1/?)) ? JC1 e?C2 n 2 2J ? . We expect that g?J will be a good population level summary for many models. In general, it is hard to show an explicit separation result for g?J . However, in simple models, we can do explicit computations to show separation. For example, in a two parameter blockmodel B = (p?q)Im +qEm , with equal block sizes, we have Em2 (A) = (p/m+(m?1)q/m)(1+o(1)), Em3 (A) = (p3 /m2 + (m ? 1)pq 2 /m2 + (m ? 1)(m ? 2)q 3 /6m2 )(1 + o(1)) and so on. Thus we see that if m = 2, then g?2 should be able to distinguish between such blockmodels (i.e. different p, q). Note: After this paper was submitted, we came to know of a concurrent work [20] that provides a topological/combinatorial perspective on the expected graph moments Emk (A). Theorem 1 in [20] shows that under some mild assumptions on the model (satisfied, for example, by generalized random graphs with bounded kernels as long as the average degree grows to infinity), Etrace(Ak ) = E(# of closed k-walks) will be asymptotic to E(# of closed k-walks that trace out a k-cycle) plus 1{k even} E(# of closed k-walks that trace out a (k/2+1)-tree). For even k, if the degree grows fast enough k-cycles tend to dominate, whereas for sparser graphs trees tend to dominate. From this and our concentration results, we can expect NCLM to be able to tell apart graphs which are different in terms the counts of these simpler closed k-walks. Incidentally, the authors of [20] also show that the expected count of closed non-backtracking walks of length k is dominated by walks tracing out k-cycles. Thus if one uses counts of closed non-backtracking k-walks (i.e. moments of the non-backtracking matrix) instead of just closed k-walks as features, one would expect similar performance on denser networks, but in sparser settings it may lead to improvements because of the absence of the non-informative trees in lower order even moments. 5 Simulation study and data analysis In this section, we describe the results of our experiments with simulated and real data to evaluate the performance of NCGE and NCLM. We measure performance in terms of 6 clustering error which is the minimum normalized hamming distance between the estimated label vector and all K! permutations of the true label assignment. Clustering accuracy is one minus clustering error. Node correspondence present: We provide two simulated data experiments1 for clustering networks with node correspondence. In each experiment twenty 150-node networks were generated from a mixture of two graphons, 13 networks from the first and the other 7 from the second. We also used a scalar multiplier with the graphons to ensure that the networks are not too dense. The average degree for all these experiments were around 20-25. We report the average error bars from a few random runs. First we generate a mixture of graphons from two blockmodels, with probability matrices (pi ? qi )Im + qi Em with i ? {1, 2}. We use p2 = p1 (1 + ) and q2 = q1 (1 + ) and measure clustering accuracy as the multiplicative error  is increased from 0.05 to 0.15. We compare CL-USVT, CL-NBS and CL-NAIVE and the results are summarized in Figure 1(A). We have observed two things. First, CL-USVT and CL-NBS start distinguishing the graphons better as  increases (as the theory suggests). Second, the naive approach does not do a good job even when  increases. Figure 1: We show the behavior of the three algorithms when  increases, when the underlying network is generated from (A) a blockmodel and (B) a smooth graphon. (A) (B) In the second simulation, we generate the networks from two smooth graphons ?1 and ?2 , where ?2 = ?1 (1 + ) (here ?1 corresponds to the graphon 3 appearing in Table 1 of [29]). As is seen from Figure 1(B), here also CL-USVT and CL-NBS outperform the naive algorithm by a huge margin. Also, CL-NBS is consistently better than CL-USVT. This may have happened because we did our experiments on somewhat sparse networks, where USVT is known to struggle. Node correspondence absent: We show the efficacy of our approach via two sets of experiments. We compare our log-moment based method NCLM with three other methods. The first is Graphlet Kernels [26] with 3, 4 and 5 graphlets, denoted by GK3, GK4 and GK5 respectively. In the second method, we use six different network-based statistics to summarize each graph; these statistics are the algebraic connectivity, the local and global clustering coefficients [23], the distance distribution [19] for 3 hops, the Pearson correlation coefficient [22] and the rich-club metric [30]. We also compare against graphs summarized by the top J eigenvalues of A/n (TopEig). These are detailed in the supplementary article [1]. ? we compute with NCLM, GraphStats and TopEig, we calculate a For each distance matrix D ? where t is learned by picking the value within a range which similarity matrix K = exp(?tD) maximizes the relative eigengap (?K (K) ? ?K+1 (K))/?K+1 (K). It would be interesting to have a data dependent range for t. We are currently working on cross-validating the range using the link prediction accuracy on held out edges. 1 Code used in this paper is publicly available at https://github.com/soumendu041/ clustering-network-valued-data. 7 For each matrix K we calculate the top T eigenvectors, and do K-means on them to get the final clustering. We use T = K; however, as we will see later in this subsection, for GK3, GK4, and GK5 we had to use a smaller T which boosted their clustering accuracy. First we construct four sets of parameters for the two parameter blockmodel (also known as the planted partition model): ?1 = {p = 0.1, q = 0.05, K = 2, ? = 0.6}, ?2 = {p = 0.1, q = 0.05, K = 2, ? = 1}, ?3 = {p = 0.1, q = 0.05, K = 8, ? = 0.6}, and ?4 = {p = 0.2, q = 0.1, K = 8, ? = 0.6}. Note that the first two settings differ only in the density parameter ?. The second two settings differ in the within and across cluster probabilities. The first two and second two differ in K. For each parameter setting, we generate two sets of 20 graphs, one with n = 500 and the other with n = 1000. For choosing J, we calculate the moments for a large J; compute a kernel similarity matrix for each choice of J and report the one with largest relative eigengap between the K th and (K + 1)th eigenvalue. We show these plots in the supplementary article [1]. We see that the eigengap increases and levels off after a point. However, as J increases, the computation time increases. We report the accuracy of J = 5, whereas J = 8 also returns the same in 48 seconds. Table 2: Error of 6 different methods on the simulated networks. Error Time (s) NCLM (J = 5) 0 25 GK3 0.5 14 GK4 0.36 16 GK5 0.26 38 GraphStats (J = 6) 0.37 94 TopEig (J = 5) 0.18 8 We see that NCLM performs the best. For GK3, GK4 and GK5, if one uses the top two eigenvectors , and clusters those into 4 clusters (since there are four parameter settings), the errors are respectively 0.08, 0.025 and 0.03. This means that for clustering one needs to estimate the effective rank of the graphlet kernels as well. TopEig performs better than GraphStats, which has trouble separating out ?2 and ?4 . Note: Intuitively one would expect that, if there is node correspondence between the graphs, clustering based on graphon estimates would work better, because it aims to estimate the underlying probabilistic model for comparison. However, in our experiments we found that a properly tuned NCLM matched the performance of NCGE. This is probably because a properly tuned NCLM captures the global features that distinguish two graphons. We leave it for future work to compare their performance theoretically. Real Networks: We cluster about fifty real world networks. We use 11 co-authorship networks between 15,000 researchers from the High Energy Physics corpus of the arXiv, 11 co-authorship networks with 21,000 nodes from Citeseer (which had Machine Learning in their abstracts), 17 co-authorship networks (each with about 3000 nodes) from the NIPS conference and finally 10 Facebook ego networks2 . The average degrees vary between 0.2 to 0.4 for co-authorship networks and are around 10 for the ego networks. Each co-authorship network is dynamic, i.e. a node corresponds to an author in that corpus and this node index is preserved in the different networks over time. The ego networks are different in that sense, since each network is the subgraph of Facebook induced by the neighbors of a given central or ?ego? node. The sizes of these networks vary between 350 to 4000. Table 3: Clustering error of 6 different methods on a collection of real world networks consisting of co-authorship networks from Citeseer, High Energy Physics (HEP-Th) corpus of arXiv, NIPS and ego networks from Facebook. Error Time (s) NCLM (J = 8) 0.1 2.7 GK3 0.6 45 GK4 0.6 50 GK5 0.6 60 GraphStats (J = 8) 0.16 765 TopEig (J = 30) 0.32 14 Table 3 summarizes the performance of different algorithms and their running time to compute distance between the graphs. We use the different sources of networks as labels, i.e. HEP-Th will be one cluster, etc. We explore different choices of J, and see that the best 2 https://snap.stanford.edu/data/egonets-Facebook.html 8 performance is from NCLM, with J = 8, followed closely by GraphStats. TopEig (J in this case is where the eigenspectra of the larger networks have a knee) and the graph kernels do not perform very well. GraphStats take 765 seconds to complete, whereas NCLM finishes in 2.7 seconds. This is because the networks are large but extremely sparse, and so calculation of matrix powers is comparatively cheap. Figure 2: Kernel matrix for NCLM on 49 real networks. In Figure 2 we plot the kernel similarity matrix obtained using NCLM on the real networks (higher the value, more similar the points are). The first 11 networks are from HEP-Th, whereas the next 11 are from Citeseer. The next 16 are from NIPS and the remaining ones are the ego networks from Facebook. First note that {HEP-Th, Citeseer}, NIPS and Facebook are well separated. However, HEP-Th and Citeseer are hard to separate out. This is also verified by the bad performance of TopEig in separating out the first two (shown in Section 5). However, in Figure 2, we can see that the Citeseer networks are different from HEP-Th in the sense that they are not as strongly connected inside as HEP-Th. 6 Discussion We consider the problem of clustering network-valued data for two settings, both of which are prevalent in practice. In the first setting, different network objects have node correspondence. This includes clustering brain networks obtained from FMRI data where each node corresponds to a specific region in the brain, or co-authorship networks between a set of authors where the connections vary from one year to another. In the second setting, node correspondence is not present, e.g., when one wishes to compare different types of networks: co-authorship networks, Facebook ego networks, etc. One may be interested in seeing if co-authorship networks are more ?similar? to each other than ego or friendship networks. We present two algorithms for these two settings based on a simple general theme: summarize a network into a possibly high dimensional feature vector and then cluster these feature vectors. In the first setting, we propose NCGE, where each network is represented using its graphon-estimate. We can use a variety of graphon estimation algorithms for this purpose. We show that if the graphon estimation is consistent, then NCGE can cluster networks generated from a finite mixture of graphons in a consistent way, if those graphons are sufficiently different. In the second setting, we propose to represent a network using an easy-to-compute summary statistic, namely the vector of the log-traces of the first few powers of a suitably normalized version of the adjacency matrix. We call this method NCLM and show that the summary statistic concentrates around its expectation, and argue that this expectation should be able to separate networks generated from different models. Using simulated and real data experiments we show that NCGE is vastly superior to the naive but often-used method of comparing adjacency matrices directly, and NCLM outperforms most computationally expensive alternatives for differentiating networks without node correspondence. In conclusion, we believe that these methods will provide practitioners with a powerful and computationally tractable tool for comparing network-structured data in a range of disciplines. 9 Acknowledgments We thank Professor Peter J. Bickel for helpful discussions. SSM was partially supported by NSF-FRG grant DMS-1160319 and a Lo?ve Fellowship. PS was partially supported by NSF grant DMS 1713082. LL was partially supported by NSF grants IIS 1663870, DMS 1654579 and a DARPA grant N-66001-17-1-4041. References [1] Supplement to ?On clustering network-valued data?. 2017. [2] David J. Aldous. Representations for partially exchangeable arrays of random variables. 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A nonparametric view of network models and newman girvan and other modularities. Proceedings of the National Academy of Sciences of the Unites States of America, 106(50):21068?21073, 2009. [9] Sourav Chatterjee. Matrix estimation by universal singular value thresholding. Ann. Statist., 43(1):177?214, 02 2015. [10] Hao Chen and Nancy Zhang. Graph-based change-point detection. Ann. Statist., 43(1):139?176, 02 2015. [11] David S Choi, Patrick J Wolfe, and Edoardo M Airoldi. Stochastic blockmodels with a growing number of classes. Biometrika, page asr053, 2012. [12] Paul Erd?s and Alfr?d R?nyi. On random graphs i. Publicationes Mathematicae (Debrecen), 6:290?297, 1959 1959. [13] Chao Gao, Yu Lu, and Harrison H. Zhou. Rate-optimal graphon estimation. Ann. Statist., 43(6):2624?2652, 12 2015. [14] C. E. Ginestet, P. Balanchandran, S. Rosenberg, and E. D. Kolaczyk. Hypothesis Testing For Network Data in Functional Neuroimaging. ArXiv e-prints, July 2014. [15] Paul W Holland, Kathryn Blackmond Laskey, and Samuel Leinhardt. Stochastic blockmodels: First steps. Social networks, 5(2):109?137, 1983. [16] D.N. Hoover. Relations on probability spaces and arrays of random variables. Technical report, Institute of Advanced Study, Princeton., 1979. [17] Brian Karrer and M. E. J. Newman. Stochastic blockmodels and community structure in networks. Phys. Rev. E, 83:016107, Jan 2011. [18] Jing Lei, Alessandro Rinaldo, et al. Consistency of spectral clustering in stochastic block models. The Annals of Statistics, 43(1):215?237, 2015. 10 [19] Priya Mahadevan, Dmitri Krioukov, Kevin Fall, and Amin Vahdat. Systematic topology analysis and generation using degree correlations. SIGCOMM Comput. Commun. Rev., 36(4):135?146, August 2006. [20] Pierre-Andr? G Maugis, Sofia C Olhede, and Patrick J Wolfe. Topology reveals universal features for network comparison. arXiv preprint arXiv:1705.05677, 2017. [21] M. E. J. Newman. Modularity and community structure in networks. Proceedings of the National Academy of Sciences, 103(23):8577?8582, 2006. [22] Mark E. Newman. Assortative mixing in networks. Phys. Rev. Lett., 89(20):208701, 2002. [23] M.E.J. Newman. The structure and function of complex networks. SIAM review, 45(2):167?256, 2003. [24] Sofia C. Olhede and Patrick J. Wolfe. Network histograms and universality of blockmodel approximation. Proceedings of the National Academy of Sciences of the Unites States of America, 111(41):14722?14727, 2014. [25] Karl Rohe, Sourav Chatterjee, and Bin Yu. Spectral clustering and the high-dimensional stochastic block model. Annals of Statistics, 39:1878?1915, 2011. [26] N. Shervashidze, SVN. Vishwanathan, TH. Petri, K. Mehlhorn, and KM. Borgwardt. Efficient graphlet kernels for large graph comparison. In JMLR Workshop and Conference Proceedings Volume 5: AISTATS 2009, pages 488?495, Cambridge, MA, USA, April 2009. Max-Planck-Gesellschaft, MIT Press. [27] S. V. N. Vishwanathan, Nicol N. Schraudolph, Risi Kondor, and Karsten M. Borgwardt. Graph kernels. J. Mach. Learn. Res., 11:1201?1242, August 2010. [28] P. J. Wolfe and S. C. Olhede. Nonparametric graphon estimation. ArXiv e-prints, September 2013. [29] Y. Zhang, E. Levina, and J. Zhu. Estimating network edge probabilities by neighborhood smoothing. ArXiv e-prints, September 2015. [30] Shi Zhou and Raul J. Mondrag?n. The rich club phenomenon in the internet topology. IEEE Communications Letters, 8(3):180?182, 2004. 11 

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A General Framework for Robust Interactive Learning? Ehsan Emamjomeh-Zadeh? David Kempe? Abstract We propose a general framework for interactively learning models, such as (binary or non-binary) classifiers, orderings/rankings of items, or clusterings of data points. Our framework is based on a generalization of Angluin?s equivalence query model and Littlestone?s online learning model: in each iteration, the algorithm proposes a model, and the user either accepts it or reveals a specific mistake in the proposal. The feedback is correct only with probability p > 21 (and adversarially incorrect with probability 1 ? p), i.e., the algorithm must be able to learn in the presence of arbitrary noise. The algorithm?s goal is to learn the ground truth model using few iterations. Our general framework is based on a graph representation of the models and user feedback. To be able to learn efficiently, it is sufficient that there be a graph G whose nodes are the models, and (weighted) edges capture the user feedback, with the property that if s, s? are the proposed and target models, respectively, then any (correct) user feedback s0 must lie on a shortest s-s? path in G. Under this one assumption, there is a natural algorithm, reminiscent of the Multiplicative Weights Update algorithm, which will efficiently learn s? even in the presence of noise in the user?s feedback. From this general result, we rederive with barely any extra effort classic results on learning of classifiers and a recent result on interactive clustering; in addition, we easily obtain new interactive learning algorithms for ordering/ranking. 1 Introduction With the pervasive reliance on machine learning systems across myriad application domains in the real world, these systems frequently need to be deployed before they are fully trained. This is particularly true when the systems are supposed to learn a specific user?s (or a small group of users?) personal and idiosyncratic preferences. As a result, we are seeing an increased practical interest in online and interactive learning across a variety of domains. A second feature of the deployment of such systems ?in the wild? is that the feedback the system receives is likely to be noisy. Not only may individual users give incorrect feedback, but even if they do not, the preferences ? and hence feedback ? across different users may vary. Thus, interactive learning algorithms deployed in real-world systems must be resilient to noisy feedback. Since the seminal work of Angluin [2] and Littlestone [14], the paradigmatic application of (noisy) interactive learning has been online learning of a binary classifier when the algorithm is provided with feedback on samples it had previously classified incorrectly. However, beyond (binary or other) classifiers, there are many other models that must be frequently learned in an interactive manner. Two ? A full version is available on the arXiv at https://arxiv.org/abs/1710.05422. The present version omits all proofs and several other details and discussions. ? Department of Computer Science, University of Southern California, emamjome@usc.edu ? Department of Computer Science, University of Southern California, dkempe@usc.edu 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. particularly relevant examples are the following: (1) Learning an ordering/ranking of items is a key part of personalized Web search or other information-retrieval systems (e.g., [12, 18]). The user is typically presented with an ordering of items, and from her clicks or lack thereof, an algorithm can infer items that are in the wrong order. (2) Interactively learning a clustering [6, 5, 4] is important in many application domains, such as interactively identifying communities in social networks or partitioning an image into distinct objects. The user will be shown a candidate clustering, and can express that two clusters should be merged, or a cluster should be split into two. In all three examples ? classification, ranking, and clustering ? the interactive algorithm proposes a model4 (a classifier, ranking, or clustering) as a solution. The user then provides ? explicitly or implicitly ? feedback on whether the model is correct or needs to be fixed/improved. This feedback may be incorrect with some probability. Based on the feedback, the algorithm proposes a new and possibly very different model, and the process repeats. This type of interaction is the natural generalization of Angluin?s equivalence query model [2, 3]. It is worth noting that in contrast to active learning, in interactive learning (which is the focus of this work), the algorithm cannot ?ask? direct questions; it can only propose a model and receive feedback in return. The algorithm should minimize the number of user interactions, i.e., the number of times that the user needs to propose fixes. A secondary goal is to make the algorithm?s internal computations efficient as well. The main contribution of this article is a general framework for efficient interactive learning of models (even with noisy feedback), presented in detail in Section 2. We consider the set of all N models as nodes of a positively weighted undirected or directed graph G. The one key property that G must satisfy is the following: (*) If s is a proposed model, and the user (correctly) suggests changing it to s0 , then the graph must contain the edge (s, s0 ); furthermore, (s, s0 ) must lie on a shortest path from s to the target model s? (which is unknown to the algorithm). We show that this single property is enough to learn the target model s? using at most log N queries5 to the user, in the absence of noise. When the feedback is correct with probability p > 12 , the required number of queries gracefully deteriorates to O(log N ); the constant depends on p. We emphasize that the assumption (*) is not an assumption on the user. We do not assume that the user somehow ?knows? the graph G and computes shortest paths in order to find a response. Rather, (*) states that G was correctly chosen to model the underlying domain, so that correct answers by the user must in fact have the property (*). To illustrate the generality of our framework, we apply it to ordering, clustering, and classification: 1. For ordering/ranking, each permutation is a node in G; one permutation is the unknown target. If the user can point out only adjacent elements that are out of order, then G is an adjacent transposition ?B UBBLE S ORT? graph, which naturally has the property (*). If the user can pick any element and suggest that it should precede an entire block of elements it currently follows, then we can instead use an ?I NSERSION S ORT? graph; interestingly, to ensure the property (*), this graph must be weighted. On the other hand, as we show in Section 3, if the user can propose two arbitrary elements that should be swapped, there is no graph G with the property (*). Our framework directly leads to an interactive algorithm that will learn the correct ordering of n items in O(log(n!)) = O(n log n) queries; we show that this bound is optimal under the equivalence query model. 2. For learning a clustering of n items, the user can either propose merging two clusters, or splitting one cluster. In the interactive clustering model of [6, 5, 4], the user can specify that a particular cluster C should be split, but does not give a specific split. We show in Section 4 that there is a weighted directed graph with the property (*); then, if each cluster is from a ?small? concept class of size at most M (such as having low VC-dimension), there is an algorithm finding the true clustering in O(k log M ) queries, where k is number of the clusters (known ahead of time). 3. For binary classification, G is simply an n-dimensional hypercube (where n is the number of sample points that are to be classified). As shown in Section 5, one immediately recovers a close variant of standard online learning algorithms within this framework. An extension to classification with more than two classes is very straightforward. 4 We avoid the use of the term ?concept,? as it typically refers to a binary function, and is thus associated specifically with a classifier. 5 Unless specified otherwise, all logarithms are base 2. 2 Due to space limits, all proofs and several other details and discussions are omitted. A full version is available on the arXiv at https://arxiv.org/abs/1710.05422. 2 Learning Framework We define a framework for query-efficient interactive learning of different types of models. Some prototypical examples of models to be learned are rankings/orderings of items, (unlabeled) clusterings of graphs or data points, and (binary or non-binary) classifiers. We denote the set of all candidate models (permutations, partitions, or functions from the hypercube to {0, 1}) by ?, and individual models6 by s, s0 , s? , etc. We write N = |?| for the number of candidate models. We study interactive learning of such models in a natural generalization of the equivalence query model of Angluin [2, 3]. This model is equivalent to the more widely known online learning model of Littlestone [14], but more naturally fits the description of user interactions we follow here. It has also served as the foundation for the interactive clustering model of Balcan and Blum [6] and Awasthi et al. [5, 4]. In the interactive learning framework, there is an unknown ground truth model s? to be learned. In each round, the learning algorithm proposes a model s to the user. In response, with probability p > 12 , the user provides correct feedback. In the remaining case (i.e., with probability 1 ? p), the feedback is arbitrary; in particular, it could be arbitrarily and deliberately misleading. Correct feedback is of the following form: if s = s? , then the algorithm is told this fact in the form of a user response of s. Otherwise, the user reveals a model s0 6= s that is ?more similar? to s? than s was. The exact nature of ?more similar,? as well as the possibly restricted set of suggestions s0 that the user can propose, depend on the application domain. Indeed, the strength of our proposed framework is that it provides strong query complexity guarantees under minimal assumptions about the nature of the feedback; to employ the framework, one merely has to verify that the the following assumption holds. Definition 2.1 (Graph Model for Feedback) Define a weighted graph G (directed or undirected) that contains one node for each model s ? ?, and an edge (s, s0 ) with arbitrary positive edge length ?(s,s0 ) > 0 if the user is allowed to propose s0 in response to s. (Choosing the lengths of edges is an important part of using the framework.) G may contain additional edges not corresponding to any user feedback. The key property that G must satisfy is the following: (*) If the algorithm proposes s and the ground truth is s? 6= s, then every correct user feedback s0 lies on a shortest path from s to s? in G with respect to the lengths ?e . If there are multiple candidate nodes s0 , then there is no guarantee on which one the algorithm will be given by the user. 2.1 Algorithm and Guarantees Our algorithms are direct reformulations and slight generalizations of algorithms recently proposed by Emamjomeh-Zadeh et al. [10], which itself was a significant generalization of the natural ?Halving Algorithm? for learning a classifier (e.g., [14]). They studied the search problem as an abstract problem they termed ?Binary Search in Graphs,? without discussing any applications. Our main contribution here is the application of the abstract search problem to a large variety of interactive learning problems, and a framework that makes such applications easy. We begin with the simplest case p = 1, i.e., when the algorithm only receives correct feedback. Algorithm 1 gives essentially best-possible general guarantees [10]. To state the algorithm and its guarantees, we need the notion of an approximate median node of the graph G. First, we denote by  {s} if s0 = s N (s, s0 ) := 0 {? s | s lies on a shortest path from s to s?} if s0 6= s the set of all models s? that are consistent with a user feedback of s0 to a model s. In anticipation of the noisy case, we allow models to be weighted7 , and denote the node weights or likelihoods by 6 When considering specific applications, we will switch to notation more in line with that used for the specific application. 7 Edge lengths are part of the definition of the graph, but node weights will be assigned by our algorithm; they basically correspond to likelihoods. 3 ?(s) ? 0. If feedback is not noisy (i.e., Pp = 1), all the non-zero node weights are equal. For every subset of models S, we write ?(S) := s?S ?(s) for the total node weight of the models in S. Now, for every model s, define 1 ?? (s) := ? max ?(N (s, s0 )) ?(?) s0 6=s,(s,s0 )?G to be the largest fraction (with respect to node weights) of models that could still be consistent with a worst-case response s0 to a proposed model of s. For every subset of models S, we denote by ?S the likelihood function that assigns weight 1 to every node s ? S and 0 elsewhere. For simplicity of notation, we use ?S (s) when the node weights are ?S . The simple key insight of [10] can be summarized and reformulated as the following proposition: Proposition 2.1 ([10], Proofs of Theorems 3 and 14) Let G be a (weighted) directed graph in which each edge e with length ?e is part of a cycle of total edge length at most c ? ?e . Then, for every node weight function ?, there exists a model s such that ?? (s) ? c?1 c . When G is undirected (and hence c = 2), for every node weight function ?, there exists an s such that ?? (s) ? 12 . In Algorithm 1, we always have uniform node weight for all the models which are consistent with all the feedback received so far, and node weight 0 for models that are inconsistent with at least one response. Prior knowledge about candidates for s? can be incorporated by providing the algorithm with the input Sinit 3 s? to focus its search on; in the absence of prior knowledge, the algorithm can be given Sinit = ?. Algorithm 1 L EARNING A MODEL WITHOUT F EEDBACK E RRORS (Sinit ) 1: S ? Sinit . 2: while |S| > 1 do 3: Let s be a model with a ?small? value of ?S (s). 4: Let s0 be the user?s feedback model. 5: Set S ? S ? N (s, s0 ). 6: return the only remaining model in S. Line 3 is underspecified as ?small.? Typically, an algorithm would choose the s with smallest ?S (s). But computational efficiency constraints or other restrictions (see Sections 2.2 and 5) may preclude this choice and force the algorithm to choose a suboptimal s. The guarantee of Algorithm 1 is summarized by the following Theorem 2.2. It is a straightforward generalization of Theorems 3 and 14 from [10] Theorem 2.2 Let N0 = |Sinit | be the number of initial candidate models. If each model s chosen in Line 3 of Algorithm 1 has ?S (s) ? ?, then Algorithm 1 finds s? using at most log1/? N0 queries. Corollary 2.3 When G is undirected and the optimal s is used in each iteration, ? = Algorithm 1 finds s? using at most log2 N0 queries. 1 2 and In the presence of noise, the algorithm is more complicated. The algorithm and its analysis are given in the full version. The performance of the robust algorithm is summarized in Theorem 2.4. Theorem 2.4 Let ? ? [ 12 , 1), define ? = ?p + (1 ? ?)(1 ? p), and let N0 = |Sinit |. Assume that log(1/? ) > H(p) where H(p) = ?p log p ? (1 ? p) log(1 ? p) denotes the entropy. (When ? = 21 , this holds for every p > 12 .) If in each iteration, the algorithm can find a model s with ?? (s) ? ?, then with probability at least (1??) 2 1 ? ?, the robust algorithm finds s? using at most log(1/? )?H(p) log N0 + o(log N0 ) + O(log (1/?)) queries in expectation. Corollary 2.5 When the graph G is undirected and the optimal s is used in each iteration, then with (1??) probability at least 1 ? ?, the robust algorithm finds s? using at most 1?H(p) log2 N0 + o(log N0 ) + O(log2 (1/?)) queries in expectation. 4 2.2 Computational Considerations and Sampling Corollaries 2.3 and 2.5 require the algorithm to find a model s with small ?? (s) in each iteration. In most learning applications, the number N of candidate models is exponential in a natural problem parameter n, such as the number of sample points (classification), or the number of items to rank or cluster. If computational efficiency is a concern, this precludes explicitly keeping track of the set S or the weights ?(s). It also rules out determining the model s to query by exhaustive search over all models that have not yet been eliminated. In some cases, these difficulties can be circumvented by exploiting problem-specific structure. A more general approach relies on Monte Carlo techniques. We show that the ability to sample models s with probability (approximately) proportional to ?(s) (or approximately uniformly from S in the case of Algorithm 1) is sufficient to essentially achieve the results of Corollaries 2.3 and 2.5 with a computationally efficient algorithm. Notice that both in Algorithm 1 and the robust algorithm with noisy feedback (omitted from this version), the node weights ?(s) are completely determined by all the query responses the algorithm has seen so far and the probability p. Theorem 2.6 Let n be a natural measure of the input size and assume that log N is polynomial in n. Assume that G = (V, E) is undirected8 , all edge lengths are integers, and the maximum degree and diameter (both with respect to the edge lengths) are bounded by poly(n). Also assume w.l.o.g. that ? is normalized to be a distribution over the nodes9 (i.e., ?(?) = 1). Let 0 ? ? < 14 be a constant, and assume that there is an oracle that ? given a set of query responses ? runs in polynomial time in n and returns a model s drawn from a distribution ?0 with dTV (?, ?0 ) ? ?. Also assume that there is a polynomial-time algorithm that, given a model s, decides whether or not s is consistent with every given query response or not. Then, for every  > 0, in time poly(n, 1 ), an algorithm can find a model s with ?? (s) ? with high probability. 3 1 2 + 2? + , Application I: Learning a Ranking As a first application, we consider the task of learning the correct order of n elements with supervision in the form of equivalence queries. This task is motivated by learning a user?s preference over web search results (e.g., [12, 18]), restaurant or movie orders (e.g., [9]), or many other types of entities. Using pairwise active queries (?Do you think that A should be ranked ahead of B??), a learning algorithm could of course simulate standard O(n log n) sorting algorithms; this number of queries is necessary and sufficient. However, when using equivalence queries, the user must be presented with a complete ordering (i.e., a permutation ? of the n elements), and the feedback will be a mistake in the proposed permutation. Here, we propose interactive algorithms for learning the correct ranking without additional information or assumptions.10 We first describe results for a setting with simple feedback in the form of adjacent transpositions; we then show a generalization to more realistic feedback as one is wont to receive in applications such as search engines. 3.1 Adjacent Transpositions We first consider ?B UBBLE S ORT? feedback of the following form: the user specifies that elements i and i + 1 in the proposed permutation ? are in the wrong relative order. An obvious correction for an algorithm would be to swap the two elements, and leave the rest of ? intact. This algorithm would exactly implement B UBBLE S ORT, and thus require ?(n2 ) equivalence queries. Our general framework allows us to easily obtain an algorithm with O(n log n) equivalence queries instead. We define the undirected and unweighted graph GBS as follows: ? GBS contains N = n! nodes, one for each permutation ? of the n elements; ? it contains an edge between ? and ? 0 if and only if ? 0 can be obtained from ? by swapping two adjacent elements. 8 It is actually sufficient that for every node weight function ? : V ? R+ , there exists a model s with ?? (s) ? 12 . 9 For Algorithm 1, ? is uniform over all models consistent with all feedback up to that point. 10 For example, [12, 18, 9] map items to feature vectors and assume linearity of the target function(s). 5 Lemma 3.1 GBS satisfies Definition 2.1 with respect to B UBBLE S ORT feedback. Hence, applying Corollary 2.3 and Theorem 2.4, we immediately obtain the existence of learning algorithms with the following properties: Corollary 3.2 Assume that in response to each equivalence query on a permutation ?, the user responds with an adjacent transposition (or states that the proposed permutation ? is correct). 1. If all query responses are correct, then the target ordering can be learned by an interactive algorithm using at most log N = log n! ? n log n equivalence queries. 2. If query responses are correct with probability p > 12 , the target ordering can be learned (1??) by an interactive algorithm with probability at least 1 ? ? using at most 1?H(p) n log n + o(n log n) + O(log2 (1/?)) equivalence queries in expectation. Up to constants, the bound of Corollary 3.2 is optimal: Theorem 3.3 shows that ?(n log n) equivalence queries are necessary in the worst case. Notice that Theorem 3.3 does not immediately follow from the classical lower bound for sorting with pairwise comparisons: while the result of a pairwise comparison always reveals one bit, there are n ? 1 different possible responses to an equivalence query, so up to O(log n) bits might be revealed. For this reason, the proof of Theorem 3.3 explicitly constructs an adaptive adversary, and does not rely on a simple counting argument. Theorem 3.3 With adversarial responses, any interactive ranking algorithm can be forced to ask ?(n log n) equivalence queries. This is true even if the true ordering is chosen uniformly at random, and only the query responses are adversarial. 3.2 Implicit Feedback from Clicks In the context of search engines, it has been argued (e.g., by [12, 18, 1]) that a user?s clicking behavior provides implicit feedback of a specific form on the ranking. Specifically, since users will typically read the search results from first to last, when a user skips some links that appear earlier in the ranking, and instead clicks on a link that appears later, her action suggests that the later link was more informative or relevant. Formally, when a user clicks on the element at index i, but did not previously click on any elements at indices j, j + 1, . . . , i ? 1, this is interpreted as feedback that element i should precede all of elements j, j + 1, . . . , i ? 1. Thus, the feedback is akin to an ?I NSERSION S ORT? move. (The B UBBLE S ORT feedback model is the special case in which j = i ? 1 always.) To model this more informative feedback, the new graph GIS has more edges, and the edge lengths are non-uniform. It contains the same N nodes (one for each permutation). For a permutation ? and indices 1 ? j < i ? n, ?j?i denotes the permutation that is obtained by moving the ith element in ? before the j th element (and thus shifting elements j, j + 1, . . . , i ? 1 one position to the right). In GIS , for every permutation ? and every 1 ? j < i ? n, there is an undirected edge from ? to ?j?i with length i ? j. Notice that for i > j + 1, there is actually no user feedback corresponding to the edge from ?j?i to ?; however, additional edges are permitted, and Lemma 3.4 establishes that GIS does in fact satisfy the ?shortest paths? property. Lemma 3.4 GIS satisfies Definition 2.1 with respect to I NSERSION S ORT feedback. As in the case of GBS , by applying Corollary 2.3 and Theorem 2.4, we immediately obtain the existence of interactive learning algorithms with the same guarantees as those of Corollary 3.2. Corollary 3.5 Assume that in response to each equivalence query, the user responds with a pair of indices j < i such that element i should precede all elements j, j + 1, . . . , i ? 1. 1. If all query responses are correct, then the target ordering can be learned by an interactive algorithm using at most log N = log n! ? n log n equivalence queries. 2. If query responses are correct with probability p > 12 , the target ordering can be learned (1??) by an interactive algorithm with probability at least 1 ? ? using at most 1?H(p) n log n + o(n log n) + O(log2 (1/?)) equivalence queries in expectation. 6 3.3 Computational Considerations While Corollaries 3.2 and 3.5 imply interactive algorithms using O(n log n) equivalence queries, they do not guarantee that the internal computations of the algorithms are efficient. The na??ve implementation requires keeping track of and comparing likelihoods on all N = n! nodes. When p = 1, i.e., the algorithm only receives correct feedback, it can be made computationally efficient using Theorem 2.6. To apply Theorem 2.6, it suffices to show that one can efficiently sample a (nearly) uniformly random permutation ? consistent with all feedback received so far. Since the feedback is assumed to be correct, the set of all pairs (i, j) such that the user implied that element i must precede element j must be acyclic, and thus must form a partial order. The sampling problem is thus exactly the problem of sampling a linear extension of a given partial order. This is a well-known problem, and a beautiful result of Bubley and Dyer [8, 7] shows that the Karzanov-Khachiyan Markov Chain [13] mixes rapidly. Huber [11] shows how to modify the Markov Chain sampling technique to obtain an exactly (instead of approximately) uniformly random linear extension of the given partial order. For the purpose of our interactive learning algorithm, the sampling results can be summarized as follows: Theorem 3.6 (Huber [11]) Given a partial order over n elements, let L be the set of all linear extensions, i.e., the set of all permutations consistent with the partial order. There is an algorithm that runs in expected time O(n3 log n) and returns a uniformly random sample from L. The maximum node degree in GBS is n ? 1, while the maximum node degree in GIS is O(n2 ). The diameter of both GBS and GIS is O(n2 ). Substituting these bounds and the bound from Theorem 3.6 into Theorem 2.6, we obtain the following corollary: Corollary 3.7 Both under B UBBLE S ORT feedback and I NSERSION S ORT feedback, if all feedback is correct, there is an efficient interactive learning algorithm using at most log n! ? n log n equivalence queries to find the target ordering. The situation is significantly more challenging when feedback could be incorrect, i.e., when p < 1. In this case, the user?s feedback is not always consistent and may not form a partial order. In fact, we prove the following hardness result. Theorem 3.8 There exists a p (depending on n) for which the following holds. GivenPa set of user responses, let ?(?) be the likelihood of ? given the responses, and normalized so that ? ?(?) = 1. Let 0 < ? < 1 be any constant. There is no polynomial-time algorithm to draw a sample from a distribution ?0 with dTV (?, ?0 ) ? 1 ? ? unless RP = NP. It should be noted that the value of p in the reduction is exponentially close to 1. In this range, incorrect feedback is so unlikely that with high probability, the algorithm will always see a partial order. It might then still be able to sample efficiently. On the other hand, for smaller values of p (e.g., constant p), sampling approximately from the likelihood distribution might be possible via a metropolized Karzanov-Khachiyan chain or a different approach. This problem is still open. 4 Application II: Learning a Clustering Many traditional approaches for clustering optimize an (explicit) objective function or rely on assumptions about the data generation process. In interactive clustering, the algorithm repeatedly proposes a clustering, and obtains feedback that two proposed clusters should be merged, or a proposed cluster should be split into two. There are n items, and a clustering C is a partition of the items into disjoint sets (clusters) C1 , C2 , . . .. It is known that the target clustering has k clusters, but in order to learn it, the algorithm can query clusterings with more or fewer clusters as well. The user feedback has the following semantics, as proposed by Balcan and Blum [6] and Awasthi et al. [5, 4]. 1. M ERGE(Ci , Cj ): Specifies that all items in Ci and Cj belong to the same cluster. 2. S PLIT(Ci ): Specifies that cluster Ci needs to be split, but not into which subclusters. 7 Notice that feedback that two clusters be merged, or that a cluster be split (when the split is known), can be considered as adding constraints on the clustering (see, e.g., [21]); depending on whether feedback may be incorrect, these constraints are hard or soft. We define a weighted and directed graph GUC on all clusterings C. Thus, N = Bn ? nn is the nth Bell number. When C 0 is obtained by a M ERGE of two clusters in C, GUC contains a directed edge (C, C 0 ) of length 2. If C = {C1 , C2 , . . .} is a clustering, then for each Ci ? C, the graph GUC contains a directed edge of length 1 from C to C \ {Ci } ? {{v} | v ? Ci }. That is, GUC contains an edge from C to the clustering obtained from breaking Ci into singleton clusters of all its elements. While this may not be the ?intended? split of the user, we can still associate this edge with the feedback. Lemma 4.1 GUC satisfies Definition 2.1 with respect to M ERGE and S PLIT feedback. 1 GUC is directed, and every edge makes up at least a 3n fraction of the total length of at least one cycle it participates in. Hence, Proposition 2.1 gives an upper bound of 3n?1 3n on the value of ? in each iteration. A more careful analysis exploiting the specific structure of GUC gives us the following: Lemma 4.2 In GUC , for every non-negative node weight function ?, there exists a clustering C with ?? (C) ? 21 . In the absence of noise in the feedback, Lemmas 4.1 and 4.2 and Theorem 2.2 imply an algorithm that finds the true clustering using log N = log B(n) = ?(n log n) queries. Notice that this is worse than the ?trivial? algorithm, which starts with each node as a singleton cluster and always executes the merge proposed by the user, until it has found the correct clustering; hence, this bound is itself rather trivial. Non-trivial bounds can be obtained when clusters belong to a restricted set, an approach also followed by Awasthi and Zadeh [5]. If there are at most M candidate clusters, then the number of clusterings is N0 ? M k . For example, if there is a set system F of VC dimension at most d such that each cluster is in the range space of F, then M = O(nd ) by the Sauer-Shelah Lemma [19, 20]. Combining Lemmas 4.1 and 4.2 with Theorems 2.2 and 2.4, we obtain the existence of learning algorithms with the following properties: Corollary 4.3 Assume that in response to each equivalence query, the user responds with M ERGE or S PLIT. Also, assume that there are at most M different candidate clusters, and the clustering has (at most) k clusters. 1. If all query responses are correct, then the target clustering can be learned by an interactive algorithm using at most log N = O(k log M ) equivalence queries. Specifically when M = O(nd ), this bound is O(kd log n). This result recovers the main result of [5].11 2. If query responses are correct with probability p > 21 , the target clustering can be learned log M with probability at least 1 ? ? using at most (1??)k + o(k log M ) + O(log2 (1/?)) 1?H(p) equivalence queries in expectation. Our framework provides the noise tolerance ?for free;? [5] instead obtain results for a different type of noise in the feedback. 5 Application III: Learning a Classifier Learning a binary classifier is the original and prototypical application of the equivalence query model of Angluin [2], which has seen a large amount of follow-up work since (see, e.g., [16, 17]). Naturally, if no assumptions are made on the classifier, then n queries are necessary in the worst case. In general, applications therefore restrict the concept classes to smaller sets, such as assuming that they have bounded VC dimension. We use F to denote the set of all possible concepts, and write M = |F|; when F has VC dimension d, the Sauer-Shelah Lemma [19, 20] implies that M = O(nd ). Learning a binary classifier for n points is an almost trivial application of our framework12 . When the algorithm proposes a candidate classifier, the feedback it receives is a point with a corrected label (or the fact that the classifier was correct on all points). 11 12 In fact, the algorithm in [5] is implicitly computing and querying a node with small ? in GUC The results extend readily to learning a classifier with k ? 2 labels. 8 We define the graph GCL to be the n-dimensional hypercube13 with unweighted and undirected edges between every pair of nodes at Hamming distance 1. Because the distance between two classifiers C, C 0 is exactly the number of points on which they disagree, GCL satisfies Definition 2.1. Hence, we can apply Corollary 2.3 and Theorem 2.4 with Sinit equal to the set of all M candidate classifiers, recovering the classic result on learning a classifier in the equivalence query model when feedback is perfect, and extending it to the noisy setting. Corollary 5.1 1. With perfect feedback, the target classifier is learned using log M queries14 . 2. When each query response is correct with probability p > 12 , there is an algorithm learning log M the true binary classifier with probability at least 1?? using at most (1??) 1?H(p) +o(log M )+ O(log2 (1/?)) queries in expectation. 6 Discussion and Conclusions We defined a general framework for interactive learning from imperfect responses to equivalence queries, and presented a general algorithm that achieves a small number of queries. We then showed how query-efficient interactive learning algorithms in several domains can be derived with practically no effort as special cases; these include some previously known results (classification and clustering) as well as new results on ranking/ordering. Our work raises several natural directions for future work. Perhaps most importantly, for which domains can the algorithms be made computationally efficient (in addition to query-efficient)? We provided a positive answer for ordering with perfect query responses, but the question is open for ordering when feedback is imperfect. For classification, when the possible clusters have VC dimension d, the time is O(nd ), which is unfortunately still impractical for real-world values of d. Maass and Tur?an [15] show how to obtain better bounds specifically when the sample points form a d-dimensional grid; to the best of our knowledge, the question is open when the sample points are arbitrary. The Monte Carlo approach of Theorem 2.6 reduces the question to the question of sampling a uniformly random hyperplane, when the uniformity is over the partition induced by the hyperplane (rather than some geometric representation). For clustering, even less appears to be known. It should be noted that our algorithms may incorporate ?improper? learning steps: for instance, when trying to learn a hyperplane classifier, the algorithm in Section 5 may propose intermediate classifiers that are not themselves hyperplanes (though the final output is of course a hyperplane classifier). At an increase of a factor O(log d) in the number of queries, we can ensure that all steps are proper for hyperplane learning. An interesting question is whether similar bounds can be obtained for other concept classes, and for other problems (such as clustering). Finally, our noise model is uniform. An alternative would be that the probability of an incorrect response depends on the type of response. In particular, false positives could be extremely likely, for instance, because the user did not try to classify a particular incorrectly labeled data point, or did not see an incorrect ordering of items far down in the ranking. Similarly, some wrong responses may be more likely than others; for example, a user proposing a merge of two clusters (or split of one) might be ?roughly? correct, but miss out on a few points (the setting that [5, 4] studied). We believe that several of these extensions should be fairly straightforward to incorporate into the framework, and would mostly lead to additional complexity in notation and in the definition of various parameters. But a complete and principled treatment would be an interesting direction for future work. Acknowledgments Research supported in part by NSF grant 1619458. We would like to thank Sanjoy Dasgupta, Ilias Diakonikolas, Shaddin Dughmi, Haipeng Luo, Shanghua Teng, and anonymous reviewers for useful feedback and suggestions. 13 14 When there are k labels, GCL is a graph with kn nodes. With k labels, this bound becomes (k ? 1) log M . 9 References [1] E. Agichtein, E. Brill, S. Dumais, and R. Ragno. Learning user interaction models for predicting web search result preferences. In Proc. 29th Intl. Conf. on Research and Development in Information Retrieval (SIGIR), pages 3?10, 2006. [2] D. Angluin. Queries and concept learning. Machine Learning, 2:319?342, 1988. [3] D. Angluin. Computational learning theory: Survey and selected bibliography. In Proc. 24th ACM Symp. on Theory of Computing, pages 351?369, 1992. [4] P. Awasthi, M.-F. Balcan, and K. Voevodski. Local algorithms for interactive clustering. Journal of Machine Learning Research, 18:1?35, 2017. [5] P. Awasthi and R. B. Zadeh. Supervised clustering. In Proc. 24th Advances in Neural Information Processing Systems, pages 91?99. 2010. [6] M.-F. Balcan and A. Blum. Clustering with interactive feedback. In Proc. 19th Intl. Conf. on Algorithmic Learning Theory, pages 316?328, 2008. [7] R. Bubley. Randomized Algorithms: Approximation, Generation, and Counting. Springer, 2001. [8] R. Bubley and M. Dyer. Faster random generation of linear extensions. Discrete Mathematics, 201(1):81?88, 1999. [9] K. Crammer and Y. Singer. Pranking with ranking. In Proc. 16th Advances in Neural Information Processing Systems, pages 641?647, 2002. [10] E. Emamjomeh-Zadeh, D. Kempe, and V. Singhal. Deterministic and probabilistic binary search in graphs. In Proc. 48th ACM Symp. on Theory of Computing, pages 519?532, 2016. [11] M. Huber. Fast perfect sampling from linear extensions. Discrete Mathematics, 306(4):420?428, 2006. [12] T. Joachims. Optimizing search engines using clickthrough data. In Proc. 8th Intl. Conf. on Knowledge Discovery and Data Mining, pages 133?142, 2002. [13] A. Karzanov and L. Khachiyan. On the conductance of order Markov chains. Order, 8(1):7?15, 1991. [14] N. Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning, 2:285?318, 1988. [15] W. Maass and G. Tur?an. On the complexity of learning from counterexamples and membership queries. In Proc. 31st IEEE Symp. on Foundations of Computer Science, pages 203?210, 1990. [16] W. Maass and G. Tur?an. Lower bound methods and separation results for on-line learning models. Machine Learning, 9(2):107?145, 1992. [17] W. Maass and G. Tur?an. Algorithms and lower bounds for on-line learning of geometrical concepts. Machine Learning, 14(3):251?269, 1994. [18] F. Radlinski and T. Joachims. Query chains: Learning to rank from implicit feedback. In Proc. 11th Intl. Conf. on Knowledge Discovery and Data Mining, pages 239?248, 2005. [19] N. Sauer. On the density of families of sets. Journal of Combinatorial Theory, Series A, 13(1):145?147, 1972. [20] S. Shelah. A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific Journal of Mathematics, 41(1):247?261, 1972. [21] K. L. Wagstaff. Intelligent Clustering with Instance-Level Constraints. PhD thesis, Cornell University, 2002. 10 

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Multi-view Matrix Factorization for Linear Dynamical System Estimation Mahdi Karami, Martha White, Dale Schuurmans, Csaba Szepesv?ri Department of Computer Science University of Alberta Edmonton, AB, Canada {karami1, whitem, daes, szepesva}@ualberta.ca Abstract We consider maximum likelihood estimation of linear dynamical systems with generalized-linear observation models. Maximum likelihood is typically considered to be hard in this setting since latent states and transition parameters must be inferred jointly. Given that expectation-maximization does not scale and is prone to local minima, moment-matching approaches from the subspace identification literature have become standard, despite known statistical efficiency issues. In this paper, we instead reconsider likelihood maximization and develop an optimization based strategy for recovering the latent states and transition parameters. Key to the approach is a two-view reformulation of maximum likelihood estimation for linear dynamical systems that enables the use of global optimization algorithms for matrix factorization. We show that the proposed estimation strategy outperforms widely-used identification algorithms such as subspace identification methods, both in terms of accuracy and runtime. 1 Introduction Linear dynamical systems (LDS) provide a fundamental model for estimation and forecasting in discrete-time multi-variate time series. In an LDS, each observation is associated with a latent state; these unobserved states evolve as a Gauss-Markov process where each state is a linear function of the previous state plus noise. Such a model of a partially observed dynamical system has been widely adopted, particularly due to its efficiency for prediction of future observations using Kalman filtering. Estimating the parameters of an LDS?sometimes referred to as system identification?is a difficult problem, particularly if the goal is to obtain the maximum likelihood estimate of parameters. Consequently, spectral methods from the subspace identification literature, based on moment-matching rather than maximum likelihood, have become popular. These methods provide closed form solutions, often involving a singular value decomposition of a matrix constructed from the empirical moments of observations (Moonen and Ramos, 1993; Van Overschee and De Moor, 1994; Viberg, 1995; Katayama, 2006; Song et al., 2010; Boots and Gordon, 2012). The most widely used such algorithms for parameter estimation in LDSs are the family of N4SID algorithms (Van Overschee and De Moor, 1994), which are computationally efficient and asymptotically consistent (Andersson, 2009; Hsu et al., 2012). Recent evidence, however, suggests that these moment-matching approaches may suffer from weak statistical efficiency, performing particularly poorly with small sample sizes (Foster et al., 2012; Zhao and Poupart, 2014). Maximum likelihood for LDS estimation, on the other hand, has several advantages. For example, it is asymptotically efficient under general conditions (Cram?r, 1946, Ch.33), and this property often translates to near-minimax finite-sample performance. Further, maximum likelihood is amenable to coping with missing data. Another benefit is that, since the likelihood for exponential families 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. and corresponding convex losses (Bregman divergences) are well understood (Banerjee et al., 2005), maximum likelihood approaches can generalize to a broad range of distributions over the observations. Similarly, other common machine learning techniques, such as regularization, can be naturally incorporated in a maximum likelihood framework, interpretable as maximum a posteriori estimation. Unfortunately, unlike spectral methods, there is no known efficient algorithm for recovering parameters that maximize the marginal likelihood of observed data in an LDS. Standard iterative approaches are based on EM (Ghahramani and Hinton, 1996; Roweis and Ghahramani, 1999), which are computationally expensive and have been observed to produce locally optimal solutions that yield poor results (Katayama, 2006). A classical system identification method, called the prediction error method (PEM), is based on minimization of prediction error and can be interpreted as maximum likelihood estimation under certain distributional assumptions (e.g., Ch. 7.4 of Ljung 1999, ?str?m 1980). PEM, however, is prone to local minima and requires selection of a canonical parameterization, which can be difficult in practice and can result in ill-conditioned problems (Katayama, 2006). In this paper, we propose an alternative approach to LDS parameter estimation under exponential family observation noise. In particular, we reformulate the LDS as a two-view generative model, which allows us to approximate the estimation task as a form of matrix factorization, and apply recent global optimization techniques for such models (Zhang et al., 2012; Yu et al., 2014). To extend these previous algorithms to this setting, we provide a novel proximal update for the two-view approach that significantly simplifies the algorithm. Finally, for forecasting on synthetic and real data, we demonstrate that the proposed algorithm matches or outperforms N4SID, while scaling better with increasing sample size and data dimension. 2 Linear dynamical systems We address discrete-time, time-invariant linear dynamical systems, specified as ?t+1 = A?t + ?t xt = C?t + t (1) where ?t ? Rk is the hidden state at time t; xt ? Rd is the observation vector at time t; A ? Rk?k is the dynamics matrix; C ? Rd?k is the observation matrix; ? is the state evolution noise; and  is the observation noise. The noise terms are assumed to be independent. As is common, we assume that the state evolution noise is Gaussian: ? ? N (0, ?? ). We additionally allow for general observation noise to be generated from an exponential family distribution (e.g., Poisson). The graphical representation for this LDS is shown in Figure 1. An LDS encodes the intuition that a latent state is driving the dynamics, which can significantly simplify estimation and forecasting. The observations typically contain only partial information about the environment (such as in the form of limited sensors), and further may contain noisy or even irrelevant observations. Learning transition models for such observations can be complex, particularly if the observations are high-dimensional. For example, in spatiotemporal processes, the data is typically extremely high-dimensional, composed of structured grid data; however, it is possible to extract a low-rank state-space that significantly simplifies analysis (Gelfand et al., 2010, Chapter 8). Further, for forecasting, iterating transitions for such a low-rank state-space can provide longer range predictions with less error accumulation than iterating with the observations themselves. The estimation problem for an LDS involves extracting the unknown parameters, given a time series of observations x1 , . . . , xT . Unfortunately, jointly estimating the parameters A, C and ?t is difficult because the multiplication of these variables typically results in a nonconvex optimization. Given the latent states ?t , estimation of A and C is more straightforward, though there are still some issues with maintaining stability (Siddiqi et al., 2007). There are some recent advances improving estimation in time series models using matrix factorization. White et al. (2015) provide a convex formulation for auto-regressive moving average models?although related to state-space models, these do not permit a straightforward conversion between the parameters of one to the other. Yu et al. (2015) factorize the observation into a hidden state and dictionary, using a temporal regularizer on the extracted hidden state?the resulting algorithm, however, is not guaranteed to provide an optimal solution. 2 ?1 A ?2 ?3 ... E C ... x1 3 x2 Figure 1: Graphical representation for the standard LDS formulation and the corresponding two-view model. The two-view formulation is obtained by a linear transformation of the LDS model. The LDS model includes only parameters C and A and the two-view model includes parameters C and E = CA, where A can be extracted from E after C and E are estimated. x3 Two-view Formulation of LDS In this section, we reformulate the LDS as a generative two-view model with a shared latent factor. In the following section, we demonstrate how to estimate the parameters of this reformulation optimally, from which parameter estimates of the original LDS can be recovered. To obtain a two-view formulation, we re-express the two equations for the LDS as two equations for pairs of sequential observations. To do so, we multiply the state evolution equation in (1) by C and add t+1 to obtain C?t+1 + t+1 = CA?t + C?t + t+1 ; representing the LDS model as xt+1 = E?t + 0t+1 xt = C?t + t (2) where we refer to E := CA as the factor loading matrix and 0t+1 := C?t + t+1 as the noise of the second view. We then have a two-view problem where we need to estimate parameters E and C. Since the noise components t and 0t are independent, the two views xt and xt+1 are conditionally independent given the shared latent state ?t . The maximum log likelihood problem for the two-view formulation then becomes max log p(x1 , . . . , xT |?0 , ?1 , . . . , ?T , C, E) = max C,E,? C,E,? T X log p(xt |?t?1 , ?t , C, E) (3) t=1 where, given the hidden states, the observations are conditionally independent. The log-likelihood (3) is equivalent to the original LDS, but is expressed in terms of the distribution p(xt |?t?1 , ?t , C, E), where the probability of an observation increases if it has high probability under both ?t?1 and ?t . The graphical depiction of the LDS and its implied two-view model is illustrated in Figure 1. 3.1 Relaxation To tackle the estimation problem, we reformulate the estimation problem for this equivalent two-view model of the LDS. Note that according to the two-view model (2), the conditional distribution (3) can be expressed as p(xt |?t?1 , ?t , C, E) = p(xt |E?t?1 ) = p(xt |C?t ). Substituting each of these in the summation (3) would result in a factor loading model that ignores the temporal correlation among data; therefore, to take the system dynamics into account we choose a balanced averaging of both as log p(xt |?t?1 , ?t , C, E) = 12 log p(xt |E?t?1 ) + 21 log p(xt |C?t ), where the likelihood of an observation increases if it has high conditional likelihood given both ?t?1 and ?t .1 With this choice and the exponential family specified by the log-normalizer (also called potential function) F : Rd ? R, with the corresponding Bregman divergence defined as DF (? zkz) := F (? z) ? F (z) ? f (z)> (? z ? z) 2 using transfer function f = ?F , the log-likelihood separates into the two components argmax C,E,? T X t=1 log p(xt |?t?1 , ?t , C, E) = argmax 21 C,E,? = argmin T X log p(xt |E?t?1 ) + log p(xt |C?t ) t=1 T X DF (E?t?1 ||f ?1 (xt )) + DF (C?t ||f ?1 (xt )) C,E,? t=1 1 The balanced averaging can be generalized to a convex combination of the log-likelihood which adds a flexibility to the problem that can be tuned to improve performance. However, we found that the simple balanced combination renders the best experimental performance in most cases. 2 Consult Banerjee et al. (2005) for a complete overview of this correspondence. 3 Each Bregman divergence term can be interpreted as the fitness measure for each view. For example, a Gaussian distribution can be expressed by an exponential family defined by F (z) = 21 kzk22 . The above derivation could be extended to different variance terms for  and 0 , which would result in different weights on the two Bregman divergences above. Further, we could also allow different exponential families (hence different Bregman divergences) for the two distributions; however, there is no clear reason why this would be beneficial over simply selecting the same exponential family, since both describe xt . In this work, therefore, we will explore a balanced loss, with the same exponential family for each view. In order to obtain a low rank solution, one can relax the hard rank constraint and employ the block Pk norm k?k2,1 = j=1 k?j: k2 as the rank-reducing regularizer on the latent state.3 This regularizer offers an adaptive rank reducing scheme that zeros out many of the rows of the latent states and hence results a low rank solution without knowing the rank a priori. For the reconstruction models C and E, we need to specify a prior that respects the conditional independence of the views xt and xt+1 given ?t . This goal can be achieved if C and E are constrained individually so that they do not compete against each other to reconstruct their respective views (White et al., 2012). Incorporating the regularizer and constraints, the resulting optimization problem has the form argmin T X L1 (E?t?1 ; xt ) + L2 (C?t ; xt ) + ? C,E,? t=1 k X k?j: k2 (4) j=1 s.t.kC:j k2 ? ?1 , kE:j k2 ? ?2 ?j ? (1, k). The above constrained optimization problem is convex in each of the factor loading matrices {C, E} and the state matrix ?, but not jointly convex in terms of all these variables. Nevertheless, the following lemma show that (4) admits a convex reformulation by change of variable.  (1)  ? ? (1) := C? and Z ? (2) := E? with their concatenated matrix Z ? := Z Lemma 1 Let Z ? (2) and Z     1 0 (1) (2) (1) (2) Z := [x1:T ?1 ], Z := [x2:T ]. In addition, let?s define I := diag( ), I := diag( ), 0 1 then the multi-view optimization problem (4) can be reformulated in the following convex form min kC:j k2 ??1 kE:j k2 ??2 "min # C ? ?: ?=Z E L1 (C?; Z(1) ) + L2 (E?; Z(2) ) + ?k?k2,1 ? (1) ; Z(1) ) + L2 (Z ? (2) ; Z(2) ) + ? max kU?1 Zk ? tr = min L1 (Z ? 0???1 ? Z where U? = ? = PT Li (yt ; y?t ). Moreover, we can show ??2 I(2) and Li (Y; Y) t=1 1?? ? tr is concave in ?. The trace norm induces a low rank result. kU?1 Zk ? ? ?1 I(1) ? the regularizer term + Proof: The proof can be readily derived from the results of White et al. (2012). that  In the next section, we demonstrate how to obtain globally optimal estimates of E, C and ?. Remark 1: This maximum likelihood formulation demonstrates how the distributional assumptions on the observations xt can be generalized to any exponential family. Once expressed as the above optimization problem, one can further consider other losses and regularizers that may not immediately have a distributional interpretation, but result in improved prediction performance. This generalized formulation of maximum likelihood for LDS, therefore, has the additional benefit that it can flexibly incorporate optimization improvements, such as robust losses.4 Also a regularizer can be designed to control overfitting to noisy observation, which is an issue in LDS that can result in an unstable latent dynamics estimate (Buesing et al., 2012a). Therefore, by controlling undesired overfitting to noisy samples one can also prevent unintended unstable model identification. 3 Throughout this paper, Xi: (X:i ) is used to denote the ith row (ith column) of matrix X and also [X; Y] ([x; y]) denotes the matrix (vector) concatenation operator which is equal to [X> , Y> ]> ([x> , y> ]> ). 4 Thus, we used L1 and L2 in (4) to generally refer to any loss function that is convex in its first argument. 4 Remark 2: We can generalize the optimization further to learn an LDS with exogenous input: a control vector ut ? Rd that impacts both the hidden state and observations. This entails adding some new variables to the general LDS model that can be expressed as ?t+1 = A?t + But + ?t xt = C?t + Dut + t with additional matrices B ? Rk?d and D ? Rd?d . Again by multiplying the state evolution equation by matrix C the resulting equations are xt+1 = E?t + Fut + Dut+1 + 0t+1 xt = C?t + Dut + t where F := CB. Therefore, the loss can be generally expressed as L1 (E?t?1 + Fut?1 + Dut ; xt ) + L2 (C?t + Dut ; xt ). The optimization would now be over the variables C, E, ?, D, F, where the optimization could additionally include regularizers on D and F to control overfitting. Importantly, the addition of these variables D, F does not modify the convexity properties of the loss, and the treatment for estimating E, C and ? in section 4 directly applies. The optimization problem is jointly convex in D, F and any one of E, C or ? and jointly convex in D and F. Therefore, an outer minimization over D and F can be added to Algorithm 1 and we will still obtain a globally optimal solution. 4 LDS Estimation Algorithm To learn the optimal parameters for the reformulated two-view model, we adopt the generalized conditional gradient (GCG) algorithm developed by Yu et al. (2014). GCG is designed for optimization problems of the form l(x) + f (x) where l(x) is convex and continuously differentiable with Lipschitz continuous gradient and f (x) is a (possibly non-differentiable) convex function. The algorithm is computationally efficient, as well providing a reasonably fast O(1/t) rate of convergence to the global minimizer. Though we have a nonconvex optimization problem, we can use the convex reformulation for two-view low-rank matrix factorization and resulting algorithm in (Yu et al., 2014, Section 4). This algorithm includes a generic local improvement step, which significantly accelerates the convergence of the algorithm to a global optimum in practice. We provide a novel local improvement update, which both speeds learning and enforces a sparser structure on ?, while maintaining the same theoretical convergence properties of GCG. In our experiments, we specifically address the setting when the observations are assumed to be Gaussian, giving an `2 loss. We also prefer the unconstrained objective function that can be efficiently minimized by fast unconstrained optimization algorithms. Therefore, using the well-established equivalent form of the regularizer (Bach et al., 2008), the objective (4) can be equivalently cast for the Gaussian distributed time series xt as min C,E,? T X kE?t?1 ? xt k22 + kC?t ? xt k22 + ? t=1 k X k?j: k2 max( ?11 kC:j k2 , ?12 kE:j k2 ). (5) j=1 This product form of the regularizer is also preferred over the square form used in (Yu et al., 2014), since it induces row-wise sparsity on ?. Though the square form k?k2F admits efficient optimizers due to its smoothness, it does not prefer to zero out rows of ? while with the regularizer of the form (5), the learned hidden state will be appropriately projected down to a lower-dimensional space where many dimensions could be dropped from ?, C and E giving a low rank solution. In practice, we found that enforcing this sparsity property on ? significantly improved stability.5 Consequently, we need optimization routines that are appropriate for the non smooth regularizer terms. The local improvement step involves alternating block coordinate descent between C, E and ?, with an accelerated proximal gradient algorithm (FISTA) (Beck and Teboulle, 2009) for each descent step. To use the FISTA algorithm we need to provide a proximal operator for the non-smooth regularizer in (5). 5 This was likely due to a reduction in the size of the transition parameters, resulting in improved re-estimation of A and a corresponding reduction in error accumulation when using the model for forecasting. 5 Algorithm 1 LDS-DV Input: training sequence {xt , t ? [1, T ]} Output: C, A, ?t , ?? , ? Initialize C0 , E0 , ?0 > > U1 ? [C> V1 ? ?> 0 ; E0 ] , 0 for i = 1, . . . do  (ui , vi ) ? arg minuv> ?A ?`(Ui , Vi ), uv> // compute polar (?i , ?i ) ? arg min `((1 ? ?)Ui Vi> + ?ui vi> ) + ?((1 ? ?)?i + ?) // partially corrective up0???1,??0 date (PCU) ? ? ? ? Uinit ? [ 1 ? ?i Ui , ?i ui ], Vinit ? [ 1 ? ?i Vi , ?i vi ] (Ui+1 , Vi+1 ) ? FISTA(Uinit Vinit ) P 2 2 ?i = 12 i+1 j=1 (k(Ui+1 ):i k2v + k(Vi+1 ):i k2 ) end for > (C; E) ? Ui+1 , ? ? Vi+1 ? A ? ?2:T ? ?1:T ?1 estimate ?? , ? by sample covariances Let the proximal operator of a convex and possibly non-differentiable function ?f (y) be defined as prox?f (x) = arg min ?f (y) + 21 kx ? yk22 . y FISTA is an accelerated version of ISTA (Iterative Shrinkage-Thresholding Algorithm) that iteratively performs a gradient descent update with the smooth component of the objective, and then applies the proximal operator
as a projection step. Each iteration updates the variable x as xk+1 = prox?k f xk ? ?k ?l(xk ) , which converges to a fixed point. If there is no known form for the proximal operator, as is the case for our non-differentiable regularizer, a common strategy is to numerically calculate the proximal update. This approach, however, can be prohibitively expensive, and an analytic (closed) form is clearly preferable. We derive such a closed form for (5) in Theorem 1. h i Theorem 1 For a vector v = vv12 composed of two subvectors v1 , v2 , define f (v) = ?kvk2v := ? max(kv1 k2 , kv2 k2 ). The proximal operator for this function is # ?" v1 max{1 ? kv?1 k , 0} ? ? ? if kv1 k ? kv2 k ? ? v2 max{1 ? ??? , 0} kv2 k # proxf (v) = " ??? ? v1 max{1 ? kv , 0} ? k ? 1 ? if kv2 k ? kv1 k ? v max{1 ? ? , 0} 2 kv2 k where ? := max{.5(kv1 k ? kv2 k + ?), 0} and ? := max{.5(kv2 k ? kv1 k + ?), 0}.  Proof: See Appendix A. This result can be further generalized to enable additional regularization components on C and E, such as including an `1 norm on each column to further enforce sparsity (such as in the elastic net). There is no closed form for the proximal operator of the sum of two functions in general. We prove, however, that for special case of a linear combination of the two-view norm with any norms on the columns of C and E, the proximal mapping reduces to a simple composition rule. Theorem 2 For norms R1 (v1 ) and R2 (v2 ), the proximal operator of the linear combination Rc (v) = ?kvk2v + ? 1 R1 (v1 ) + ?2 R 2 (v2 ) for ?1 , ?2 ? 0 admits the simple composition prox?1 R1 (v1 ) proxRc (v) = prox?k.k2v . prox?2 R2 (v2 )  Proof: See Appendix A. 4.1 Recovery of the LDS model parameters The above reformulation provides a tractable learning approach to obtain the optimal parameters for the two-view reformulation of LDS; given this optimal solution, we can then estimate the parameters 6 to the original LDS. The first step is to estimate the transition matrix A. A natural approach is to ? =C ? ?E ? for pseudoinverse C ? ? . This A, ? however, might be sensitive to inaccurate use (2), and set A estimation of the (effective) hidden state dimension k. We found in practice that modifications from the optimal choice of k might result in unstable solutions and produce unreliable forecasts. Instead, ? can be learned from the hidden states themselves. This approach also focuses a more stable A estimation of A on the forecasting task, which is our ultimate aim. Given the sequence of hidden states, ?1 , . . . , ?T , there are several strategies that could be used to estimate A, including simple autoregressive models to more sophisticated strategies (Siddiqi et al., ? = arg minA PT ?1 k?t+1 ? A?t k2 which 2007). We opt for a simple linear regression solution A 2 t=1 ? we found produced stable A. ? t , t = xt ? C?t . Having obtained To estimate the noise parameters ?? , ? , recall ?t = ?t+1 ? A? ? A, therefore, we can estimate the noise covariance matrices by computing their sample covariances PT PT 1 > ? > ?? = 1 as ? t=1 ?t ?t , ? = T ?1 t=1 t t . The final LDS learning procedure is outlined in T ?1 Algorithm 1. For more details about polar computation and partially corrective subroutine see (Yu et al., 2014, Section 4). 5 Experimental results We evaluate the proposed algorithm by comparing one step prediction performance and computation speed with alternative methods for real and synthetic time series. We report the normalized mean PTtest PTtest kyt ?? yt k2 1 square error (NMSE) defined as NMSE = PTt=1 where ?y = Ttest test t=1 yt . 2 t=1 kyt ??y k Algorithms: We compared the proposed algorithm to a well-established method-of moment-based algorithm, N4SID (Van Overschee and De Moor, 1994), Hilbert space embeddings of hidden Markov models (HSE-HMM) (Song et al., 2010), expectation-maximization for estimating the parameters of a Kalman filter (EM) (Roweis and Ghahramani, 1999) and PEM (Ljung, 1999). These are standard baseline algorithms that are used regularly for LDS identification. The estimated parameters by N4SID were used as the initialization point for EM and PEM algorithms in our experiments. We used the built-in functions, n4sid and pem, in Matlab, with the order selected by the function, for the subspace identification method and PEM, respectively. For our algorithm, we select the regularization parameter ? using cross-validation. For the time series, the training data is split by performing the learning on first 80% of the training data and evaluating the prediction performance on the remaining 20%. Real datasets: For experiments on real datasets we select the climate time series from IRI data library that recorded the surface temperature on the monthly basis for tropical Atlantic ocean (ATL) and tropical Pacific ocean (CAC). In CAC we selected first 30 ? 30 grids out of the total 84 ? 30 locations with 399 monthly samples, while in ATL the first 9 ? 9 grids out of the total 38 ? 25 locations are selected each with timeseries of length 564. We partitioned each area to smaller areas of size 3 ? 3 and arrange them to vectors of size 9, then seasonality component of the time series are removed and data is centered to have zero mean. We ran two experiments for each dataset. For the first, the whole sequence is sliced into 70% training and 30% test. For the second, a short training set of 70 samples is selected, with a test sequence of size 50. Synthetic datasets: In the synthetic experiments, the datasets are generated by an LDS model (1) of different system orders, k, and observation sizes, d. For each test case, 100 data sequences of length 200 samples are generated and sliced to 70%, 30% ratios for training set and test set, respectively. The dynamics matrix A is selected to produce a stable system: {|?i (A)| = s : s ? 1, ?i ? (1, k)} where ?i (A) is the ith eigen value of matrix A. The noise components are drawn from Gaussian distributions and scaled so that p? := E{? > ?}/m and p := E{> }/n. Each test is repeated with the following settings: {S1: s = 0.970, p? = 0.50 and p = 0.1}, {S2: s = 0.999, p? = 0.01 and p = 0.1}. Results: The NMSE and run-time results obtained on real and synthetic datasets are shown in Table 1 and Table 2, respectively. In terms of NMSE, LDS-DV outperforms and matches the alternative methods. In terms of algorithm speed, the LDS-DV learns the model much faster than the competitors and scales well to larger dimension models. The speed improvement is more significant for larger datasets and observations with higher dimensions. 7 LDS-MV N4SID EM HSE-HMM PEM-SSID ATL(Long) NMSE Time 0.45?0.03 0.26 0.52?0.04 2.34 0.64?0.04 7.87 675.87?629.46 0.79 0.71?0.08 20.00 Table 1: Real time series ATL(Short) CAC(Long) NMSE Time NMSE Time 0.54?0.05 0.22 0.58?0.02 0.28 0.59?0.05 0.95 0.61?0.02 1.23 0.88?0.07 3.92 0.81?0.02 5.70 0.97?0.01 0.16 11.24?8.23 0.39 1.52?0.66 16.38 1.38?0.15 19.67 CAC(Short) NMSE Time 0.63?0.03 0.14 0.84?0.07 1.08 1.02?0.08 4.12 2.82?1.60 0.17 2.68?0.78 20.58 Table 2: Synthetic time series (S1) d=5 , k=3 NMSE LDS-MV 0.12?0.01 N4SID 0.12?0.01 EM 0.18?0.01 HSE-HMM 2.4e+4?1.7e+4 PEM-SSID 0.14?0.01 (S2) d=5 , k=3 Time NMSE 0.49 0.17?0.02 0.81 0.42?0.04 4.99 0.15?0.02 0.48 2.2e+7?2.2e+7 10.72 0.25?0.03 (S1) d=8 , k=6 Time NMSE 0.36 0.08?0.00 0.76 0.11?0.00 4.62 0.14?0.01 0.50 7.8e+03?7.7e+03 9.08 0.12?0.01 Time 0.66 1.45 6.01 0.49 15.22 (S2) d=8 , k=6 NMSE 0.04?0.00 0.39?0.04 0.04?0.00 0.65?0.02 0.08?0.01 (S1) d=16 , k=9 Time 0.52 1.38 5.03 0.55 13.97 NMSE 0.07?0.00 0.10?0.00 0.13?0.00 22.92?21.83 0.09?0.01 Time 1.01 4.29 19.21 0.53 38.39 (S2) d=16 , k=9 NMSE 0.03?0.00 0.42?0.04 0.03?0.00 0.71?0.01 0.06?0.02 Time 1.72 4.40 19.83 0.61 41.10 1 10 Seconds NMSE 12 LDS-DV N4SID EM 1.2 0.8 8 Prediction MSE of LDS-MV Results for real and synthetic datasets are listed in Table 1 and Table 2, respectively. The first column of each dataset is the average normalized MSE with standard error and the second column is the algorithm runtime in CPU seconds. The best NMSE according to pairwise t-test with significance level of 5% is highlighted. LDS-DV N4SID EM HSE-HMM 6 4 2 0.6 100 200 300 400 500 600 Training Sequence Length (T) (a) NMSE 0 100 200 300 400 500 Training Sequence Length(T) (b) Runtime 600 3.5 3 2.5 2 1.5 1 0.5 0.5 1 1.5 2 2.5 3 Prediction MSE of n4SID 3.5 (c) Scatter plot of MSE Figure 2: a) NMSE of the LDS-DV for increasing length of training sequence. The difference between LDS-DV and N4SID is more significant in shorter training length, while both converge to the same accuracy in large T . HSE-HMM is omitted due to its high error. b) Runtime in CPU seconds for increasing length of training sequence. LDS-DV scales well with large sample length. c) MSE of the LDS-DV versus MSE of N4SID. In higher values of MSE, the points are below identity function line and LDS-DV is more likely to win. For test cases with |?i (A)| ' 1, designed to evaluate the prediction performance of the methods for marginally stable systems, LDS-DV still can learn a stable model while the other algorithms might not learn a stable model. The proposed LDS-DV method does not explicitly impose stability, but the regularization favors A that is stable. The regularizer on latent state encourages smooth dynamics and controls overfitting: overfitting to noisy observations can lead to unstable estimate of the model (Buesing et al., 2012a), and a smooth latent trajectory is a favorable property in most real-world applications. Figure 2(c) shows the MSE of LDS-DV versus N4SID, for all the CAC time-series. This figure illustrates that for easier problems, LDS-DV and N4SID are more comparable. However, as the difficulty increase, and MSE increases, LDS-DV begins to consistently outperform N4SID. Figures 2(a) and 2(b) illustrate the accuracy and runtime respectively of the algorithms versus training length. We used the synthetic LDS model under condition S1 with n = 8, m = 6. Values are averaged over 20 runs with a test length of 50 samples. LDS-DV has better early performance, for smaller sample sizes. At larger sample sizes, they reach approximately the same error level. 6 Conclusion In this paper, we provided an algorithm for optimal estimation of the parameters for a time-invariant, discrete-time linear dynamical system. More precisely, we provided a reformulation of the model as a two-view objective, which allowed recent advances for optimal estimation for two-view models to be applied. The resulting algorithm is simple to use and flexibly allows different losses and regularizers 8 to be incorporated. Despite this simplicity, significant improvements were observed over a widely accepted method for subspace identification (N4SID), both in terms of accuracy for forecasting and runtime. The focus in this work was on forecasting, therefore on optimal estimation of the hidden states and transition matrices; however, in some settings, estimation of noise parameters for LDS models is also desired. An unresolved issue is joint optimal estimation of these noise parameters. Though we do explicitly estimate the noise parameters, we do so only from the residuals after obtaining the optimal hidden states and transition and observation matrices. Moreover, consistency of the learned parameters by the proposed procedure of this paper is still an open problem and will be an interesting future work. The proposed optimization approach for LDSs should be useful for applications where alternative noise assumptions are desired. A Laplace assumption on the observations, for example, provides a more robust `1 loss. A Poisson distribution has been advocated for count data, such as for neural activity, where the time series is a vector of small integers (Buesing et al., 2012b). The proposed formulation of estimation for LDSs easily enables extension to such distributions. An important next step is to investigate the applicability to a wider range of time series data. Acknowledgments This work was supported in part by the Alberta Machine Intelligence Institute and NSERC. During this work, M. White was with the Department of Computer Science, Indiana University. References Andersson, S. (2009). Subspace estimation and prediction methods for hidden Markov models. The Annals of Statistics. ?str?m, K. (1980). Maximum likelihood and prediction error methods. Automatica, 16(5):551?574. Bach, F., Mairal, J., and Ponce, J. (2008). Convex sparse matrix factorizations. arXiv:0812.1869v1. 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Efficient Simulation of Biological Neural Networks on Massively Parallel Supercomputers with Hypercube Archi tect ure Ernst Niebur Computation and Neural Systems California Institute of Technology Pasadena, CA 91125, USA Dean Brettle Booz, Allen and Hamilton, Inc. 8283 Greensboro Drive McLean, VA 22102-3838, USA Abstract We present a neural network simulation which we implemented on the massively parallel Connection Machine 2. In contrast to previous work, this simulator is based on biologically realistic neurons with nontrivial single-cell dynamics, high connectivity with a structure modelled in agreement with biological data, and preservation of the temporal dynamics of spike interactions. We simulate neural networks of 16,384 neurons coupled by about 1000 synapses per neuron, and estimate the performance for much larger systems. Communication between neurons is identified as the computationally most demanding task and we present a novel method to overcome this bottleneck. The simulator has already been used to study the primary visual system of the cat. 1 INTRODUCTION Neural networks have been implemented previously on massively parallel supercomputers (Fujimoto et al., 1992, Zhang et al., 1990). However, these are implementations of artificial, highly simplified neural networks, while our aim was explicitly to provide a simulator for biologically realistic neural networks. There is also at least one implementation of biologically realistic neuronal systems on a moderately 904 Efficient Simulation of Biological Neural Networks parallel but powerful machine (De Schutter and Bower, 1992) , but the complexity of the used neuron model makes simulation of larger numbers of neurons impractical. Our interest here is to provide an efficient simulator of large neural networks of cortex and related subcortical structures. The most important characteristics of the neuronal systems we want to simulate are the following: ? Cells are highly interconnected (several thousand connections per cell) but far from fully interconnected. ? Connections do not follow simple deterministic rules (like, e.g., nearest neighbor connections). ? Cells communicate with each other via delayed spikes which are binary events ("all-or-nothing"). ? Such communication events are short (1 ms) and infrequent (1 to 100 per second). ? The temporal fine structure of the spike trains may be an important information carrier (Kreiter and Singer, 1992, Richmond and Optican , 1990, Softky and Koch, 1993). 2 IMPLEMENTATION The biological network was modelled as a set of improved integrate-and-fire neurons which communicate with each other via delayed impulses (spikes). The single-cell model and details of the connectivity have been described in refs. (Wehmeier et al., 1989, Worgotter et al., 1991). Despite the rare occurrence of action potentials, their processing accounts for the major workload of the machine. The efficient implementation of inter-neuron communication is therefore the decisive factor which determines the efficacy of the simulator implementation. By "spike propagation" we denote the process by which a neuron communicates the occurrence of an action potential to all its postsynaptic partners. While the most efficient computation of the neuronal equations is obtained by mapping each neuron on one processor, this is very inefficient for spike propagation. This is due to the fact that spikes are rare events and that in the SIMD architecture used, each processor has to wait for the completion of the current tasks of all other processors. Therefore, only very few processors are active at any given time step. A more efficient data representation than provided by this "direct" algorithm is shown in Fig. 1. In this "transposed" scheme, a processor changes its role from simulating one of the neurons to simulating one synapse, which is, in general, not a synapse of the neuron simulated by the processor (see legend of Fig. 1). At any given time step, the addresses of the processors representing spiking neurons are broadcast along binary trees which are implemented efficiently (in time wmplexity log2M for M processors) in a hypercube architecture such as the CM-2. We obtain further computational efficiency by dividing the processor array into "partitions" of size M and by implementing partially parallel I/O scheduling (both not discussed here). 905 906 Niebur and Brettle 1 2 3 4 1 ,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 ? ? ? ? _ ' - _ _ ..IIl.I ++++ M,1 M,2 M,3 M,4 1,i 2, i 3, i ? _.........,....... .........-.. _.............._.. i,1 i,2 i,3 i,4 ? ........-- --_ ..... .........- r.-.......... ---~r' J...-............- M-1 5 ....."". -- ? ? ? ? ? ? ......... .._\,a.... ,. --_...... r-? i, i t--...... .. .......,.- 1----~ + M,i t - ............. M 1,M 2,M 3,M 1----- ._-i, ................. M --- .........- . ..--... --- + M,M Figure 1: Transposed storage method for connections. The storage space for each of the N processors is represented by a vertical column. A small part of this space is used for the time-dependent variables describing each of the N neurons (upper part of each column, "Cell data"). The main part of the storage is used for datasets consisting of the addresses, weights and delays of the synapses ("Synapse data"), represented by the indices i, j in the figure. For instance, "1, I" stands for the first synapse of neuron 1, "1,2" for the second synapse of this neuron and so on. Note that the storage space of processor i does not hold the synapses of neuron i. If neuron i generates a spike, all M processors are used for propagating the spike (black arrows) Efficient Simulation of Biological Neural Networks 3 PERFORMANCE ANALYSIS In order to accurately compare the performance of the described spike propagation algorithms, we implemented both the direct algorithm and the transposed algorithm and compared their performances with analytical estimates. 10 -::s 1 Cf) ........ E-t 0.1 0.01 0.001 0.0001 0.001 0.01 0.1 1 p Figure 2: Execution time for the direct algorithm (diamonds) and the transposed algorithm (crosses) as function of the spiking probability p for each cell. If all cells fire at each time step, there is no advantage for the transposed algorithm; in fact, it is at a disadvantage due to the overhead discussed in the text. Therefore, the two curves cross at a value just below p = 1. As expected, the largest difference between them is found for the smallest values of p. Figure 2 compares the time required for the direct algorithm to the time required for the transposed algorithm as a function of p, the average number of spikes per neuron per time step. Note that while the time required rises much more rapidly for the transposed algorithm than the direct algorithm, it takes significantly less time for p < 0.5. The peak speedup was a factor of 454 which occurred at p = 0.00012 (or 1.2 impulses per second at a timestep of O.lms, corresponding approximately to spontaneous spike rates). The absolutely highest possible speedup, obtained if there is exactly one spike in every partition at every time step, is equal to M (M == 1024 in this simulation). The average speedup is determined by the maximal number of spiking neurons per time step in any partition, since the processors in all partitions have to wait until the last partition has propagated all of its spikes. The average maximal number of spikes in a system of N partitions, each one consisting of M 907 908 Niebur and Brettle neurons IS M Nmar(p, M, N) N ={; k J; ( ~ ) TI(k)mft(k)N-m (1) where p is the spiking probability of one cell, II(k) is the probability that a given partition has k spikes and k-l IT(k) = L II(i) (2) i=O 1000 100 10 1 0.0001 0.001 0.01 0.1 1 p Figure 3: Speedup of the transposed algorithm over the direct algorithm as a function of p for different VP ratios; M = 1024. The ideal speedup (uppermost curve; diamonds), computed in eq. 3 essentially determines the observed speedup. (lower curves; "+" signs: VP-ratio=1, diamonds: VP-raio=2, crosses: VP-ratio=4.). The difference between the ideal and the effectively obtained speedup is due to communication and other overhead of the transposed algorithm. Note that the difference in speedup for different VP ratios (difference between lower curves) is relatively small, which shows that the penalty for using larger neuron numbers is not large. As expected, the speedup approaches unity for p ~ 1 in all cases. It can be shown that for independent neurons and for low spike rates, II( k) is the Poisson distribution and IT(k) the incomplete r function. The average maximal Efficient Simulation of Biological Neural Networks number of spikes for M = 1024 and different values of P (eq. 1) can be shown to be a mildly growing function of the number of partitions which shows that the performance will not be limited crucially by changing the number of partitions. Therefore, the algorithm scales well with increasing network size and the performance-limiting factor is the activity level in the network and not the size of the network. This is also evident in Fig. 3 which shows the effectively obtained speedup compared to the ideal speedup, which would be obtained if the transposed algorithm were limited only by eq. 1 and would not require any additional communication or other overhead. Using Nmax(P, M, N) from eq. 1 it is clear that this ideal speedup is given by M Nmax(P, M, N) (3) The difference between theory and experiment can be attributed to the time required for the spread operation and other additional overhead associated with the transposed algorithm. At P = 0.0010 (or 10 ips) the obtained speedup is a factor of 106. 4 VERY LARGE SYSTEMS Using the full local memory of the machine and the "Virtual Processor" capability of the CM-2, the maximal number of neurons that can be simulated without any change of algorithm is as high as 4,194,304 ("4M"). Figure 3 shows that the speedup is reduced only slightly as the number of neurons increases, when the additional neurons are simulated by virtual processors . The performance is essentially limited by the mean network activity, whose effect is expressed by eq. 3, and the additional overhead originating from the higher "VP ratio" is small. This corroborates our earlier conclusion that the algorithm scales well with the size of the simulated system. Although we did not study the scaling of execution time with the size of the simulated system for more than 16,384 real processors, we expect the total execution time to be basically independent of the number of neurons, as long as additional neurons are distributed on additional processors. Acknowlegdements We thank U. Wehmeier and F . Worgotter who provided us with the code for generating the connections, and G. Holt for his retina simulator. Discussions with C. Koch and F. W orgotter were very helpful. We would like to thank C. Koch for his continuing support and for providing a stimulating research atmosphere. We also acknowledge the Advanced Computing Laboratory of Los Alamos National Laboratory, Los Alamos, NM 87545. Some of the numerical work was performed on computing resources located at this facility. This work was supported by the National Science Foundation, the Office of Naval Research, and the Air Force Office of Scientific Research. 909 910 Niebur and Brettle References De Schutter E. and Bower J .M. (1992). Purkinje cell simulation on the Intel Touchstone Delta with GENESIS. In Mihaly T. and Messina P., editors, Proceedings of the Grand Challenge Computing Fair, pages 268-279. CCSF Publications, Caltech, Pasadena CA. Fujimoto Y., Fukuda N., and Akabane T. (1992). Massively parallel architectures for large scale neural network simulations. IEEE Transactions on Neural Networks, 3(6):876-888. Kreiter A.K. and Singer W. (1992). Oscillatory neuronal responses in the visualcortex of the awake macaque monkey. Europ. 1. Neurosci., 4(4):369-375. Richmond B.J. and Optican L.M. (1990). Temporal encoding of two-dimensional patterns by single units in primate primary visual cortex. II: Information transmission. J. Neurophysiol., 64:370-380. Softky W. and Koch C. (1993). The highly irregular firing of cortical-cells is inconsistent with temporal integration of random epsps. 1. Neurosci., 13(1):334-350. Wehmeier U., Dong D., Koch C., and van Essen D. (1989). Modeling the visual system. In Koch C. and Segev I., editors, Methods in Neuronal Modeling, pages 335-359. MIT Press, Cambridge, MA. Worgotter F., Niebur E., and Koch C. (1991). Isotropic connections generate functional asymmetrical behavior in visual cortical cells. J. Neurophysiol., 66(2):444-459. Zhang X., Mckenna M., Mesirov J., and Waltz D. (1990). An efficient implementation of the back-propagation algorithm on the Connection Machine CM-2. In Touretzky D.S., editor, Neural Information Processing Systems 2, pages 801-809. Morgan-Kaufmann, San Mateo, CA. 

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211 HIGH DENSITY ASSOCIATIVE MEMORIES! A"'ir Dembo Information Systems Laboratory, Stanford University Stanford, CA 94305 Ofer Zeitouni Laboratory for Information and Decision Systems MIT, Cambridge, MA 02139 ABSTRACT A class of high dens ity assoc iat ive memories is constructed, starting from a description of desired properties those should exhib it. These propert ies include high capac ity, controllable bas ins of attraction and fast speed of convergence. Fortunately enough, the resulting memory is implementable by an artificial Neural Net. I NfRODUCTION Most of the work on assoc iat ive memories has been structure oriented, i.e.. given a Neural architecture, efforts were directed towards the analysis of the resulting network. Issues like capacity, basins of attractions, etc. were the main objects to be analyzed cf., e.g. [1], [2], [3], [4] and references there, among others. In this paper, we take a different approach, we start by explicitly stating the desired properties of the network, in terms of capacity, etc. Those requirements are given in terms of axioms (c.f. below). Then, we bring a synthesis method which enables one to design an architecture which will yield the desired performance. Surprisingly enough, it turns out that one gets rather easily the following properties: (a) High capacity (unlimited in the continuous state-space case, bounded only by sphere-packing bounds in the discrete state case). (b) Guaranteed basins of attractions in terms of the natural metric of the state space. (c) High speed of convergence in the guaranteed basins of attraction. Moreover, it turns out that the architecture suggested below is the only one which satisfies all our axioms (-desired properties-)I Our approach is based on defining a potential and following a descent algorithm (e.g., a gradient algorithm). The main design task is to construct such a potential (and, to a lesser extent, an implementat ion of the descent algorithm via a Neural network). In doing so, it turns out that, for reasons described below, it is useful to regard each des ired memory locat ion as a -part icle- in the state space. It is natural to require now the following requirement from a IAn expanded version of this work has been submitted to Phys. Rev. A. This work was carried out at the Center for Neural Sc ience, Brown University. ? American Institute of Physics 1988 212 .eJlOry: (Pl) The potential should be linear w.r.t. adding partic les in the sense that the potential of two particles should be the sum of the potentials induced by the individual particles (i.e ?? we do not allow interparticles interaction). (P2) Part icle locat ions are the only poss ible sites of stable .emory locations. (P3) The system should be invariant to translations and rotations of the coordinates. We note that the last requirement is made only for the sake of simplicity. It is not essential and may be dropped without affecting the results. In the sequel. we construct a potential which satisfies the above requirements. We refer the reader to [5] for details of the proofs. etc. Acknowledgements. We would like to thank Prof. L.N. Cooper and C.M. Bachmann for many fruitful discussions. In particular. section 2 is part of a joint work with them ([6]). 2. HIGH DENSIlY STORAGE MODEL In what follows we present a particular case of a method for the construct ion of a high storage dens ity neural memory. We define a function with an arbitrary number of minima that lie at preassigned points and define an appropriate relaxat ion procedure. The general case in presented in [5]. Let i 1 ..... i m be a set of m arb itrary d ist inct memories in RN. The ?energy? function we will use is: m ~ =- i2 Qi Iii - ii I-L (1) i=l where we assume throughout that N ~ 3. L ~ (N - 2). and Qi > 0 and use 1??? 1 to denote the Euclidean distance. Note that for L = 1. NF3. ~ is the electrostat ic potent ial induced by negat ive fixed part ic les with charges -Qi. This ?energy? funct ion possesses global minima at i 1 ????? i m (where ~(ii) .. - ) and has no local minima except at these points. A rigorous proof is presented in [5] together with the complete characterization of functions having this property. As a relaxation procedure. we can choose any dynamical system for which ~ is strictly decreasing. uniformly in compacts. In this instance. the theory of dynamical systems guarantees that for almost any initial data. the trajectory of the system converges to one of the desired points i 1 ????? i m? However. to give concrete results and to further exploit the resemblance to electrostatic. consider the relaxation: 213 . .. -= II E- -= Il (2) i=1 where for N=3. L=1. equation (2) describes the motion of a positive t~st particle in the electrost!tic f~eld ~ generated by the negative f1xed charges -Q1 ???? L -~ at xl ????? x m? Since the field E;i is just minus the gradient of e. it is clear that along trajectories of (2). de/dt ~ O. with equality only at the fbed points of (2). which are exactly the stat ionary po ints of e. Therefore. using (2) as the relaxation procedure. we can conclude that entering at any ~(O). the system converges to a stationary point of e. The space of inputs is partitioned into m domains of attraction. each one corresponding to a different memory. and the boundaries (a set of measure zero). on which p(O) will converge to a saddle point of e. We can now explain why e~ has no spurious local minima. at least for L=1. N=3. using elementary physical arguments. Suppose e has a spurious local minima at y ~ xl ????? x m? then in a s!!all neighborhood of y which does not include any of the xi. the field ~ points towards y. Thus. on any closed surface in that neighborhood. the integral of the normal inward component of ~ is positive. However. this integral is just the total charge included inside the surface. which is zero. Thus we arrive at a contradiction. so y can not be a local minimum. We now have a relaxation procedure. such that almost any ~(O) is attracted by one of the xi. but we have not yet spec ified the shapes of the basins of attraction. By varying the charges Qi. we can enlarge one basin of attraction at the expense of the others (and vice versa). Even when all of the Qi are eqmal. the position of the xi might cause ~(O) not to converge to the closest memory. as emphasized in the example in fig. 1. However. let r = min1~i~j~mlxi - i j 1 be the minimal distance between any two memoriesJ then if I~(O) - ii I~ it can be shown that ~(O) ,[? .,lIk) L +! N+i will converge to xi. (provided that k = - - 11). Thus. i f thamemories are densely packed in a hypersphere. by choosing k large enough (i.e. enlarging the parameter L). convergence to the closest memory for any -interesting- input. that is an input i;:(O) with a distinct closest memory. is guaranteed. The detailed proof of the above property is given in [5]. It is based on bound ing the number of x j ? j~i. in a hypersphere of radius R(Rlr) around xi. by [2R/r + 1]N. tlien bounding the magnitude of the field induced any Xj. j~i. on the boundar, of such a hypersphere by (R-li;:(O)-xiP- +1). 'I. and finally integrat ing to show that for I~(O)-ii 15. (i~~I/~ ,with e<1. the convergence of ~(O) to xi is within finite time T. which behaves like e L+2 for L 1 and e < 1 and fixed. Intuitively the reason for ? 214 this behaviour is the short-range nature of the fields used in Because of this. we also expect extremely low equat ion (2) ? convergence rate for inputs ~(O) far away from all of the xi. The radial nature of these fields suggests a way to overcome this difficulty. that is to increase the convergence rate from points very far away. without disturbing all of the aforementioned desirable properties of the model. Assume that we '. know in advance that all of the xi lie inside some large hypersphere S around the origin. Then. at any point ~ outside S. the field ~ has a positive projection radially into S. By adding a longrange force to B-. effective only outside of S. we can hasten the mgvement towards S. from points far away, without creating additional minima inside of S. As an example the force (-~ for ji , S, 0 for ji 8 S) will pull any test input ji(O) to the boundary of S within the small finite time T ~ I 1/1SI. and from then on the system wil} behave " inside S according to the original field Up to this point. our derivations have been for a continuous system. but from it we can deduce Figure 1 a discrete system. We shall do this mainly for a R ? I and 0 ? 1 clearer comparison between our high density memory model and the discrete version of Hopfield's model. Before continuing in that direction. note that our continuous system has unl imited storage capacity unlike Hopfield's continuous system. which like his discrete model, has limited capac ity. For the discrete system, assume that the Xi are composed of elements ?1 and replace the Euclid\an dJstance in (1) with the normal ized Hamming 4 istance lii1 - ~21 = 1; I '=1111~ - 11~ I. This places the vec tors :i i on the un it hypersphere. J J J The relaxation process for the discrete system will be of the type defined in Hopfield's model in [11 Choose at random a component to be updated (that is, a neighbor ~' of ii such that Iii' - iii = 2/N). calculate the "energy" difference. r.e = ~(ii~-~(ii). and only if r.e < O. change this component, that is: ,I Bu. 11?1 ~f.l.1 sign(~(~~ - ~(ji?, (3) where e(ii) is the potent ial energN in (1). Since there is a finite number of possible ~ vectors (2), convergence in finite time is guaranteed. This relaxation procedure is rigid since the movement is limited to points with components +1. Therefore. although the local minima of ~(ii) defined in (2) are only at the desired points Xi' the relaxation may get stuck at some ii which is not a stationary point of ~(ii). However, the short range behaviour of the potential e(~), unlike the long-range behavior of the quadratic potential used by Hopfield, gives 215 rise to results similar to those we have quoted for the continuous ll10del (equation (1?. Specifically. let the stored me~ories i 1 ????? i m be separated from one another by having at least pN different components (0 < p i 1/2 and p fixed), and let ~(O) agree up to at least one ii with at most epN errors between them (0 i e < 1/2. with e fixed), then jHO) converges monotonically to i i by the relaxat ion procedure given in equat ion (3). This result holds independently of m. provided that N is large enough (typically. Np In(1~e) L 1) and L is chosen so that fi In(!~e) The proof is constructed by bounding the cummulative effect of terms - Se I~ ii rL. j;&i. to t~e energy difference and showing that it is dominafed by I~ ii 1 L. For details. we refer the reader again to [5]. - Note the importance of this property: unlike the Hopfield model which is limited to miN. the suggested system is optimal in the sense of Information Theory. since for every set of memories i 1 ????? i m separated from each other by a Hamming distance pN. up to 1/2 pN errors in the input can be corrected. provided that N is large and L properly chosen. As for the complexity of the system. we note that the nonlinear operat ion a -L. for a}O and L integer (which is at (the heart of our system computationally)' is equivalent to e-Lln a) and can be implemented. therefore. by a simple electrical circuit composed of diodes. which have exponential input-output characteristics. and resistors. which can carry out the necessary multiplications (cf. the implementation of section 3). Further. since both liil and I~I are held fixed in the discrete system. where all states are on the unit hypersphere. I~ ii 12 is equivalent to the inner product of ~ and ii' up to a constant. To conclude. the suggested model involves about m'N multiplications. followed by m nonlinear operations. and then m'N additions. The original model of Hopfield involves multiplications and additions. and then N nonlinear operations. but is limited to miN. Therefore. whenever the Hopfield model is applicable the complexity of both ll10dels is comparable. - Nf 3. IMPLEMENI'ATION We propose below one possible network which implements the discrete time and space version of the model described above. An implementation for the ocntinuous time case. which is even simpler. is also hinted. We point out that the implementation described below is by no means unique. (and maybe even not the simplest one). Moreover. the -neurons? used are artificial neurons which perform various tasks. as follows: There are (N+1) neurons which are delay elements. and pOintwise non-linear functions (which may be interpreted as delayless. intermediate neurons). There are ~N synaptic connections between those two layers of neurons. In addition. as in the Hopfield \'l'\. 216 model, we have at each iteration to specify (either deterministically or stochastically) which coordinate are we updating. To do that, we use an N dimensional ?control register? whose content is always a unit vector of {O, l}N (and the location of the '1' will denote the next coordiante to be changed). This vector may be varied from instant n to n + 1 either by shift (?sequential coordinate update?) or at random. Let Ai' UUN be the i-th output of the ?co1!,trol? register, xi' l~UN and V be the (N+1) I!eurons inputs and xi = xi (l-2A i ) the corresponding outputs (where xi' xi8{+1,-1), Ai 8{0,1}, but V is a real number), _j' l~j~ be the input of the j-th inte;medi~te neuron (-1~_ ~1), ~j = -(1-_ )-L be its output, and 'ji = ui j IN be the synaptiC weight of thJ ij - th synapsis, where u~j) refers here to the i-th element of the j-th memory. The system's equations are: <i ~ N (4a) 1 ~ j <m (4b) 1 ~ "" -(1 __ )-L j (4c) j (4d) S 1 - - V? = i"(l-sign(V (4e) 1 V ~V + SV < i ~ N (4f) (4g) The system is initialized by xi = xi (0) (the probe vector), and V = + CD. A block diagram of this sytem appears in Fig. 2. Note that we made use of N + m + 1 neurons and O(Nm) connections. As for the continuous time case (with memories on the unit sphere) we will get the equations: 217 m - 2 1 ~ i ~ N (Sa) x21.? 1 ~ j ~ m (Sb) ? 1 Xi + 2m VX i = LN "jil1 j. j=l N -j =N 2 N 6 " J. i X1.? = 2 i .. l i=l _(L + 1) l1j = (1 + 6 - 2_ j) -= 2 'I" <j ~ m (Sc) ~ V (Sd) l1j j=l with similar interpretation (here there is no all components are updated continuously). 'control' register as s Legend @] i Deloy Unit (Neuron) , Synoptic Switch ( 0= { .. '2 Figure 2 c=O) C =I _~o fc Synoptic Switch (0 =Zi, t c =0) c=I Computation UnIt (0= 1/2(1-sign(i2-i, Il) Neural Network Implementotion 218 REFERENCES 1. 2. 3. 4. 5". 6. Bopfield. -Neural Networks and Physical Systems with Emergent Collective Computational Abilities-. Proc. Nat. Acad. Sci. U.S.A ?? Vol. 79 (1982). pp. 2554-2558. R.I. McEliece. et al ?? -The Capacity of the Hopfield Associative Memory-. IEEE" Trans. on Inf. Theory. Vol. IT-33 (1987). pp. 461482. A. Dembo. -On the Capac ity of the Hopfield Memory-. submitted. IEEE Trans. on Inf. Theory. Kohonen. T ?? Self Organization and Associative Memory. Springer. Berlin. 1984. Dembo. A. and Ze itouni. 0 ?? General Potent ial Surfaces and Neural Networks. submitted. Phys. Rev. A. Bachmann. C.M.. Cooper. L.N., Dembo. A. and Zeitouni. 0.. A relazation Model for Memory with high storage density. to appear. Proc. Natl. Ac. Science. 1.1. 

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Optimal signalling in Attractor Neural Networks Isaac Meilijson Eytan Ruppin . . School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University, 69978 Tel-Aviv, Israel. Abstract In [Meilijson and Ruppin, 1993] we presented a methodological framework describing the two-iteration performance of Hopfieldlike attractor neural networks with history-dependent, Bayesian dynamics. We now extend this analysis in a number of directions: input patterns applied to small subsets of neurons, general connectivity architectures and more efficient use of history. We show that the optimal signal (activation) function has a slanted sigmQidal shape, and provide an intuitive account of activation functions with a non-monotone shape. This function endows the model with some properties characteristic of cortical neurons' firing. 1 Introduction It is well known that a given cortical neuron can respond with a different firing pat- tern for the same synaptic input, depending on its firing history and on the effects of modulator transmitters (see [Connors and Gutnick, 1990] for a review). The time span of different channel conductances is very broad, and the influence of some ionic currents varies with the history of the membrane potential [Lytton, 1991]. Motivated by the history-dependent nature of neuronal firing, we continue .our previous investigation [Meilijson and Ruppin, 1993] (henceforth, M & R) describing the performance of Hopfield-like attract or neural networks (ANN) [Hopfield, 1982] with history-dependent dynamics. ?Currently in the Dept. of Computer science, University of Maryland 485 486 Meilijson and Ruppin Building upon the findings presented in M & R, we now study a more general framework: ? We differentiate between 'input' neurons receiving the initial input signal with high fidelity and 'background' neurons that receive it with low fidelity. ? Dynamics now depend on the neuron's history of firing, in addition to its history of input fields. ? The dependence of ANN performance on the network architecture can be explicitly expressed. In particular, this enables the investigation of corticallike architectures, where neurons are randomly connected to other neurons, with higher probability of connections formed between spatially proximal neurons [Braitenberg and Schuz, 1991]. Our goal is twofold: first, to search for the computationally most efficient historydependent neuronal signal (firing) function, and study its performance with relation to memoryless dynamics. As we shall show, optimal history-dependent dynamics are indeed much more efficient than memory less ones. Second, to examine the optimal signal function from a biological perspective. As will shall see, it shares some basic properties with the firing of cortical neurons. 2 The model e, ... Our framework is an ANN storing m + 1 memory patterns ~1, ,~m+l, each an N-dimensional vector. The network is composed of N neurons, each of which is randomly connected to K other neurons. The (m + l)N memory entries are independent with equally likely ?1 values. The initial pattern X, signalled by L(~ N) initially active neurons, is a vector of ?l's, randomly generated from one of the memory patterns (say ~ = m +!) such that P(Xi = ~i) = for each of the L initially active neurons and P(Xi ~i) for each initially quiescent (non-active) neuron. Although f,6 E [0,1) are arbitrary, it is useful to think of f as being 0.5 (corresponding to an initial similarity of 75%) and of 6 as being zero - a quiescent neuron has no prior preference for any given sign. Let al = mlnl denote the initial memory load, where nl = LK I N is the average number of signals received by each neuron. e = Ii! = li6 The notion of 'iteration' is viewed as an abstraction of the overall dynamics for some length of time, during which some continuous input/output signal function (such as the conventional sigmoidal function) governs the firing rate of the neuron. We follow a Bayesian approach under which the neuron's signalling and activation decisions are based on the a-posteriori probabilities assigned to its two possible true memory states, ?1. Initially, neuron i is assigned a prior probability or 1?2 6 which is conveniently expressed as Ai(O) Ai(O) = P(~ = = 1+e-1 2g j !2 log l?1 l-t' if i is active if i is silent (0)' 11 X i, 1/1? = l~! where, letting get) = Optimal Signalling in Attractor Neural Networks The input field observed by neuron i as a result of the initial activity is N ~ 'LJ " w,IJ . .J.. 1?(1) X ]' IJ ] -JloCI) 1 (1) nl j=1 where 1/ 1) = 0, 1 indicates whether neuron j has fired in the first iteration, lij = 0,1 indicates whether a connection exists from neuron j to neuron i and Wij denotes its magnitude, given by the Hopfield prescription m+l Wij = L: e JJ ie JJ j (2) . JJ=1 As a result of observing the input field fi(I), which is approximately normally distributed (given ei, Xi and 1/ 1 ?), neuron i changes its opinion about {ei = I} from Ai(O) to A?(1) -- P 1 (e? - llX? 1 - I , J.(1) 1 t .(I?) -_ 1 + e-1 (1) 2gi ,I expressed in terms of the ( additive) generalized field 9i( 1) We now neurons neuron i neurons (3) , = gi(O) + :1 f/l). get to the second iteration, in which, as in the first iteration, some of the become active and signal to the network. We model the signal function emits as h(9i(1), Xi, li(l?). The field observed by neuron i (with n2 updating per neuron) is N 2- "'W, LJ '].. I .. h(g.(I) X' 1.(2) -- JI I] n2 ]= . 1 ] , ], / J?(1?) (4) , on the basis of which neuron i computes its posterior belief Ai(2) p(ei = 2 1 llXi, li(I), f/ ), f/ ?) and expresses its final choice of sign as Xi(2) sign(A/ 2 ) 0.5). The two-iteration performance of the network is measured by the final similarity 1 1 ",N X (2)c Sf 1 + ff p(X/2) ei) + N L-j=1 j '-oj (5) = = 3 2 = = = 2 Analytical results The goals of our analysis have been: A. To present an expression for the performance under arbitrary architecture and activity parameters, for general signal functions ho and hi. B. Use this expression to find the best choice of signal functions which maximize performance. We show the following: The neuron's final decision is given by Xi(2) = Sign [(Ao + Boli(1?)Xi + Al fi(l) + A 2 fi(2)] for some constants AD, Bo, At and A 2 ? (6) 487 Meilijson and Ruppin 488 The performance achieved is (7) where, for some A3 >0 a '" =m n'" nl m +mA3 ' (8) (Q"'~)(x, t) = 1; t ~ (x + g~?) + 1 2 t ~ (x _ g~?) and ~ (9) is the standard normal cumulative distribution function. The optimal analog signal function, illustrated in figure 1, is ho = h(g/1),+1,0) = R(gj(l),O) hI = h(gj(1), +1,1) where, for some A4 ? (10) = R(gi(l), ?) - 1 > and A5 > 0, R(s, t) = A4 tanh(s) - A5(S - g(t?. (b) Silent neurons - - - - Active neurons 2.0 v I, Signal 0.0 -V 1 ~.O ---~~--~~--n-~------g 1 ? 4 5' Input field . . , ----- -1 \ \ \ ....0 L--~---'-~_ _- ' - - - ' - - - - ' -_ _~~~-----' -5.0 -3.0 -1.0 1.0 ItllUt field 3.0 5.0 Figure 1: (a) A typical plot of the slanted sigmoid, Network parameters are N = 5000, K 3000, nl 200 and m = 50. (b) A sketch of its discretized version. = = The nonmonotone form of these functions, illustrated in figure 1, is clear. Neurons that have already signalled +1 in the first iteration have a lesser tendency to send positive signals than quiescent neurons. The signalling of quiescent neurons which receive no prior information (6 = 0) has a symmetric form. The optimal signal is shown to be essentially equal to the sigmoid modified by a correction term depending only on the current input field. In the limit of low memory load (f./ fol ~ 00), the best signal is simply a sigmoidal function of the generalized input field. Optimal Signalling in Attractor Neural Networks To obtain a discretized version of the slanted sigmoid, we let the signal be sign(h(y)) as long as Ih(y)1 is big enough - where h is the slanted sigmoid. The resulting signal, as a function of the generalized field, is (see figure la and lb) < {3I (j) or {34 (j) < y < {3s (j) y > {36 (j) or {32 (j) < y < (33 (j) y (11) otherwise < (3l(D) < (32(O) ~ (33(O) < (34(O) < (3s(O) < (36(D) < 00 and -00 < (31(l) < ~ (3/I) < (34(l) ~ (3S(l) < (36(1) < 00 define, respectively, the firing pattern of where (32(l) -00 the neurons that were silent or active in the first iteration. To find the best such discretized version of the optimal signal, we search numerically for the activity level v which maximizes performance. Every activity level v, used as a threshold on Ih(y) I, defines the (at most) twelve parameters (3/j) (which are identified numerically via the Newton-Raphson method) as illustrated in figure lb. 4 Numerical Results 1.00 ,;;-; ;,-:..: ~ ...- _ ._....... _....... . . .. _-----_ .. .. .. -..- - -_ .... ....... ..... .. ..- ...... 1.000 //'--- :'~' .:: I I 0.95 ,/ 0.980 / I Posterior-probability-bned signalling -- -- . DI_tlzad slgnalDng .... . Analog optmal signaling 0.90 I Discrete signalling .. ....... Analog signalUng t 0.980 I ! 0.85 '-~---'-_~--"-~_-'--~~_~...J 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 0.940 '-~---'-_~....l.-~--:-'---~---'-_~...J 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 K K Figure 2: Two-iteration performance as a function of connectivity K. (a) Network parameters are N 5000, nl 200, and m 50. All neurons receive their input state with similar initial overlap f 6 0.5. (b) Network parameters are N 5000, m 50, ni 200, f 0.5 and 6 O. = = = = = = = = = = Using the formulation presented in the previous section, we investigated numerically the two-iteration performance achieved in several network architectures with optimal analog signalling and its discretization. Already in small scale networks of a few hundred neurons our theoretical calculations correspond fairly accurately with 489 490 Meilijson and Ruppin simulation results. First we repeat the example of a cortical-like network investigated in M & R, but now with optimal analog and discretized signalling. The nearly identical marked superiority of optimal analog and discretized dynamics over the previous, posterior-probability-based signalling is evident, as shown in figure 2 (a). While low activity is enforced in the first iteration, the number of neurons allowed to become active in the second iteration is not restricted, and best performance is typically achieved when about 70% of the neurons in the network are active (both with optimal signalling and with the previous, heuristic signalling). Figure 2 (b) displays the performance achieved in the same network, when the input signal is applied only to the small fraction (4%) of neurons which are active in the first iteration (expressing possible limited resources of input information). We see that (for 1< > 1000) near perfect final similarity is achieved even when the 96% initially quiescent neurons get no initial clue as to their true memory state, if no restrictions are placed on the second iteration activity level. fit Next we have fixed the value of w = = 1, and contrasted the case (nl = 200, f = 0.5) of figure 2 (b) with (nl = 50, f = 1). The overall initial similarity under (nl = 50, f = 1) is only half its value under (nl = 200, f = 0.5). In spite of this, we have found that it achieves a slightly higher final similarity. This supports the idea that the input pattern should not be applied as the conventional uniformly distorted version of the correct memory, but rather as a less distorted pattern applied only to a small subset of the neurons. (b) (a) 1.00 /7;;;;':::"?~""'~ . , ..... -_ .. --_ . --- --------- 0.970 ,,- /' /.-:,' I ",-'-' f , ,, ,,, : :! DIscrete signaling - - - - Analog slgraliing ,: ~' ,.~/ .; i' ! 0.98 ,.' . /./ ( I I { I I / i "/ II 0.98 ~ df~1 !~ " Upper bound pertlnNnce - . - 3--0 GausaIan connec1IvIty - - - - 2-D GaussIan connectIvl1y - - - - Mulll-layered nelWodc _ ..'-' Lower bound perlonrence 1'1 ;'~ f,'I, 0.950 0.94 I,Ii '" i ~II 'I ,I I I. 0.11400 .0 2000.0 4000.0 eooo.O N 8000.0 10000.0 0.920 .0 1000.0 2000.0 K Figure 3: (a) Two-iteration performance in a full-activity network as a function of network size N. Network parameters are nl = I{ = 200, m = 40 and f = 0.5. (b) Two-iteration performance achieved with various network architectures, as a function of the network connectivity K. Network parameters are N = 5000, nl = 200, m = 50, f = 0.5 and 6 = O. Figure 3 (a) illustrates the performance when connectivity and the number of sig- Optimal Signalling in Attractor Neural Networks nals received by each neuron are held fixed, but the network size is increased. A region of decreased performance is evident at mid-connectivity (K ~ N /2) values, due to the increased residual variance. Hence, for neurons capable of forming K connections on the average, the network should either be fully connected or have a size N much larger than K. Since (unavoidable eventually) synaptic deletion would sharply worsen the performance of fully connected networks, cortical ANNs should indeed be sparsely connected. The final similarity achieved in the fully connected network (with N K 200) should be noted. In this case, the memory load (0.2) is significantly above the critical capacity of the Hopfield network, but optimal history-dependent dynamics still manage to achieve a rather high two-iterations similarity (0.975) from initial similarity 0.75. This is in agreement with the findings of [Morita, 1993, Yoshizawa et a/., 1993], who show that nonmonotone dynamics increase capacity. = = Figure 3 (b) illustrates the performance achieved with various network architectures, all sharing the same network parameters N, K, m and input similarity parameters nl, f, 0, but differing in the spatial organization of the neurons' synapses. As evident, even in low-activity sparse-connectivity conditions, the decrease in performance with Gaussian connectivity (in relation, say, to the upper bound) does not seem considerable. Hence, history-dependent ANNs can work well in a cortical-like architecture. 5 Summary The main results of this work are as follows: ? The Bayesian framework gives rise to the slanted-sigmoid as the optimal signal function, displaying the non monotone shape proposed by [Morita, 1993]. It also offers an intuitive explanation of its form. ? Martingale arguments show that similarity under Bayesian dynamics persistently increases. This makes our two-iteration results a lower bound for the final similarity achievable in ANNs. ? The possibly asymmetric form of the function, where neurons that have been silent in the previous iteration have an increased tendency to fire in the next iteration versus previously active neurons, is reminiscent of the bi-threshold phenomenon observed in biological neurons [Tam, 1992]. ? In the limit of low memory load the best signal is simply a sigmoidal function of the generalized input field. ? In an efficient associative network, input patterns should be applied with high fidelity on a small subset of neurons, rather than spreading a given level of initial similarity as a low fidelity stimulus applied to a large subset of neurons. ? If neurons have some restriction on the number of connections they may form, such that each neuron forms some K connections on the average, then efficient ANNs, converging to high final similarity within few iterations, should be sparsely connected. 491 492 Meilijson and Ruppin ? With a properly tuned signal function, cortical-like Gaussian-connectivity ANNs perform nearly as well as randomly-connected ones . ? Investigating the 0,1 (silent, firing) formulation, there seems to be an interval such that only neurons whose field values are greater than some low threshold and smaller than some high threshold should fire. This seemingly bizarre behavior may correspond well to the behavior of biological neurons; neurons with very high field values have most probably fired constantly in the previous 'iteration', and due to the effect of neural adaptation are now silenced. References [Braitenberg and Schuz, 1991] V. Braitenberg and A. Schuz. Anatomy of the Cortex: Statistics and Geometry. Springer-Verlag, 1991. [Connors and Gutnick, 1990] B.W. Connors and M.J. Gutnick. Intrinsic firing patterns of diverse neocortical neurons. TINS, 13(3):99-104, 1990. [Hopfield, 1982] J.J. Hopfield. Neural networks and physical systems with emergent collective abilities. Proc. Nat. Acad. Sci. USA, 79:2554, 1982. [Lytton, 1991] W. Lytton. Simulations of cortical pyramidal neurons synchronized by inhibitory interneurons. J. Neurophysiol., 66(3):1059-1079, 1991. [Meilijson and Ruppin, 1993] I. Meilijson and E. Ruppin. History-dependent attractor neural networks. Network, 4:1-28, 1993. [Morita, 1993] M. Morita. Associative memory with nonmonotone dynamics. Neural Networks, 6:115-126, 1993. [Tam, 1992] David C. Tam. Signal processing in multi-threshold neurons. In T. McKenna, J. Davis, and S.F. Zornetzer, editors, Single neuron computation, pages 481-501. Academic Press, 1992. [Yoshizawa et al., 1993] S. Yoshizawa, M. Morita, and S.-I. Amari. Capacity of associative memory using a nonmonotonic neuron model. Neural Networks, 6:167176, 1993. 

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High Performance Neural Net Simulation on a Multiprocessor System with "Intelligent" Communication Urs A. Miiller, Michael Kocheisen, and Anton Gunzinger Electronics Laboratory, Swiss Federal Institute of Technology CH-B092 Zurich, Switzerland Abstract The performance requirements in experimental research on artificial neural nets often exceed the capability of workstations and PCs by a great amount. But speed is not the only requirement. Flexibility and implementation time for new algorithms are usually of equal importance. This paper describes the simulation of neural nets on the MUSIC parallel supercomputer, a system that shows a good balance between the three issues and therefore made many research projects possible that were unthinkable before. (MUSIC stands for Multiprocessor System with Intelligent Communication) 1 Overview of the MUSIC System The goal of the MUSIC project was to build a fast parallel system and to use it in real-world applications like neural net simulations, image processing or simulations in chemistry and physics [1, 2]. The system should be flexible, simple to program and the realization time should be short enough to not have an obsolete system by the time it is finished. Therefore, the fastest available standard components were used. The key idea of the architecture is to support the collection and redistribution of complete data blocks by a simple, efficient and autonomously working communication network realized in hardware. Instead of considering where to send data and where from to receive data, each processing element determines which part of a (virtual) data block it has produced and which other part of the same data block it wants to receive for the continuation of the algorithm. 888 Parallel Neural Net Simulation Host computer (Sun, PC, Macintosh) - user terminal - mass storage SCSI r??-..????_??. . _??? __??__???_?????_?????_????_??_-???_?1 ! MUSIC board I ! Bo~ I iI manager I: I ..I . PE : _ ??? __ ?? _ ..._ ??_ .:.! Board II' manager ?? _ - ._ ????? _ ??? _ ?? _ ??? _ MUSIC board ?? 110 board I - ....- - - - - - . 1 Transputer links vo PE 32+8 bit, 5 MHz Outside world Figure 1: Overview of the MUSIC hardware Figure 1 shows an overview of the MUSIC architecture. For the realization of the communication paradigm a ring architecture has been chosen. Each processing element has a communication interface realized with a XILINX 3090 programmable gate array. During communication the data is shifted through a 40-bit wide bus (32 bit data and 8 bit token) operated at a 5-MHz clock rate. On each clock cycle, the processing elements shift a data value to their right neighbors and receive a new value from their left neighbors. By counting the clock cycles each communication interface knows when to copy data from the stream passing by into the local memory of its processing element and, likewise, when to insert data from the local memory into the ring. The tokens are used to label invalid data and to determine when a data value has circulated through the complete ring. Three processing elements are placed on a 9 x 8.5-inch board, each of them consisting of a Motorola 96002 floating-point processor, 2 Mbyte video (dynamic) RAM, 1 Mbyte static RAM and the above mentioned communication controller. The video RAM has a parallel port which is connected to the processor and a serial port which is connected to the communication interface. Therefore, data processing is almost not affected by the communication network's activity and communication and processing can overlap in time. This allows to use the available communication bandwidth more efficiently. The processors run at 40 MHz with a peak performance of 60 MFlops. Each board further contains an Inmos T425 transputer as a board 889 890 Milller, Kocheisen, and Gunzinger Number of processing elments: Peak performance: Floating-point format: Memory: Programming language: Cabinet: Cooling: Total power consumption: Host computer: 60 3.6 Gflops 44 bit IEEE single extended precision 180 Mbyte C, Assembler 19-inch rack forced air cooling less than 800 Watt Sun workstation, PC or Macintosh Table 1: MUSIC system technical data manager, responsible for performance measurements and data communication with the host (a Sun workstation, PC or Macintosh). In order to provide the fast data throughput required by many applications, special I/O modules (for instance for real-time video processing applications) can be added which have direct access to the fast ring bus. An SCSI interface module for four parallel SCSI-2 disks, which is currently being developed, will allow the storage of huge amount of training data for neural nets. Up to 20 boards (60 processing elements) fit into a standard 19-inch rack resulting in a 3.6-Gflops system. MUSIC's technical data is summarized in Table 1. For programming the communication network just three library functions are necessary: Init_commO to specify the data block dimensions and data partitioning, Data.IeadyO to label a certain amount of data as ready for communication and Wait...ciataO to wait for the arrival of the expected data (synchronization). Other functions allow the exchange and automatic distribution of data blocks between the host computer and MUSIC and the calling of individual user functions. The activity of the transputers is embedded in these functions and remains invisible for the user. Each processing element has its own local program memory which makes MUSIC a MIMD machine (multiple instructions multiple data). However, there is usually only one program running on all processing elements (SPMD single program multiple data) which makes programming as simple or even simpler as programming a SIMD computer (single instruction multiple data). The difference to SIMD machines is that each processor can take different program pathes on conditional branches without the performance degradation that occurs on SIMD computers in such a case. This is especially important for the simulation of neural nets with nonregular local structures. = 2 Parallelization of Neural Net Algorithms The first implemented learning algorithm on MUSIC was the well-known backpropagation applied to fully connected multilayer perceptrons [3]. The motivation was to gain experience in programming the system and to demonstrate its performance on a real-world application. All processing elements work on the same layer a time, each of them producing an individual part of the output vector (or error vector in the backward path) [1]. The weights are distributed to the processing elements accordingly. Since a processing element needs different weight subsets in Parallel Neural Net Simulation 200.-----.-----~----._----._----~----_n .. 900-600-30 ,.../-:------:- v' 300-200-10 ~~...........;. -..~.-.~....:....:;.-.;...: ....+....~.... ~ ....~. ................ + 50 .....? ???? ~ O~ o ...... II 203-80-26 + I!JI!JI!IDI!JI!JIII!JIiIIiII!JIDIiI ____L -_ _ _ _L -_ _ _ _ 10 20 ~ 30 ____ ~ ____ 40 ~ 50 _ _ _ _-U 60 Number of processing elements Figure 2: Estimated (lines) and measured (points) back-propagation performance for different neural net sizes. the forward and in the backward path, two subsets are stored and updated on each processing element. Each weight is therefore stored and updated twice on different locations on the MUSIC system [1]. This is done to avoid the communication of the weights during learning what would cause a saturation of the communication network. The estimated and experimentally measured speedup for different sizes of neural nets is illustrated in Figure 2. Another frequently reported parallelization scheme is to replicate the complete network on all processing elments and to let each of them work on an individual subset of the training patterns [4, 5, 6]. The implementation is simpler and the communication is reduced. However, it does not allow continuous weight update, which is known to converge significantly faster than batch learning in many cases. A comparison of MUSIC with other back-propagation implementations reported in the literature is shown in Table 2. Another category of neural nets that have been implemented on MUSIC are cellular neural nets (CNNs) [10]. A CNN is a two-dimensional array of nonlinear dynamic cells, where each cell is only connected to a local neighborhood [11, 12]. In the MUSIC implementation every processing elment computes a different part of the array. Between iteration steps only the overlapping parts of the neighborhoods need to be communicated. Thus, the computation to communication ratio is very high resulting in an almost linear speedup up to the maximum system size. CNNs are used in image processing and for the modeling of biological structures. 3 A Neural Net Simulation Environment After programming all necessary functions for a certain algorithm (e.g. forward propagate, backward propagate, weight update, etc.) they need to be combined 891 892 Muller, Kocheisen, and Gunzinger System No. of PEs 1 1 1 64 10 64K 1 40 1 60 256 356 PC (80486, 50 MHz)_* Sun (Sparcstation 10)* Alpha Station (150 MHz)* Hypercluster [7] Warp [4] CM-2** [6] Cray Y-MP C90*** RAP [8] NEC SX-3*** MUSIC* Sandy /8** [9] GFll [5] *Own measurements **Estimated numbers ***No published reference available. Performance forward Learmng [MCPS] 1.1 3.0 8.3 27.0 - 180.0 220.3 574.0 - 504.0 - Peak (McuPS] (%) 0.47 1.1 3.2 9.9 17.0 40.0 65.6 106.0 130.0 247.0 583.0 901.0 38.0 43_0 8.6 - - 50.0 9.6 28.0 31.0 54.0 Cont. weight update Yes Yes Yes No No Yes Yes Yes Yes Yes No Table 2: Comparison of floating-point back-propagation implementations. "PEs" means processing elements, "MCPS" stands for millions of connections per second in the forward path and "MCUPS" is the number of connection updates per second in the learning mode, including both forward and backward path. Note that not all implementations allow continuous weight update. in order to construct and train a specific neural net or to carry out a series of experiments. This can be done using the same programming language that was used to program the neural functions (in case of MUSIC this would be C). In this case the programmer has maximum flexibility but he also needs a good knowledge of the system and programming language and after each change in the experimental setup a recompilation of the program is necessary. Because a set of neural functions is usually used by many different researchers who, in many cases, don't want to be involved in a low-level (parallel) programming of the system, it is desirable to have a simpler front-end for the simulator. Such a front-end can be a shell program which allows to specify various parameters of the algorithm (e.g. number of layers, number of neurons per layer, etc.). The usage of such a shell can be very easy and changes in the experimental setup don't require recompilation of the code. However, the flexibility for experimental research is usually too much limited with a simple shell program. We have chosen a way in between: a command language to combine the neural functions which is interactive and much simpler to learn and to use than an ordinary programming language like C or Fortran. The command language should have the following properties: - interactive easy to learn and to use flexible loops and conditional branches variables transparent interface to neural functions. Parallel Neural Net Simulation Instead of defining a new special purpose command language we decided to consider an existing one. The choice was Basic which seems to meet the above requirements best. It is easy to learn and to use, it is widely spread, flexible and interactive. For this purpose a Basic interpreter, named Neuro-Basic, was written that allows the calling of neural (or other) functions running parallel on MUSIC. From the Basic level itself the parallelism is completely invisible. To allocate a new layer with 300 neurons, for instance, one can type a = new_layer(300) The variable a afterwards holds a pointer to the created layer which later can be used in other functions to reference that layer. The following command propagates layer a to layer b using the weight set w propagate (a, b, w) Other functions allow the randomization of weights, the loading of patterns and weight sets, the computation of mean squared errors and so on. Each instruction can be assigned to a program line and can then be run as a program. The sequence 10 a 20 b 30 w = new_layer(300) = new_layer(10) = new_weights(a, b) for instance defines a two-layer perceptron with 300 input and 10 output neurons being connected with the weights w. Larger programs, loops and conditional branches can be used to construct and train complete neural nets or to automatically run complete series of experiments where experimental setups depend on the result of previous experiments. The Basic environment thus allows all kinds of gradations in experimental research, from the interactive programming of small experiments till large off-line learning jobs. Extending the simulator with new learning algorithms means that the programmer just has to write the parallel code of the actual algorithm. It can then be controlled by a Basic program and it can be combined with already existing algorithms. The Basic interpreter runs on the host computer allowing easy access to the input/output devices of the host. However, the time needed for interpreting the commands on the host can easily be in the same order of magnitude as the runtime of the actual functions on the attached parallel processor array. The interpretation of a Basic program furthermore is a sequential part of the system (it doesn't run faster if the system size is increased) which is known to be a fundamental limit in speedup (Amdahls law [13]). Therefore the Basic code is not directly interpreted on the host but first is compiled to a simpler stack oriented meta-code, named b-code, which is afterwards copied and run on all processing elements at optimum speed. The compilation phase is not really noticeable to the user since compiling 1000 source lines takes less than a second on a workstation. Note that Basic is not the programming language for the MUSIC system, it is a high level command language for the easy control of parallel algorithms. The actual programming language for MUSIC is C or Assembler. 893 894 Muller, Kocheisen, and Gunzinger Of course, Neuro-Basic is not restricted to the MUSIC system. The same principle can be used for neural net simulation on conventional workstations, vector computers or other parallel systems. Furthermore, the parallel algorithms of MUSIC also run on sequential computers. Simulations in Neuro-Basic can therefore be executed locally on a workstation or PC as well. 4 Conclusions Neuro-Basic running on MUSIC proved to be an important tool to support experimental research on neural nets. It made possible to run many experiments which could not have been carried out otherwise. An important question, however, is, how much more programming effort is needed to implement a new algorithm in the Neuro-Basic environment compared to an implementation on a conventional workstation and how much faster does it run. Algorithm Back-propagation ~ C) Back-propagation (Assembler) Cellular neural nets (CNN) additional programming x 2 x 8 x 3 speedup 60 240 60 Table 3: Implementation time and performance ratio of a 60-processor MUSIC system compared to a Sun Sparcstation-10 Table 3 contains these numbers for back-propagation and cellular neural nets. It shows that if an additional programming effort of a factor two to three is invested to program the MUSIC system in C, the return of investment is a speedup of approximately 60 compared to a Sun Sparcstation-10. This means one year of CPU time on a workstation corresponds to less than a week on the MUSIC system. Acknowledgements We would like to express our gratitude to the many persons who made valuable contributions to the project, especially to Peter Kohler and Bernhard Baumle for their support of the MUSIC system, Jose Osuna for the CNN implementation and the students Ivo Hasler, Bjorn Tiemann, Rene Hauck, Rolf Krahenbiihl who worked for the project during their graduate work. This work was funded by the Swiss Federal Institute of Technology, the Swiss National Science Foundation and the Swiss Commission for Support of Scientific Research (KWF). References [1] Urs A. Miiller, Bernhard Baumle, Peter Kohler, Anton Gunzinger, and Walter Guggenbiihl. Achieving supercomputer performance for neural net simulation with an array of digital signal processors. IEEE Micro Magazine, 12(5):55-65, October 1992. Parallel Neural Net Simulation [2] Anton Gunzinger, Urs A. Miiller, Walter Scott, Bernhard Bliumle, Peter Kohler, Hansruedi Vonder Miihll, Florian Miiller-Plathe, Wilfried F. van Gunsteren, and Walter Guggenbiihl. Achieving super computer performance with a DSP array processor. In Robert Werner, editor, Supercomputing '92, pages 543-550. IEEEj ACM, IEEE Computer Society Press, November 16-20, 1992, Minneapolis, Minnesota 1992. [3] D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representation by error propagation. In David E. Rumelhart and James L. McClelland, editors, Parallel Distributet Processing: Explorations in the Microstructure of Cognition, volume 1, pages 318-362. Bradford Books, Cambridge MA, 1986. [4] Dean A. Pomerleau, George L. Gusclora, David S. Touretzky, and H. T. Kung. Neural network simulation at Warp speed: How we got 17 million connections per second. In IEEE International Conference on Neural Networks, pages 11.143-150, July 24-27, San Diego, California 1988. [5] Michael Witbrock and Marco Zagha. An implementation of backpropagation learning on GF11, a large SIMD parallel computer. Parallel Computing, 14(3):329-346, 1990. [6] Xiru Zhang, Michael Mckenna, Jill P. Mesirov, and David L. Waltz. An efficient implementation of the back-propagation algorithm on the Connection Machine CM-2. In David S. Touretzky, editor, Advances in Neural Information Processing Systems (NIPS-89), pages 801-809,2929 Campus Drive, Suite 260, San Mateo, CA 94403, 1990. Morgan Kaufmann Publishers. [7] Heinz Miihlbein and Klaus Wolf. Neural network simulation on parallel computers. In David J. Evans, Gerhard R. Joubert, and Frans J. Peters, editors, Parallel Computing-89, pages 365-374, Amsterdam, 1990. North Holland. [8] Phil Kohn, Jeff Bilmes, Nelson Morgan, and James Beck. Software for ANN training on a Ring Array Processor. In John E. Moody, Steven J. Hanson, and Richard P. Lippmann, editors, Advances in Neural Information Processing Systems 4 (NIPS-91), 2929 Campus Drive, Suite 260, San Mateo, California 94403, 1992. Morgan kaufmann. [9] Hideki Yoshizawa, Hideki Kato Hiroki Ichiki, and Kazuo Asakawa. A highly parallel architecture for back-propagation using a ring-register data path. In 2nd International Conference on Microe/ectrnics for Neural Networks (ICMNN-91), pages 325-332, October 16-18, Munich 1991. [10] J. A. Osuna, G. S. Moschytz, and T. Roska. A framework for the classification of auditory signals with cellular neural networks. In H. Dedieux, editor, Procedings of 11. European Conference on Circuit Theory and Design, pages 51-56 (part 1). Elsevier, August 20 - Sept. 3 Davos 1993. [11] Leon O. Chua and Lin Yang. Cellular neural networks: Theory. IEEE Transactions on Circuits and Systems, 35(10):1257-1272, October 1988. [12] Leon O. Chua and Lin Yang. Cellular neural networks: Applications. IEEE Transactions on Circuits and Systems, 35(10):1273-1290, October 1988. [13] Gene M. Amdahl. Validity of the single processor approach to achieving large scale computing capabilities. In AFIPS Spring Computer Conference Atlantic City, NJ, pages 483-485, April 1967. 895 

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What Does the Hippocampus Compute?: A Precis of the 1993 NIPS Workshop Mark A. Gluck Center for Molecular and Behavioral Neuroscience Rutgers University Newark, NJ 07102 gluck@pavlov.rutgers.edu Computational models of the hippocampal-region provide an important method for understanding the functional role of this brain system in learning and memory . The presentations in this workshop focused on how modeling can lead to a unified understanding of the interplay among hippocampal physiology, anatomy, and behavior. Several approaches were presented. One approach can be characterized as "top-down" analyses of the neuropsychology of memory, drawing upon brain-lesion studies in animals and humans. Other models take a "bottom-up" approach, seeking to infer emergent computational and functional properties from detailed analyses of circuit connectivity and physiology (see Gluck & Granger, 1993, for a review). Among the issues discussed were: (1) integration of physiological and behavioral theories of hippocampal function, (2) similarities and differences between animal and human studies, (3) representational vs. temporal properties of hippocampaldependent behaviors, (4) rapid vs. incremental learning, (5) mUltiple vs. unitary memory systems, (5) spatial navigation and memory, and (6) hippocampal interaction with other brain systems. Jay McClelland, of Carnegie-Mellon University, presented one example of a topdown approach to theory development in his talk, "Complementary roles of neocortex and hippocampus in learning and memory" McClelland reviewed findings indicating that the hippocampus appears necessary for the initial acquisition of some forms of memory, but that ultimately all forms of memory are stored independently of the hippocampal system. Consolidation in the neocortex appears to occur over an extended period -- in humans the process appears to extend over several years. McClelland suggested that the cortex uses interleaved learning to extract the structure of events and experiences while the hippocampus provides a special system for the rapid initial storage of traces of specific events and experiences in a form that minimizes mutual interference between memory traces. According to this view, the hippocampus is necessary to avoid the catastrophic 1173 1174 Gluck interference that would result if memories were stored directly in the neocortex. Consolidation is slow to allow the gradual integration of new knowledge via continuing interleaved learning (McClelland, 1994/in press). In another example of top-down modeling, Mark Gluck of Rutgers University discussed "Stimulus representation and hippocampal function in animal and human learning." He described a computational account of hippocampal-region function in classical conditioning (Gluck & Myers, 1993; Myers & Gluck, 1994). In this model, the hippocampal region constructs new stimulus representations biased by two opponent constraints: first, to differentiate representations of stimuli which predict different future events, and second, to compress together representations of cooccurring or redundant stimuli. This theory accurately describe the role of the hippocampal region in a wide range of conditioning paradigms. Gluck also presented an extension of this theory which suggests that stimulus compression may arise from the operation of circuitry in the superficial layers of entorhinal cortex, whereas stimulus differentiation may arise from the operation of constituent circuits of the hippocampal formation. Discussing more physiologically-motivated "bottom-up" research, Michael Hasselmo, of Harvard University, talked about "The septohippocampal system: Feedback regulation of cholinergic modulation." Hasselmo presented a model in which feedback regulation sets appropriate dynamics for learning of novel input or recall of familiar input. This model extends previous work on cholinergic modulation of the piriform cortex (Hasselmo, 1993; Hasselmo, 1994). This model depends upon a comparison in region CAl between self-organized input from entorhinal cortex and recall of patterns of activity associated with CA3 input. When novel afferent input is presented, the inputs to CA 1 do not match, and cholinergic modulation remains high, allowing storage of a new association. For familiar input, the match between input patterns suppresses modulation, allowing recall dynamics dominated by input from CA3. Michael Recce and Neil Burgess, from England, presented their work on "Using phase coding and wave packets to represent places." They are attempting to model the spatial behavior of rats in terms of the firing of single cells in the hippocampus. A reinforcement signal enables a set of "goal cells" to learn a population vector encoding the direction of the rat from the goal. This is achieved by exploiting the apparent phase-coding of place cell firing, and the presence of head-direction cells. The model shows rapid latent-learning and robust navigation to previously encountered goal locations (Burgess, O'Keefe, & Recce, 1993; Burgess, Recce, & 0' Keefe, 1994). Spatial trajectories and cell firing characteristics compare well with experimental data. Richard Granger, of U .C. Irvine, was originally scheduled to talk on "Distinct biology and computation of entorhinal, dentate, CA3 and CAl." Granger and colleagues have noted that synaptic changes in each component of the hippocampus (i.e., DG, CA3 and CAl) exhibit different time courses, specificities, and reversibility. As such, they suggest that subtypes of memory operate serially, in an What Does the Hippocampus Compute?: A Precis of the 1993 NIPS Workshop "assembly line" of specialized functions, each of which adds a unique aspect to the processing of memories (Granger et al, 1994). In other talks, Bruce McNaughton of the University of Arizona discussed models of spatial navigation (McNaughton et aI, 1991) and William Levy from the University of Virginia presented a theory of how sparse recurrence of CA3 and several other, less direct feedback systems, leads to an ability to learn and compress sequences (Levy, 1989). Mathew Shapiro, of McGill University, had been scheduled to talk on computing locations and trajectories with simulated hippocampal place fields. References Burgess N, O'Keefe 1 & Recce M (1993) Using hippocampal "place cells" for navigation, exploiting phase coding, in: Hanson S 1, Giles C L & Cowan 1 D (eds.) Advances in Neural Information Processing Systems 5. San Mateo, CA: Morgan Kaufmann. Burgess N, Recce M and O'Keefe 1 (1994) A model of hippocampal function, Neural Networks, Special Issue on Neurodynamics and Behavior, to be published. Gluck, M. and Granger, R. (1993). Computational models of the neural bases of learning and memory. Annual Review of Neuroscience. 16, 667-706. Gluck, M., & Myers, C . (1993). Hippocampal mediation of stimulus representation: A computational theory. Hippocampus, 3., 491-516. Granger, R., Whitson, 1., Larson, 1. and Lynch, G. (1994). Non-Hebbian properties of LTP enable high-capacity encoding of temporal sequences. Proc. Nat'l. Acad. Sci., (in press). Hasselmo, M.E. (1993) Acetylcholine and learning in a cortical associative memory. Neural Computation 5,32-44. Hasselmo, M.E. (1994) Runaway synaptic modification in models of cortex: Implications for Alzheimer's disease. Neural Networks, in press. Levy, W. B (1989) A computational approach to hippocampal function. In: Computational Models of Learning in Simple Neural Systems. (R.D. Hawkins and G.H. Bower, Eds.), New York: Academic Press, pp. 243-305. McClelland, 1. L. (1994/in press). The organization of memory: A parallel distributed processing perspective. Revue Neurologique, Masson, Paris McNaughton, B., Chen, L., & Markus, E. (1991). "Dead reckoning", landmark learning, and the sense of direction: A neurophysiological and computational hypothesis. 10urnal of Cognitive Neuroscience, 3.(2), 190-202. 1175 

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Stability and Observability Max Garzon Fernanda Botelho garzonmGhermea.maci.memat.edu botelhofGhermea.maci.memat.edu Institute for Intelligent Systems Department of Mathematical Sciences Memphis State University Memphis, TN 38152 U.S.A. The theme was the effect of perturbations of the defining parameters of a neural network due to: 1) mea"urement" (particularly with analog networks); 2) di"cretization due to a) digital implementation of analog nets; b) bounded-precision implementation of digital networks; or c) inaccurate evaluation of the transfer function(s}; 3) noise in or incomplete input and/or output of the net or individual cells (particularly with analog networks). The workshop presentations address these problems in various ways. Some develop models to understand the influence of errors/perturbation in the output, learning and general behavior of the net (probabilistic in Piche and TresPi optimisation in Rojas; dynamical systems in Botelho k Garson). Others attempt to identify desirable properties that are to be preserved by neural network solutions (equilibria under faster convergence in Peterfreund & Baram; decision regions in Cohen). Of particular interest is to develop networks that compute robustly, in the sense that small perturbations of their parameters do not affect their dynamical and observable behavior (stability in biological networks in Chauvet & Chauvet; oscillation stability in learning in Rojas; hysterectic finite-state machine simulation in Casey). In particular, understand how biological networks cope with uncertainty and errors (Chauvet & Chauvet) through the type of stability that they exhibit. QUESTIONS AND ANSWERS Some questions served to focus the presentations and discussion. Some were (partially) answered, and others were barely touched: <> What are the mod "ignificant error" in defining parameter" with re"pect to output behavior? By evidence presented, i/o and weights seem to be the most sensitive. <> Is there an essential difference between perturbations in weights (long-term memory) and inputs (short-memory)? They seem to playa symmetric role in feedforward and, to some extent, recurrent nets. But evidence is not conclusive. <> How can the effects of perturbation" be kept under control or eliminated altogether'! If one is only interested in dynamical qualitative features, small enough errors of any kind (as incurred in digital implementations for example) are not relevant for most nets (What you see on the screen is what should be happening). <> Are they architecture (in)dependentf On the other hand, they spread rapidly under iteration and exact quantification varies with the architecture. <> Are stability and implementation based on dynamical features the only ways to 1171 1172 Garzon and Botelho cope with error!/perturbatiofU f The difficulty to quantify (perhaps due to lack of research) seems to indicate so. Stability worth a closer look for its own sake. <> Doe, requiring robud computation really redrict the capabilitie, of neural network, f Apparently not, since in all likelihood there exist universal neural nets which tolerate small errors (see talk by Botelho & Garlon). Wide open. TALKS AND SHORT ABSTRACTS ? TraJ~tory Control of Convergent Networks, Natan Peterfreund and Y. Baram. We present a class of feedback control functions which accelerate convergence rates of autonomous nonlinear dynamical systems such as neural network models, without affecting the basic convergence properties (e.g. equilibrium points). natanOtx.technion.ac.il ? Sensitivity of Neural Network to Errors, Steven Piche. Using stochastic models, analytic expressions for the effects of such errors are derived for arbitrary feedforward neural networks. Both, the degree of nonlinearity and the relationship between input correlation and the weight vectors, are found to be important in determining the effects of errors. picheOlllcc. COm ? Stability of Learning in Neural Networks, Raul Roja!. Finding optimal combinations of learning and momentum rates for the standard backpropagation involves difficult tradeoffs across fractal boundaries. We show that statistic preprocessing can bring error functions under control. rOjaaOinf. fu-berlin.de ? Stability of Purklnje Cells in Cerebellar Cortex, Gilbert Ohauvet and Pierre Ohauvet. The cerebellar cortex (involved in learning and retrieving) is a hierarchical functional unit built around a Purkinje cell, which has its own functional properties. We have shown experimentally that Purkinje dynamical systems have a unique solution, which is asymptotically stable. It seems possible to give a general explanation of stability in biological systems. chauvetOibt. uni v-angers. fr. ? Recall and Learning with Deficient Data, Volker Tresp, Subutai Ahmad, Ralph Neuneier. Mean values and maximum likelihood estimators are not the best ways to cope with noisy data. See their LA:5 poster summary in these proceedings for an extended abstract. treapOzfe. aiemena. de ? Computation Dynamics in Discrete-Time Recurrent Networks, Mike Oasey. We consider training recurrent higher-order neural networks to recognize regular languages, using the cycles in their diagrams for hysterectic simulation of finite state machines. The latter suggests a general logical approach to solving the 'neural code' problem for living organisms, necessary for understanding information processing in the nervous system. mcaseyOsdcc. ucsd. edu ? Synthesis of Decision Regions in Dynamical Systems, Mike Oohen. As a first step toward a representation theory of decision functions via neural nets, he presented a method which enables the construction of a system of differential equations exhibiting a given finite set of decision regions and equilibria with a very large class of indices consistent with the Morse inequalities. mikeOpark. bu. edu ? Observability of Discrete and Analog Networks, F. Botelho and M. Garzon. We show that most networks (with finitely many analog or infinitely many boolean neurons) are observable (i.e., all their corrupted pseudo-orbits actually reflect true orbits). See their DS:2 poster summary in these proceedings. 

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Unsupervised Learning of Mixtures of Multiple Causes in Binary Data Eric Saund Xerox Palo Alto Research Center 3333 Coyote Hill Rd., Palo Alto, CA, 94304 Abstract This paper presents a formulation for unsupervised learning of clusters reflecting multiple causal structure in binary data. Unlike the standard mixture model, a multiple cause model accounts for observed data by combining assertions from many hidden causes, each of which can pertain to varying degree to any subset of the observable dimensions. A crucial issue is the mixing-function for combining beliefs from different cluster-centers in order to generate data reconstructions whose errors are minimized both during recognition and learning. We demonstrate a weakness inherent to the popular weighted sum followed by sigmoid squashing, and offer an alternative form of the nonlinearity. Results are presented demonstrating the algorithm's ability successfully to discover coherent multiple causal representat.ions of noisy test data and in images of printed characters. 1 Introduction The objective of unsupervised learning is to identify patterns or features reflecting underlying regularities in data. Single-cause techniques, including the k-means algorithm and the standard mixture-model (Duda and Hart, 1973), represent clusters of data points sharing similar patterns of Is and Os under the assumption that each data point belongs to, or was generated by, one and only one cluster-center; output activity is constrained to sum to 1. In contrast, a multiple-cause model permits more than one cluster-center to become fully active in accounting for an observed data vector. The advantage of a multiple cause model is that a relatively small number 27 28 Saund of hidden variables can be applied combinatorially to generate a large data set. Figure 1 illustrates with a test set of nine 121-dimensional data vectors. This data set reflects two independent processes, one of which controls the position of the black square on the left hand side, the other controlling the right. While a single cause model requires nine cluster-centers to account for this data, a perspicuous multiple cause formulation requires only six hidden units as shown in figure 4b. Grey levels indicate dimensions for which a cluster-center adopts a "don't-know /don't-care" assertion . ????????? Figure 1: Nine 121-dimensional test data samples exhibiting multiple cause structure. Independent processes control the position of the black rectangle on the left and right hand sides. While principal components analysis and its neural-network variants (Bourlard and Kamp, 1988; Sanger, 1989) as well as the Harmonium Boltzmann Machine (Freund and Haussler, 1992) are inherently multiple cause models, the hidden representations they arrive at are for many purposes intuitively unsatisfactory. Figure 2 illustrates the principal components representation for the test data set presented in figure 1. Principal components is able to reconstruct the data without error using only four hidden units (plus fixed centroid), but these vectors obscure the compositional structure of the data in that they reveal nothing about the statistical independence of the left and right hand processes. Similar results obtain for multiple cause unsupervised learning using a Harmonium network and for a feedforward network using the sigmoid nonlinearity. We seek instead a multiple cause formulation which will deliver coherent representations exploiting "don't-know/don't-care" weights to make explicit the statistical dependencies and independencies present when clusters occur in lower-dimensional subspaces of the full J -dimensional data space. Data domains differ in ways that underlying causal processes interact. The present discussion focuses on data obeying a WRITE-WHITE-AND-BLACK model, under which hidden causes are responsible for both turning "on" and turning "off" the observed variables. a b Figure 2: Principal components representation for the test data from figure 1. (a) centroid (white: -1, black: 1). (b) four component vectors sufficient to encode the nine data points. (lighter shadings: Cj,k < 0; grey: Cj,k 0; darker shading: Cj,/.: > 0). = Unsupervised Learning of Mixtures of Multiple Causes in Binary Data 2 Mixing Functions A large class of unsupervised learning models share the architecture shown in figure 3. A binary vector Di (d i ,l,di ,2, ... di,j, ... di,J) is presented at the data layer, and a measurement, or response vector mi (mi ,l, mi,2, ... mi ,k, ... mi ,K) is computed at the encoding layer using "weights" Cj,k associating activity at data dimension j with activity at hidden cluster-center k. Any activity pattern at the encoding layer can be turned around to compute a prediction vector ri (ri,l" ri,2, ... ri,j, ... ri,J) at the data layer. Different models employ different functions for performing the measurement and prediction mappings, and give different interpretations to the weights. Common to most models is a learning procedure which attempts to optimize an objective function on errors between data vectors in a training set, and predictions of these data vectors under their respective responses at the encoding layer. = = = encoding layer (cluster-centers) pMietion data layer d j (observed data) r. J (predicted) Figure 3: Architecture underlying a large class of unsupervised learning models. The key issue is the mixing function which specifies how sometimes conflicting predictions from individual hidden units combine to predict values on the data dimensions. Most neural-network formulations, including principal components variants and the Boltzmann Machine, employ linearly weighted sum of hidden unit activity followed by a squashing, bump, or other nonlinearity. This form of mixing function permits an error in prediction by one cluster center to be cancelled out by correct predictions from others without consequence in terms of error in the net prediction . As a result, there is little global pressure for cluster-centers to adopt don't-know values when they are not quite confident in their predictions. Instead, a mult.iple cause formulation delivering coherent cluster-centers requires a form of nonlinearit.y in which active disagreement must result in a net "uncertain" or neutral prediction that results in nonzero error. 29 30 Saund 3 Multiple Cause Mixture Model Our formulation employs a zero-based representation at the data layer to simplify the mathematical expression for a suitable mixing function. Data values are either 1 or -1; the sign of a weight Cj ,k indicates whether activity in cluster-center k predicts a 1 or -1 at data dimension j, and its magnitude (ICj,kl ~ 1) indicates strength of belief; Cj ,k 0 corresponds to "don't-know /don't-care" (grey in figure 4b). = The mixing function takes the form, r.,) = L k <".<0 II mi ,k(-c),k) k (1 + m"kCj,k) - 1 <". <0 + L mi,kc) ,k k <".>0 I- II (1 - m"kCj,k) k <".>0 This formula is a computationally tractable approximation to an idealized mixing function created by linearly interpolating boundary values on the extremes of mi,k E {O, I} and Cj,k E {-I, 0, I} rationally designed to meet the criteria outlined above. Both learning and measurement operate in the context of an objective function on predictions equivalent to log-likelihood. The weights Cj,k are found through gradient ascent in this objective function, and at each training step the encoding mi of an observed data vector is found by gradient ascent as well. 4 Experimental Results Figure 4 shows that the model converges to the coherent multiple cause representation for the test data of figure 1 starting with random initial weights. The model is robust with respect to noisy training data as indicated in figure 5. In figure 6 the model was trained on data consisting of 21 x 21 pixel images of registered lower case characters. Results for J( = 14 are shown indicating that the model has discovered statistical regularities associated with ascenders, descenders, circles, etc. a b ...----.-Figure 4: Multiple Cause Mixture Model representation for the test data from figure 1. (a) Initial random cluster-centers. (b) Cluster-centers after seven training iterations (white: Cj,k -1; grey: Cj,k 0; black: Cj,k 1). = = = Unsupervised Learning of Mixtures of Multiple Causes in Binary Data 5 Conclusion Ability to compress data, and statistical independence of response activities (Barlow, 1989), are not the only criteria by which to judge the success of an encoder network paradigm for unsupervised learning. For many purposes, it is equally important that hidden units make explicit statistically salient structure arising from causally distinct processes. The difficulty lies in getting the internal knowledge-bearing entities sensibly to divvy up responsibility for training data not just pointwise, but dimensionwise. Mixing functions based on linear weighted sum of activities (possibly followed by a nonlinearity) fail to achieve this because they fail to pressure the hidden units into giving up responsibility (adopting "don't know" values) for data dimensions on which they are prone to be incorrect. We have outlined criteria, and offered a specific functional form, for nonlinearly combining beliefs in a predictive mixing function such that statistically coherent hidden representations of multiple causal structure can indeed be discovered in binary data. References Barlow, H.; [1989], "Unsupervised Learning," Neural Computation, 1: 295-31l. Bourlard, H., and Kamp, Y.; [1988], Auto-Association by Multilayer Perceptrons and Singular Value Decomposition," Biological Cybernetics, 59:4-5, 291-294. Duda, R., and Hart, P.; [1973], Pattern Classification and Scene Analysis, Wiley, New York. Foldiak, P.; [1990], "Forming sparse representations by local anti-Hebbian learning," Biological Cybernetics, 64:2, 165-170. Freund, Y., and Haussler, D.; [1992]' "Unsupervised learning of distributions on binary vectors using two-layer networks," in Moody, J., Hanson, S., and Lippman, R., eds, Advances in Neural Information Processing Systems 4, Morgan Kauffman, San Mateo, 912-919. Nowlan, S.; [1990], "Maximum Likelihood Competitive Learning," in Touretzky, D., ed., Advances in Neural Information Processing Systems 2, Morgan Kauffman, San Mateo, 574-582. Sanger, T.; [1989], "An Optimality Principle for Unsupervised Learning," in Touretzky, D., ed., Advances in Neural Information Processing Systems, Morgan Kauffman, San Mateo, 11-19. 31 32 Saund a b observpd data d, c ? ? ? ? ? nlf' a~ l1I cme nt s 1n " k ? ? ? ? ? predictions r, '., . . , .;.' x .~ ::.:,;,; .. ,. . . ., .., , ' ;, . '.' .:: ' .;;., Figure 5: Multiple Cause Mixture Model results for noisy training data. (a) Five test data sample suites with 10% bit-flip noise. Twenty suites were used to train from random initial cluster-centers, resulting in the representation shown in (b) . (c) Left: Five test data samples di ; Middle: Numerical activities mi,k for the most active cluster-centers (the corresponding cluster-center is displayed above each mi,k value); Right: reconstructions (predictions) ri based on the activities . Not.e how these "clean up" the noisy samples from which they were computed. Unsupervised Learning of Mixtures of Multiple Causes in Binary Data a b Figure 6: (a) Training set of twenty-six 441-dimensional binary vectors. (b) Multiple Cause Mixt.ure Model representation at J{ = 14. (c) Left: Five test data samples di ; Middle: Numerical activities mi,k for the most active cluster-centers (the corresponding cluster-center is displayed above each mi,k value); Right: reconstructions (predictions) ri based on the activities. 33 34 Saund observed data d; measurements predictions m;,k c ri 

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Synchronization, oscillations, and 1/ f noise in networks of spiking neurons Martin Stemmler, Marius Usher, and Christof Koch Computation and Neural Systems, 139-74 California Institute of Technology Pasadena, CA 91125 Zeev Olami Dept. of Chemical Physics Weizmann Institute of Science Rehovot 76100, Israel Abstract We investigate a model for neural activity that generates long range temporal correlations, 1/ f noise, and oscillations in global activity. The model consists of a two-dimensional sheet of leaky integrateand-fire neurons with feedback connectivity consisting of local excitation and surround inhibition. Each neuron is independently driven by homogeneous external noise. Spontaneous symmetry breaking occurs, resulting in the formation of "hotspots" of activity in the network. These localized patterns of excitation appear as clusters that coalesce, disintegrate, or fluctuate in size while simultaneously moving in a random walk constrained by the interaction with other clusters. The emergent cross-correlation functions have a dual structure, with a sharp peak around zero on top of a much broader hill . The power spectrum associated with single units shows a 1/ f decay for small frequencies and is flat at higher frequencies, while the power spectrum of the spiking activity averaged over many cells-equivalent to the local field potential-shows no 1/ f decay but a prominent peak around 40 Hz. 629 630 Stemmler, Usher, Koch, and Olami 1 The model The model consists of a 100-by-l00 lattice of integrate-and-fire units with cyclic lattice boundary conditions. Each unit represents the nerve cell membrane as a 20 msec) with the addition of a reset mechanism; the simple RC circuit (r refractory period TreJ is equal to one iteration step (1 msec). = Units are connected to each other within the layer by local excitatory and inhibitory connections in a center-surround pattern. Each unit is excitatorily connected to N = 50 units chosen from a Gaussian probability distribution of u = 2.5 (in terms of the lattice constant), centered at the unit's position N inhibitory connections per unit are chosen from a uniform probability distribution on a ring eight to nine lattice constants away. Once a unit reaches the threshold voltage, it emits a pulse that is transmitted in one iteration (1 msec) to connected neighboring units, and the potential is reset by subtracting the threshold from resting potential. \Ii(t + 1) = (exp( -l/r)\Ii(t) + h (t)) O[vth - V(t)]. (1) Ii is the input current, which is the sum of lateral currents from presynaptic units and external current. The lateral current leads to an increase (decrease) in the membrane potential of excitatory (inhibitorily ) connected cells. The weight of the excitation and inhibition, in units of voltage threshold, is ~ and J3 ~. The values a = 1.275 and J3 = 0.67 were used for simulations. The external input is modeled independently for each cell as a Poisson process of excitatory pulses of magnitude 1/ N, arriving at a mean rate "ext. Such a simple cellular model mimics reasonably well the discharge patterns of cortical neurons [Bernander et al., 1994, Softky and Koch, 1993]. 2 Dynamics and Pattern Formation In the mean-field approximation, the firing rate of an integrate-and-fire unit is a function of the input current [Amit and Tsodyks, 1991] given by f(I) = (TreJ - r In[l - 1/(1 r)])-l, (2) where Tref is the refractory period and r the membrane time constant. In this approximation, the dynamics associated with eq. 1 simplify to ~i = -Ii + L j Wijf(Ij) + It xt , (3) where Wij represents the connection strength matrix from unit j to unit i. Homogeneous firing activity throughout the network will result as long as the connectivity pattern satisfies W(k)-l < 0 for all k, where W(k) is the Fourier transform of Wij . As one increases the total strength of lateral connectivity, clusters of high firing activity develop. These clusters form a hexagonal grid across the network; for even stronger lateral currents, the clusters merge to form stripes. The transition from a homogeneous state to hexagonal clusters to stripes is generic to many nonequilibrium systems in fluid mechanics, nonlinear optics, reactiondiffusion systems, and biology. (The classic theory for fluid mechanics was first Synchronization, Oscillations, and IlfNoise in Networks of Spiking Neurons developed by [Newell and Whitehead, 1969], see [Cross and Hohenberg, 1993] for an extensive review. Cowan (1982) was the first to suggest applying the techniques of fluid mechanics to neural systems.) The richly varied dynamics of the model, however, can not be captured by a meanfield description. Clusters in the quasi-hexagonal state coalesce, disintegrate, or fluctuate in size while simultaneously moving in a random walk constrained by the interaction with other clusters. R~ndom Walk of Clusters 16 14 E ... "" t'.: B ; 12 10 8 6 o~~--~~--~~--~~--~~ o 2 6 x 8 (latt~ce 10 12 14 16 18 un~t~) Figure 1: On the left, the summed firing activity for the network over 50 msec of simulation is shown. Lighter shades denote higher firing rates (maximum firing rate 120 Hz). Note the nearly hexagonal pattern of clusters or "hotspots" of activity. On the right, we illustrate the motion of a typical cluster. Each vertex in the graph represents a tracked cluster's position averaged over 50 msec. Repulsive interactions with surrounding clusters generally constrain the motion to remain within a certain radius. This vibratory motion of a cluster is occasionally punctuated by longerrange diffusion. Statistical fluctuations, diffusion and synchronization of clusters, and noise in the external input driving the system lead to 1/ I-noise dynamics, long-range correlations, and oscillations in the local field potential. These issues shall be explored next. 3 1/ f Noise The power spectra of spike trains from individual units are similar to those published in the literature for nonbursting cells in area MT in the behaving monkey [Bair et al., 1994]. Power spectra were generally flat for all frequencies above 100 Hz. The effective refractory period present in an integrate-and-fire model introduces a dip at low frequencies (also seen in real data). Most noteworthy is the l/lo.s component in the power spectrum at low frequencies. Notice that in order to see such a decay for very low frequencies in the spectrum, single units must be recorded for on the order of 10-100 sec, longer than the recording time for a typical trial in neurophysiology. We traced a cluster of neuronal activity as it diffused through the system, and 631 632 Stemmler. Usher. Koch. and Olami Spike Tra~n Power Spectrum lSI distribution 3r-----~----~------~----_r----~ 2.5 0.7 0.5 2 ... 1.5 0.3 1 0.2 0.15 0.5 0.1 20 40 60 80 100 Hz 30. 50. 70. 100. 150. 200. msec Figure 2: Typical power spectrum and lSI distribution of single units over 400 sec of simulation. At low frequencies, the power spectrum behaves as f- O.S ?O.017 up to a cut-off frequency of ~ 8 Hz. The lSI distribution on the right is shown on a log-log scale. The lSI histogram decays as a power law pet) ex t-1.70?O.02 between 25 and 300 msec. In contrast, a system with randomized network connections will have a Poisson-distributed lSI histogram which decays exponentially. measured the lSI distribution at a fixed point relative to the cluster center. In the cluster frame of reference, activity should remain fairly constant, so we expect and do find an interspike interval (lSI) distribution with a single characteristic relaxation time: Pr(t) = A(r)exp(-tA(r)) , where the firing rate A(r) is now only a function of the distance r in cluster coordinates. Thus Pr(t) is always Poisson for fixed r. If a cluster diffuses slowly compared to the mean interspike interval, a unit at a fixed position samples many lSI distributions of varying A(r) as the cluster moves. The lSI distribution in the fixed frame reference is thus pet) = j A(r)2 exp( -t A(r?)dr. (4) Depending on the functional form of A(r), pet) (the lSI distribution for a unit at a fixed position) will decay as a power law, and not as an exponential. Empirically, the distribution of firing rates as a function of r can be approximated (roughly) by a Gaussian. A Gaussian A(r) in eq. 4 leads to pet) t- 2 for t at long times. In turn, a power-law (fractal) pet) generates 1/ f noise (see Table 1). f'oi 4 Long-Range Cross-Correlations Excitatory cross-correlation functions for units separated by small distances consist of a sharp peak at zero mean time delay followed by a slower decay characterized by a power law with exponent -0.21 until the function reaches an asymptotic level. Nelson et al. (1992) found this type of cross-correlation between neurons-a "castle on a hill" -to be the most common form of correlation in cat visual cortex. Inhibitory Synchronization, Oscillations, and lifNoise in Networks of Spiking Neurons cross-correlations show a slight dip that is much less pronounced than the sharp excitatory peak at short time-scales. Cross-Correlation at d 1 1000 750 500 250 -300 -200 -100 o 100 200 300 msec Cross-Correlation at d 9 1000 750 500 250 -300 -200 -100 o 100 200 300 msec Figure 3: Cross-correlation functions between cells separated by d units of the lattice. Given the center-surround geometry of connections, the upper curve corresponds to mutually excitatory coupling and the lower to mutually inhibitory coupling. Correlations decay as l/t O. 21 , consistent with a power spectrum of single spike trains that behaves as 1/ fo .8. Since correlations decay slowly in time due to the small exponent of the power, long temporal fluctuations in the firing rate result, as the 1/ f-type power spectra of single spike trains demonstrate. These fluctuations in turn lead to high variability in the number of events over a fixed time period. In fact, the decay in the auto-correlation and power spectrum, as well as the rise in the variability in the number of events, can be related back to the slow decay in the interspike interval (lSI) distribution. If the lSI distribution decays as a power law pet) ,...., t- II , then the point process giving rise to it is fractal with a dimension D = v - I [Mandelbrot, 1983]. Assuming that the simulation model can be described as a fully ergodic renewal process, all these quantities will scale together [Cox and Lewis, 1966, Teich, 1989, Lowen and Teich, 1993, Usher et al., 1994]: 633 634 Stemmler, Usher, Koch, and Olami Table 1: Scaling Relations and Empirical Results Auto-correlation Var(N) Var(N) "-J Var(N) '" Nil N1.54 A(t) A(t) "-J "-J t ll - 2 t- 0 .21 Power Spectrum S(I) "-J /-11+1 S(I) ""' /-0.81 lSI Distribution pet) ""' t- II pet) "-J c1. 7O These relations will be only approximate if the process is nonrenewal or nonergodic, or if power-laws hold over a limited range. The process in the model is clearly nonrenewal, since the presence of a cluster makes consecutive short interspike intervals for units within that cluster more likely than in a renewal process. Hence, we expect some (slight) deviations from the scaling relations outlined above. Cluster Oscillations and the Local Field Potential 5 The interplay between the recurrent excitation that leads to nucleation of clusters and the "firewall" of inhibition that restrains activity causes clusters of high activity to oscillate in size. Fig 4 is the power spectrum of ensemble activity over the size of a typical cluster. Power Spectrum of Cluster ActlVlty withln radlus d=9 25 20 10-4 (lJ 15 :J: 0 P... 10 5 0 0 20 40 60 80 100 Hz Figure 4: Power spectrum of the summed spiking activity over a circular area the size of a single cluster (with a radius of 9 lattice constants) recorded from a fixed point on the lattice for 400 seconds. Note the prominent peak centered at 43 Hz and the loss of the 1// component seen in the single unit power spectra (Fig. 2). These oscillations can be understood by examining the cross-correlations between cells. By the Wiener-Khinchin theorem, the power spectrum of cluster activity is the Fourier transform of the signal's auto-correlation. Since the cluster activity is the sum of all single-unit spiking activity within a cluster of N cells, the autocorrelation of the cluster spiking activity will be the sum of N auto-correlations functions of the Synchronization, Oscillations, and lifNoise in Networks of Spiking Neurons individual cells and N x (N - 1) cross-correlation functions among individual cells within the cluster. The ensemble activity is thus dominated by cross-correlations. In general, the excitatory "castles" are sharp relative to the broad dip in the crosscorrelation due to inhibition (see Fig. 3). In Fourier space, these relationships are reversed: broader Fourier transforms of excitatory cross-correlations are paired with narrower Fourier transforms of inhibitory cross-correlations. Superposition of such transforms leads to a peak in the 30-70 Hz range and cancellation of the 1/ f component which was present the single unit power spectrum. Interestingly, the power spectra of spike trains of individual cells within the network (Fig. 2) show no evidence of a peak in this frequency band. Diffusion of clusters disrupts any phase relationship between single unit firing and ensemble activity. The ensemble activity corresponds to the local field potential in neurophysiological recordings. While oscillations between 30 and 90 Hz have often been seen in the local field potential (or sometimes even in the EEG) measured in cortical areas in the anesthetized or awake cat and monkey, these oscillations are frequently not or only weakly visible in multi- or single-unit data (e.g., [Eeckman and Freeman, 1990, Kreiter and Singer, 1992, Gray et al., 1990, Eckhorn et al., 1993]). We here offer a general explanation for this phenomenon. Acknowledgments: We are indebted to William Softky, Wyeth Bair, Terry Sejnowski, Michael Cross, John Hopfield, and Ernst Niebur, for insightful discussions. Our research was supported by a Myron A. Bantrell Research Fellowship, the Howard Hughes Medical Institute, the National Science Foundation, the Office of Naval Research and the Air Force Office of Scientific Research. References [Amit and Tsodyks, 1991] Amit, D. J. and Tsodyks, M. V. (1991). Quantitative study of attractor neural network retrieving at low rates: 1. substrate spikes, rates and neuronal gain. Network Com., 2(3):259-273. [Bair et al., 1994] Bair, W., Koch, C., Newsome, W., and Britten, K. (1994). Power spectrum analysis of MT neurons in the behaving monkey. J. Neurosci., in press. [Bernander et al., 1994] Bernander, 0., Koch, C., and Usher, M. (1994). The effect of synchronized inputs at the single neuron level. Neural Computation, in press. [Cowan, 1982] Cowan, J. D. (1982). Spontaneous symmetry breaking in large scale nervous activity. Int. J. Quantum Chemistry, 22:1059-1082. [Cox and Lewis, 1966] Cox, D. and Lewis, P. A. W. (1966). The Statistical Analysis of Series of Events. Chapman and Hall, London. [Cross and Hohenberg, 1993] Cross, M. C. and Hohenberg, P. C. (1993). Pattern formation outside of equilibrium. Rev. Mod. Phys., 65(3):851-1112. [Eckhorn et al., 1993] Eckhorn, R., Frien, A., Bauer, R., Woelbern, T., and Harald, K. (1993). High frequency (60-90 hz) oscillations in primary visual cortex of awake monkey. Neuroreport, 4:243-246. 635 636 Stemmler, Usher, Koch, and Olami [Eeckman and Freeman, 1990] Eeckman, F . and Freeman, W. (1990). Correlations between unit firing and EEG in the rat olfactory system. Brain Res., 528(2):238244. [Grayet al., 1990] Gray, C. M., Engel, A. K., Konig, P., and Singer, W. (1990) . Stimulus dependent neuronal oscillations in cat visual cortex: receptive field properties and feature dependence. Europ. J. Neurosci., 2:607-619. [Kreiter and Singer, 1992] Kreiter, A. K. and Singer, W. (1992). Oscillatory neuronal responses in the visual cortex of the awake macaque monkey. Europ. J. Neurosci., 4:369-375. [Lowen and Teich, 1993] Lowen, S. B. and Teich, M. C. (1993). Fractal renewal processes generate Iff noise. Phys. Rev. E, 47(2):992-1001. [Mandelbrot, 1983] Mandelbrot, B. B. (1983). The fractal geometry of nature. W. H. Freeman, New York. [Nelson et al., 1992] Nelson, J. I., Salin, P. A., Munk, M. H.-J., Arzi, M., and Bullier, J. (1992). Spatial and temporal coherence in cortico-cortical connections: A cross-correlation study in areas 17 and 18 in the cat . Visual Neuroscience, 9:21-38. [Newell and Whitehead, 1969] Newell, A. C. and Whitehead, J. A. (1969). Finite bandwidth, finite amplitude convection. J. Fluid Mech ., 38:279-303. [Softky and Koch, 1993] Softky, W. R. and Koch , C. (1993). The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. J. Neurosci., 13(1):334-350. [Teich, 1989] Teich, M. C. (1989). Fractal character of the auditory neural spike train. IEEE Trans. Biomed. Eng., 36(1):150-160. [Usher et al., 1994] Usher, M., Stemmler, M., Koch, C ., and Olami, Z. (1994). Network amplification of local fluctuations causes high spike rate variability, fractal firing patterns, and oscillatory local field potentials. Neural Computation, in press. PART V CONTROL, NAVIGATION, AND PLANNING 

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The Statistical Mechanics of k-Satisfaction Scott Kirkpatrick* Racah Institute for Physics and Center for Neural Computation Hebrew University Jerusalem, 91904 Israel kirk@fiz.huji.ac .il Geza Gyorgyi Institute for Theoretical Physics Eotvos University 1-1088 Puskin u. 5-7 Budapest, Hungary gyorgyi@ludens.elte.hu, N aft ali Tishby and Lidror Troyansky Institute of Computer Science and Center for Neural Computation The Hebrew University of Jerusalem 91904 Jerusalem, Israel {tishby, lidrort }@cs.huji.ac.il Abstract The satisfiability of random CNF formulae with precisely k variables per clause ("k-SAT") is a popular testbed for the performance of search algorithms. Formulae have M clauses from N variables, randomly negated, keeping the ratio a = M / N fixed . For k = 2, 1 this model has been proven to have a sharp threshold at a between formulae which are almost aways satisfiable and formulae 00 . Computer experiwhich are almost never satisfiable as N ments for k = 2, 3, 4, 5 and 6, (carried out in collaboration with B. Selman of ATT Bell Labs). show similar threshold behavior for each value of k. Finite-size scaling, a theory of the critical point phenomena used in statistical physics, is shown to characterize the size dependence near the threshold. Annealed and replica-based mean field theories give a good account of the results. = --jo "Permanent address: IBM TJ Watson Research Center, Yorktown Heights, NY 10598 USA. (kirk@watson.ibm.com) Portions of this work were done while visiting the Salk Institute, with support from the McDonnell-Pew Foundation. 439 440 Kirkpatrick, Gyorgyi, Tishby, and Troyansky 1 Large-scale computation without a length scale It is increasingly possible to model the natural world on a computer. Condensed matter physics has strategies to manage the complexities of such calculations, usually depending on a characteristic length. For example, molecules or atoms with finite ranged interactions can be broken down into weakly interacting smaller parts. We may also use symmetry to identify natural modes of the system as a whole. Even in the most difficult case, continuous phase transitions correlated over a wide range of scales, the renormalization group provides a way of collapsing the problem down to its "relevant" parts by providing a generator of behavior on all scales in terms of the critical point itself. But length scales are not much help in organizing another sort of large calculation. Examples include large rule-based "expert systems" that model the particulars of complex industrial processes. Digital Equipment, for example, has used a network of three or more expert systems (originally called "R1/XCON") to check computer orders for completeness and internal consistency, to schedule production and shipping, and to aid a salesman to anticipate customers' needs. This very detailed set of tasks in 1979 required 2 programmers and 250 rules to deal with 100 parts. In the ten years described by Barker (1989), it grew 100X, employing 60 programmers and nearly 20,000 rules to deal with 30,000 part numbers. 100X in ten years is only moderate growth, and it would be valuable to understand how technical, social, and business factors have constrained it. Many important commercial and scientific problems without length scales are ready for attack by computer modelling or automatic classification, and lie within a few decades of XCON's size. Retail industries routinely track 10 5 - 10 6 distinct items kept in stock. Banks, credit card companies, and specialized information providers are building models of what 10 8 Americans have bought and might want to buy next. In biology, human metabolism is currently described in terms of > 1000 substances coupled through> 10,000 reactions, and the data is doubling yearly. Similarly, amino acid sequences are known for> 60,000 proteins. A deeper understanding of the computational cost of these problems of order 10 6 ?2 is needed to see which are practical and how they can be simplified. We study an idealization of XC ON-style resolution search, and find obvious collective effects which may be at the heart of its computational complexity. 2 Threshold Phenomena and Random k-SAT Properties of randomly generated combinatorial structures often exhibit sharp threshold phenomena analogous to the phase transitions studied in condensed matter physics. Recently, thresholds have been observed in randomly generated Boolean formulae. Mitchell et al. (1992) consider the k-satisfiability problem (k-SAT). An instance of k-SAT is a Boolean formula in conjunctive normal form (CNF), i.e., a conjunction (logical AND) of disjunctions or clauses (logical ORs), where each disjunction contains exactly k literals. A literal is a Boolean variable or, with equal probability, its negation. The task is to determine whether there is an assignment to the variables such that all clauses evaluate to true. Here, we will use N to denote the number of variables and M for the number of clauses in a formula. The Statistical Mechanics of k-Satisfaction For randomly generated 2-SAT instances, it has been shown analytically that for large N, when the ratio a: = M / N is less than 1 the instances are almost all satisfiable, whereas for ratios larger than 1, almost all instances are unsatisfiable (Chvatal and Reed 1992; Goerdt 1992). For k ~ 3, a rigorous analysis has proven to be elusive. Experimental evidence, however, strongly suggests a threshold with a: ~ 4.3 for 3SAT (Mitchell et al. 1992; Crawford and Auton 1993; Larrabee 1993). One of the main reasons for studying randomly generated 3CNF formulae is for their use in the empirical evaluation of combinatorial search algorithms. 3CNF formulae are good candidates for the evaluation of such algorithms because determining their satisfiability is an NP-complete problem. This also holds for larger values of k. For k = 1 or 2, the satisfiability problem can be solved efficiently (Aspvall et al. 1979) . Despite the worst-case complexity, simple heuristic methods can usually determine the satisfiability of random formulae. However, computationally challenging test instances are found by generating formulae at or near the threshold (Mitchell et al. 1992). Cheeseman (1991) has made a similar observation of increased computational cost for heuristic search at a boundary between two distinct phases or behaviors of a combinatorial model. We will provide a precise characterization of the N -dependence of the threshold phenomena for k-SAT with k ranging from 2 to 6. We will employ finite size scaling, a method from statistical physics in which direct observation of the width of the threshold , or "critical region" of a transition is used to characterize the "universal" behavior of quantities across the entire critical region, extending the analysis to combinatorial problems in which N characterizes the size of the model observed. For discussion of the applicability of finite-size scaling to systems without a metric, see Kirkpatrick and Selman (1993). Thr ? ? ho~d. 1. ,': i rOr 2SAT. .' 1(/ if II! !i i! if ill ~~ '" ~... O. B !i 0 . 6 1,1 ' g 0 . 2 0 and 6SAT /'<> .'" :' .... Ii' // 11/ // ~i f/ }' 0 ~ 5SAT , I: 0 . 4 ~ ',j ~ 4SAT, !/ ii ill ? / 3SAT , 0 J :1 . .....; ' ",' 1.0 20 J.. 30 MI N 40 so Fig. 1: Fraction of unsatisfiable formulae for 2-, 3- 4-, 5- and 6-SAT. 60 441 442 Kirkpatrick, Gyorgyi, Tishby, and Troyansky 3 Experimental data We have generated extensive data on the satisfiability of randomly generated kCNF formulae with k ranging from 2 to 6. Fig. 1 shows the fraction of random k-SAT formulae that is unsatisfiable as a function of the ratio, a. For example, the left-most curve in Fig. 1 shows the fraction of formulae that is unsatisfiable for random 2CNF formulae with 50 variables over a range of values of a. Each data point was generated using 10000 randomly generated formulae, giving 1% accuracy. We used a highly optimized implementation of the Davis-Putnam procedure (Crawford and Auton 1993). The procedure works best on formulae with smaller k . Data was obtained for k = 2 on samples with N ~ 500, for k = 3 with N ~ 100, and for k = 5 with N ~ 40, all at comparable computing cost. Fig. 1 (for N ranging from 10 to 50) shows a threshold for each value of k. Except for the case k = 2, the curves cross at a single point and sharpen up with increasing N. For k = 2, the intersections between the curves for the largest values of N seem to be converging to a single point as well, although the curves for smaller N deviate. The point where 50% of the formulae are unsatisfiable is thought to be where the computationally hardest problems are found (Mitchell et al. 1992; Cheeseman et al. 1991). The 50% point lies consistently to the right of the scale-invariant point (the point where the curves cross each other), and shifts with N. There is a simple explanation for the rapid shift of the thresholds to the right with increasing k . The probability that a given clause is satisfied by a random input configuration is (2k - 1)/2k = (1 - 2- k ) _ 'k. If we treat the clauses as independent, the probability that all clauses are satisfied is ,~ = ,k N . We define the entropy, 5, per in~ut as l/N times the log2 of the expected number of satisfying configurations,2 N 'k . 5 = 1 + alog2(,k) 1- a/aann, and the vanishing of the entropy gives an estimate of the threshold, identical to the upper bound derived by several workers (see Franco (1983) and citations in Chvatal (1992)): aann = -(log2(1 - 2- k))-1 ~ (ln2)2k. This is called an annealed estimate for C?c, because it ignores the interactions between clauses, just as annealed theories of materials (see Mezard 1986) average over many details of the disorder. We have marked aann with an arrow for each k in the figures, and tabulate it in Table 1. = 4 Results of Finite-Size Scaling Analysis From Fig . 1, it is clear that the threshold "sharpens up" for larger values of N. Both the threshold shift and the increasing slope in the curves of Fig. 1 can be accounted for by finite size scaling. (See Stauffer and Aharony (1992) or Kirkpatrick and Swendsen (1985).) We plot the fraction of samples unsatisfied against the dimensionless rescaled variable, y = Nl/V(a - c?c)/a c . Values for a c and 1I must be derived from the experimental data. First a c is determined as the crossing point of the curves for large N in Fig. 1. Then 1I is determined to make the slopes match up through the critical region. In Fig. 2 (for k = 3) we find that these two parameters capture both the threshold shift and the steepening of the curves, using a c 4.17 and 1I 1.5. We see that F, the fraction = = The Statistical Mechanics of k-Satisfaction _>SAT.,. scakMf CFOuover functton, III SAT modele ",.12 ? "=20 ? N=24 a N=tO Il N. 50 a. N.. 100 .... . 01 J i a f '0 01 ..? .fi' i. Of I 02 -2 -\ 2 3 Y = = Fig. 2: Rescaled 3-SAT data using a c 4.17, lJ 1.5. Fig. 3: Rescaled data for 2-, 3-, 4-, 5-, and 6-SAT approach annealed limit. of unsatisfiable formulae, is given by F(N, a) = I(y) , where the invariant function, I, is that graphed in Fig. 2. A description of the 50% threshold shift follows immediately. If we define y' by I(y') = 0.5, then a50 = a c (1 + y' N- 1 / V ) . From Fig. 2 we find that a50 ~ 4.17 + 3.1N- 2 / 3 . Crawford and Auton (1993) fit their data on the 50% point as a function of N by arbitrarily assuming that the leading correction will be O(I/N) . They obtain a50 = 4.24 + 6/N. However, the two expressions differ by only a few percent as N ranges from 10 to 00. We also obtained good results in rescaling the data for the other values of k . In Table 1 we give the critical parameters obtained from this analysis. The error bars are subjective, and show the range of each parameter over which the best fits were obtained. Note that v appears to be tending to 1, and aann becomes an increasingly good approximation to a c as k increases. The success of finite-size scaling with different powers, v, is strong evidence for criticality, i.e., diverging correlations, even in the absence of any length . Finally, we found that all the crossovers were similar in shape. In fact, combining the various rescaled curves in figure 3 shows that the curves for k ~ 3 all coincide in the vicinity of the 50% point, and tend to a limiting form, which can be obtained by extending the annealed arguments of the previous section. If we define then the probability that a formula remains unsatisfied for all 2N configurations is The curve for k = 2 is similar in form, but shifted to the right from the other ones. 443 444 Kirkpatrick, Gyorgyi, Tishby, and Troyansky k 2 3 4 5 6 O'ann 0'2 O'c 2.41 5.19 10.74 21.83 44.01 1.38 4.25 9.58 20.6 42.8 1.0 4.17?.03 9.75?.05 20.9?.1 43.2?.2 0" 2.25 0.74 0.67 0.71 0.69 V 2.6?.2 1.5?.1 1.25?.05 1.1?.O5 1.05?.05 Table 1: Critical parameters for random k-SAT. 5 Outline of Statistical Mechanics Analysis Space permits only a sketch of our analysis of this model. Since the N inputs are binary, we may represent them as a vector, X, of Ising spins: X={xi=?l} i=l, ... N. Each random formula, F, can be written as a sum of its M clauses, Cj, M F = LCj, j=1 where k Cj = II (1 - Jj 1X)/2. 1=1 where the vector, Jj,l, has only one non-zero element, ?1, at the input which it selects. F evaluates to the number of clauses left unsatisfied by a particular configuration. It is natural to take the value of F to be the energy. The partition function, z = tr{x.}e.6.r = tr{x.} e.6 Cj , II j where f3 is the inverse of a fictitious temperature, factors into contributions from each clause. The "annealed" approximation mentioned above consists simply of taking the trace over each subproduct individually, neglecting their interactions. In this construction, we expect both energy and entropy, S, to be extensive quantities, that is, proportional to N. Fig. 4 shows that this is indeed the case for S( a). The lines in Fig. 4 are the annealed predictions S( a, k) = 1 - 0'/ aann. Expressions for the energy can also be obtained from the annealed theory, and used to compare the specific heat observed in numerical experiments with the simple limit in which the clauses do not interact. This gives evidence supporting the identification of the unsatisfied phase as a spin glass. Finally, a plausible phase diagram for the 0 at finite spin glass-like "unsatisfied" phase is obtained by solving for S(T) temperatures. = To perform the averaging over the random clauses correctly requires introducing replicas (see Mezard 1986), which are identical copies of the random formula, and defining q, the overlap between the expectation values of the spins in any two replicas, as the new order parameter. The results appear to be capable of accounting The Statistical Mechanics of k-Satisfaction for the difference between experiment and the annealed predictions at finite k. For example, an uncontrolled approximation in which we consider just two replicas gives the values of a2 in Table 1, and accounts rather closely for the average overlap found experimentally between pairs of lowest energy states, as shown in Fig. 5. The 2-replica theory gives q as the solution of a(k, q) = 2k(1 + q)k-l(4k - 2k+l + (1 - ql)/ln?l + q)(l - q)) for q as a function of a. This gives the lines in Fig 5. We defined a2 (in Table 1) as the point of inflection, or the maximum in the slope of q(a). Entropy tor It- SAT. l = 2. 3, t. S ,~~:- ' l i.frlk.ll'Sp?' n6k2 p' ? ' n2 f.k2 'qob.I2CM<A' 'qob.l2_ 'qobNek3' ',,1211<2' 'qob.t2Ok2' 'qob1121c2' 07 p'_ ' nlO, p' D ? nH .p? ....... ' n12U p2 ' ? o. 'n20 kf, p ' 0.1 ..-....t ? 0 x ? ? ? ' nlO kS p' ? ' n20kS p ' - 0.5 0' ~ 04 o. 03 02 o1 01 10 15 25 30 u t loO H/ N 111M 20 2S Fig. 4: Entropy as function of a for k = 2, 3, 4, and 5. Fig. 5: q calculated from 2-replica theory vs experimental ground state overlaps. Arrows pointing up are O'ann, arrows pointing down are a2. 6 Conclusions We have shown how finite size scaling methods from statistical physics can be used to model the threshold in randomly generated k-SAT problems. Given the good fit of our scaling analysis, we conjecture that this method can also give useful models of phase transitions in other combinatorial problems with a control parameter. Several authors have attempted to relate NP-hardness or NP-completeness to the characteristics of phase transitions in models of disordered systems. Fu and Anderson (see Fu 1989) have proposed spin glasses (magnets with 2-spin interactions of random sign) as having inherent exponential complexity. Huberman and colleagues (see Clearwater 1991) were first to focus on the diverging correlation length seen at continuous phase transitions as the root of computational complexity. In fact, both effects can play important roles, but are not sufficient and may not even be necessary. There are NP-complete problems (e.g. travelling salesman, or max-clique) which lack a phase boundary at which "hard problems" cluster. Percolation thresholds are phase transitions, yet the cost of exploring the largest cluster never exceeds N steps, Exponential search cost in k-SAT comes from the random signs of the inputs, which require that the space be searched repeatedly. Note that a satisfying 445 446 Kirkpatrick, Gyorgyi, TIshby, and Troyansky input configuration in 2-SAT can be determined, or its non-existence proven, in polynomial time, because it can be reduced to a percolation problem on a random directed graph (Aspvall 1979). The spin glass Hamiltonians studied by Fu and Anderson have a form close to our 2-SAT formulae, but the questions studied are different. Finding an input configuration which falsifies the minimum number of clauses is like finding the ground state in a spin glass phase, and is NP-hard when a > a c , even for k = 2. Therefore, if both diverging correlations (diverging in size if no lengths are defined) and random sign or "spin-glass" effects are present, we expect a local search like Davis-Putnam to be exponentially difficult on average. But these characteristics do not imply NP-completeness. 7 References Aspvall, B., Plass, M.F., and Tarjan, R.E. (1979) A linear-time algorithm for testing the truth of certain quantified Boolean formulae. Inform. Process. Let., Vol. 8., 1979, 289-314. Barker, V. E., and O'Connor, D. (1989). 32(3), 1989, 298-318. Commun. Assoc. for Computing Machinery, Cheeseman, P., Kanefsky, B., and Taylor, W.M. (1991). Where the really hard problems are. Proceedings IJCAI-91, 1991, 163-169. Clearwater, S.H., Huberman B.A., Hogg, T. (1991) Cooperative Solution of Constraint Satisfaction Problems. Science, Vol. 254, 1991, 1181-1183 Crawford, J.M. and Auton L.D. (1993). Experimental Results on the Crossover Point in Satisfiability Problems. Proc. of AAAI-99, 1993. Chvatal, V. and Reed, B. (1992) Mick Gets Some: The Odds are on his Side. Proc. of STOC, 1992, 620-627. Fu, Y. (1989). The Uses and Abuses of Statistical Mechanics in Computational Complexity. in Lectures in the Sciences of Complexity, ed. D. Stein, pp. 815-826, Addison-Wesley, 1989. Franco, J. and Paull, M. (1988). Probabilistic Analysis of the Davis-Putnam Procedure for solving the Satisfiability Problem. Discrete Applied Math., Vol. 5, 77-87, 1983. Goerdt, A. (1992). A threshold for unsatisfiability. Proc. 17th Int. Symp. on the Math. Foundations of Compo Sc., Prague, Czechoslovakia, 1992. Kirkpatrick, S. and Swendsen, R.H. (1985). Statistical Mechanics and Disordered Systems. CA CM, Vol. 28, 1985, 363-373. Kirkpatrick, S., and Selman, B. (1993), submitted for publication. Larrabee, T. and Tsuji, Y. (1993) Evidence for a Satisfiability Threshold for Random 3CNF Formulas, Proc. of the AAAI Spring Symposium on AI and NP-hard problems, Palto Alto, CA, 1993. Mezard, M., Parisi, G., Virasoro, M.A. (1986). Spin Glass Theory and Beyond, Singapore: World Scientific, 1986. Mitchell, D., Selman, B., and Levesque, H.J. (1992) Hard and Easy Distributions of SAT problems. Proc. of AAAI-92, 1992, 456-465. Stauffer, D. and Aharony, A. (1992) Introduction to Percolation Theory. London: Taylor and Francis, 1992. See especially Ch. 4. 

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Feature Densities are Required for Computing Feature Correspondences Subutai Ahmad Interval Research Corporation 1801-C Page Mill Road, Palo Alto, CA 94304 E-mail: ahmadCDinterval.com Abstract The feature correspondence problem is a classic hurdle in visual object-recognition concerned with determining the correct mapping between the features measured from the image and the features expected by the model. In this paper we show that determining good correspondences requires information about the joint probability density over the image features. We propose "likelihood based correspondence matching" as a general principle for selecting optimal correspondences. The approach is applicable to non-rigid models, allows nonlinear perspective transformations, and can optimally deal with occlusions and missing features. Experiments with rigid and non-rigid 3D hand gesture recognition support the theory. The likelihood based techniques show almost no decrease in classification performance when compared to performance with perfect correspondence knowledge. 1 INTRODUCTION The ability to deal with missing information is crucial in model-based vision systems. The feature correspondence problem is an example where the correct mapping between image features and model features is unknown at recognition time. For example, imagine a network trained to map fingertip locations to hand gestures. Given features extracted from an image, it becomes important to determine which features correspond to the thumb, to the index finger, etc. so we know which input units to clamp with which numbers. Success at the correspondence matching step 961 962 Ahmad / Class bouudary I Class 2 Class 1 o ? P2 L -_ _ _ _ _ _ _ _~_ _ _ _ _ _ _ _ _ _ _ _ _ __ o I Xl Figure 1: An example 2D feature space. Shaded regions denote high probability. Given measured values of 0.2 and 0.9, the points PI and P2 denote possible instantiations but PI is much more likely. is vital for correct classification. There has been much previous work on this topic (Connell and Brady 1987; Segen 1989; Huttenlocher and Ullman 1990; Pope and Lowe 1993) but a general solution has eluded the vision community. In this paper we propose a novel approach based on maximizing the probability of a set of models generating the given data. We show that neural networks trained to estimate the joint density between image features can be successfully used to recover the optimal correspondence. Unlike other techniques, the likelihood based approach is applicable to non-rigid models, allows perspective 3D transformations, and includes a principled method for dealing with occlusions and missing features. 1.1 A SIMPLE EXAMPLE Consider the idealized example depicted in Figure 1. The distribution of features is highly non-uniform (this is typical of non-rigid objects). The classification boundary is in general completely unrelated to the feature distribution. In this case, the class (posterior) probability approaches 1 as feature Xl approaches 0, and 0 as it approaches 1. Now suppose that two feature values 0.2 and 0.9 are measured from an image. The task is to decide which value gets assigned to X I and which value gets assigned to X2. A common strategy is to select the correspondence which gives the maximal network output (i.e. maximal posterior probability). In this example (and in general) such a strategy will pick point P2, the wrong correspondence. This is because the classifer output represents the probability of a class given a specific feature assignment and specific values. The correspondence problem however, is something completely different: it deals with the probability of getting the feature assignments and values in the first place. Feature Densities Are Required for Computing Feature Correspondences 2 LIKELIHOOD BASED CORRESPONDENCE MATCHING We can formalize the intuitive arguments in the previous section. Let C denote the set of classes under consideration. Let X denote the list of features measured from the image with correspondences unknown. Let A be the set of assignments of the measured values to the model features. Each assignment a E A reflects a particular choice of feature correspondences. \Ve consider two different problems: the task of choosing the best assignment a and the task of classifying the object given X. Selecting the best correspondence is equivalent to selecting the permutation that maximizes p(aIX, C). This can be re-written as: C)p(aIC) ( I X C) = p(Xla, pa- , p(XIC) (1) p(XIC) is a normalization factor that is constant across all a and can be ignored. Let Xa denote a specific feature vector constructed by applying permutation a to X. Then (1) is equivalent to maximizing: p(aIX, C) = p(xaIC)p(aIC) (2) p(aIC) denotes our prior knowledge about possible correspondences. (For example the knowledge that edge features cannot be matched to color features.) When no prior knowledge is available this term is constant. We denote the assignment that maximizes (2) the maximum likelihood correspondence match. Such a correspondence maximizes the probability that a set of visual models generated a given set of image features and will be the optimal correspondence in a Bayesian sense. 2.1 CLASSIFICATION In addition to computing correspondences, we would like to classify a model from the measured image features, i.e. compute p( CdX, C). The maximal-output based solution is equivalent to selecting the class Ci that maximizes p(Cilxa, C) over all assignments a and all classes Ci. It is easy to see that the optimal strategy is actually to compute the following weighted estimate over all candidate assignments: p (C.IX C) = Lap(CiIX, a, C)p(Xla, C)p(aIC) , , p(XIC) (3) Classification based on (3) is equivalent to selecting the class that maximizes: (4) a Note that the network output based solution represents quite a degraded estimate of (4). It does not consider the input density nor perform a weighting over possible 963 964 Ahmad correspondences. A reasonable approximation is to select the maximum likelihood correspondence according to (2) and then use this feature vector in the classification network. This is suboptimal since the weighting is not done but in our experience it yields results that are very close to those obtained with (4). 3 COMPUTING CORRESPONDENCES WITH GBF NETWORKS In order to compute (2) and (4) we consider networks of normalized Gaussian basis functions (GBF networks). The i'th output unit is computed as: (5) with: Here each basis function j is characterized by a mean vector f.lj and by oJ, a vector representing the diagonal covariance matrix. Wji represents the weight from the j'th Gaussian to the i'th output. 7rj is a weight attached to each basis function. Such networks have been popular recently and have proven to be useful in a number of applications (e.g. (Roscheisen et al. 1992; Poggio and Edelman 1990). For our current purpose, these networks have a number of advantages. Under certain training regimes such as EM or "soft clustering" (Dempster et al. 1977; Nowlan 1990) or an approximation such as K-11eans (Neal and Hinton 1993), the basis functions adapt to represent local probability densities. In particular p(xaIC) :::::: E j bj(x a). If standard error gradient training is used to set the weights Wij then Yi(X a ) :::::: p( Cilxa, C) Thus both (2) and (4) can be easilty computed.(Ahmad and Tresp 1993) showed that such networks can effectively learn feature density information for complex visual problems. (Poggio and Edelman 1990) have also shown that similar networks (with a different training regime) can learn to approximate the complex mappings that arise in 3D recognition. 3.1 OPTIMAL CORRESPONDENCE MATCHING WITH OCCLUSION An additional advantage of G BF networks trained in this way is that it is possible to obtain closed form solutions to the optimal classifier in the presence of missing or noisy features. It is also possible to correctly compute the probability of feature vectors containing missing dimensions. The solution consists of projecting each Gaussian onto the non-missing dimensions and evaluating the resulting network. Note that it is incorrect to simply substitute zero or any other single value for the missing dimensions. (For lack of space we refer the reader to (Ahmad and Tresp Feature Densities Are Required for Computing Feature Correspondences '"five" "four" '1hree" "two" "one" ..tlumbs _up " )Jointing ,. Figure 2: Classifiers were trained to recognize these 7 gestures. a 3D computer model of the hand is used to generate images of the hand in various poses. For each training example, we randomly choose a 3D orientation and depth, compute the 3D positions of the fingertips and project them onto 2D. There were 5 features yielding a lOD input space. 1993) for further details.) Thus likelihood based approaches using GBF networks can simultaneously optimally deal with occlusions and the correspondence problem. 4 EXPERIMENTAL RESULTS We have used the task of 3D gesture recognition to compare likelihood based methods to the network output based technique. (Figure 2 describes the task.) "\rVe considered both rigid and non-rigid gesture recognition tasks. We used a GBF network with 10 inputs, 1050 basis functions and 7 output units. For comparision we also trained a standard backpropagation network (BP) with 60 hidden units on the task. For this task we assume that during training all feature correspondences are known and that during training no feature values are noisy or missing. For this task we assume that during training all feature correspondences are known and that during training no feature values are noisy or missing. Classification performance with full correspondence information on an independent test set is about 92% for the GBF network and 93% for the BP network. (For other results see (\Villiams et al. 1993) who have also used the rigid version of this task as a benchmark.) 4.1 EXPERIMENTS WITH RIGID HAND POSES Table 1 plots the ability of the various methods to select the correct correspondence. Random patterns were selected from the test set and all 5! = 120 possible combinations were tried. MLCM denotes the percentage of times the maximum likelihood method (equation (2)) selected the correct feature correspondence. GBFM and BP-M denotes how often the maximal output method chooses the correct correspondence using GBF nets and BP. "Random" denotes the percentage if correspondences are chosen randomly. The substantially better performance of MLCM suggests that, at least for this task, density information is crucial. It is also interesting to examine the errors made by MLCM. A common error is to switch the features for the pinky and the adjacent finger for gestures "one", "two", "thumbs-up" and "pointing". These two fingertips often project very close to one another in many poses; such a mistake usually do not affect subsequent classification. 965 966 Ahmad Selection Method Random GBF-NI BP-M MLCM Percentage Correct 1.2% 8.8% 10.3% 62.0% Table 1: Percentage of correspondences selected correctly. Classifier BP-Random BP-11ax GBF-Max GBF-vVLC GBF-Known Classification Performance 28.0% 39.2% 47.3% 86.2% 91.8% Table 2: Classification without correspondence information. Table 2 shows classification performance when the correspondence is unknown. GBF-WLC denotes weighted likelihood classification using GBF networks to compute the feature densities and the posterior probabilities. Performance with the output based techniques are denoted with GBF-M and BP-M. BP-R denotes performance with random correspondences using the back propagation network. GBFknown plots the performance of the G BF network when all correspondences are known. The results are quite encouraging in that performance is only slightly degraded with WLC even though there is substantially less information present when correspondences are unknown. Although not shown, results with MLCM (i.e. not doing the weighting step but just choosing the correspondence with highest probability) are about 1% less than vVLC. This supports the theory that many of the errors of MLCM in Table 1 are inconsequential. 4.1.1 Missing Features and No Correspondences Figure 3 shows error as a function of the number of missing dimensions. (The missing dimensions were randomly selected from the test set.) Figure 3 plots the average number of classes that are assigned higher probability than the correct class. The network output method and weighted likelihood classification is compared to the case where all correspondences are known. In all cases the basis functions were projected onto the non-missing dimensions to approximate the Bayes-optimal condition. As before, the likelihood based method outperforms the output based method. Surprisingly, even with 4 of the 10 dimensions missing and with correspondences unknown, \VLC assigns highest probability to the correct class on average (performance score < 1.0). Feature Densities Are Required for Computing Feature Correspondences Error vs missing features without correspondence 3.5 3 2.5 GBF-M +-WLC -eG BF-Known +-- 2 Error 1.5 1 0.5 rL-_-~-::r- o L -_ _ _ _ _ _ o ~ ________ 1 ~ _ _ _ _ _ _ _ __ L_ _ _ _ _ _ _ _ 2 3 ~ ______ 4 ~ 5 No. of missing features Figure 3: Error with mIssmg features when no correspondence information is present. The y-axis denotes the average number of classes that are assigned higher probability than the correct class. 4.2 EXPERIMENTS WITH NON-RIGID HAND POSES In the previous experiments the hand configuration for each gesture remained rigid. Correspondence selection with non-rigid gestures was also tried out. As before a training set consisting of examples of each gesture was constructed. However, in this case, for each sample, a random perturbation (within 20 degrees) was added to each finger joint. The orientation of each sample was allowed to randomly vary by 45 degrees around the x, y, and z axes. When viewed on a screen the samples give the appearance of a hand wiggling around. Surprisingly, GBF networks with 210 hidden units consistently selected the correct correspondences with a performance of 94.9%. (The performance is actually better than the rigid case. This is because in this training set all possible 3D orientations were not allowed.) 5 DISCUSSION We have shown that estimates of joint feature densities can be used to successfully deal with lack of correspondence information even when some input features are missing. We have dealt mainly with the rather severe case where no prior information about correspondences is available. In this particular case to get the optimal correspondece, all n! possibilities must be considered. However this is usually not necessary. Useful techniques exist for reducing the number of possible correspondences. For example, (Huttenlocher and Ullman 1990) have argued that three fea- 967 968 Ahmad ture correspondences are enough to constrain the pose of rigid objects. In this case only O(n 3 ) matches need to be tested. In addition features usually fall into incompatible sets (e.g. edge features, corner features: etc.) further reducing the nWllber of potential matches. Finally: with ullage sequences one can use correspondence ulformation from the previous frame to constraul the set of correspondences in the current frame. \\llatever the situation, a likelihood based approach is a prulcipled method for evaluatulg the set of available matches. Acknowledgements 1'luch of this research was conducted at Siemens Central Research in :Munich, Germany. I would like to thank Volker Tresp at Siemens for many interesting discussions and Brigitte \Virtz for providulg the hand model. References Ahmad, S. and V. Tresp (1993). Some solutions to the missing feature problem Ul vision. In S. Hanson, J. Cowan: and C. Giles (Eds.), Advances in Neural Information Processing Systems 5, pp. 393 400. 110rgan Kaufmann Publishers. Connell, J. and 1'1. Brady (1987). Generating and generalizing models of visual objects. A1?tificial Intelligence 31: 159 183. Dempster, A.: N. Laird: and D. Rubin (1977). 1tlaximwu-likelihood fromulcomplete data via the E1'1 algorithm. J. Royal Statistical Soc. Ser. B 39, 1 38. Huttenlocher: D. and S. Ullman (1990). Recognizing solid objects by alignment with an ullage. International Journal of Computer Vision 5(2), 195 212. Neal, R. and G. HiIlton (1993). A new view of the E1tl algorithm that justifies incremental and other variants. Biometrika, submitted. Nowlan: S. (1990). 1/1aximwll likelihood competitive learnulg. III D. Touretzky (Ed.), Advances in Neural Information Processing Systems 2: pp. 574 582. San 1tlateo: CA: 1tlorgan Kaufmann Publishers. Poggio, T. and S. Edelman (1990). A network that learns to recognize threediIllensional objects. Nature 343(6225), 1 3. Pope: A. and D. Lowe (1993: 1tIay). Learuulg object recognition models from llllages. In Fourth International Confe1'ence on Computer Vision, Berlin. IEEE Computer Society Press. Roscheisen: ~I.: R. Hofmann, and V. Tresp (1992). Neural control for rolling mills: Incorporating domain theories to overcome data deficiency. In 1?1. J., H. S.J.: and L. R. (Eds.), Advances in Neural Information Processing Systems 4. 1'Iorgan Kaufman. Segen: J. (1989). 1:1odel learning and recognition of nonrigid objects. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition: San Diego: CA. \Villiams: C. K.: R. S. Zemel: and ~I. C. 1/10zer (1993). Unsupervised learning of object models. In AAAI Fall 1993 Symposium on Machine Lea?ming in Computer Vision: pp. 20 24. Proceedings available as AAAI Tech Report FSS-93-04. 

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Counting function theorem for multi-layer networks Adam Kowalczyk Telecom Australia, Research Laboratories 770 Blackburn Road, Clayton, Vic. 3168, Australia (a.kowalczyk@trl.oz.au) Abstract x We show that a randomly selected N-tuple of points ofRn with probability> 0 is such that any multi-layer percept ron with the first hidden layer composed of hi threshold logic units can implement exactly 2 2:~~~ ( Nil) different dichotomies of x. If N > hin then such a perceptron must have all units of the first hidden layer fully connected to inputs. This implies the maximal capacities (in the sense of Cover) of 2n input patterns per hidden unit and 2 input patterns per synaptic weight of such networks (both capacities are achieved by networks with single hidden layer and are the same as for a single neuron). Comparing these results with recent estimates of VC-dimension we find that in contrast to the single neuron case, for sufficiently large nand hl, the VC-dimension exceeds Cover's capacity. 1 Introduction In the course of theoretical justification of many of the claims made about neural networks regarding their ability to learn a set of patterns and their ability to generalise, various concepts of maximal storage capacity were developed. In particular Cover's capacity [4] and VC-dimension [12] are two expressions of this notion and are of special interest here. We should stress that both capacities are not easy to compute and are presen tly known in a few particular cases of feedforward networks only. VC-dimension, in spite of being introduced much later, has been far 375 376 Kowalczyk more researched, perhaps due to its significance expressed by a well known relation between generalisation and learning errors [12, 3]. Another reason why Cover's capacity gains less attention, perhaps, is that for the single neuron case it is twice higher than VC-dimension. Thus if one would hypothesise a similar relation to be true for other feedforward networks, he would judge Cover's capacity to be quite an unattractive parameter for generalisation estimates, where VC-dimension is believed to be unrealistically big. One of the aims of this paper is to show that this last hypothesis is not true, at least for some feedforward networks with sufficiently large number of hidden units. In the following we will always consider multilayer perceptrons with n continuously-valued inputs, a single binary output, and one or more hidden layers, the first of which is made up of threshold logic units only. The derivation of Cover's capacity for a single neuron in [4] is based on the so-called Function Counting Theorem, proved for the linear function in the sixties (c.f. [4]), which states that for an N -tuple i of points in general position one can implement (Nil) 2 2::~=o different dichotomies of i. Extension of this result to the multilayer case is still an open problem (c.f. T. Cover's address at NIPS'92). One of the complications arising there is that in contrast to the single neuron case even for perceptrons with two hidden units the number of implementable dichotomies may be different for different N -tuples in general position [8]. Our first main result states that this dependence on i is relatively weak, that for a multilayer perceptron the number of implementable dichotomies (counting function) is constant on each of a finite number of connected components into which the space of N-tuples in general position can be decomposed. Then we show that for one of these components C(N, nh 1 ) different dichotomies can be implemented, where hl is the number of hidden units in the first hidden layer (all assumed to be linear threshold logic units). This leads to an upper bound on Cover's capacity of 2n input patterns per (hidden) neuron and 2 patterns per adjustable synaptic weight, the same as for a single neuron. Comparing this result with a recent lower bound on VC-dimension of multilayer perceptrons [10] we find that for for sufficiently large nand hl the VC-dimension is higher than Cover's capacity (by a factor log2(h 1 )). C( N, n) deC The paper extends some results announced in [5] and is an abbreviated version of a forthcoming paper [6J. 2 2.1 Results Standing assumptions and basic notation We recall that in this paper a multilayer perceptron means a layered feedforward network with one or more hidden layers, and the first hidden layer built exclusively from threshold logic units. A dichotomy of an N-tuple i = (Xl, ... , XN) E (Rn)N is a function 6: {Xl, ... , XN} {0,1}. For a multilayer perceptron F : Rn - {O,l} let i ~ CF(i) denote the number of different dichotomies of i which can be implemented for all possible selections of synaptic weights and biases. We shail call CF(i) a counting function following the terminology used in [4]. Counting Function Theorem for Multi-Layer Networks Example 1. ? : Rn -+ C?(x) = C(N, n) def 2 :E?=o (Nil) for a single threshold logic unit {O, 1} [4]. 0 Points of an N-tuple x E (Rn)N are said to be in general po&ition if there does not exist an 1 ~r min(N, n - l)-dimensional affine hyperplane in R n containing (l + 2) of them. We use a symbol gP(n, N) C (Rn)N to denote that set of all N-tuples x in general position. Throughout this paper we assume to be given a probability measure dlJ Rn such that the density f : Rn -+ R is a continuous function. 2.2 def f dx on Counting function is locally constant We start with a basic characterisations of the subset gP(n, N) C (Rn)N. Theorem 1 (i) gP(n, N) is an open and dense subset of (Rn)N with a finite number of connected components. (ii) Any of these components is unbounded, has an infinite Lebesgue measure and has a positive probability measure. Proof outline. (i) The key point to observe is that gP(n, N) = {x : p(x) =I- O}, where p : (Rn)N -+ R is a polynomial on (Rn)N. This implies immediately that gP( n, N) is open and dense in (Rn)N. The finite number of connected components follows from the results of Milnor [7] (c.f. [2]). (ii) This follows from an observation that each of the connected components Ci has the property that if (Xl, ... , XN) E Ci and a > 0, then (ax!, ... ,axN) E C,. 0 As Example 1 shows, for a single neuron the counting function is constant on gP(n, N). However, this may not be the case even for perceptrons with two hidden units and two inputs (c.f. [8, 6] for such examples and Corollary 8). Our first main result states that this dependence on x is relatively weak. Theorem 2 CF(X) is constant on connected components ofgP(n, N). Proof outline. The basic heuristic behind the proof of this theorem is quite simple. If we have an N-tuple E (Rn)N which is split into two parts by a hyperplane, then this split is preserved for any sufficiently small perturbation Y E (Rn)N of x, and vice versa, any split of y corresponds to a split of X. The crux is to show that if x is in general position, then a minute perturbation y of x cannot allow a bigger number of splits than is possible for x. We refer to [6] for details. 0 x The following corollary outlines the main impact of Theorem 2 on the rest of the paper. It reduces the problem of investigation of the function CF(X) on gP(n, N) to a consideration of a set of individual, special cases of N-tuples which, in particular, are amenable to be solved analytically. x Corollary 3 If E gP(n, N), then CF(X) tuple f E (Rn)N with a probability> O. = CF(f) for a randomly &elected N- 377 378 Kowalczyk 2.3 A case of special component of gP( n, N) The following theorem is the crux of the paper. Theorem 4 There exists a connected component CC C gP( n , N) C (Rn)N such that CF(i) h1n (N - 1) = C(N, nh = 2 t; i 1) (for i E CC) with equality iff the input and first hidden layer are fully connected. The synaptic weights to units not in the first hidden layer can be constant. Using now Corollary 3 we obtain: Corollary 5 CF(i) = C(N, nh 1 ) for i E (Rn)N with a probability> O. The component CC C gP(n, N) in Theorem 4 is defined as the connected component containing (1) where c : R __ R n is the curve defined as c(t) de! (t, t 2, ... ,tn ) for t E Rand o < tt < t2 < ... < tN are some numbers (this example has been considered previously in [11]). The essential part of the proof of Theorem 4 is showing the basic properties of the N-tuple PN which will be described by the Lemma below. Any dichotomy h of the N-tuple fiN (c.f. 1) is uniquely defined by its value at C(tl) (2 options) and the set of indices 1 :s; il < i2 < ... < ile < N of all transitional pairs (C(ti;), C(ti;+I)), i.e. all indices i j such that h(C(ti;)) =f: h(C(ti;+I)), where j = 1, "'1 k, (additional (N;l) options). Thus it is easily seen that there exist altogether 2 (N;I) different dichotomies of where 0 5 k PN 5 k of transitional pairs, < N. Lemma 6 Given integers n, N, h transitional pairs. (i) If k for any given number > 0, k nh, then there exist hyperplanes .(pj) = 9 ~ 0 and a dichotomy h of PN with k H(Wi,bi)' (Wi, bd E Rn x R, such that (bo + t,.,9(W'. P; + b,?) , (2) (3) for i = 1, ... , hand j = 1, "', N; here Vi de! 1 ifn is even and Vi del (_l)i ifn is odd, bo de! -0.5 if n is odd, h is even and h(po) = 1, and bo de! 0.5, otherwise. (ii) If k (Will = nh, then Wij =f: 0 for j = 1, ... , nand i = 1, "'1 h, where Wi = Wi2, ... ,Win)' (iii) If k > nh, then (2) and (3) cannot be satisfied. The proof of Lemma 6 relies on usage of the Vandermonde determinant and its derivatives. It is quite technical and thus not included here (c.f. [6] for details). Counting Function Theorem for Multi-Layer Networks Theorem 7 (Mitchison & Durbin [lO]f../?? Huang & Huang [6] .' ... Baum [2]. Sakurai [11] .... ......... 10 2 1 -t--~~-.I--~~~I~I~~~I~~~-.I--~.~ 1 2 5 10 102 10 3 10 4 Number of hidden units(h 1) Figure 1: Some estimates of capacity. 3 3.1 Discussion An upper bound on Cover's capacity The Cover's capacity (or just capacity) of a neural network F : R n -+ {O,1}, G ap( F), is defined as the maximal N such that for a randomly selected N - tuple i = (Xl, ... ,XN) E (Rn)N of points of Rn, the network can implement 1/2 of all dichotomies of with probability 1 [4, 8]. x Corollary 5 implies that Gap(F) is not greater than maximal N such that Gp(PN )/2 N = G(N, nhl) ~ 1/2. (4) since any property which holds with probability 1 on (Rn)N must also hold probability 1 on GG (c.f Theorem 4). The left-hand-side of the above equation is just the sum of the binomial expansion of (1/2 + 1/2)N-l up to hln-th term, so, using the symmetry argument, we find that it is ~ 1/2 if and only if it has at least half of the all terms, i.e. when N - 1 + 1 ::; 2(hln + 1). Thus the 2(hln + 1) is the maximal value of N satisfying (4). 1 Now let us recall that a multilayer perceptron as in this paper can implement any dichotomy of any N-tuple in general position if N < nhl + 1 [I, 11]. This leads to the following result: x Theorem 7 nhl + 1 ~ Gap(F) ::; 2(nhl + 1). lNote that for large N the choice of cutoff value 1/2 is not critical, since the probability of a dichotomy being implementable drops rapidly as hi n a.pproa.ches 2N /2. 379 380 Kowalczyk N I #w 10 I/) dVC<F)/#w (Sakurai [11]) ~ 0) ?iii :it .2 8 i5.. co c>.. -... I/) 0 6 CD .a E ~ Z ..... 4 CIl E ..---(Cap(F)/#W ) (Theorem 7) g 15 c.. "5 2 c.. .s 0 Figure 2: Comparison of estimates of the ratios of Cover's capacity per synaptic weight (Cap(F)/#w) and VC-dimension per synaptic weight (dvc(F)/#w). (Note that the upper bound for VC-dimension has so far been proved for low number of hidden layers [9,10].) for any multilayer perceptron F : R n -+ {O, I} with the first hidden layer built from the hi threshold logic units. For the most efficient networks in this class, with a single hidden layer, we thus obtain the following result: 1 - O(I/nhl) ::; Cap(F)/#w ::; 2, where #w denotes the number of synaptk weights and biases. 3.2 A relation to VC-dimension The VC-dimension, dvc(F), is defined as the largest N such that there exists an N-tuple i = (Xl, ... ,XN) E (Rn)N for which the network can implement all possible 2N dichotomies. Recent results of Sakurai [10] imply (5) For sufficiently large nand hl this estimate exceeds 2(nhl + 1) which is an upper bound on Cap(F). Thus, in contrast to the single threshold logic unit case we have the following (c.f. Fig. 3): Corollary 8 Cap(F) 3.3 < dvc(F) if hi ? 1. Memorisation ability of multilayer perceptron Corollary 8 combined with Theorem 7 and Figure 2 imply that for some cases of patters in general position multilayer perceptron can memorise and reliably retrieve Counting Function Theorem for Multi-Layer Networks (even with 100% accuracy) much more (~ log2(h 1 ) times more) than 2 patterns per connection, as is the case for a single neuron [4]. This proves that co-operation between hidden units can significantly improve the storage efficiency of neural networks. 3.4 A relation to PAC learning Vapnik's estimate of generalisation error [12] (an error rate on independent test set) EG(F) ~ EL(F) + D(N, dvc(F), EL, '1) (6) holds for N > dvc(F) with probability larger that (1 - '1). It contains two terms: (i) learning error E L( F) and (ii) confidence interval D(p, dvc, EL, '1) where del 2W(p, dvc, '1) [1 + ,,11 + EL!W(p, dvc, '1)] , 2N w(N, dvc, 11) = (In dvc dvc + 1) 2N - In '1 N? The ability of obtaining small learning error EL(F) is, in a sense, controlled by Cap(F), while the size of the confidence interval D is controlled by both dvc(F) and Cap(F) (through EL(F)). For a multilayer perceptron as in Theorem 7 when dvc(F) ? Cap(F) (Fig. 2) it can turn out that actually the capacity rather than the VC-dimension is the most critical factor in obtaining low generalisation error EG(F). This obviously warrants further research into the relation between capacity and generalisation. The theoretical estimates of generalisation error based on VC-dimension are believed to be too pessimistic in comparison with some experiments. One may hypothesise that this is caused by too high values of dvc(F) used in estimates such as (6). Since Cover's capacity in the case multilayer perceptron with hl ? 1 turned up to be much lower than VC-dimension, one may hope that more realistic estimates could be achieved with generalisation estimates linked directly to capacity. This subject will obviously require further research. Note that some results along these lines can be found in Cover's paper [4]. 3.5 Some open problems Theorem 7 gives estimates of capacity per variable connection for a network with the minimal number of neurons in the first hidden layer showing that these neurons have to be fully connected. The natural question arises at this point as to whether a network with a bigger number but not fully connected neurons in the first hidden layer can achieve a better capacity (per adjustable synaptic weight). The values of the counting function i f-t Cp(i) are provided in this paper for the particular class of points in general position, for i E CC C (Rn)N. The natural question is whether they may be by chance a lower or upper bound for the counting function for the general case of i E (Rn)N ? The results of Sakurai [11] seem to point to the former case: in his case, the sequences PN (p!, ... , PN) turned out to be "the hardest" in terms of hidden units required to implement 100% of = 381 382 Kowalczyk dichotomies. Corollary 8 and Figure 1 also support this lower bound hypothesis. They imply in particular that there exists a N'-tuple Y = (Yl, Yl, ... , YN') E (Rfl.)N', where N' deC VC-dimension sufficiently large nand h. 4 > N, such that CF(Y) = 2N' ? 2N > CF(PN) for Acknowledgement The permission of Managing Director, Research and Information Technology, Telecom Australia, to publish this paper is gratefully acknowledged. References [1] E. Baum. On the capabilities of multilayer perceptrons. Journal of Complezity, 4:193-215, 1988. [2] S. Ben-David and M. Lindenbaum. Localization VS. identification of semialgebraic sets. In Proceedings of the Sizth Annual Workshop on Computational Learning Theory (to appear), 1993. [3] A. Blumer, A. Ehrenfeucht, D. Haussler, and M.K. Warmuth. Learnability and the Vapnik-Chernovenkis dimensions. Journal of the ACM, 36:929-965, (Oct. 1989). [4] T.M. Cover. Geometrical and statistical properties of linear inequalities with applications to pattern recognition. IEEE Trans. Elec. Comp., EC-14:326334, 1965. [5) A. Kowalczyk. Some estimates of necessary number of connections and hidden units for feed-forward networks. In S.l. Hanson et al., editor, Advances in Neural Information Processing Systems, volume 5. Morgan Kaufman Publishers, Inc., 1992. [6] A. Kowalczyk. Estimates of storage capacity of multi-layer perceptron with threshold logic hidden units. In preparation, 1994. [7) J. Milnor. On Betti numbers of real varieties. Proceedings of AMS, 15:275-280, 1964. [8] G.J. Mitchison and R.M. Durbin. Bounds on the learning capacity of some multi-layer networks. Biological Cybernetics, 60:345-356, (1989). [9] A. Sakurai. On the VC-dimension of depth four threshold circuits and the complexity of boolean-valued functions. Manuscript, Advanced Research Laboratory, Hitachi Ltd., 1993. [10] A. Sakurai. Tighter bounds of the VC-dimension of three-layer networks. In WCNN93, 1993. [11] A. Sakurai. n-h-1 networks store no less n? h + 1 examples but sometimes no more. In Proceedings of IJCNN9~, pages 111-936-111-941. IEEE, June 1992. [12] V. Vapnik. Estimation of Dependences Based on Empirical Data. SpringerVerlag, 1982. 

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760 A NOVEL NET THAT LEARNS SEQUENTIAL DECISION PROCESS G.Z. SUN, Y.C. LEE and H.H. CHEN Department of PhYJicJ and AJtronomy and InJtitute for Advanced Computer StudieJ UNIVERSITY OF MARYLAND,COLLEGE PARK,MD 20742 ABSTRACT We propose a new scheme to construct neural networks to classify patterns. The new scheme has several novel features : 1. We focus attention on the important attributes of patterns in ranking order. Extract the most important ones first and the less important ones later. 2. In training we use the information as a measure instead of the error function. 3. A multi-percept ron-like architecture is formed auomatically. Decision is made according to the tree structure of learned attributes. This new scheme is expected to self-organize and perform well in large scale problems. ? American Institute of Physics 1988 761 1 INTRODUCTION It is well known that two-layered percept ron with binary connections but no hidden units is unsuitable as a classifier due to its limited power [1]. It cannot solve even the simple exclusive-or problem. Two extensions have been prop'osed to remedy this problem. The first is to use higher order connections l2]. It has been demonstrated that high order connections could in many cases solve the problem with speed and high accuracy [3], [4]. The representations in general are more local than distributive. The main drawback is however the combinatorial explosion of the number of high-order terms. Some kind of heuristic judgement has to be made in the choice of these terms to be represented in the network. A second proposal is the multi-layered binary network with hidden units r5]. These hidden units function as features extracted from the bottom input layer to facilitate the classification of patterns by the output units. In order to train the weights, learning algorithms have been proposed that backpropagate the errors from the visible output layer to the hidden layers for eventual adaptation to the desired values. The multi-layered networks enjoy great popularity in their flexibility. However, there are also problems in implementing the multi-layered nets. Firstly, there is the problem of allocating the resources. Namely, how many hidden units would be optimal for a particular problem. If we allocate too many, it is not only wasteful but also could negatively affect the performance of the network. Since too many hidden units implies too many free parameters to fit specifically the training patterns. Their ability to generalize to noval test patterns would be adversely affected. On the other hand, if too few hidden units were allocated then the network would not have the power even to represent the trainig set. How could one judge beforehand how many are needed in solving a problem? This is similar to the problem encountered in the high order net in its choice of high order terms to be represented. Secondly, there is also the problem of scaling up the network. Since the network represents a parallel or coorperative process of the whole system, each added unit would interact with every other units. This would become a serious problem when the size of our patterns becomes large. Thirdly, there is no sequential communication among the patterns in the conventional network. To accomplish a cognitive function we would need the patterns to interact and communicate with each other as the human reasoning does. It is difficult to envision such an interacton in current systems which are basically input-output mappings. 2 THE NEW SCHEME In this paper, we would like to propose a scheme that constructs a network taking advantages of both the parallel and the sequential processes. We note that in order to classify patterns, one has to extract the intrinsic features, which we call attributes. For a complex pattern set, there may be a large number of attributes. But differnt attributes may have different 762 ranking of importance. Instead of ext racing them all simultaneously it may be wiser to extract them sequentially in order of its importance [6], [7]. Here the importance of an attribute is determined by its ability to partition the pattern set into sub-categories. A measure of this ability of a processing unit should be based on the extracted information. For simplicity, let us assume that there are only two categories so that the units have only binary output values 1 and but the input patterns may have analog representations). We call these units, including their connection weights to the input layer, nodes. For given connection weights, the patterns that are classified by a node as in category 1 may have their true classifications either 1 or 0. Similarly, the patterns that are classified by a node as in category 0 may also have their true classifications either 1 or o. As a result, four groups of patterns are formed: (1,1), (0,0), (1,0), (0,1). We then need to judge on the efficiency of the node by its ability to split these patterns optimally. To do this we shall construct the impurity fuctions for the node. Before splitting, the impurity of the input patterns reaching the node is given by ?( (1) where pt = Nf / N is the probability of being truely classified as in category 1, and P~ = N~/N is the probability of being truely classified as in category o. After splitting, the patterns are channelled into two branches, the impurity becomes 1(1 = -Pt L j=O,1 P(j, 1) logP(j, 1) - P; L P(j, O)logP(j, 0) (2) j=O,1 where Pi = Ni / N is the probability of being classified by the node as in category 1, P; = N8/N is the probability of being classified by the node as in category 0, and P(j, i) is the probability of a pattern, which should be in category j, but is classified by the node as in category i. The difference (3) represents the decrease of the impurity at the node after splitting. It is the quantity that we seek to optimize at each node. The logarithm in the impurity function come from the information entropy of Shannon and Weaver. For all practical purI?ose, we found the. optimization of (3) the same as maximizing the entropy l6] where Ni is the number of training patterns classified by the node as in category i, N ij is the number of training patterns with true classification in category i but classified by the node as in category j. Later we shall call the terms in the first bracket SI and the second S2. Obviously, we have i = 0,1 763 After we trained the first unit, the training patterns were split into two branches by the unit. If the classificaton in either one of these two branches is pure enough, or equivalently either one of Sl and S2 is fairly close to 1, then we would terminate that branch ( or branches) as a leaf of the decision tree, and classify the patterns as such. On the other hand, if either branch is not pure enough, we add additional node to split the pattern set further. The subsequent unit is trained with only those patterns channeled through this branch. These operations are repeated until all the branches are terminated as leaves. 3 LEARNING ALGORITHM We used the stochastic gradient descent method to learn the weights of each node. The training set for each node are those patterns being channeled to this node. As stated in the previous section, we seek to maximize the entropy function S. The learning of the weights is therefore conducted through oS 1:::. Wj = 11 ow- (5) J Where 11 is the learning rate. The gradient of S can be calculated from the following equation oS = ~ [(1 _ 2NJ1) oNn oWj N Nl oW; (1 _ 2NJo) ONIO NJ oWj + (1 _ 2 Nil ) oNOl + Nl oWj + (1 _ 2N'fO) ONoo] NJ oWj (6) Using analog units or = 1 + exp( - we have oor = ow- 1 Lj WjII) orC1 _ or)!'; J J Furthermore, let Ar then N;; = t. = 1 or 0 being the [iA' + (1 - (7) (8) true answer for the input pattern r , i)(1 - A') 1[i O' + (1 - j)(1 - 0') 1 (9) Substituting these into equation (5), we get 1:::.Wj = 2T} :L[2Ar(NU - NlO) r Nl No + Ni~ - Ni;]or(l - or)IJ (10) No Nl In applying the formula (10),instead of calculating the whole summation at once, we update the weights for each pattern individually. Meanwhile we update N ij in accord with equation (9). 764 Figure 1: The given classification tree, where 01 , O'l and 03 are chosen to be all zeros in the numerical example. 4 AN EXAMPLE To illustrate our method, we construct an example which is itself a decision tree. Assuming there are three hidden variables ai, a'l, a3, a pattern is given by a ten-dimensional vector II, I'l, ... , 110 , constructed from the three hidden variables as follows + a3 II - al 1'l - 2al - a'l 16 17 - a3 - 2a'l 18 - 2al 19 - 4a3 - 3a l 13 I" Is + 2a'l + 3a3 - al - 5al - 4a" 110 2a3 a3 - al 2al + 3a3 + 2a'l + 2 a 3? A given pattern is classified as either 1 (yes) or 0 (no) according to the corresponding values of the hidden variables ai, a'l, a3. The actual decision is derived from the decision tree in Fig.1. In order to learn this classification tree, we construct a training set of 5000 patterns generated by randomly chosen values ai, a'l, a3 in the interval -1 to +1. We randomly choose the initial weights for each node, and terminate 765 5=0.79 G 51 =0.60/ ~=0.87 " G G 51 =0.65/ VI 51 = 0.85/ (SS/S)W i (fIg (2519/35) ,S2= 0.88 ~OCE:S] (16171114) ~= 0.73 5. =0.90/ (92/S)rul 52=0.96 ffQ](548/12) Figure 2: The learned classification tree structure a branch as a leaf whenever the branch entropy is greater than 0.80. The entropy is started at S = 0.65, and terminated at its maximum value S = 0.79 for the first node. The two branches of this node have the entropy fuction valued at SI = 0.61, S2 = 0.87 respectively. This corrosponds to 2446 patterns channeled to the first branch and 2554 to the second. Since S2 > 0.80 we terminate the second branch. Among 2554 patterns channeled to the second branch there are 2519 patterns with true classification as no and 35 yes which are considered as errors. After completing the whole training process, there are totally four nodes automatically introduced. The final result is shown in a tree structure in Fig.2. The total errors classified by the learned tree are 3.4 % of the 5000 trainig patterns. After trainig we have tested the result using 10000 novel patterns, the error among which is 3.2 %. 5 SUMMARY We propose here a new scheme to construct neural network that can automatically learn the attributes sequentially to facilitate the classification of patterns according to the ranking importance of each attribute. This scheme uses information as a measure of the performance of each unit. It is 766 self-organized into a presumably optimal structure for a specific task. The sequential learning procedure focuses attention of the network to the most important attribute first and then branches out' to the less important attributes. This strategy of searching for attributes would alleviate the scale up problem forced by the overall parallel back-propagation scheme. It also avoids the problem of resource allocation encountered in the high-order net and the multi-layered net. In the example we showed the performance of the new method is satisfactory. We expect much better performance in problems that demand large size of units. 6 acknowledgement This work is partially supported by AFOSR under the grant 87-0388. References [1] M. Minsky and S. Papert, Perceptron, MIT Press Cambridge, Ma(1969). [2] Y.C. Lee, G. Doolen, H.H. Chen, G.Z. Sun, T. Maxwell, H.Y. Lee and C.L. Giles, Machine Learning Using A High Order Connection Netweork, Physica D22,776-306 (1986). [3] H.H. Chen, Y.C. Lee, G.Z. Sun, H.Y. Lee, T. Maxwell and C.L. Giles, High Order Connection Model For Associate Memory, AlP Proceedings Vol.151,p.86, Ed. John Denker (1986). [4] T. Maxwell, C.L. Giles, Y.C. Lee and H.H. Chen, Nonlinear Dynamics of Artificial Neural System, AlP Proceedings Vol.151,p.299, Ed. John Denker(1986). [5] D. Rummenlhart and J. McClelland, Parallel Distributit'e Processing, MIT Press(1986). [6] L. Breiman, J. Friedman, R. Olshen, C.J. Stone, Classification and Regression Trees,Wadsworth Belmont, California(1984). [7] J.R. Quinlan, Machine Learning, Vol.1 No.1(1986). 

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Estimating analogical similarity by dot-products of Holographic Reduced Representations. Tony A. Plate Department of Computer Science, University of Toronto Toronto, Ontario, Canada M5S 1A4 email: tap@ai.utoronto.ca Abstract Models of analog retrieval require a computationally cheap method of estimating similarity between a probe and the candidates in a large pool of memory items. The vector dot-product operation would be ideal for this purpose if it were possible to encode complex structures as vector representations in such a way that the superficial similarity of vector representations reflected underlying structural similarity. This paper describes how such an encoding is provided by Holographic Reduced Representations (HRRs), which are a method for encoding nested relational structures as fixed-width distributed representations. The conditions under which structural similarity is reflected in the dot-product rankings of HRRs are discussed. 1 INTRODUCTION Gentner and Markman (1992) suggested that the ability to deal with analogy will be a "Watershed or Waterloo" for connectionist models. They identified "structural alignment" as the central aspect of analogy making. They noted the apparent ease with which people can perform structural alignment in a wide variety of tasks and were pessimistic about the prospects for the development of a distributed connectionist model that could be useful in performing structural alignment. In this paper I describe how Holographic Reduced Representations (HRRs) (Plate, 1991; Plate, 1994), a fixed-width distributed representation for nested structures, can be used to obtain fast estimates of analogical similarity. A HRR is a high dimensional vector, 1109 1110 Plate and the vector dot-product of two HRRs is an efficiently computable estimate of the overall similarity between the two structures represented. This estimate reflects both surface similarity and some aspects of structural similarity, l even though alignments are not explicitly calculated. I also describe contextualization, an enrichment ofHRRs designed to make dot-product comparisons of HRRs more sensitive to structural similarity. 2 STRUCTURAL ALIGNMENT & ANALOGICAL REMINDING People appear to perform structural alignment in a wide variety of tasks, including perception, problem solving, and memory recall (Gentner and Markman, 1992; Markman, Gentner and Wisniewski, 1993). One task many researchers have investigated is analog recall. A subject is shown a number of stories and later is shown a probe story. The task is to recall stories that are similar to the probe story (and sometimes evaluate the degree of similarity and perform analogical reasoning). MACIFAC, a computer models of this process, has two stages(Gentner and Forbus, 1991). The first stage selects a few likely analogs from a large number of potential analogs. The second stage searches for an optimal (or at least good) mapping between each selected story and the probe story and outputs those with the best mappings. Two stages are necessary because it is too computationally expensive to search for an optimal mapping between the probe and all stories in memory. An important requirement for a first stage is that its performance scale well with both the size and number of episodes in long-term memory. This prevents the first stage of MACIFAC from considering any structural features. Large pool of items in memory 0000 0 0 0 0 000 0 08 0 0 0 0 0000 0 0 0 0 000 00 0 o:J0 0 0 0 3 o0 ~~ 0 Probe Good analogies analogies 00 o 0 00 o 0 0 Expensive selection pro- Cheap filtering process based on surface features cess based on structural features Figure 1: General architecture of a two-stage retrieval model. While it is indisputable that people take structural correspondences into account when evaluating and using analogies (Gentner, Rattermann and Forbus, 1993), it is less certain whether structural similarity influences access to long term memory (i.e., the first-stage reminding process). Some studies have found little effect of analogical similarity on reminding (Gentner and Forbus, 1991; Gentner, Rattermann and Forbus, 1993), while others have found some effect (Wharton et aI., 1994). l"Surface features" of stories are the features of the entities and relations involved, and "structural features" are the relationships among the relations and entities. Estimating Analogical Similarity by Dot-Products of Holographic Reduced Representations In any case, surface features appear to influence the likelihood of a reminding far more than do structural features. Studies that have found an effect of structural similarity on reminding seem to indicate the effect only exists, or is greater, in the presence of surface similarity (Gentner and Forbus, 1991; Gentner, Rattermann and Forbus, 1993; Thagard et al., 1990). 2.1 EXAMPLES OF ANALOGY BETWEEN NESTED STRUCTURES. To test how well the HRR dot-product works as an estimate of analogical similarity between nested relational structures I used the following set of simple episodes (see Plate (1993) for the full set). The memorized episodes are similar in different ways to the probe. These examples are adapted from (Thagard et al., 1990). Probe: Spot bit Jane, causing Jane to flee from Spot. Episodes in long-term memory: El (L8) Fido bit John, causing John to flee from Fido. E2 (ANcm) Fred bit Rover, causing Rover to flee from Fred. E3 (AN) Felix bit Mort, causing Mort to flee from Felix. E6 (88) John fled from Fido, causing Fido to bite John. E7 (FA) Mort bit Felix, causing Mort to flee from Felix. In these episodes Jane, John, and Fred are people, Spot, Fido and Rover are dogs, Felix is a cat, and Mort is a mouse. All of these are objects, represented by token vectors. Tokens of the same type are considered to be similar to each other, but not to tokens of other types. Bite, flee, and cause are relations. The argument structure of the cause relation, and the patterns in which objects fill multiple roles constitutes the higher-order structure. The second column classifies the relationship between each episode and the probe using Gentner et aI's types of similarity: LS (Literal Similarity) shares relations, object features, and higher-order structure; AN (Analogy, also called True Analogy) shares relations and higher-order structure, but not object features; SS (Surface Similarity, also called Mere Appearance) shares relations and object features, but not higher-order structure; FA (False Analogy) shares relations only. ANcm denotes a cross-mapped analogy - it involves the same types of objects as the probe, but the types of corresponding objects are swapped. 2.2 MACIFAC PERFORMANCE ON TEST EXAMPLES The first stage of MACIFAC (the "Many Are Called" stage) only inspects object features and relations. It uses a vector representation of surface features. Each location in the vector corresponds to a surface feature of an object, relation or function, and the value in the location is the number of times the feature occurs in the structure. The first-stage estimate of the similarity between two structures is the dot-product of their feature-count vectors. A threshold is used to select likely analogies. It would give El (L8), E2 (ANcm), and E6 (88) equal and highest scores, i.e., (L8, ANcm, 88) > (AN, FA) The Structure Mapping Engine (SME) (Falkenhainer, Forbus and Gentner, 1989) is used as the second stage of MACIFAC (the "Few Are Chosen" stage). The rules of SME are that mapped relations must match, all the arguments of mapped relations must be mapped consistently, and mapping of objects must be one-to-one. SME would detect structural correspondences between each episode and the probe and give the literally similar and analogous episodes the highest rankings, i.e., LS > AN > (SS, FA). 1111 1112 Plate A simplified view of the overall similarity scores from MAC and the full MACIFAC is shown in Table 1. There are four conditions - the two structures being compared can be similar in structure and/or in object attributes. In all four conditions, the structures are assumed to involve similar relations - only structural and object attribute similarities are varied. Ideally, the responses to the mixed conditions should be flexible, and controlled by which aspects of similarity are currently considered important. Only the relative values of the scores are important, the absolute values do not matter. Structural Similarity YES NO Object Attribute Similarity YES NO (LS) High (AN) Low (SS) High (FA) Low (a) Scores from MAC. Structural Similarity YES NO Object Attribute Similarity YES NO (LS) High (AN) tMed-High Low (SS) :J:Med-Low (FA) (b) Ideal similarity scores. Table 1: (a) Scores from the fast (MAC) similarity estimator in MACIFAC. (b) Scores from an ideal structure-sensitive similarity estimator, e.g., SME as used in MACIFAC. In the remainder of this paper I describe how HRRs can be used to compute fast similarity estimates that are more like ratings in Table 1b, i.e., estimates that are flexible and sensitive to structure. 3 HOLOGRAPHIC REDUCED REPRESENTATIONS A distributed representation for nested relational structures requires a solution to the binding problem. The representation of a relation such as bite (spot, jane) ("Spot bit Jane.") must bind 'Spot' to the agent role and 'Jane' to the object role. In order to represent nested structures it must also be possible to bind a relation to a role, e.g., bite (spot, jane) and the antecedent role of the cause relation. n-l Zi = 2.:= XkYj-k Zo = k=O (Subscript are modulo-n) (a) Zl = Z2 = + X2Yl + XIY2 + XOYI + X2Y2 X2YO + XIYl + XOY2 XoYo XIYO (b) Figure 2: (a) Circular convolution. (b) Circular convolution illustrated as a compressed outer product for n = 3. Each of the small circles represents an element of the outer product of x and Y, e.g., the middle bottom one is X2Yl. The elements of the circular convolution of x and yare the sums of the outer product elements along the wrapped diagonal lines. Holographic Reduced Representations (HRRs) (Plate, 1994) use circular convolution to solve the binding problem. Circular convolution (Figure 2a) is an operation that maps two n-dimensional vectors onto one n-dimensional vector. It can be viewed as a compressed outer product, as shown in Figure 2b. Algebraically, circular convolution behaves like multiplication - it is commutative, associative, and distributes over addition. Circular Estimating Analogical Similarity by Dot-rroducts of Holographic Reduced Representations convolution is similarity preserving: if ~ ~ ~' then ~ ? b ~ ~' ? b. Associations can be decoded using a stable approximate inverse: ~ * ? (~ ? b) ~ b (provided that the vector elements are normally distributed with mean zero and variance lin). The approximate inverse is a permutation of vector elements: = an-i. The dot-product of two vectors, a similarity measure, is: ~. b = L~:Ol aibi. High dimensional vectors (n in the low thousands) must be used to ensure reliable encoding and decoding. ar The HRR for bi te (spot, jane) is: F =< bite + biteagt ? spot + biteobj ? jane>, where < . > is a normalization operation ? ~ >= ~I V!! . ~). Multiple associations are superimposed in one vector and the representations for the objects (spot and jane) can also be added into the HRR in order to make it similar to other HRRs involving Spot and Jane. The HRR for a relation is the same size as the representation for an object and can be used as the filler for a role in another relation. 4 EXPT. 1: HRR DOT-PRODUCT SIMILARITY ESTIMATES Experiment 1 illustrates the ways in which the dot-products of ordinary HRRs reflect, and fail to reflect, the similarity of the underlying structure of the episodes. Base vectors Token vectors person, dog, cat, mouse jane =< person + idjane > bite, flee, cause john =< person + idjohn > biteagt, fleeagt, causeantc fred =< person + idfred > biteobj, flee from, causecnsq mort =< mouse + idntort > spot =< dog + id spot > fido =< dog + idfido > rover =< dog + idrover > felix =< cat + id felix> The set of base and tokens vectors used in Experiments 1, 2 and 3 is shown above. All base and id vectors had elements independently chosen from a zero-mean normal distribution with variance lin. The HRR for the probe is constructed as follows. and the HRRs for the other episodes are constructed in the same manner. Pbite =< bite + biteagt ? spot + biteobj ? jane> P flee =< flee + fleeagt ? jane + flee from ? spot> P objects =< jane + spot> P =< cause + P objects + Pbite + P flee + causeantc ? Pbite + causecnsq ? P flee> Experiment 1 was run 100 times, each time with a new choice of random base vectors. The vector dimension was 2048. The means and standard deviations of the HRR dot-products of the probe and each episode are shown in Table 2. Probe: Spot bit Jane. causing Jane to flee from Spot. Episodes in long-term memory: El LS Fido bit John, causing John to flee from Fido. E2 AN Cnt Fred bit Rover, causing Rover to flee from Fred. E3 AN Felix bit Mort, causing Mort to flee from Felix. E6 SS John fled from Fido, causing Fido to bite John. E7 FA Mort bit Felix, causing Mort to flee from Felix. Dot-product with probe Exptl Expt2 Expt3 Avg Sd 0.70 0.016 0.63 0.81 0.47 0.022 0.47 0.69 0.39 0.024 0.39 0.61 0.47 0.018 0.44 0.53 0.39 0.024 0.39 0.39 Table 2: Results of Experiments 1,2 and 3. In 94 out of 100 runs, the ranking of the HRR dot-products was consistent with LS > (ANcm, SS) > (FA, AN) 1113 1114 Plate (where the ordering within the parenthesis varies). The order violations are due to "random" fluctuations of dot-products, whose variance decreases as the vector dimension increases. When the experiment was rerun with vector dimension 4096 there was only one violation of this order out of 100 runs. These results represent an improvement over the first stage of MACIFAC - the HRR dotproduct distinguishes between literal and surface similarity. However, when the episodes do not share object attributes, the HRR dot-product is not affected by structural similarity and the scores do not distinguish analogy from false analogy or superficial similarity. 5 EXPERIMENTS 2 AND 3: CONTEXTUALIZED HRRS Dot-product comparisons ofHRRs are not sensitive to structural similarity in the absence of similar objects. This is because the way in which objects fill multiple roles is not expressed as a surface feature in HRRs. Consequently, the analogous episodes E2 (ANcm) and E3 (AN) do not receive higher scores than the non analogous episodes E6 (SS) and E7 (FA). We can force role structure to become a surface feature by "contextualizing" the representations of fillers. Contextualization involves incorporating information about what other roles an object fills in the representation of a filler. This is like thinking of Spot (in the probe) as an entity that bites (a biter) and an entity that is fled from (a "fled-from"). In ordinary HRRs the filler alone is convolved with the role. In contextualized HRRs a blend of the filler and its context is convolved with the role. The representation for the context of object in a role is the typical fillers of the other roles the object fills. The context for Spot in the flee relation is represented by typ~~; and the context in the bite relation is represented by typ~~eo:n (where typ~~; = bite ? bite~gt and typ~~:em = flee ? fleejrom). The degree of contextualization is governed by the mixing proportions ""0 (object) and ""c (context). The contextualized HRR for the probe is constructed as follows: Pbite =< bite + biteagt ? (X:ospot + X:ctyp~~:eTn) + biteobj ? (X:ojane + X:ctyp!~~e) > P flee =< flee + fleeagt ? (X:ojane + X:ctyp~tn + fleefroTn ? (X:ospot + X:ctyp~~n > P objects =< jane + spot> P =< cause + P objects + P bite + P flee + causeantc ? Pbite + causecnsq ? P flee> A useful similarity estimator must be flexible and able to adjust salience of different aspects of similarity according to context or command. The degree to which role-alignment affects the HRR dot-product can be adjusted by changing the degree of contextualization in just one episode of a pair. Hence, the items in memory can be encoded with a fixed ,.., values (,..,-: and ,..,;;-) and the salience of role alignment can be changed by altering the degree of contextualization in the probe (,..,~ and This is fortunate as it would be impractical to recode all items in memory in order to alter the salience of role alignment in a particular comparison. The same technique can be used to adjust the importance of other features. ,..,n. Two experiments were performed with contextualized HRRs, with the same episodes as used in Experiment 1. In Experiment 2 the probe was non-contextualized (,..,~ = 1, ,..,~ = 0), and in Experiment 3 the probe was contextualized (,..,~ = 1/~,,..,~ = 1/~). For both Experiments 2 and 3 the episodes in memory were encoded with the same degree of contextualization (,..,-: = 1/~,,..,;;- = 1/ ~). As before, each set of comparisons was run 100 times, and the vector dimension was 2048. The results are shown in Table 2. Estimating Analogical Similarity by Dot-Products of Holographic Reduced Representations The scores in Experiment 2 (non-contextualized probe) were consistent (in 95 out of 100 runs) with the same order as given for Experiment 1: L8 > (AN cm ,88) > (FA, AN) The scores in Experiment 3 (contextualized probe) were consistent (in all 100 runs) with an ordering that ranks analogous episodes as strictly more similar than non-analogous ones: L8 > AN cm > AN > 88 > FA 6 DISCUSSION The dot-product of HRRs provides a fast estimate of the degree of analogical match and is sensitive to various structural aspects of the match. It is not intended to be a model of complex or creative analogy making, but it could be a useful first stage in a model of analogical reminding. Structural Similarity YES NO Object Attribute Similarity NO YES (LS) High (AN) Low (SS) Med (FA) Low (a) Ordinary-HRR dot-products. Structural Similarity YES NO Object Attribute Similarity NO YES (LS) High (AN) tMed-High (SS) tMed-Low (FA) Low (b) Contextualized-HRR dot-products. Table 3: Similarity scores from ordinary and contextualized HRR dot-product comparisons. The flexibility comes adjusting the weights of various components in the probe. The dot-product of ordinary HRRs is sensitive to some aspects of structural similarity. It improves on the existing fast similarity matcher in MACIFAC in that it discriminates the first column of Table 3 - it ranks literally similar (LS) episodes higher than superficially similar (88) episodes. However, it is insensitive to structural similarity when corresponding objects are not similar. Consequently, it ranks both analogies (AN) and false analogies (FA) lower than superficially similar (S8) episodes. The dot-product of contextualized HRRs is sensitive to structural similarity even when corresponding objects are not similar. It ranks the given examples in the same order as would the full MACIFAC or ARCS system. Contextualization does not cause all relational structure to be expressed as surface features in the HRR vector. It only suffices to distinguish analogous from non-analogous structures when no two entities fill the same set of roles. Sometimes, the distinguishing context for an object is more than the other roles that the object fills. Consider the situation where two boys are bitten by two dogs, and each flees from the dog that did not bite him. With contextualization as described above it is impossible to distinguish this from the situation where each boy flees from the dog that did bite him. HRR dot-products are flexible - the salience of various aspects of similarity can be adjusted by changing the weights of various components in the probe. This is true for both ordinary and contextualized HRRs. HRRs retain many of the advantages of ordinary distributed representations: (a) There is a simple and computationally efficient measure of similarity between two representations - 1115 1116 Plate the vector dot-product. Similar items can be represented by similar vectors. (b) Items are represented in a continuous space. (c) Information is distributed and redundant. Hummel and Biederman (1992) discussed the binding problem and identified two main problems faced by conjunctive coding approaches such as Tensor Products (Smolensky, 1990). These are exponential growth of the size of the representation with the number of associated objects (or attributes), and insensitivity to attribute structure. HRRs have much in common with conjunctive coding approaches (they can be viewed as a compressed conjunctive code), but do not suffer from these problems. The size of HRRs remains constant with increasing numbers of associated objects, and sensitivity to attribute structure has been demonstrated in this paper. The HRR dot-product is not without its drawbacks. Firstly, examples for which it will produce counter-intuitive rankings can be constructed. Secondly, the scaling with the size of episodes could be a problem - the sum of structural-feature matches becomes a less appropriate measure of similarity as the episodes get larger. A possible solution to this problem is to construct a spreading activation network of HRRs in which each episode is represented as a number of chunks, and each chunk is represented by a node in the network. The software used for the HRR calculations is available from the author. References Falkenhainer, B., Forbus, K. D., and Gentner, D. (1989). The Structure-Mapping Engine: Algorithm and examples. Artificial Intelligence, 41: 1-63. Gentner, D. and Forbus, K. D. (1991). MAC/FAC: A model of similarity-based retrieval. In Proceedings of the Thirteenth Annual Cognitive Science Society Conference, pages 504-509, Hillsdale, NJ. Erlbaum. Gentner, D. and Markman, A. B. (1992). Analogy - Watershed or Waterloo? Structural alignment and the development of connectionist models of analogy. In Giles, C. L., Hanson, S. J., and Cowan, J. D., editors, Advances in Neural Information Processing Systems 5 (NIPS*92), pages 855-862, San Mateo, CA. Morgan Kaufmann. Gentner, D., Rattermann, M. J., and Forbus, K. D. (1993). The roles of similarity in transfer: Separating retrievability from inferential soundness. Cognitive Psychology, 25:431-467. Hummel, J. E. and Biederman, I. (1992). Dynamic binding in a neural network for shape recognition. Psychological Review, 99(3):480-517. Markman, A. B., Gentner, D., and Wisniewski, E. J. (1993). Comparison and cognition: Implications of structure-sensitive processing for connectionist models. Unpublished manuscript. Plate, T. A. (1991). Holographic Reduced Representations: Convolution algebra for compositional distributed representations. In Mylopoulos, J. and Reiter, R., editors, Proceedings of the 12th International loint Conference on Artificial Intelligence, pages 30-35, San Mateo, CA. Morgan Kaufmann. Plate, T. A. (1993). Estimating analogical similarity by vector dot-products of Holographic Reduced Representations. Unpublished manuscript. Plate, T. A. (1994). Holographic reduced representations. IEEE Transactions on Neural Networks. To appear. Smolensky, P. (1990) . Tensor product variable binding and the representation of symbolic structures in connectionist systems. Artificial Intelligence, 46(1-2):159-216. Thagard, P., Holyoak, K. J., Nelson, G., and Gochfeld, D. (1990). Analog Retrieval by Constraint Satisfaction. Artificial Intelligence, 46:259-310. Wharton, C. M., Holyoak, K. J., Downing, P. E., Lange, T. E., Wickens, T. D., and Melz, E. R. (1994). Below the surface: Analogical similarity and retrieval competition in reminding. Cognitive Psychology. To appear. 

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Identifying Fault-Prone Software Modules Using Feed-Forward Networks: A Case Study N. Karunanithi Room 2E-378, Bellcore 435 South Street Morristown, NJ 07960 E-mail: karun@faline.bellcore.com Abstract Functional complexity of a software module can be measured in terms of static complexity metrics of the program text. Classifying software modules, based on their static complexity measures, into different fault-prone categories is a difficult problem in software engineering. This research investigates the applicability of neural network classifiers for identifying fault-prone software modules using a data set from a commercial software system. A preliminary empirical comparison is performed between a minimum distance based Gaussian classifier, a perceptron classifier and a multilayer layer feed-forward network classifier constructed using a modified Cascade-Correlation algorithm. The modified version of the Cascade-Correlation algorithm constrains the growth of the network size by incorporating a cross-validation check during the output layer training phase. Our preliminary results suggest that a multilayer feed-forward network can be used as a tool for identifying fault-prone software modules early during the development cycle. Other issues such as representation of software metrics and selection of a proper training samples are also discussed. 793 794 Karunanithi 1 Problem Statement Developing reliable software at a low cost is an important issue in the area of software engineering (Karunanithi, Whitley and Malaiya, 1992). Both the reliability of a software system and the development cost can be reduced by identifying troublesome software modules early during the development cycle. Many measurable program attributes have been identified and studied to characterize the intrinsic complexity and the fault proneness of software systems. The intuition behind software complexity metrics is that complex program modules tend to be more error prone than simple modules. By controlling the complexity of software modules during development, one can produce software systems that are easy to maintain and enhance (because simple program modules are easy to understand). Static complexity metrics are measured from the passive program texts early during the development cycle and can be used as a valuable feedback for allocating resources in future development efforts (future releases or new projects). Two approachs can be applied to relate static complexity measures with faults found or program changes made during testing. In the estimative approach regressions models are used to predict the actual number of faults that will be disclosed during testing (Lipow, 1982; Gaffney, 1984; Shen et al., 1985; Crawford et al., 1985; Munson and Khoshgoftaar, 1992). Regression models assume that the metrics that constitute independent variables are independent and normally distributed. However, most practical measures often violate the normality assumptions and exhibit high correlation with other metrics (i.e., multicollinearity). The resulting fit of the regression models often tend to produce inconsistent predictions. Under the classification approach software modules are categorized into two or more fault-prone classes (Rodriguez and Tsai, 1987; Munson and Khoshgoftaar, 1992; Karunanithi, 1993; Khoshgoftaar et al., 1993). A special case of the classification approach is to classify software modules into either low-fault (non-complex) or high-fault (complex) categories. The main rationale behind this approach is that the software managers are often interested in getting some approximate feedback from this type of models rather than accurate predictions of the number of faults that will be disclosed. Existing two-class categorization models are based on linear discriminant principle (Rodriguez and Tsai, 1987; Munson and Khoshgoftaar, 1992). Linear discriminant models assume that the metrics are orthogonal and that they follow a normal distribution. To reduce multicollinearity, researchers often use principle component analysis or some other dimensionality reduction techniques. However, the reduced metrics may not explain all the variability if the original metrics have nonlinear relationship. In this paper, the applicability of neural network classifiers for identifying fault proneness of software modules is examined. The motivation behind this research is to evaluate whether classifiers can be developed without usual assumptions about the input metrics. In order to study the usefulness of neural network classifiers, a preliminary comparison is made between a simple minimum distance based Gaussian classifier, a single layer perceptron and a multilayer feed-forward network developed using a modified version of Fahlman's Cascade Correlation algorithm (Fahlman and Lebiere, 1990). The modified algorithm incorporates a cross-validation for constraining the growth of the size of the network. In this investigation, other issues Identifying Fault-Prone Software Modules Using Feed-Forward Networks: A Case Study such as selection of proper training samples and representation of metrics are also considered. 2 Data Set Used The metrics data used in this study were obtained from a research conducted by Lind and Vairavan (Lind and Vairavan, 1989) for a Medical Imaging System software. The complete system consisted of approximately 4500 modules amounting to about 400,000 lines of code written in Pascal, FORTRAN, PL/M and assembly level. From this set, a random sample of 390 high level language routines was selected for the analysis. For each module in the sample, program changes were recorded as an indication of software fault. The number of changes in the program modules varied from zero to 98. In addition to changes, 11 software complexity metrics were extracted from each module. These metrics range from total lines of code to Belady's bandwidth metric. (Readers curious about these metrics may refer to Table I of Lind and Vairavan, 1989.) For the purpose of our classification study, these metrics represent 11 input (both real and integer) variables of the classifier. A software module is considered as a low fault-prone module (Category I) if there are 0 or 1 changes and as a high fault-prone module (Category II) if there are 10 or more changes. The remaining modules are considered as medium fault category. For the purpose of this study we consider only the low and high fault-prone modules. Our extreme categorization and deliberate discarding of program modules is similar to the approach used in other studies (Rodriguez and Tsai, 1987; Munson and Khoshgoftaar, 1992). After discarding medium fault-prone modules, there are 203 modules left in the data set. Of 203 modules, 114 modules belong to the low fault-prone category while the remaining 89 modules belong to the high fault-prone category. The output layer of the neural nets had two units corresponding to two fault categories. 3 Training Data Selection We had two objectives in selecting training data: 1) to evaluate how well a neural network classifier will perform across different sized training sets and 2) to select the training data as much unbiased as possible. The first objective was motivated by the need to evaluate whether a neural network classifier can be used early in the software development cycle. Thus the classification experiments were conducted ~, ~, ~, ~, 190 fraction of 203 samples belonging using training samples of size S to Categories I and II. The remaining ~ 1-S) fraction of the samples were used for testing the classifiers. In order to avoid bias in the training data, we randomly selected 10 different training samples for each fraction S. This resulted in 6 X 10 (=60) different training and test sets. = }, 795 796 Karunanithi 4 4.1 Classifiers Compared A Minimum Distance Classifier In order to compare neural network classifiers and linear discriminant classifiers we implemented a simple minimum distance based two-class Gaussian classifier of the form (Nilsson, 1990): IX - Gi l = ((X - Gi)(X - Gi)t)1/2 where Gi, i = 1, 2 represent the prototype points for the Categories I and II, X is a 11 dimensional metrics vector, and t is the transpose operator. The prototype points G1 and G2 are calculated from the training set based on the normality assumption. In this approach a given arbitrary input vector X is placed in Category I if IX - G11< IX - G21 and in Category II otherwise. All raw component metrics had distributions that are asymmetric with a positive skew (i .e., long tail to the right) and they had different numerical ranges. Note that asymmetric distributions do not conform to the normality assumption of a typical Gaussian classifier . First, to remove the extreme asymmetry of the original distribution of the individual metric we transformed each metric using a natural logarithmic base. Second, to mask the influence of individual component metric on the distance score, we divided each metric by its standard deviation of the training set. These transformations considerably improved the performance of the Gaussian classifier. To be consistent in our comparison we used the log transformed inputs for other classifiers also. 4.2 A Perceptron Classifier A perceptron with a hard-limiting threshold can be considered as a realization of a non-parametric linear discriminant classifier. If we use a sigmoidal unit, then the continuous valued output of the perceptron can be interpreted as a likelihood or probability with which inputs are assigned to different classes. In our experiment we implemented a perceptron with two sigmoidal units (outputs 1 and 2) corresponding to two categories. A given arbitrary vector X is assigned to Category I if the value of the output unit 1 is greater than the output of the unit 2 and to Category II otherwise. The weights of the network are determined iteratively using least square error minimization procedure. In almost all our experiments, the perceptron learned about 75 to 80 percentages of the training set. This implies that the rest of the training samples are not linearly separable. 4.3 A Multilayer Network Classifier To evaluate whether a multilayer network can perform better than the other two classifiers, we repeated the same set of experiments using feed-forward networks constructed by Fahlman's Cascade-Correlation algorithm. The Cascade-Correlation algorithm is a constructive training algorithm which constructs a suitable network architecture by adding one hidden (layer) unit at a time. (Refer to Fahlman and Lebiere, 1990 for more details on the Cascade-Correlation algorithm.) Our initial results suggested that the multilayer layer networks constructed by the CascadeCorrelation algorithm are not capable of producing a better classification accuracy Identifying Fault-Prone Software Modules Using Feed-Forward Networks: A Case Study than the other two classifiers. An analysis of the network suggested that the resulting networks had too many free variables (i.e., due to too many hidden units). A further analysis of the rate of decrease of the residual error versus the number of hidden units added to the networks revealed that the Cascade-Correlation algorithm is capable of adding more hidden units to learn individual training patterns at the later stages of the training phase than in the earlier stages. This happens if the training set contains patterns that are interspersed across different decision regions or what might be called "border patterns" (Ahmed, S. and Tesauro, 1989). In an effort to constrain the growth of the size of the network, we modified the Cascade-Correlation algorithm to incorporate a cross-validation check during the output layer training phase. For each training set of size S, one third was used for cross-validation and the remaining two third was used to train the network. The network .construction was stopped as soon as the residual error of the crossvalidation set stopped decreasing from the residual error at the end of the previous output layer training phase. The resulting network learned about 95% of the training patterns. However, the cross-validated construction considerably improved the classification performance of the networks on the test set. Table 1 presented in the next section provides a comparison between the networks developed with and without cross-validation. Training Set Size Sin% 25 33 50 67 75 90 25 33 50 67 75 90 Error Statistics Hidden Unit Type I Error Type II Statistics Mean I Std Mean I Std Mean I Without Cross-Validation 5.1 24.64 7.2 16.38 1.5 20.24 6.2 8.4 17.27 1.8 7.4 18.30 7.4 18.65 1.8 9.7 1.7 15.78 6.5 18.05 14.54 7.6 10.4 1.8 16.85 10.33 7.2 17.73 11.2 1.6 With Cross-Validation 5.4 1.9 1.3 20.19 12.11 2.2 5.5 12.40 1.0 18.24 2.0 0.9 17.41 5.6 15.04 5.8 2.7 14.32 14.08 1.1 13.27 7.0 13.84 2.7 1.3 9.77 9.4 2.9 1.2 15.47 Error Std 6.4 5.5 6.4 7.1 7.3 8.3 4.7 4.1 5.2 5.5 5.4 5.1 Table 1: A Comparison of Nets With and Without Cross-Validation. 5 Results In this section we present some preliminary results from our classification experiments. First, we provide a comparison between the multilayer networks developed with and without cross-validation. Next, we compare different classifiers in terms of their classification accuracy. Since a neural network's performance can be affected by the weight vector used to initialize the network, we repeated the training experiment 25 times with different initial weight vectors for each training set. This 797 798 Karunanithi resulted in a total of 250 training trials for each value of S. The results reported here for the neural network classifiers represent a summary statistics for 250 experiments. The performance of the classifiers are reported in terms of classification errors. There are two type of classification errors that a classifier can make: a Type I error occurs when the classifier identifies a low fault-prone (Category I) module as a high fault-prone (Category II) module; a Type II error is produced when a high faultprone module is identified as a low fault-prone module. From a software manager's point of view, these classification errors will have different implications. Type I misclassification will result in waste of test resources (because modules that are less fault-prone may be tested longer than what is normally required). On the other hand, Type II misclassification will result in releasing products that are of inferior quality. From reliability point of view, a Type II error is a serious error than a Type I error. No. of Patterns S Training Test % Set Set I I 25 33 50 67 75 90 50 66 101 136 152 182 86 77 57 37 28 12 25 33 50 67 75 90 50 66 101 136 152 182 67 60 45 30 23 9 Error Statistics Gaussian Perceptron Multilayer Nets Mean 1 Std Mean 'I Std Mean r Std Type I Error Statistics 13.16 4.7 16.17 5.5 20.19 5.4 11.44 4.0 11.74 3.9 18.24 5.5 12.45 3.2 3.2 17.41 5.6 11.58 14.32 5.8 9.46 4.1 10.14 3.9 8.57 5.4 9.15 5.8 13.27 7.0 14.17 7.9 4.03 4.3 9.77 9.4 Type 11 Error Statistics 15.61 4.2 15.98 7.8 12.11 4.7 15.46 4.6 15.78 6.6 12.40 4.1 16.97 6.8 15.04 5.2 16.01 5.1 16.00 5.4 7.6 14.08 5.5 16.11 17.39 5.8 18.39 6.3 13.84 5.4 6.3 5.6 15.47 5.1 19.11 21.11 I I Table 2: A Summary of Type I and Type II Error Statistics. Table 1 compares the complexity and the performance of the multilayer networks developed with and without cross-validation. Columns 2 through 7 represent the size and the performance of the networks developed by the Cascade-Correlation without cross-validation. The remaining six columns correspond to the networks constructed with cross-validation. Hidden unit statistics for the networks suggest that the growth of the network can be constrained by adding a cross-validation during the output layer training. The corresponding error statistics for both the Type I and Type II errors suggest that an improvement classification accuracy can be achieved by cross-validating the size of the networks. Table 2 illustrates the preliminary results for different classifiers. The first two columns in Table 2 represent the size of the training set in terms of S as a percentage of all patterns and the number of patterns respectively. The third column represents the number oft est patterns in Categories I (1st half) and the II (2nd half). The remaining six columns represent the error statistics for the three classifiers in Identifying Fault-Prone Software Modules Using Feed-Forward Networks: A Case Study terms of percentage mean errors and standard deviations. The percentages errors were obtained by dividing the number of misclassifications by the total number of test patterns in that Category. The Type I error statistics in the first half of the table suggest that the Gaussian and the Perceptron classifiers may be better than multilayer networks at early stages of the software development cycle. However, the difference in performance of the Gaussian classifier is not consistent across all values of S. The neural network classifiers seem to improve their performance with an increase in the size of the training set. Among neural networks, the perceptron classifier seems to perform classification than a multilayer net. However, the Type II error statistics in the second half of the table suggest that a multilayer network classifier may provide a better classification of Category II modules than the other two classifiers. This is an important results from the reliability perspective. 6 Conclusion and Work in Progress We demonstrated the applicability of neural network classifiers for identifying faultprone software modules. We compared the classification efficacy of three different pattern classifiers using a data set from a commercial software system. Our preliminary empirical results are encouraging in that there is a role for multilayer feed-forward networks either during the software development cycle of a subsequent release or for a similar product. The cross-validation implemented in our study is a simple heuristics for constraining the size of the networks constructed by the Cascade-Correlation algorithm. Though this improved the performance of the resulting networks, it should be cautioned that cross-validation may be needed only if the training patterns exhibit certain characteristics. In other circumstances, the networks may have to be constructed using the entire training set. At this stage we have not performed complete analysis on what characteristics of the training samples would require cross-validation for constraining the network growth. Also we have not used other sophisticated structure reduction techniques. We are currently exploring different loss functions and structure reduction techniques. The Cascade-Correlation algorithm always constructs a deep network. Each additional hidden unit develops an internal representation that is a higher order sigmoidal computation than those of previously added hidden units. Such a complex internal representation may not be appropriate in a classification application such as the one studied here. We are currently exploring alternatives to construct shallow networks within the Cascade-Correlation frame work. At this stage, we have not performed any analysis on how the internal representations of a multilayer network correlate with the input metrics. This is currently being studied. References Ahmed, S. and G. Tesauro (1989). "Scaling and Generalization in Neural Networks: A Case Study", Advances in Neural Information Processing Systems 1, pp 160-168, D. Touretzky, ed. Morgan Kaufmann. 799 800 Karunanithi Crawford, S. G., McIntosh, A. A. and D. Pregibon (1985). "An Analysis of Static Metrics and Faults in C Software", The Journal of Systems and Software, Vol. 5, pp. 37-48. Fahlman, S. E. and C. Lebiere (1990). "The Cascaded-Correlation Learning Architecture," Advances in Neural Information Processing Systems 2, pp 524-532, D. Touretzky, ed. Morgan Kaufmann. Gaffney Jr., J. E. (1984). "Estimating the Number of Faults in Code", IEEE Trans. on Software Eng., Vol. SE-lO, No.4, pp. 459-464. Karunanithi, N, Whitley, D. and Y. K. Malaiya (1992). "Prediction of Software Reliability Using Connectionist Models" , IEEE Trans. on Software Eng., Vol. 18, No.7, pp. 563-574. Karunanithi, N. (1993). "Identifying Fault-Prone Software Modules Using Connectionist Networks", Proc. of the 1st Int'l Workshop on Applications of Neural Networks to Telecommunications, (IWANNT'93), pp. 266-272, J. Alspector et al., ed., Lawrence Erlbaum, Publisher. Khoshgoftaar, T. M., Lanning, D. L. and A. S. Pandya (1993). "A Neural Network Modeling Methodology for the Detection of High-Risk Programs" , Proc. of the 4th Int'l Symp. on Software Reliability Eng. pp. 302-309. Lind, R. K. and K. Vairavan (1989). "An Experimental Investigation of Software Metrics and Their Relationship to Software Development Effort", IEEE Trans. on Software Eng., Vol. 15, No.5, pp. 649-653. Lipow, M. (1982). "Number of Faults Per Line of Code", IEEE Trans. on Software Eng., Vol. SE-8, No.4, pp. 437-439. Munson, J. C. and T. M. Khoshgoftaar (1992). "The Detection of Fault-Prone Programs", IEEE Trans. on Software Eng., Vol. 18, No.5, pp. 423-433. Nilsson, J. Nils (1990). The Mathematical Foundations of Learning Machines, Morgan Kaufmann, Chapters 2 and 3. Rodriguez, V. and W. T. Tsai (1987). "A Tool for Discriminant Analysis and Classification of Software Metrics", Information and Software Technology, Vol. 29, No.3, pp. 137-149. Shen, V. Y., Yu, T., Thebaut, S. M. and T. R. Paulsen (1985). "Identifying ErrorProne Software: An Empirical Study", IEEE Trans. on Software Eng., Vol. SE-ll, No.4, pp. 317-323. 

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The Parti-game Algorithm for Variable Resolution Reinforcement Learning in Multidimensional State-spaces Andrew W. Moore School of Computer Science Carnegie-Mellon University Pittsburgh, PA 15213 Abstract Parti-game is a new algorithm for learning from delayed rewards in high dimensional real-valued state-spaces. In high dimensions it is essential that learning does not explore or plan over state space uniformly. Part i-game maintains a decision-tree partitioning of state-space and applies game-theory and computational geometry techniques to efficiently and reactively concentrate high resolution only on critical areas. Many simulated problems have been tested, ranging from 2-dimensional to 9-dimensional state-spaces, including mazes, path planning, non-linear dynamics, and uncurling snake robots in restricted spaces. In all cases, a good solution is found in less than twenty trials and a few minutes. 1 REINFORCEMENT LEARNING Reinforcement learning [Samuel, 1959, Sutton, 1984, Watkins, 1989, Barto et al., 1991] is a promising method for control systems to program and improve themselves. This paper addresses its biggest stumbling block: the curse of dimensionality [Bellman, 1957], in which costs increase exponentially with the number of state variables. Some earlier work [Simons et al., 1982, Moore, 1991, Chapman and Kaelbling, 1991, Dayan and Hinton, 1993] has considered recursively partitioning state-space while learning from delayed rewards. The new ideas in the parti-game algorithm in- 711 712 Moore clude (i) a game-theoretic splitting criterion to robustly choose spatial resolution (ii) real-time incremental maintenance and planning with a database of all previous experIences, and (iii) using local greedy controllers for high-level "funneling" actions. 2 ASSUMPTIONS The parti-game algorithm applies to difficult learning control problems in which: 1. State and action spaces are continuous and multidimensional. 2. "Greedy" and hill-dim bing techniques would become stuck, never attaining the goal. 3. Random exploration would be hopelessly time-consuming. 4. The system dynamics and control laws can have discontinuities and are unknown: they must be learned. The experiments reported later all have properties 1-4. However, the initial algorithm, described and tested here, has the following restrictions: 5. 6. 7. 8. Dynamics are deterministic. The task is specified by a goal, not an arbitrary reward function. The goal state is known. A "good" solution is required, not necessarily the optimal path. This nation of goodness can be formalized as "the optimal path to within a given resolution of state space". 9. A local greedy controller is available, which we can ask to move greedily towards any desired state. There is no guarantee that a request to the greedy controller will succeed. For example, in a maze a greedy path to the goal would quickly hit a wall. Future developments may include relatively straightforward additions to the algorithm that would remove the need for restrictions 6-9. Restriction 5 is harder to remove. 3 ESSENTIALS OF THE PARTI-GAME ALGORITHM The state space is broken into partitions by a kd-tree [Friedman et al., 1977]. The controller can always sense its current (continuous valued) state, and can cheaply compute which partition it is in. The space of actions is also discretized so that in a partition with N neighboring partitions , there are N high-level actions. Each high level action corresponds to a local greedy controller, aiming for the center of the corresponding neighboring partition. Each partition keeps records of all the occasions on which the system state has passed through it. Along with each record is a memory of which high level action was used (i.e. which neighbor was aimed for) and what the outcome was. Figure 1 provides an illustration. Given this database of (partition, high-level-action, outcome) triplets, and our knowledge of the partition containing the goal state, we can try to compute the The Parti-Game Algorithm for Variable Resolution Reinforcement Learning Partition I Partition 2 Figure 1: Three trajectories starting in partition 1, using high-level action "Aim at partition 2". Partition 1 remembers three outcomes. ................... (Part 1, Aim 2 (Part 1, Aim 2 (Part 1, Aim 2 , I I I --+ --+ --+ Part 2) Part 1) Part 3) Partition 3 best route to the goal. The standard approach would be to model the system as a Markov Decision Task in which we empirically estimate the partition transition probabilities. However, the probabilistic interpretation of coarse resolution partitions can lead to policies which get stuck. Instead, we use a game-theoretic approach, in which we imagine an adversary. This adversary sees our choice of high-level action, and is allowed to select any of the observed previous outcomes of the action in this partition. Partitions are scored by minimaxing: the adversary plays to delay or prevent us getting to the goal and we play to get to the goal as quickly as possible. Whenever the system's continuous state passes between partitions, the database of state transitions is updated and, if necessary, the minimax scores of all partitions are updated. If real-time constraints do not permit full recomputation, the updates take place incrementally in a manner similar to prioritized sweeping [Moore and Atkeson, 1993]. As well as being robust to coarseness, the game-theoretic approach also tells us where we should increase the resolution . Whenever we compute that we are in a losing partition we perform resolution increase. We first compute the complete set of connected partitions which are also losing partitions. We then find the subset of these partitions which border some non-losing region. We increase the resolution of all these border states by splitting them along their longest axes 1 . 4 INITIAL EXPERIMENTS Figure 2 shows a 2-d continuous maze. Figure 3 shows the performance of the robot during the very first trial. It begins with intense exploration to find a route out of the almost entirely enclosed start region. Having eventually reached a sufficiently high resolution, it discovers the gap and proceeds greedily towards the goal, only to be stopped by the goal's barrier region. The next barrier is traversed at a much lower resolution, mainly because the gap is larger. Figure 4 shows the second trial, started from a slightly different position. The policy derived from the first trial gets us to the goal without further exploration. The trajectory has unnecessary bends. This is because the controller is discretized according to the current partitioning. If necessary, a local optimizer could be used 1 More intelligent splitting criteria are under investigation. 713 714 Moore Start I? Figure 2: A 2-d maze problem. The point robot must find a path from start to goal without crossing any of the barrier lines. Remember that initially it does not know where any obstacles are, and must discover them by finding impassable states. Figure 3: The path taken during the entire first trial. See text for explanation. to refine this trajectory2. The system does not explore unnecessary areas. The barrier in the top left remains at low resolution because the system has had no need to visit there . Figures 5 and 6 show what happens when we now start the system inside this barrier. Figure 7 shows a 3-d state space problem. If a standard grid were used, this would need an enormous number of states because the solution requires detailed threepoint-turns. Parti-game's total exploration took 18 times as much movement as one run of the final path obtained. Figure 8 shows a 4-d problem in which a ball rolls around a tray with steep edges. The goal is on the other side of a ridge. The maximum permissible force is low, and so greedy strategies, or globally linear control rules, get stuck in a limit cycle. Parti-game's solution runs to the other end of the tray, to build up enough velocity to make it over the ridge. The exploration-length versus final-path-Iength ratio is 24. Figure 9 shows a 9-joint snake-like robot manipulator which must move to a specified configuration on the other side of a barrier. Again, no initial model is given: the controller must learn it as it explores. It takes seven trials before fixing on the solution shown. The exploration-length versus final-path-length ratio is 60. 2 Another method is to increase the resolution along the trajectory [Moore, 1991]. The Parti-Game Algorithm for Variable Resolution Reinforcement Learning ,.rn c-f- 1/ (') I /- 1-1 f-H - I 11 j ~ .f"" I-~ v'\ f- "- r--- I- ,r- Il L~ 1 1 IT'" 1-+' 1 ./ 1-1 f-H V~ r l.J J "- r--- ./ / ~ J /--. /'-- ) ) -1 --1 Figure 4: The second trial. Figure 5: Starting inside the top left barrier. Figure 6: that. The trial after Figure 7: A problem with a planar rod being guided past obstacles. The state space is three-dimensional: two values specify the position of the rod's center, and the third specifies the rod's angle from the horizontal. The angle is constrained so that the pole's dotted end must always be below the other end. The pole's center may be moved a short distance (up to 1/40 of the diagram width) and its angle may be altered by up to 5 degrees, provided it does not hit a barrier in the process. Parti-game converged to the path shown below after two trials. The partitioning lines on the solution diagram only show a 2-d slice of the full kd-tree. Trials Steps Partitions 10 no 149 149 149 Change 715 716 Moore Figure 8: A puck sliding over a hilly surface (hills shown by contours below: the surface is bowl shaped, with the lowest points nearest the center, rising steeply at the edges). The state space is four-dimensional: two position and two velocity variables. The controls consist of a force which may be applied in any direction, but with bounded magnitude. Convergence time was two trials. lu..&.I.I.WI'U'I.? ? ? ? ? Trials Steps Partitions 1 2 3 2609 13 115 13 no change 10 Figure 9: A nine-degree-of-freedom planar robot must move from the shown start configuration to the goal. The solution entails curling, rotating and then uncurling. It may not intersect with any of the barriers, the edge of the workspace, or itself. Convergence occurred after seven trials. f-Fixed base Trials Steps Partitions 1 2 1090 430 41 66 3 353 67 4 330 69 5 739 78 6 200 85 7 52 85 8 The Parti-Game Algorithm for Variable Resolution Reinforcement Learning 5 DISCUSSION Possible extensions include: ? Splitting criteria that lay down splits between trajectories with spatially distinct outcomes. ? Allowing humans to provide hints by permitting user-specified controllers ("behaviors") as extra high-level actions. ? Coalescing neighboring partitions that mutually agree. We finish by noting a promising sign involving a series of snake robot experiments with different numbers of links (but fixed total length). Intuitively, the problem should get easier with more links, but the curse of dimensionality would mean that (in the absence of prior knowledge) it becomes exponentially harder. This is borne out by the observation that random exploration with the three-link arm will stumble on the goal eventually, whereas the nine link robot cannot be expected to do so in tractable time. However, Figure 10 indicates that as the dimensionality rises, the amount of exploration (and hence computation) used by parti-game does not rise exponentially. Real-world tasks may often have the same property as the snake example: the complexity of the ultimate task remains roughly constant as the number of degrees of freedom increases. If so, we may have uncovered the Achilles' heel of the curse of dimensionality. ~ ell "'" ~ 180 ~ 160 = 140 Q CJ ~ "'" ~ 120 ~ 100 ~ 80 .CI ~ ~ e 4060 .......= 20 fI.l Q := "'~" ~ 0 I 3 4 5 6 7 8 9 Figure 10: The number of partitions finally created against degrees of freedom for a set of snakelike robots. The kd-trees built were all highly non-uniform, typically having maximum depth nodes of twice the dimensionality. The relation between exploration time and dimensionality (not shown) had a similar shape. Dimensionality References [Barto et ai., 1991] A. G. Barto, S. J. Bradtke, and S. P. Singh. Real-time Learning and Control using Asynchronous Dynamic Programming. Technical Report 91-57, University of Massachusetts at Amherst, August 1991. [Bellman, 1957] R. E . Bellman. Dynamic Programming. Princeton University Press, Princeton, N J, 1957. [Chapman and Kaelbling, 1991) D. Chapman and L. P. Kaelbling. Learning from Delayed Reinforcement In a Complex Domain. Technical Report, Teleos Research, 1991. 717 718 Moore [Dayan and Hinton, 1993] P. Dayan and G. E. Hinton. Feudal Reinforcement Learning. In S. J. Hanson, J. D Cowan, and C. L. Giles, editors, Advances in Neural Information Processing Systems 5. Morgan Kaufmann, 1993. [Friedman et al., 1977) J. H. Friedman, J. L. Bentley, and R. A. Finkel. An Algorithm for Finding Best Matches in Logarithmic Expected Time. ACM Trans. on Mathematical Software, 3(3):209-226, September 1977. [Moore and Atkeson, 1993] A. W. Moore and C. G. Atkeson. Prioritized Sweeping: Reinforcement Learning with Less Data and Less Real Time. Machine Learning, 13, 1993. [Moore, 1991] A. W. Moore. Variable Resolution Dynamic Programming: Efficiently Learning Action Maps in Multivariate Real-valued State-spaces . In L. Birnbaum and G. Collins, editors, Machine Learning: Proceedings of the Eighth International Workshop. Morgan Kaufman, June 1991. [Samuel, 1959] A. L. Samuel. Some Studies in Machine Learning using the Game of Checkers. IBM Journal on Research and Development, 3, 1959. Reprinted in E. A. Feigenbaum and J. Feldman, editors, Computers and Thought, McGraw-Hill, 1963. [Simons et al., 1982) J. Simons, H. Van Brussel, J. De Schutter, and J. Verhaert. A SelfLearning Automaton with Variable Resolution for High Precision Assembly by Industrial Robots. IEEE Trans. on Automatic Control, 27(5):1109-1113, October 1982. [Singh, 1993] S. Singh. Personal Communication. ,1993. [Sutton, 1984) R. S. Sutton. Temporal Credit Assignment in Reinforcement Learning. Phd. thesis, University of Massachusetts, Amherst, 1984. [Watkins, 1989] C . J. C. H. Watkins . Learning from Delayed Rewards. King's College, University of Cambridge, May 1989. PhD. Thesis, 

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An Analog VLSI Model of Central Pattern Generation in the Leech Micah S. Siegel* Department of Electrical Engineering Yale University New Haven, CT 06520 Abstract I detail the design and construction of an analog VLSI model of the neural system responsible for swimming behaviors of the leech. Why the leech? The biological network is small and relatively well understood, and the silicon model can therefore span three levels of organization in the leech nervous system (neuron, ganglion, system); it represents one of the first comprehensive models of leech swimming operating in real-time. The circuit employs biophysically motivated analog neurons networked to form multiple biologically inspired silicon ganglia. These ganglia are coupled using known interganglionic connections. Thus the model retains the flavor of its biological counterpart, and though simplified, the output of the silicon circuit is similar to the output of the leech swim central pattern generator. The model operates on the same time- and spatial-scale as the leech nervous system and will provide an excellent platform with which to explore real-time adaptive locomotion in the leech and other "simple" invertebrate nervous systems. 1. INTRODUCTION A Central Pattern Generator (CPG) is a network of neurons that generates rhythmic output in the absence of sensory input (Rowat and Selverston, 1991). It has been * Present address: Micah Siegel, Computation and Neural Systems, Mail Stop 139-74 California Institute of Technology, Pasadena, CA 91125. 622 An Analog VLSI Model of Central Pattern Generation in the Leech I 'oult ~ ~v I lre=ov "iuhib I Figure l. Silicon neuromime. The circuit includes tonic excitation, inhibitory synapses and an inhibitory recovery time. Note that there are two inhibitory synapses per device. Iionic sets the level of tonic excitatory input; Vinhib sets the synaptic strength; Irecov determines the inhibitor recover time. suggested that invertebrate central pattern generation may represent an excellent theatre within which to explore silicon implementations of adaptive neural systems: invertebrate CPG networks are orders of magnitude smaller than their vertebrate counterparts, much detailed information is available about them, and they guide behaviors that may be of technological interest (Ryckebusch et al., 1989). Furthermore, CPG networks are typically embedded in larger neural circuits and are integral to the neural correlates of adaptive behavior in many natural organisms (Friesen, 1989). On strategy for modeling "simple" adaptive behaviors is first to evolve a biologically plausible framework within which to include increasingly more sophisticated and verisimilar adaptive mechanisms; because the model of leech swimming presented in this paper encompasses three levels of organization in the leech central nervous system, it may provide an ideal such structure with which to explore potentially useful adaptive mechanisms in the leech behavioral repertoire. Among others, these mechanisms include: habituation of the swim response (Debski and Friesen, 1985), the local bending reflex (Lockery and Kristan, 1990), and conditioned learning of the stepping and shortening behaviors (Sahley and Ready, 1988). 623 624 Siegel A Cel Co. ~l ~, .1 l= 1= -1O? 11& 10' a", 1O' 0;-, "0' .........- -..?. .. 01-1(12 ~ I 11:1' ,ao? .0 120' IJ 150? n leo' C~;;Ip. f1' '10' p::66e B phase 113 123 ~,II ,..,. Je(l"1C" )I.IY' /0' ~JllU\ L--J e ~~U, ___I l. L_-''IIL~' ,i~ 27-..J ~ 2S J ~~I} I I Figure 2. The individual ganglion. (A) Cycle phases of the oscillator neurons in the biological ganglion (from Friesen, 1989). (B) Somatic potential of the simplified silicon ganglion. (C) Circuit diagram of silicon ganglion using cells and s na tic connections identified in the leech an lion. 2. LOCOMOTORY CPG IN THE LEECH As a first step toward modeling a full repertoire of adaptive behavior in the medicinal leech (Hirundo medicinalis), I have designed, fabricated, and successfully tested an analog silicon model of one critical neural subsystem - the coupled oscillatory central pattern generation network responsible for swimming. A leech swims by undulating its segmented body to form a rearward-progressing body wave. This wave is analogous to the locomotory undulations of most elongated aquatic animals (e.g. fish), and some terrestrial amphibians and reptiles (including salamanders and snakes) (Friesen, 1989). The moving crests and troughs in the body wave are produced by phase-delayed contractile rhythms of the dorsal and ventral body wall along successive segments (Stent and Kristan, 1981). The interganglionic neural subsystem that subserves this behavior constitutes an important modeling platform because it guides locomotion in the leech over a wide range of frequencies and adapts to varying intrinsic and extrinsic conditions (Debski and Friesen, 1985). In the medicinal leech, interneurons that coordinate the rearward-progressing swimming contractions undergo oscillations in membrane potential and fire impulses in bursts. It appears that the oscillatory activity of these intemeurons arises from a network rhythm that depends on synaptic interaction between neurons rather than from an endogenous polarization rhythm arising from inherently oscillatory membrane potentials in individual An Analog VLSI Model of Central Pattern Generation in the Leech A ganglion: 10 9 .... ~II----- 11 tail head ----i~~ 8 28 __----' 9 { 27 -------+--- 123 28 _ _- - - I 10 { ~-------------~~~~----- 27 -,...-_ _ _ _+-__ 123~ ~---r---------~ 28 ______' 11 { lOOms 27 _ _ _ _ _ _+-.--I Figure 3. The complete silicon model. (A) Coupled oscillatory ganglia. As in the leech nervous system, interganglionic connections employ conduction delays. (B) Somatic recording of cells (28, 27, 123) from three midbody ganglia (9,10,11) in the silicon model. Notice the phase-delay in homologous cells of successive ganglia. (The apparent "beat" frequencies riding on the spike bursts are an aliasing artifact of the digital oscilloscope measurement and the time-scale; all spikes are approximately the same hei ht. neurons (Friesen, 1989). The phases of the oscillatory intemeurons fonn groups clustered about three phase points spaced equally around the activity cycle. To first approximation, all midbody ganglia of the leech nerve cord express an identical activity rhythm. However, activity in each ganglion is phase-delayed with respect to more anterior ganglia (Friesen, 1989); presumably this is responsible for the undulatory body wave characteristic of leech swimming. 625 626 Siegel 3. THE SILICON MODEL The silicon analog model employs biophysically realistic neural elements (neuromimes), connected into biologically realistic ganglion circuits. These ganglion circuits are coupled together using known interganglionic connections. This silicon model thus spans three levels of organization in the nervous system of the leech (neuron, ganglion, system), and represents one of the first comprehensive models of leech swimming (see also Friesen and Stent, 1977). The hope is that this model will provide a framework for the implementation of adaptive mechanisms related to undulatory locomotion in the leech and other invertebrates. The building block of the model CPO is the analog neuromime (see figure I); it exhibits many essential similarities to its biological counterpart. Like CPO interneurons in the leech swim system, the silicon neuromime integrates current across a somatic "capacitance" and uses positive feedback to generate action potentials whose frequency is determined by the magnitude of excitatory current input (Mead, 1989). In the leech swim system. nearly tonic excitatory input is transformed by a system of inhibition to produce the swim pattern (Friesen. 1989); adjustable tonic excitation is therefore included in the individual silicon neuromime. Inhibitory synapses with adjustable weights are also implemented. Like its biological counterpart, the silicon neuromime includes a characteristic recovery time from inhibition. From theoretical and experimental studies. such inhibition recovery time is thought to play an important functional role in the interneurons that constitute the leech swim system (Friesen and Stent, 1977). Axonal delays have been demonstrated in the intersegmental interaction between ganglia in the leech. Similar axonal delays have been implemented in the silicon model using Shifting delay lines. The building block of the distributed model for the leech swim system is the ganglion. These biologically motivated silicon ganglia are constructed using only (though not all) identified cells and synaptic connections between cells in the biological system. Cells 27, 28, and 123 constitute a central inhibitory loop within each ganglion. Figure 2 exhibits the simplified diagram and the cycle phases of oscillatory interneurons in both the biological and the silicon ganglion. As in the leech ganglion, the phase relationships in the model ganglion fall into three groups, with cells 27. 28. and 123 participating each in the appropriate group of the oscillatory cycle. It is interesting that, though the silicon model captures the spirit of the tri-phasic output, the model is imprecise with respect to the exact phase locations of cells 27. 28. and 123 within their respective groups. This discrepancy between the silicon model and the biological system may point to the significance of other swim interneurons for swim pattern generation in the leech. Undoubtedly. the additional oscillatory interneurons sculpt this tri-phasic output significantly. The silicon model of coupled successive segments in the leech is implemented using these silicon neurons and biologically motivated ganglia. The model employs interganglionic connections known to exist in the biological system and generates qualitatively similar output at the same time-scale as the leech system. It appears in the leech that synchronization between ganglia is governed by the interganglionic synaptic interaction of interneurons involved in the oscillatory pattern rather than by autonomous An Analog VLSI Model of Central Pattern Generation in the Leech coordinating neurons (Friesen. 1989). In the silicon model. interganglionic interaction is represented by a projection from more anterior cell 123 to more posterior cell 28; this A B ----.1UJV'lJNil Wf;tU12i'l .. lOOms Figure 4. Phase lag between more anterior and more posterior segments in both systems. (A) Intersegmental phase lag in (B) the leech swim system (from Friesen. 1989). Intersegmental phase lag in the silicon model. Though not shown in the figure, this cycle repeats at the same frequency as ote chan e of time scale. the c cle in A. projection is also observed between cells 123 and 28 of successive ganglia in the leech (Friesen, 1989). however it is by no means the only such interganglionic connection. In addition, the biological system utilizes conduction delays in its interganglionic projections; each of these is modeled in the silicon system by a delay line (Friesen and Stent. 1977) analogous to an active cable with adjustable propagation speed. Figure 3 demonstrates the silicon model of three coupled ganglia with transmission delays. Notice that neuromimes in each successive ganglion are phase-delayed from homologous neuromimes in more anterior ganglia. Figure 4 shows this phase delay more explicitly. 4. DISCUSSION The analog silicon model of central pauern generation in the leech successfully captures design principles from three levels of organization in the leech nervous system and has been tested over a wide range of network parameter values. It operates on the same timescale as its biological counterpart and gives rise to ganglionic activity that is qualitatively similar to activity in the leech ganglion. Furthermore. it maintains biologically plausible phase relationship between homologous elements of successive ganglia. The design of the silicon model is intentionally compatible with analog Very Large Scale Integration (VLSI) technology. making its integrated spatial-scale close to that of the leech nervous system. It is interesting that this highly simplified model captures qualitatively the output both within and between ganglia of the leech; it may be illuminating to explore the functional significance of other swim interneurons by their inclusion in similar silicon networks. The current model provides an important platform for future implementations of invertebrate adaptive behaviors, especially those behaviors related to swim and other locomotory pattern generation. The hope is that such behaviors 627 628 Siegel can be evolved incrementally using neuromime models of identified adaptive interneurons to modulate the swim central pattern generating network. Acknowledgments I would like to thank the department of Electrical Engineering at Yale University for encouraging and generously supporting independent undergraduate research. References Rowat, P.P. and Selverston, A.I. (1991). Network, 2, 17-41. Ryckebusch, S., Bower, J.M., Mead, C., (1989). In D.Touretzky (ed.), Advances in Neural Information Processing Systems, 384-393. San Mateo, CA: Morgan Kaufmann. Friesen, W.O. (1989). In J. Jacklet (ed), Neuronal and Cellular Oscillators, 269-316. New York: Marcel Dekker. Debski, E.A. and Friesen, W.O. (1985). Journal of Experimental Biology, 116, 169188. Lockery, S.R. and Kristan, W.B. (1990). Journal of Neuroscience, 10(6), 1811-1815. Sahley, C.L. and Ready, D.P. (1988). Journal of Neuroscience, 8(12), 4612-4620. Stent, G.S. and Kristan, W.B. (1981). In K.Muller, J Nicholls, and G. Stent (eds) , Neurobiology of the Leech, 113-146. Cold Spring Harbor: Cold Spring Harbor Laboratory. Mead, C.A. (1989). Analog VLSl and Neural Systems, Reading, MA: Addison-Wesley. Friesen, W.O. and Stent, G.S. (1977). Biological Cybernetics, 28,27-40. 

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Pulling It All Together: Methods for Combining Neural Networks Michael P. Perrone Institute for Brain and Neural Systems Brown University Providence, RI mpp@cns. brown. edu The past several years have seen a tremendous growth in the complexity of the recognition, estimation and control tasks expected of neural networks. In solving these tasks, one is faced with a large variety of learning algorithms and a vast selection of possible network architectures. After all the training, how does one know which is the best network? This decision is further complicated by the fact that standard techniques can be severely limited by problems such as over-fitting, data sparsity and local optima. The usual solution to these problems is a winner-take-all cross-validatory model selection. However, recent experimental and theoretical work indicates that we can improve performance by considering methods for combining neural networks. This workshop examined current neural network optimization methods based on combining estimates and task decomposition, including Boosting, Competing Experts, Ensemble Averaging, Metropolis algorithms, Stacked Generalization and Stacked Regression. The issues covered included Bayesian considerations, the role of complexity, the role of cross-validation, incorporation of a priori knowledge, error orthogonality, task decomposition, network selection techniques, overfitting, data sparsity and local optima. Highlights of each talk are given below. To obtain the workshop proceedings, please contact the author or Norma Caccia (norma_caccia@brown.edu) and ask for IBNS ONR technical report #69. M. Perrone (Brown University, "Averaging Methods: Theoretical Issues and Real World Examples") presented weighted averaging schemes [7], discussed their theoretical foundation [6], and showed that averaging can improve performance whenever the cost function is (positive or negative) convex which includes Mean Square Error, a general class of Lp-norm cost functions, Maximum Likelihood Estimation, Maximum Entropy, Maximum Mutual Information, the Kullback-Leibler Information (Cross Entropy), Penalized Maximum Likelihood Estimation and Smoothing Splines [6]. Averaging was shown to improve performance on the NIST OCR data, a human face recognition task and a time series prediction task [5]. J. Friedman (Stanford, "A New Approach to Multiple Outputs Using Stacking") presented a detailed analysis of a method for averaging estimators and noted simulations showed that averaging with a positivity constraint was better than cross- 1188 Pulling It All Together: Methods for Combining Neural Networks validation estimator selection [1]. S. Nowlan (Synaptics, "Competing Experts") emphasized the distinctions between static and dynamic algorithms and between averaged and stacked algorithms; and presented results of the mixture of experts algorithm [3] on a vowel recognition task and a hand tracking task. H. Drucker (AT&T, "Boosting Compared to Other Ensemble Methods") reviewed the boosting algorithm [2] and showed how it can improve performance for OCR data. J. Moody (OGI, "Predicting the U.S. Index ofIndustrial Production") showed that neural networks make better predictions for the US IP index than standard models [4] and that averaging these estimates improves prediction performance further. W. Buntine (NASA Ames Research Cent.er, "Averaging and Probabilistic Networks: Automating the Process") discussed placing combination techniques within the Bayesian framework. D. Wolpert (Santa Fe Institute, "Infen ing a Function vs. Inferring an Inference Algorithm") argued that theory can not, in general, identify the optimal network; so one must make assumptions in order to improve performance. H. Thodberg (Danish Meat Research Institute, "Error Bars on Predictions from Deviations among Committee Member~ (within Bayesian Backprop)") raised the provocative (and contentious) point that Bayesian arguments support averaging while Occam's Razor (seemingly?) does not. S. Hashem (Purdue University, "Merits of Combining Neural Networks: Potential Benefits and Risks") emphasized the importance of dealing with collinearity when using averaging methods. References [1] Leo Breiman. Stacked regression. Technical Report TR-367, Department of Statistics, University of California, Berkeley, August 1992. [2] Harris Drucker, Robert Schapire, and Patrice Simard. Boosting performance in neural networks. International Journal of Pattern Recognition and Artificial Intelligence, [To appear]. [3] R. A. Jacobs, M. 1. Jordan, S. J. Nowlan, and G. E. Hinton. Adaptive mixtures of local experts. Neural Computation, 3(2), 1991. [4] U. Levin, T. Leen, and J. Moody. Fa.st pruning using principal components. In Steven J. Hanson, Jack D. Cowan, and C. Lee Giles, editors, Advances in Neural Information Processing Systems 6. Morgan Kaufmann, 1994. [5] M. P. Perrone. Improving Regression Estimation: A veraging ~Methods for Variance Reduction with Eztensions to General Convez Measure Optimization. PhD thesis, Brown University, Institute for Brain and Neural Systems; Dr. Leon N Cooper, Thesis Supervisor, May 1993. [6] M. P. Perrone. General averaging results for convex optimization. In Proceedings of the 1993 Connectionist Models Su,mmer School, pages 364-371, Hillsdale, N.T, 1994. Erlbaum Associates. [7] M. P. Perrone and L. N Cooper. '!\Then networks disagree: Ensemble method for neural networks. In Artificial Neuml Networks for Speech and l!ision. ChapmanHall, 1993. Chapter 10. 1189 

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Locally Adaptive Nearest Neighbor Algorithms Dietrich Wettschereck Thomas G. Dietterich Department of Computer Science Oregon State University Corvallis, OR 97331-3202 wettscdGcs.orst.edu Abstract Four versions of a k-nearest neighbor algorithm with locally adaptive k are introduced and compared to the basic k-nearest neighbor algorithm (kNN). Locally adaptive kNN algorithms choose the value of k that should be used to classify a query by consulting the results of cross-validation computations in the local neighborhood of the query. Local kNN methods are shown to perform similar to kNN in experiments with twelve commonly used data sets. Encouraging results in three constructed tasks show that local methods can significantly outperform kNN in specific applications. Local methods can be recommended for on-line learning and for applications where different regions of the input space are covered by patterns solving different sub-tasks. 1 Introduction The k-nearest neighbor algorithm (kNN, Dasarathy, 1991) is one of the most venerable algorithms in machine learning. The entire training set is stored in memory. A new example is classified with the class of the majority of the k nearest neighbors among all stored training examples. The (global) value of k is generally determined via cross-validation. For certain applications, it might be desirable to vary the value of k locally within 184 Locally Adaptive Nearest Neighbor Algorithms different parts of the input space to account for varying characteristics of the data such as noise or irrelevant features . However, for lack of an algorithm, researchers have assumed a global value for k in all work concerning nearest neighbor algorithms to date (see, for example, Bottou, 1992, p. 895, last two paragraphs of Section 4.1). In this paper, we propose and evaluate four new algorithms that determine different values for k in different parts of the input space and apply these varying values to classify novel examples. These four algorithms use different methods to compute the k-values that are used for classification. We determined two basic approaches to compute locally varying values for k. One could compute a single k or a set of k values for each training pattern, or training patterns could be combined into groups and k value(s) computed for these groups. A procedure to determine the k to be used at classification time must be given in both approaches. Representatives of these two approaches are evaluated in this paper and compared to the global kNN algorithm. While it was possible to construct data sets where local algorithms outperformed kNN, experiments with commonly used data sets showed, in most cases, no significant differences in performance. A possible explanation for this behavior is that data sets which are commonly used to evaluate machine learning algorithms may all be similar in that attributes such as distribution of noise or irrelevant features are uniformly distributed across all patterns. In other words, patterns from data sets describing a certain task generally exhibit similar properties. Local nearest neighbor methods are comparable in computational complexity and accuracy to the (global) k-nearest neighbor algorithm and are easy to implement. In specific applications they can significantly outperform kNN. These applications may be combinations of significantly different subsets of data or may be obtained from physical measurements where the accuracy of measurements depends on the value measured. Furthermore, local kNN classifiers can be constructed at classification time (on-line learning) thereby eliminating the need for a global cross-validation run to determine the proper value of k . 1.1 Methods compared The following nearest neighbor methods were chosen as representatives of the possible nearest neighbor methods discussed above and compared in the subsequent experiments: ? k-nearest neighbor (kNN) This algorithm stores all of the training examples. A single value for k is determined from the training data. Queries are classified according to the class of the majority of their k nearest neighbors in the training data. ? localKNN 1:11 unrelltricted This is the basic local kNN algorithm. The three subsequent algorithms are modifications of this method. This algorithm also stores all of the training examples. Along with each training example, it stores a list of those values of k that correctly classify that example under leave-one-out cross-validation. To classify a query q, the M nearest neighbors of the query are computed, and that k which classifies correctly most of these M 185 186 Wettschereck and Dietterich neighbors is determined. Call this value kM,q. The query q is then classified with the class of the majority of its kM,q nearest neighbors. Note that kM,q can be larger or smaller than M. The parameter M is the only parameter of the algorithm, and it can be determined by cross-validation. ? localKNN kI pruned The list of k values for each training example generally contains many values. A global histogram of k values is computed, and k values that appear fewer than L times are pruned from all lists (at least one k value must, however, remain in each list). The parameter L can be estimated via crossvalidation. Classification of queries is identical to localKNN kI unrestricted. ? localKNN one 1: per clau For each output class, the value of k that would result in the correct (leaveone-out) classification of the maximum number of training patterns from that class is determined. A query q is classified as follows: Assume there are two output classes, C1 and C2 ? Let kl and k2 be the k value computed for classes Cl and C2, respectively. The query is assigned to class C1 if the percentage of the kl nearest neighbors of q that belong to class C1 is larger than the percentage of the k2 nearest neighbors of q that belong to class C2. Otherwise, q is assigned to class C2. Generalization of that procedure to any number of output classes is straightforward. ? localKNN one 1: per cluster An unsupervised cluster algorithm (RPCL, l Xu et al., 1993) is used to determine clusters of input data. A single k value is determined for each cluster. Each query is classified according to the k value of the cluster it is assigned to. 2 Experimental Methods and Data sets used To measure the performance of the different nearest neighbor algorithms, we employed the training set/test set methodology. Each data set was randomly partitioned into a training set containing approximately 70% of the patterns and a test set containing the remaining patterns. After training on the training set, the percentage of correct classifications on the test set was measured. The procedure was repeated a total of 25 times to reduce statistical variation. In each experiment, the algorithms being compared were trained (and tested) on identical data sets to ensure that differences in performance were due entirely to the algorithms. Leave-one-out cross-validation (Weiss & Kulikowski, 1991) was employed in all experiments to estimate optimal settings for free parameters such as k in kNN and M in localKNN. 1 Rival Penalized Competitive Learning is a straightforward modification of the well known k-means clustering algorithm. RPCL's main advantage over k-means clustering is that one can simply initialize it with a sufficiently large number of clusters. Cluster centers are initialized outside of the input range covered by the training examples. The algorithm then moves only those cluster centers which are needed into the range of input values and therefore effectively eliminates the need for cross-validation on the number of clusters in k-means. This paper employed a simple version with the number of initial clusters always set to 25, O'c set to 0.05 and O'r to 0.002. Locally Adaptive Nearest Neighbor Algorithms We report the average percentage of correct classifications and its standard error. Two-tailed paired t-tests were conducted to determine at what level of significance one algorithm outperforms the other. We state that one algorithm significantly outperforms another when the p-value is smaller than 0.05. 3 3.1 Results Experiments with Constructed Data Sets Three experiments with constructed data sets were conducted to determine the ability of local nearest neighbor methods to determine proper values of k . The data sets were constructed such that it was known before experimentation that varying k values should lead to superior performance. Two data sets which were presumed to require significantly different values of k were combined into a single data set for each of the first two experiments. For the third experiment, a data set was constructed to display some characteristics of data sets for which we assume local kNN methods would work best. The data set was constructed such that patterns from two classes were stretched out along two parallel lines in one part of the input space. The parallel lines were spaced such that the nearest neighbor for most patterns belongs to the same class as the pattern itself, while two out of the three nearest neighbors belong to the other class. In other parts of the input space, classes were well separated, but class labels were flipped such that the nearest neighbor of a query may indicate the wrong pattern while the majority of the k nearest neighbors (k > 3) would indicate the correct class (see also Figure 4). Figure 1 shows that in selected applications, local nearest neighbor methods can lead to significant improvements over kNN in predictive accuracy. Letter Experiment 2 Sine-21 Experiment 3 Wave-21 Combined Constructed 70 .0?O.6 -4~~~~~~~~~~~~~~~~~ I. ks pruned ? ks unrestricted Q one k per class 0 one k per cluster 1 Figure 1: Percent accuracy of local kNN methods relative to kNN on separate test sets. These differences (*) were statistically significant (p < 0.05). Results are based on 25 repetitions. Shown at the bottom of each graph are sizes of training sets/sizes of test sets/number of input features. The percentage at top of each graph indicates average accuracy of kN N ? standard error. The best performing lqcal methods are locaIKNNl;, pruned, localKNNl;8 unre,tricted, 187 188 Wettschereck and Dietterich and 10calKNN one k per cluster. These methods were outperformed by kNN in two of the original data sets. However, the performance of these methods was clearly superior to kNN in all domains where data were collections of significantly distinct subsets. 3.2 Experiments with Commonly Used Data Sets Twelve domains of varying sizes and complexities were used to compare the performance of the various nearest neighbor algorithms. Data sets for these domains were obtained from the UC-Irvine repository of machine learning databases (Murphy & Aha, 1991, Aha, 1990, Detrano et al., 1989). Results displayed in Figure 2 indicate that in most data sets which are commonly used to evaluate machine learning algorithms, local nearest neighbor methods have only minor impact on the performance of kNN. The best local methods are either indistinguishable in performance from kNN (localKNN one k per cluster) or inferior in only one domain (localKNN k, pruned). 105150/4 16 150/64/9 ~"""T'"-f&C:NN * -2 I. ks pruned ? ks unrestricted Iilll one k per class 0 one k per cluster 1 Figure 2: Percent accuracy of local kNN methods relative to kNN on separate test sets. These differences (*) were statistically significant (p < 0.05). Results are based on 25 repetitions. Shown at the bottom of each graph are sizes of training sets/sizes of test sets/number of input features. The percentage at top of each graph indicates average accuracy of kNN ? standard error. The number of actual k values used varies significantly for the different local methods (Table 1). Not surprisingly, 10calKNNks unrestricted uses the largest number of distinct k values in all domains. Pruning of ks significantly reduced the number of values used in all domains. However, the method using the fewest distinct k values is 10calKNN one k per cluster, which also explains the similar performance of kNN and 10calKNN one k per cluster in most domains. Note that several clusters computed by 10calKNN one k per cluster may use the same k. Locally Adaptive Nearest Neighbor Algorithms Table 1: Average number of distinct values for k used by local kNN methods. Task Letter recos. Led-16 CombinedLL Sine-21 Waveform-21 Combined SW Constructed Iris Glasd Wine Hunsarian Cleveland Votins Led-7 Display Led-24. Display Waveform-2I Waveform-4.0 Iaolet Letter Letter reco6' kNN 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 k! J2runed 7.6?1.1 I6.4.?2.5 52.0?3.8 6.6?l.O 9.1?1.4. 13 .5?1.5 1l .8?O.9 1.6?O.2 7.7?O.8 2 .2?O.4. 4..I?O.6 8.0?l.O 4..I?O .4. 5.6?O.4. 16.0?2.9 9.7?1.3 8.4.?2.0 1l.5?2.1 9.4.?1.9 local kNN methods one k per k! unredricted cia!! 6 .4. ?O.3 10.8?1.5 4.3.3?O.9 9.2?O .1 H .7?O.4. 71.4.?1.2 27.5?1.1 2.0?O.O 28.0?1.5 2 .9?O.1 3.0?O.O 30.8?1.6 2.0?O.O 15. 7?O .5 2 .4.?O.1 2.0?O. 2 1l . 2?O.7 3. 3?O.2 3.8?O. 4. 2.0?O.1 2 .0?O.O 12.6?O.6 17.2?1.1 1.8?O. 1 2.0?O.O 6.4.?O. 3 7.6?O.4. 6.1?O.2 37.4.?1.6 9 .0?O.2 27 .8?1.2 3 .0?O.O 3.0?O .O 29.9?1.5 4.3.9?O.6 16.5?O .5 6.0?O.3 I7.0?2.3 one k per clu!ter 1.8?O.2 9 .2?O .5 3.0?O.2 l.O?O.O 4. .2?O .2 4. .8?O.2 5.4.?O.2 2 .3 ?O.1 1.9?O.2 2 .6?O .1 l.O?O.O 4. .6?O.2 1.3?O.1 1.0?O.O 1 .6?O.2 4..3?O.1 4..8?O.1 7.1?O. 3 2 .4.?O.2 Figure 3 shows, for one single run of Experiment 2 (data sets were combined as described in Figure 1), which k values were actually used by the different local 1, methods. Three clusters of k values can be seen in this graph, one cluster at k one at k = 7,9,11,12 and the third at k = 19,20,21. It is interesting to note that the second and the third cluster correspond to the k values used by kNN in the separate experiments. Furthermore, kNN did not use k = 1 in any of the separate runs. This gives insight into why kNN's performance was inferior to that of the local methods in this experiment: Patterns in the combined data set belong to one of three categories as indicated by the k values used to classify them (k = 1, k ~ 10, k ~ 20). Hence, the performance difference is due to the fact that kNN must estimate at training time which single category will give the best performance while the local methods make that decision at classification time for each query depending on its local neighborhood. = ? 13 kvalues (bars) ? 30 k values (bars) El 3 k values (bars) o Sk values (bars) one k per class 0 one k per cluster I Figure 3: Bars show number of times local kNN methods used certain k values to classify test examples in Experiment 2 (Figure 1 (Combined), numbers based on single run). KNN used k = 1 in this experiment. 189 190 Wettschereck and Dietterich 4 Discussion Four versions of the k-nearest neighbor algorithm which use different values of k for patterns which belong to different regions of the input space were presented and evaluated in this paper. Experiments with constructed and commonly used data sets indicate that local nearest neighbor methods may have superior classification accuracy than kNN in specific domains. Two methods can be recommended for domains where attributes such as noise or relevance of attributes vary significantly within different parts of the input space. The first method, called localKNN 1:" pruned, computes a list of "good" k values for each training pattern, prunes less frequent values from these lists and classifies a query according to the list of k values of a pre-specified number of neighbors of the query. Leave-one-out cross-validation is used to estimate the proper amount of pruning and the size of the neighborhood that should be used. The other method, localKNN one k per du,ter, uses a cluster algorithm to determine clusters of input patterns. One k is then computed for each cluster and used to classify queries which fall into this cluster. LocalKNN one k per du,ter performs indistinguishable from kNN in all commonly used data sets and outperforms kNN on the constructed data sets. This method compared with all other local methods discussed in this paper introduces a lower computational overhead at classification time and is the only method which could be modified to eliminate the need for leave-one-ou t cross-validation. The only purely local method, localKNN k. unre,tricted, performs well on constructed data sets and is comparable to kNN on non-constructed data sets. Sensitivity studies (results not shown) showed that a constant value of 25 for the parameter M gave results comparable to those where cross-validation was used to determine the value of M. The advantage of localKNN k, unrestricted over the other local methods and kNN is that this method does not require any global information whatsoever (if a constant value for M is used). It is therefore possible to construct a localKNN k6 unre,tricted classifier for each query which makes this method an attractive alternative for on-line learning or extremely large data sets. If the researcher has reason to believe that the data set used is a collection of subsets with significantly varying attributes such as noise or number of irrelevant features, we recommend the construction of a classifier from the training data using localKNN on e k per du,ter and comparison of its performance to kNN. If the classifier must be constructed on-line then localKNNk, unre,tricted should be used instead of kNN. We conclude that there is considerable evidence that local nearest neighbor methods may significantly outperform the k-nearest neighbor method on specific data sets. We hypothesize that local methods will become relevant in the future when classifiers are constructed that simultaneously solve a variety of tasks. Acknowledgements This research was supported in part by NSF Grant IRI-8657316, NASA Ames Grant NAG 2-630, and gifts from Sun Microsystems and Hewlett-Packard. Many thanks Locally Adaptive Nearest Neighbor Algorithms to Kathy Astrahantseff and Bill Langford for helpful comments during the revision of this manuscript. References Aha, D.W. (1990). A Study of Instance-Based Algorithms for Supervised Learning Tasks. Technical Report, University of California, Irvine. Bottou, L., Vapnik, V. (1992). Local Learning Algorithms. Neural Computation, 4(6), 888-900. Dasarathy, B.V. (1991). Nearest Neighbor(NN) Norms: NN Pattern Classification Techniques. IEEE Computer Society Press. Detrano, R., Janosi, A., Steinbrunn, W., Pfisterer, M., Schmid, K., Sandhu, S., Guppy, K., Lee, S. & Froelicher, V. (1989). Rapid searches for complex patterns in biological molecules. American Journal of Cardiology, 64, 304-310. Murphy, P.M. & Aha, D.W. (1991). UCI Repository of machine learning databases Technical Report, University of California, Irvine. {Machine-readable data repository}. Weiss, S.M., & Kulikowski, C.A. (1991). Computer Systems that learn. San Mateo California: Morgan Kaufmann Publishers, INC. Xu, L., Krzyzak, A., & Oja, E. (1993). Rival Penalized Competitive Learning for Clustering Analysis, RBF Net, and Curve Detection IEEE Transactions on Neural Networks, 4(4),636-649. I iI !: : !! .50 da.. point. ??- - - - - - Nol..,(.-da.. - - - - - _ ii ??- - - - - - Nol..,rr-da.. - - - - - .50d."polnta i ! : : ! ~ kNN correct: 69.3% local kNN correct: 66.9% Total correct: ! .51.0'11> i 84.6% I kNN:70.0% local kNN: 74.8% 77 ..5% 78.3% Size o( ttalnlnll ..,t: 480 leat act: 120 Figure 4: Data points for the Constructed data set were drawn from either of the two displayed curves (i.e. all data points lie on either of the two curves). Class labels were flipped with increasing probabilities to a maximum noise level of approximately 45% at the respective ends of the two lines. Listed at the bottom is performance of kNN and 10calKNN unre.stricted within different regions of the input space and for the entire input space. 191 

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A Local Algorithm to Learn Trajectories with Stochastic Neural Networks Javier R. Movellan? Department of Cognitive Science University of California San Diego La Jolla, CA 92093-0515 Abstract This paper presents a simple algorithm to learn trajectories with a continuous time, continuous activation version of the Boltzmann machine. The algorithm takes advantage of intrinsic Brownian noise in the network to easily compute gradients using entirely local computations. The algorithm may be ideal for parallel hardware implementations. This paper presents a learning algorithm to train continuous stochastic networks to respond with desired trajectories in the output units to environmental input trajectories. This is a task, with potential applications to a variety of problems such as stochastic modeling of neural processes, artificial motor control, and continuous speech recognition . For example, in a continuous speech recognition problem, the input trajectory may be a sequence of fast Fourier transform coefficients, and the output a likely trajectory of phonemic patterns corresponding to the input. This paper was based on recent work on diffusion networks by Movellan and McClelland (in press) and by recent papers by Apolloni and de Falco (1991) and Neal (1992) on asymmetric Boltzmann machines. The learning algorithm can be seen as a generalization of their work to the stochastic diffusion case and to the problem of learning continuous stochastic trajectories. Diffusion networks are governed by the standard connectionist differential equations plus an independent additive noise component. The resulting process is governed ?Pa.rt of this work was done while a.t Ca.rnegie Mellon University. 83 84 Movellan by a set of Langevin stochastic differential equations dai(t) = Ai dri/ti(t) dt + crdBi(t); i E {I, ... , n} (1) where Ai is the processing rate of the ith unit, cr is the diffusion constant, which controls the flow of entropy throughout the network, and dBi(t) is a Brownian motion differential (Soon, 1973). The drift function is the deterministic part of the process. For consistency I use the same drift function as in Movellan and McClelland, 1992 but many other options are possible: dri/ti(t) = E J=1 Wijaj(t) - /-l ai (t), where Wij is the weight from the jth to the ith unit, and /-1 is the inverse of a logistic function scaled in the (min - max) interval:/- 1 (a) log m-;~!~' = In practice DNs are simulated in digital computers with a system of stochastic difference equations ai(t+dt)=ai(t)+Aidri/ti(t)dt+crzi(t)../Xi; iE{I, ... ,n} (2) where Zi(t) is a standard Gaussian random variable. I start the derivations of the learning algorithm for the trajectory learning task using the discrete time process (equation 2) and then I take limits to obtain the continuous diffusion expression. To simplify the derivations I adopt the following notation: a trajectory of states -input, hidden and output units- is represented as a [a(I) ... a(t m )] [al(I) ... an (I) ... al(t m ) .. . an (t m )]. The trajectory vector can be partitioned into 3 consecutive row vectors representing the trajectories of the input, hidden and output units a [xhy). = = = The key to the learning algorithm is obtaining the gradient of the probability of specific trajectories. Once we know this gradient we have all the information needed to increase the probability of desired trajectories and decrease the probability of unwanted trajectories. To obtain this gradient we first need to do some derivations on the transition probability densities. Using the discrete time approximation to the diffusion process, it follows that the conditional transition probability density functions are multivariate Gaussian (3) From equation 2 and 3 it follows that o A' OWij log p(a(t + dt)1 a(t? = ; Zi(t) ~ V dtaj(t) (4) Since the network is Markovian, the probability of an entire trajectory can be computed from the product of the transition probabilities t ",-1 p(a) II p(a(t + dt)la(t? = p(a(to? (5) t=to The derivative of the probability of a specific trajectory follows o() A : a = p(a)~ Wij cr L t",-l t=to Zi(t)../Xi aj(t) (6) A Local Algorithm to Learn Trajectories with Stochastic Neural Networks In practice, the above rule is all is needed for discrete time computer simulations. We can obtain the continuous time form by taking limits as ~t --+ 0, in which case the sum becomes Ito's stochastic integral of aj(t) with respect to the Brownian motion differential over a {to, T} interval. op(a) aWi; =p(a)'~i U iT A similar equation may be obtained for the o;i~) = pea)! ? U aj(t)dBi(t) (7) to iT ~i parameters drifti(t)dBi(t) (8) to For notational convenience I define the following random variables and refer to them as the delta signals . 6Wij {a) = O1oga pea) = -~i Wij U iT (9) aj(t)dBi(t) to and (10) A 1 n B 1 ~ (\ " 1\ c: A 0 CIS A ;; c: > n e:( 0.5- 0 :::; ~ ~ I 0.5- en C) CIS ~ en .a: ~ 0 0 V ~ V I I 100 200 Time Steps l} V 0 300 0 V V V V I I 100 200 300 Time Steps Figure 1: A) A sample Trajectory. B) The Average Trajectory. As Time Progresses Sample Trajectories Become Statistically Independent Dampening the Average. 8S 86 Movellan The approach taken in this paper is to minimize the expected value of the error assigned to spontaneously generated trajectories 0 = E(p(a? where pea) is a signal indicating the overall error of a particular trajectory and usually depends only on the output unit trajectory. The necessary gradients follow (11) (12) Since the above learning rule does not require calculating derivatives of the p function, it provides great flexibility making it applicable to a wide variety of situations. For example pea) can be the TSS between the desired and obtained output unit trajectories or it could be a reinforcement signal indicating whether the trajectory is or is not desirable. Figure La shows a typical output of a network trained with TSS as the p signal to follow a sinusoidal trajectory. The network consisted of 1 input unit, 3 hidden units, and 1 output unit. The input was constant through time and the network was trained only with the first period of the sinusoid. The expected values in equations 11 and 12 were estimated using 400 spontaneously generated trajectories at each learning epoch. It is interesting to note that although the network was trained for a single period, it continued oscillating without dampening. However, the expected value of the activations dampened, as Figure l.b shows. The dampening of the average activation is due to the fact that as time progresses, the effects of noise accumulate and the initially phase locked trajectories become independent oscillators. 20,-------------___________ p transition 18 =0.2 -:.c== 16 >- Hidden state =0 !'-p(response 1) p transition =0.1 [RuP=. J Hidden state =1 =0.05 ~ 12 o J! p(response 1) r~~J 14 ~ - 0.10 .E g 8 g> 6 =0.8 best possible performance ...J 4 2 O~------,-------~------~ .0 900 1900 2900 Learning Epoch Figure 2: A) The Hidden Markov Emitter. B) Average Error Throughout Training. The Bayesian Limit is Achieved at About 2000 Epochs. A Local Algorithm to Learn Trajectories with Stochastic Neural Networks The learning rule is also applicable in reinforcement situations where we just have an overall measure of fitness of the obtained trajectories, but we do not know what the desired trajectory looks like. For example, in a motor control problem we could use as fitness signal (-p) the distance walked by a robot controlled by a DN network. Equations 11 and 12 could then be used to gradually improve the average distance walked by the robot. In trajectory recognition problems we could use an overall judgment of the likelihood of the obtained trajectories. I tried this last approach with a toy version of a continuous speech recognition problem. The "emitter" was a hidden Markov model (see Figure 2) that produced sequences of outputs - the equivalent of fast Fourier transform loads - fed as input to the receiver. The receiver was a DN network which received as input, sequences of 10 outputs from the emitter Markov model. The network's task was to guess the sequence of hidden states of the emitter given the sequence of outputs from the emitter. The DN outputs were interpreted as the inferred state of the emitter. Output unit activations greater than 0.5 were evaluated as indicating that the emitter was in state 1 at that particular time. Outputs smaller than 0.5 were evaluated as state O. To achieve optimal performance in this task the network had to combine two sources of information: top-down information about typical state transitions of the emitter, and bottom up information about the likelihood of the hidden states of the emitter given its responses. The network was trained with rules 11 and 12 using the negative log joint probability of the DN input trajectory and the DN output trajectory as error signal. This signal was calculated using the transition probabilities of the emitter hidden Markov model and did not require knowledge of its actual state trajectories. The necessary gradients for equations 11 and 12 were estimated using 1000 spontaneous trajectories at each learning epoch. As Figure 3 shows the network started producing unlikely trajectories but continuously improved. The figure also shows the performance expected from an optimal classifier. As training progressed the network approached optimal performance. Acknowledgements This work was funded through the NIMH grant MH47566 and a grant from the Pittsburgh Supercomputer Center. References B. Apolloni, & D. de Falco. (1991) Learning by asymmetric parallel Boltzmann machines. Neural Computation, 3, 402-408. R. Neal. (1992) Asymmetric Parallel Boltzmann Machines are Belief Networks, Neural Computation, 4, 832-834. J. Movellan & J. McClelland. (1992a) Learning continuous probability distributions with symmetric diffusion networks. To appear in Cognitive Science. T. Soon. (1973) Random Differential Equations in Science and Engineering, Academic Press, New York. 87 

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An Optimization Method of Layered Neural Networks based on the Modified Information Criterion Sumio Watanabe Information and Communication R&D Center Ricoh Co., Ltd. 3-2-3, Shin-Yokohama, Kohoku-ku, Yokohama, 222 Japan sumio@ipe.rdc.ricoh.co.jp Abstract This paper proposes a practical optimization method for layered neural networks, by which the optimal model and parameter can be found simultaneously. 'i\Te modify the conventional information criterion into a differentiable function of parameters, and then, minimize it, while controlling it back to the ordinary form. Effectiveness of this method is discussed theoretically and experimentally. 1 INTRODUCTION Learning in art.ificialneural networks has been studied based on a statist.ical framework, because the statistical theory clarifies the quantitative relation between t.he empirical error and the prediction error. Let us consider a function <p( w; x) from the input space R/\ to the out.put space R L with a paramet.er 'lV. "\i\Te assume that training samples {(.1:j, yd}~l are taken from t.he true probabilit.y density Q(x, y). Let us define the empirical error by (1) 293 294 Watanabe and the prediction error by E(w) == JJ lIy - ip(w; x)11 2 Q(x, y)dxdy. (2) If we find a parameter w* which minimizes Eemp( w), then * < E(w ) >== (1 + 2(F(w*) + 1) * 1 NL ) < Eemp{w ) > +o(N)' (3) where < . > is the average value for the training samples, o( 1/N) is a small term which satisfies No(I/N) ~ 0 when N ~ 00, and F(w*), N, and L are respectively the numbers of the effective parameters of w*, the training samples, and output units. Although the average < . > cannot be calculated in the actual application, the optimal model for the minimum prediction error can be found by choosing the model that minimizes the Akaike informat.ion crit.erion (AIC) [1], * 2(F(w*) + 1) * J(w)=(I+ NL )Eemp(w). (4) This method was generalized for arbitrary distance [2]. The Bayes informat.ion criterion (BIC) [3] and the minimum descript.ion lengt.h (MDL) [4] were proposed to overcome the inconsistency problem of AIC that the true model is not always chosen even when N ~ 00. The above information criteria have been applied to the neural network model selection problem, where the maximum likelihood estimator w* was calculated for each model, and then information criteria were compared. Nevertheless, the practical problem is caused by the fact. that we can not always find the ma..ximum likelihood estimator for each model, and even if we can. it takes long calculation time. In order to improve such model selection procedures, this paper proposes a practical learning algorithm by which the optimal model and parameter can be found simultaneously. Let us consider a modified information criterion, Ju(w) == (1+ 2(FuCw) + 1) NL )Eemp(w). (5) where a > 0 is a parameter and Fa(w) is a Cl-class function which converges to F(w) when a ~ O. \Ve minimize Ja(w), while controlling a as a ~ 0, To show effectiveness of this method, we show experimental results, and discuss the theoretical background. 2 2.1 A Modified Information Criterion A Formal Information Criterion Let us consider a conditional probability distribut.ion. P{W,O";ylx) = (2nO"12)L/2 exp (- Ily-ip(w;x)W ? 2 ), _0" (6) An Optimization Method of Layered Neural Networks where a function rp( w; x) = {rpi(W; x)} is given by the three-layered perceptron, II l\" +L rpi(W; :1:) = p(WiO Wij p(WjO +L j=1 and W wjkxd), (7) k=l = {w iO, Wij} is a set of biases and weights and p(.) is a sigmoidal function. Let A1max be the full-connected neuralnctwork model with 1'1." input units, H hidden units, and L output units, and /vt be the family of all models made from A1max by pruning weights or eliminating biases. \Vhcn a sct of training samples {(Xi, vd }[~:1 is given, we define an empirical loss and the prediction loss by 1 N - 1N' _ log P(w, 0"; vi/xd, L (8) 1=1 -JJ L(w,O") Q(:l",v) 10gP(w,0"; Vlx)d:t:dy. (9) Minimizing Lemp (w, 0") is equivalent to minimizing E elllp {w), and mIIllmlzing L(w, 0") is equivalent. to minimizing E(w). \Ye assume t.hat. t.here exists a parameter (wAI'O"AI) which minimizes Lemp{W,CT) in each modcl.H E A1. By the theory of AIC, we have the following formula, (10) Based on this property, let us define a formal information criterion I (Af) for a model Af by (11) I{Jlf) = 2N Lemp{wAI' O"~I ) + A( Fo (wAf) + 1) where A is a constant and Fo (w) is the number of nonzero parameters in w, Jl L Fo{w) = L L II fO(Wij) l\ +L i=1 j=O L fO{Wjd? (12) j=lk=O where fo (x) is 0 if x = 0, or 1 if otherwise. I{1U) is formally equal to AIC if A = 2, or l\'IDL if A = 10g{N). Notc that F(w) ~ Fo{w) for arbitrary wand that F( w AJ ) Fo (w AI) if and only if the Fisher information mat.rix of the model !II is positive definite. = 2.2 A Modified Information Criterion In order to find the optimal model and parameter simultaneously, we define a modified information critcrion. For Q' > O. 2NLemp(w,0") L Fo{w) + A{Fo{w) + 1), Jl LLfO'{Wij) i=l j=O H I{ + LLfo{wjJ.o), j=ll,?=O where fa-(x) satisfies the following two conditions. (13) (14) 295 296 Watanabe (1) 10.(x) -+ 10(x) when 0: -+ O. (2) If Ixl :::; Ivi then 0:::; 10.(.1:) :::; 10(Y) :::; 1. For example, 1- exp( _x2 /0: 2 ) and 1-1/(1 + (x/0:)2) satisfy this condition. Based on these definitions, we have the following theorem. Theorem min 1(111) = lim min 10 (w, 0'). AI EM o,~o W,CT This theorem shows that the optimal model and parameter can be found by minimizing 1a(1O, 0') while controlling 0: as 0: -+ 0 (The parameter 0: plays the same role as the temperature in the simulated annealing). As Fo.(x) -+ Fo(x) is not uniform convergence, this theorem needs the second condition on 1a (:t'). (For proof of the theorem, see [5]). If we choose a different.iable function for 10 (10), then its local minimum can be found by the steepest descent method, dw 0 dO' 0 dt =-o10 10 (w,0'), Tt=-oO'la(w,O'). (15) These equat.ions result in a learning dynamics, ~1o = N -TJ 0 2: {ow IIvi - A ';'(10; .'ri) 112 + ; A2 0F Ot;'}, (16) i=l where 0'2 = (I/NL)"'?//=lllvi - ,;,(w;:rdIl 2 . and 0: is slowly controlled as 0: -+ O. This dynamics can be understood as the (,lTor backpropagation with the added term. 3 3.1 Experimental Results The true distribution is contained in the models First, we consider a case when t.he true distribut.ion is cont.ained in the model family M. Figure 1 (1) shows the true model from which t.he training samples were taken. One thousand input samples were t.aken from the uniform probability on [-0.5,0.5] x [-0.5,0.5] x [-0.5,0.5]. The output samples were calculat.ed by the network in Figure 1 (1), and noizes were added which were taken from a normal distribution with the expectation 0 and the variance 3.33 x 10- 3 . Ten thousands testing samples were t.aken from t.he same distribut.ion. "Te used 10 ('IV) = 1 exp( _w 2 /20'2) as a soft.ener function, and t.he "annealing schedule" of 0 ' was set as 0:( n) = 0'0 (1 - n/ n max ) + ?, where 'Il is the t.raining cycle number, 0 '0 = 3.0, n max = 25000, and ? = 0.01. Figure 1 (2) shows the full-connected nlOdel Afmax with 10 hidden units, which is the initial model. In the training, the learning speed TJ was set as 0.1. We compared the empirical errors and t.he prediction errors for several cases for A (Figure 1 (5), (6)). If A = 2, the crit.erion is AIC, and if A = 10g(N) = 6.907, it is BIC or MDL. Figure 1 (3) and (4) show the optimized models and parameters for the criteria ,vith A = 2 and A = 5. \\Then .4 = 5, t.he true model could be found. An Optimization Method of Layered Neural Networks 3.2 The true distribution is not contained Second, let. us consider a case that the true distribution is not contained in the model family. For t.he training samples and the testing samples, we used the same probability density as the above case except that the function was (17) Figure 2 (1) and (2) show the training error and the prediction error, respectively. In t.his case, the best generalized model was found by AIC, shown in Figure 3. In the optimized network, Xl and X2 were almost separated from X3, which means that the network could find the structure of the true model in eq.{17.) The practical application to ultrasonic image reconstruct.ion is shown in Figure 3. 4 4.1 Discussion An information criterion and pruning weights If P(w, u; ylx) sufficiently approximates Q(YI:~~ ) and N is sufficiently large, we have (18) where Z N = Lemp{ 'LV j\f) - LC(iJ j\f) and 'IV j\f is the parameter which minimizes L( 'lV, u) in the model lIf. Although < ZN >= 0 resulting in equation (10), its standard deviation has the same order as (1/ VN). However, if 1111 C 1If2 or lIt!1 ~ lith, then 'Ii; 1111 and 'LV 1\12 expected to be almost common. and it doesn't essentially affect the model selection problem [2]. The model family made by pruning weights or by eliminating biases is not. a totally ordered set but a partially ordered set for the order "c". Therefore, if a model 111 E M is select.ed, it is the optimal model in a local model family M' = {1If' E Mj 1If' C 111 or 111' ~ Af}, but it may not be the optimal model in the global family M. Artificial neural networks have the local minimum problem not. only in the parameter space but also in the model family. 4.2 The degenerate Fisher information matrix. If the true probability is contained in the model and the number of hidden units is larger than necessary one, then the Fisher informat.ion matrix is degenerated, and consequently. the maximum likelihood est.imator is not. subject t.o the asympt.otically normal distribution [6]. Therefore, the prediction error is not given by eq.(3), or AIC cannot. be deriyed. However, by the proposed method, the selected model has the non-degenerated Fiher information matrix, because if it is degenerate then the modified information crit.erion is not. minimized. 297 298 Watanabe ~ - "~,. output unit N(O,3.33 X 10 ) 10 ~ -2.2 2.2 , 7 -0.7 -2.9 t (1) True model (2) Initial model for learning. (w*) E (3) Optimized by AIC(A=2) (4) Optimized by A=5 E (w*) = 3.29 X 10 -3 ~m'w*) = 3.31 X 10 -3 emp E(w~ = 3.39 X 10- E(w*) emp 3.35 initial 1 initial 3 3.3 initial 2 *' E(W) = 3.37XlO -3 10 -3 3.45 3 3.4 initial 3.35 A 3.25 X 3 AIC MDL (5) The emprical error A (6) The prediction error Figure I: True distribution is contained in the models. E (w*) emp ! E(w*) The empirical error 3.31 X 10- 3 The prediction error 3.41 X 10 - 3 initial 2 initial 2 initial 1 3.7 3.6 initial 3 initial 3 3.6 <'4;2' ~ WI~ 3.5 3.5 3.4 3.4 3.3 SIC AIC (1) The empirical error Figure 2: 3 --t-t--+--+--t-+-..... A 3.409 X 10'AIC SIC (2) The prediction error _;~9~65 ~/O~4' TT xl x2 x3 (3) Optimized by AI C (A=2). True distribution is not contained in the models. An Optimization Method of Layered Neural Networks (1) An Ultrasonic Imaging System (2) Sample Objects. Images for Traiuillg Images for Tcstiug Reconstructed Image Origillal Illlagcs --~~--+--~~---+----~ 15 units Restored using LS:-'I - - - - 1 - - - - - - - + - - - - - - - - 1 ": Restored using .\ I C - - -- - t - - - - - - + - - - - - - l neighborhood "" f-~ - .. '. : '. : . ~ltrasOnic Image 32X32 (3) Neural Net.works ., - ~ . Restored using :.IDL_-'--_----''---_ _ _ _-'--_ _ _ _..-.J (4 )Rcstored Images Figure 3: Practical Applicat.ion t.o Image Rest.oration The propo~ed method was applied t.o ultrasonic image rest.orat.ioll. Figure 3 (1). (2), (3), (4) respectively show an ultrasonic imaging system, the sample objects, and a neural network for image restorat.ion, and the original restored images. The number of paramet.ers optimized by LS~L AIC. and ':\IDL were respect.in-Iy 166. 138. and 57. Rather noizeless images w('re obtained using the modified AIC or 1IDL. For example, the '"Tail of R" ?was clearly restored using AIC. 299 300 Watanabe 4.3 Relation to another generalization methods In the neural information processing field, many methods have been proposed for preventing the over-fit.ting problem. One of t.he most. famous met.hods is the weight decay method, in which we assume a priori probabilit.y distribut.ion on the parameter space and minimize El (w) = Eemp( 10) + '\C( 10), (19) where ,\ and C(w) are chosen by several heuristic methods [7]. The BIC is the information criterion for such a met.hod [3], and the proposed method may be understood as a met.hod how to cont.rol ,\ and C( w). 5 Conclusion An optimization met.hod for layered neural networks was proposed ba.<;ed on the modified informat.ion criterion, and its effectiveness was discussed theoretically and experimentally. Acknowledgements The author would like to t.hank Prof. S. Amari, Prof. S. Yoshizawa, Prof. K. Aihara in University of Tokyo, and all members of the Amari seminar for their active discussions about statistical met.hods in neural net.works. References [1] H.Akaike. (1974) A New Look at the St.atistical Model Identification. it IEEE Trans. on Automatic Control, Vol.AC-19, No.6, pp.716-723. [2] N.Murata, S.Yoshizawa, and S.Amari.(1992) Learning Curves, IvIodel Sel~ction and Complexit.y of Neural Networks. Ad?lIa.nces in Neural Injorm(?tion Processing Systems 5, San Mateo, Morgan Kaufman, pp.607-614. [3] C.Schwarz (1978) Estimating the dimension of a model. Annals of St(ttistics Vo1.6, pp.461-464. [4] J .Rissanen. (1984) Universal Coding, Information, Prediction, and Estimation. IEEE Tra:ns. on Injormation Theory, Vo1.30, pp.629-636. [5] S.Watanabe. (1993) An Optimization :r.,?1 ethod of Artificial Neural Networks based on a Modified Informat.ion Criterion. IEICE technical Re1JOrt Vol.NC93-52, pp.71-78. [6] H.'iVhite. (1989) Learning in Art.ificial Neural Net.works : A Stat.istical Perspective. Neural Computation, Vol.l, pp.425-464. [7] A.S.'iVeigend, D.E.Rumelhart, and B.A.Huberman. (1991) Generalizat.ion of weight-elimination with application t.o foreca.<;t.ing. Advances in Neural Information Processing Systems, Vo1.3, pp.875-882. PART II LEARNING THEORY, GENERALIZATION, AND COMPLEXITY 

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Recovering a Feed-Forward Net From Its Output Charles Fefferman * and Scott Markel David Sarnoff Research Center CN5300 Princeton, NJ 08543-5300 e-mail: cf9imath.princeton .edu smarkel@sarnoff.com ABSTRACT We study feed-forward nets with arbitrarily many layers, using the standard sigmoid, tanh x. Aside from technicalities, our theorems are: 1. Complete knowledge of the output of a neural net for arbitrary inputs uniquely specifies the architecture, weights and thresholds; and 2. There are only finitely many critical points on the error surface for a generic training problem. Neural nets were originally introduced as highly simplified models of the nervous system. Today they are widely used in technology and studied theoretically by scientists from several disciplines. However, they remain little understood. Mathematically, a (feed-forward) neural net consists of: (1) A finite sequence of positive integers (Do, D 1 , ... , D?); (2) A family of real numbers (wJ d defined for 1 :5 e5: L, and (3) A family of real numbers (OJ) defined for 15: f 5: L, 15: j 5: Dl. 1 5: j 5: D l , 1 5: k :5 Dl-l ; The sequence (Do, D 1 , .. " D L ) is called the architecture of the neural net, while the W]k are called weights and the OJ thresholds. Neural nets are used to compute non-linear maps from }R.N to }R.M by the following construction. vVe begin by fixing a nonlinear function 0-( x) of one variable. Analogy with the nervous system suggests that we take o-(x) asymptotic to constants as x tends to ?oo; a standard choice, which we adopt throughout this paper, is o-(.r) = * Alternate address: Dept. of Mathematics. Princeton University, Princeton, NJ 08544-1000. 335 336 Fefferman and Markel tanh ax). Given an "input" (tl , ... ,tDo) E JR Do , we define real numbers Os l S L, 1 S j S De by the following induction on l . = 0 then x; = t ( 4) If l (5) If the x~-l are known with l fixed (1 SlS L), then we set x; for j . for ISjSDe. Here xf ,... , Xhl are interpreted as the outputs of Di "neurons" in the of the net. The output map of the net is defined as the map lth "layer" In practical applications , one tries to pick the neural net [(Do, Dl"'" DL), (W]k)' (OJ)] so that the output map <I> approximates a given map about which we have only imperfect information. The main result of this paper is that under generic conditions, perfect knowledge of the output map <I> uniquely specifies the architecture, the weights and the thresholds of a neural net, up to obvious symmetries. ~Iore precisely, the obvious symmetries are as follows . Let C1o, 11, . .. , ~(L) be permutations, with 11.= {I, ... , De} {I, . . . , De}; and let {e]: Os f. S L, IS j 50 De} be a collection of ? 1 'so Assume that Ii = (identity) and e] = + 1 whenever l = 0 or ? L. Then one checks easily that the neural nets -T = (7) [(Do, D 1 , .. . , DL), (wh), (eJ)] (8) [(Do , D 1 ,.?. , DL), (W]k) ' (O'J)] and have the same output map if we set (9) and This reflects the facts that the neurons in layer l are interchangeable (1 50 f. 50 L - 1) , and that the function 0'( x) is odd. The nets (7) and (8) will be called isomorphtc if they are related by (9). Note in particular that isomorphic neural nets have the same architecture. Our main theorem asserts that, under generic conditions, any two neural nets with the same output map are isomorphic. \Ve discuss the generic conditions which we impose on neural nets. \Ve have to avoid obvious counterexamples such as: (10) Suppose all the weights W]k are zero. Then the output map <I> is constant . The architecture and thresholds of the neural net are clearly not uniquely determined by <I>. (11) Fix lo, JI, h with IS fo S L - 1 and Isil < h 50 Dio ' Suppose we have lo O~o and w~o w~o for all k. Then (5) gi ves x~o = x~o Therefore , the 11 J2 11k 12k Jl J2' e = = Recovering a Feed-Forward Net from Its Output output depends on ;,J~j~l and wJj;l only through the sum i...;Jj~l the output map does not uniquely determine the weights. + wJr;-l. So Our hypotheses are more than adequate to exclude these counterexamples. Specifically, we assume that 1= (12) OJ (13) 0 and :0;1 1= I?1J/I for j 1= j'. wh 1= 0; and for j 1= j', the ratio WJdW]lk is not equal to any fraction of the form pi q with p, q integers and 1 ~ q ~ 100 Dl- Evidently, these conditions hold for generic neural nets. The precise statement of our main theorem is as follows. If two neural nets satisfy (12), (13) and ha've the same output, then the nets are isomorphic. It would be interesting to replace (12), (13) by minimal hypotheses. and to study functions O'(x) other than tanh (~x). \Ve now sketch the proof of our main result . sacrificing accuracy for simplicity. After a trivial reduction. we may assume Do = DL = 1. Thus, the outputs of the nodes xJ(t) are functions of one variable, and the output map of the neural net is t ~ xf (t). The key idea is to continue the xJ (t) analytically to complex values of t, and to read off the structure of the net from the set of singularities of the xJ, ~ote that 0'( x) = tanh Ox) is meromorphic, with poles at the points of an arithmetic progression {(2m + l);ri: mE ?:}. This leads to two crucial observations. X] (t) (14) When P. = 1, the poles of form an arithmetic progression II;. and (15) 'Vhen e. > 1, every pole of any xi-1(t) is an accumulation point of poles of any X] (t). In fact, (14) is immediate from the formula x;(t) = O'(WJlt the special case Do = 1 of (5). \Ve obtain (16) 1 _ II j - {(2m + l);ri 1 OJ . w jl . mE 2 + O}), which is merely } To see (15), fix e., j, 'It, and assume for simplicity that X~-l(t) has a simple pole at to, while xi- 1(t) (k 1= t:) is analytic in a neighborhood of to. Then (17) t. xr.- 1 (t) = t _Ato + /(t), with / analytic in a neighborhood of to. From (17) and (5), we obtain (18) xJ(t) = O'(W;t-;A(t - to)-1 (19) g(t) + g(t?, with = wJtcf(t) + LWJkX~-I(t) + ?1J analytic in a neighborhood of to. k;c~ Thus, in a neighborhood of to, the poles of X] (1) are the solutions (20) mE:: . tm of the equation 337 338 Fefferman and Markel There are infinitely many solutions of (20), accumulating at to. Hence. to is an accumulation point of poles of xJ(t), which completes the proof of (15). In view of (14), (15), it is natural to make the following definitions. The natural domain of a neural net is the largest open subset of the complex plane to which the output map t ........ xf(t) can be analytically continued. For l? 0 we define the lth singular set Singe C) by setting Sing(O) = complement of the natural domain in C, and Singe e+ 1) = the set of all accumulation points of Singe f). These definitions are made entirely in terms of the output map, without reference to the structure of the given neural net. On the other hand, the sets Sing( ?) contain nearly complete information on the architecture, weights and thresholds of the net. This will allow us to read off the structure of a neural net from the analytic continuation of its output map. To see how the sets Sing(f) reflect the structure of the net, we reason as follows. From (14) and (15) we expect that = (21) For 1 $f $ L, Sing(L -l) is the union over j 1, ... , Dl of the set of poles of xJ(t), together with their accumulation points (which we ignore here), and (22) For f? L, Sing(l) is empty. Immediately, then, we can read off the "depth"' L of the neural net; it is simply the smallest e for which Sing(l) is empty. vVe need to solve for Dt , wh, OJ. We proceed by induction on l. When f = 1, (14) and (21) show that Sing(L - 1) is the union of arithmetic progressions IT}, j == 1, ... , D 1 . Therefore, from Sing(L - 1) we can read off Dl and the IT]. (vVe will return to this point later in the introduction.) In view of (16), IT] determines the weights and thresholds at layer 1. modulo signs. Thus. we have found D I , W}k' g}. When l > 1, we may assume that (23) The D l " wJ~, Of are already known, for 1 ~ l' < f. Our task is to find De, W]k' gJ. In view of (23), we can find a pole to of xk-1(t) for our favorite k. Assume for simplicity that to is a simple pole of x~-I(tL and that the X~-l(t) (k ::j:. ~) are analytic in a neighborhood of to. Then X~-I(t) is given by (17) in a neighborhood of to, with A already known by virtue of (23). Let U be a small neighborhood of to. We will look at the image Y of U n Singe L - l) under the map t ........ t:to' Since A, to and Sing(L - e) are already known, so is Y. On the other hand, we can relate Y to De. WJk' OJ as follows. From (21) we see that Y is the union over j = 1,. ", Dl of (24) Yj = image of U n { Poles of xJ (t)} under t f---> tt:to)' Recovering a Feed-Forward Net from Its Output For fixed j, the poles of xJ(t) in a neighborhood of to are the write lm given by (20). \Ve (25) Equation (20) shows that the first expression in brackets in (25) is equal to (2m + 1 )'7ri. Also, since tm -+ to as Iml - 00 and 9 is analytic in a neighborhood of to, the second expression in brackets in (25) tends to zero. Hence, W~ leA _) tm - to = (2m+1)7ri-g(to)+o(1) forlargem. Comparing this with the definition (24), \':e see that Yj is asymptotic to the arithmetic progression (26) l _ {(2m + 1)7ri - g(to). ~} IT ] l .mEtL.. . Wjt. Thus, the known set Y is the union over j = 1... " Dl of sets Yj, with Yj asymptotic to the arithmetic progression IT~ . From Y, we can therefore read off Dl and the II~ . (\Ve will return to this point in a moment.) \Ve see at once from (26) that wJ ~ is determined up to sign by II]. Thus, we have found Dl and \Vith more work, we can also find the OJ, completing the induction on t. who The above induction shows that the structure of a neural net may be read off from the analytic continuation of its output map. \Ve believe that the analytic continuation of the output map will lead to further consequences in the study of neural nets. Let us touch briefly on a few points which we glossed over above . First of all, suppose we are given a set Y C C, and we know that Y is the union of sets Yl , ... , YD, with Yj asymptotic to an arithmetic progression IT j . vVe assumed above that III, ... , ITD are uniquely determined by Y. In fact, without some further hypothesis on the IT j, this need not be true. For instance, we cannot distinguish IT 1 U IT 2 from II3 if II 1 {odd integers}, II:! {even integers}. II3 {all integers} . On the other hand, we can clearly recognize ITl {all integers} and IT2 {mj2 : m an integer} . from their union ITI U II 2 Thus, irrational numbers enter the picture. The role of our generic hypothesis (13) is to control the arithmetic progressions that arise in our proof. = = = = = Secondly, suppose xk(t) has a pole at to. We assumed for simplicity that xt(t) is analytic in a neighborhood of to for k -::j:. k. However, one of the xk(t) (k -::j:. ft) may also have a pole at to. In that case, the X~+l (t) may all be analytic in a neighborhood of to, because the contributions of the singularities of the xf in (~WJtlxt + OJ+l) (J" may cancel. Thus, the singularity at to may disappear from the output map. \Vhile this circumstance is hardly generic, it is not ruled out by our hypotheses (12), (13). 339 340 Feffennan and Markel Because singularities can disappear, we have to make technical changes in our description of Sing(f). For example, in the discussion following (23), Y need not be the union of the sets rj. Rather, Y is their "approximate union". (See [FD, Next, we should point out that the signs of the weights and thresholds require some attention, even though we have some freedom to change signs by applying isomorphisms. (See (9).) Finally, in the definition of the natural domain, we have assumed that there is a unique maximal open set to which the output map continues analytically. This need not be true of a general real-analytic function on the line - for instance. take f(t) = (1 + t 2)1/2. Fortunately, the natural domain is well-defined for any function that continues analytically to the complement of a countable set. The defining formula (5) lets us check easily that the output map continues to the complement of a countable set, so the natural domain makes sense. This concludes our overview of the proof of our main theorem. The full proof of our results will appear in [F]. Both the uniqueness problem and the use of analytic continuation have already appeared in the neural net literature. In particular, it was R. Hecht-Nielson who pointed out the role of isomorphisms and posed the uniqueness problem. His paper with Chen and Lu [CLH] on "equioutput transformations" on the space of all neural nets influenced our work. E . Sontag [So] and H. Sussman [Su] proved sharp uniqueness theorems for one hidden layer. The proof in [So] uses complex variables. Acknow ledgements Fefferman is grateful to R. Crane, S. j\Iarkel, J. Pearson, E. Sontag, R. Sverdlove, and N. vVinarsky for introducing him to the study of neural nets. This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the Air Force Office of Scientific Research under Contract F49620-92-C-0072. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon. This work was also supported by the National Science Foundation. The following posters, presented at XIPS 93, may clarify our uniqueness theorem. References [CLH] R. Hecht-Nielson, et al., On the geometry of feedforward neural network error surfaces. (to appear). [F] C. Fefferman, Reconstructing a neural network from its output, Re\'ista Mathematica Iberoamericana. (to appear). [So] F. Albertini and E. Sontag, Uniqueness of weights for neural networks. (to appear). [Su] H. Sussman, Uniqueness of the weights JOT minimal feedforward nets u'ith a given input-output map, Neural Networks 5 (1992), pp. 589-593. Recovering a Feed-Forward Net from Its Output "......,.---.... Recovering a Feed-Forward Net from Its 0 utput Suppose an unknown neural netwOf1t i. placed in a black box. Charles Fefferman David Sarnoff Research Center and Princeton Univarsity Princeton. N_ Jersey You aren't allowed to look in the box, blA you are IIIlowed to observe the outputs produced by the network for arbitrary inputs. PI_edt., Scot! A . Markel David Sarnoff Research Center Princeton, N_ Jersey Then. in principle, you have enough information to determine the network architecture (number d layers and number of nodes in each layer) and the unique values for a. the weghts. ..........- ", The Key Question The Output Map of a Neural Network Fix a feed-forward neural network with the standard sigmoid CI (x) = tanh x. ~ When can two neural networks ~ Y, y. have the same output map? y. The map that carries input vectors (XI' ???? x.J to outputvectors (YI' ??.? Y,,) is called the OUTPUT MAP of the neural network. Obvious Examples of Two Neural Networks with the Same Output Map Unlquene.. Theorem Start with a neural network N. Thene~her 1. permlte the nodes in a hidden layer. or 2. fix a hidden node. and change the sign d evefY weight (Including the bias weght) that involves that node This yields a new neural n~ork with the same output map as N. Let N and N' be neural networks that satisfy generic conditions described below. " N and N' have the same output map. then they differ only by sign changes and permutations of hidden node?? 341 342 Fefferman and Markel .--.,.-- ............- Outline of the Proof Generic CondKlon. ? it's enough to con.ider networks with one input node and one output node (see below) We essume thet ? aI _ighl. ere non-zero ? bias weight. within each layer have distinct ebsollte values ? the ralio of weighl. from node i in layer I to nodes j and k in layer (1+ 1) is not equal to any fraction of the form ~q with p. q Integers and 1~q~100'(number of nodes in layer I) Some such assumptions are needed to avoid obvious counterexamples. ? al node output. are nt:IW functions of a .ingle. real variable t (the network input) ? analytically continue the network output to a function f of a .ingle. cofl1llex varillble t ? the qualitative geometry of the pole. of the function f determines the network architecture ( _ belCM') ? the asymptotica of the function f near its singularities determine the weights ....-.c.... .......... .. ......, Reduction to ? Network wtth Single Input and Output Node ? ? focus attention on a single output node, ignoring the others Geometric Description of the Pole. '~'Y ? ?) ? study only input data w~h a single non-zero entry . ... : ... , ?.;)/' <.~'.???-..J.' ?? . . . . .........? ....: 0 ?? ,) ?? . ... . ? ? poles (smell dols) accumulate al essenllal singularities (smell squares) ? essential singularities (small squares) accumulate at more complicated essentlal slngularitles (large dots) .. Determining the Network Architecture from the Picture ? three kinds 01 singularities (small dots, smaH squares. large dots) => thr_ layers of sigmoids, i.e. two hidden layers and an output layer ? thr_ 'spiral arms' of small squares accumulate at each large dot => three nodes in the second hidden layer ? two 'spiral arms' of small dots accumulate at each smaa square => two nodes in the first hidden layer c_ Detennlnlng the Network Architecture from the Picture (conl'd) ? from the network reduction we know thai there is one input node and one output node ? therefore. the network architech.e is es pictured 

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Optimal Brain Surgeon: Extensions and performance comparisons Babak Hassibi* David G. Stork Gregory Wolff Takahiro Watanabe Ricoh California Research Center 2882 Sand Hill Road Suite 115 Menlo Park, CA 94025-7022 and * Department of Electrical Engineering 105B Durand Hall Stanford University Stanford, CA 94305-4055 Abstract We extend Optimal Brain Surgeon (OBS) - a second-order method for pruning networks - to allow for general error measures, and explore a reduced computational and storage implementation via a dominant eigenspace decomposition. Simulations on nonlinear, noisy pattern classification problems reveal that OBS does lead to improved generalization, and performs favorably in comparison with Optimal Brain Damage (OBD). We find that the required retraining steps in OBD may lead to inferior generalization, a result that can be interpreted as due to injecting noise back into the system. A common technique is to stop training of a large network at the minimum validation error. We found that the test error could be reduced even further by means of OBS (but not OBD) pruning. Our results justify the t ~ 0 approximation used in OBS and indicate why retraining in a highly pruned network may lead to inferior performance. 263 264 Hassibi, Stork, Wolff, and Watanabe 1 INTRODUCTION The fundamental theory of generalization favors simplicity. For a given level of performance on observed data, models with fewer parameters can be expected to perform better on test data. In practice, we find that neural networks with fewer weights typically generalize better than large networks with the same training error. To this end, LeCun, Denker and Solla's (1990) Optimal Brain Damage method (OED) sought to delete weights by keeping the training error as small as possible. Hassibi and Stork (1993) extended OED to include the off-diagonal terms in the network's Hessian, which were shown to be significant and important for pruning in classical and benchmark problems. OED and Optimal Brain Surgeon (OES) share the same basic approach of training a network to (local) minimum in error at weight w*, and then pruning a weight that leads to the smallest increase in the training error. The predicted functional increase in the error for a change in full weight vector 8w is: BE 8E= ( Bw )T ?8w+-8w?-1 T B2E 2 'V' ~O Bw 2 ~ =H ?8w + O(1I8wI1 3 ) , (1 ) "-v-" ~O where H is the Hessian matrix. The first term vanishes because we are at a local minimum in error; we ignore third- and higher-order terms (Gorodkin et al., 1993). Hassibi and Stork (1993) first showed that the general solution for minimizing this function given the constraint of deleting one weight was: (2) Here, e q is the unit vector along the qth direction in weight space and Lq is the saliency of weight q - an estimate of the increase in training error if weight q is pruned and the other weights updated by the left equation in Eq. 2. 2 GENERAL ERROR MEASURES AND FISHER'S METHOD OF SCORING In this section we show that the recursive procedure for computing the inverse Hessian for sum squared errors presented in Hassibi and Stork (1993) generalizes to any twice differentiable distance norm and that the key approximation based on Fisher's method of scoring is still valid. Consider an arbitrary twice differentiable distance norm d(t, 0) where t is the desired output (teaching vector) and 0 = F(w, in) the actual output. Given a weight vector w, F maps the input vector in to the output; the total error over P patterns is E = J; 2:f=l d(tlkJ, olkJ). It is straightforward to show that for a single output unit network the Hessian is: Optimal Brain Surgeon: Extensions and Performance Comparisons _ 2. ~ 8P(w, in[kJ) . 8 2 d(t[k J,0[kJ) . 8pT(w, in[kJ) H- P L 8 8 2 8w k=1 W 0 + ~ ~ 8d(t[kJ, O[kJ) . 8 2 P(w, in[kJ) L P k=1 80 8w 2 ? (3) The second term is of order O(lIt - 011); using Fisher's method of scoring (Sever & Wild, 1989), we set this term to zero. Thus our Hessian reduces to: 2. ~ 8P(w, in[kJ) . 8 2 d(t[kJ, o[kJ) . 8pT(w, in[kJ) P L 8w 80 2 8w? (4) k=1 8F(W in[k1) 8 2 d(t[ k1 o[kl) . . .. We define Xk 8W and ak 80 2 , and followmg the lOgIC of HasSlbl and Stork (1993) we can easily show that the recursion for computing the inverse Hessian becomes: H- 1 .X .XT .H- 1 H- 1 = H- 1 _ k k+l k+l k H- 1 = a-II and Hp-l = H- 1 , k+l k .E. + XT . H- 1 . X '0 , ak k+l k k+l (5) where a is a small parameter - effectively a weight decay constant. Note how different error measures d(t,o) scale the gradient vectors X k forming the Hessian (Eq. 4). For the squared error d(t,o) = (t - 0)2, we have ak = 1, and all gradient vectors are weighted equally. The cross entropy or Kullback-Leibler distance, H = d(t, 0) yields ak = o log ~ + (1- 0) log i~ =~? ' 0::; 0, t::; 1 (6) otkl(I~O[l'l). Hence if o[kJ is close to zero or one, Xk is given a large weight in the Hessian; conversely, the smallest value of ak occurs when o[kJ = 1/2. This is desirable and makes great intuitive sense, since in the cross entropy norm the value of o[kJ is interpreted as the probability that the kth input pattern belongs to a particular class, and therefore we give large weight to Xk whose class we are most certain and small weight to those which we are least certain. 3 EIGENSPACE DECOMPOSITION Although DES has been shown to be a powerful method for small and intermediate sized networks - Hassibi, Stork and Wolff (1993) applied OES successfully to NETtaik - its use in larger problems is difficult because of large storage and computation requirements. For a network of n weights, simply storing the Hessian requires 0(n 2 /2) elements and 0(Pn 2 ) computations are needed for each pruning step. Reducing this computational burden requires some type of approximation. Since OES uses the inverse of the Hessian, any approximation to DES will at some level reduce to an approximation of H. For instance OED uses a diagonal approximation; magnitude-based methods use an isotropic approximation; and dividing the network into subsets (e.g., hidden-to-output and input-to-hidden) corresponds to the less-restrictive block diagonal approximation. In what follows we explore the dominant eigenspace decomposition of the inverse Hessian as our approximation. It should be remembered that all these are subsets of the full DBS approach. 265 266 Hassibi, Stork, Wolff, and Watanabe 3.1 Theory The dominant eigendecomposition is the best low-rank approximation of a matrix (in an induced 2-norm sense). Since the largest eigenvalues of H- 1 are the smallest eigenvalues of H, this method will, roughly speaking, be pruning weights in the approximate nullspace of H. Dealing with a low rank approximation of H-l will drastically reduce the storage and computational requirements. Consider the eigendecomposition of H: (7) where ~s contains the largest eigenvalues of H and ~N the smallest ones. (We use the subscripts Sand N to loosely connote signal and noise.) The dimension of the noise subspace is typically m? n. Us and UN are n x (n - m) and n x m matrices that span the dominant eigenspace of Hand H-l, and * denotes matrix transpose and complex conjugation. If, as suggested above, we restrict the weight prunings to lie in UN, we obtain the following saliency and full weight change when removing the qth weight: - Lq - 8w = - 1 2 w~ = - -----...:.~--- ef . UN . ~N1 . UN . e q Wq 1 1 e Tq . UN . ~N . U *N . e q ~N Uiveq , (8) (9) where we have used 'bars' to indicate that these are approximations to Eq. 2. Note now that we need only to store ~N and UN, which have roughly nm elements. Likewise the computation required to estimate ~N and UN is O(Pnm). The bound on Lq is: Lq < Lq < Lq +2 LqLq w~ 1 . a(s) , (10) where a(8) is the smallest eigenvalue of ~s. Moreover if Q:.(8) is large enough so that Q:.( 8) > [H! l)qq we have the following simpler form: (11) In either case Eqs. 10 and 11 indicate that the larger a(8) is, the tighter the bounds are. Thus if the subspace dimension m is such that the eigenvalues in Us are large, then we will have a good approximation. LeCun, Simard and Pearlmutter (1993) have suggested a method that can be used to estimate the smallest eigenvectors of the Hessian. However, for 0 BS (as we shall see) it is best to use the Hessian with the t ~ 0 approximation, and their method is not appropriate. Optimal Brain Surgeon: Extensions and Performance Comparisons 3.2 Simulations We pruned networks trained on the three Monk's problems (Thrun et al., 1991) using the full OBS and a 5-dimensional eigenspace version of OBS, using the validation error rate for stopping criterion. (We chose a 5-dimensional subspace, because this reduced the computational complexity by an order of magnitude.) The Table shows the number of weights obtained. It is clear that this eigenspace decomposition was not particularly successful. It appears as though the the off-diagonal terms in H beyond those in the eigenspace are important, and their omission leads to bad pruning. However, this warrants further study. Monk1 Monk2 Monk3 4 unpruned 58 39 39 OBS 14 16 4 5-d eigenspace 28 27 11 OBS/OBD COMPARISON General criteria for comparing pruning methods do not exist. Since such methods amount to assuming a particular prior distribution over the parameters, the empirical results usually tell us more about the problem space, than about the methods themselves. However, for two methods, such as OBS and OBD, which utilize the same cost function, and differ only in their approximations, empirical comparisons can be informative. Hence, we have applied both OBS and OBD to several problems, including an artificially generated statistical classification task, and a real-world copier voltage control problem. As we show below, the OBS algorithm usually results in better generalization performance. 4.1 MULTIPLE GAUSSIAN PRIORS We created a two-catagory classification problem with a five-dimensional input space. Category A consisted of two Gaussian distributions with mean vectors /-LA! = (1,1,0,1, .5) and /-LA2 = (0,0,1,0, .5) and covariances ~A! = Diag[0.99, 1.0, 0.88, 0.70, 0.95] and ~A2 = Diag[1.28, 0.60, 0.52, 0.93, 0.93] while category B had means /-LB! = (0,1,0,0, .5) and /-LB2 = (1,0,1,1, .5) and covariances ~Bl = Diag[0.84, 0.68, 1.28, 1.02,0.89] and ~B2 = Diag[0.52, 1.25, 1.09,0.64,1.13]. The networks were feedforward with 5 input units, 9 hidden units, and a single output unit (64 weights total). The training and the test sets consisted of 1000 patterns each, randomly chosen from the equi-probable categories. The problem was a difficult one: even with the somewhat large number of weights it was not possible to obtain less than 0.15 squared error per training pattern. We trained the networks to a local error minimum and then applied OBD (with retraining after each pruning step using backpropagation) as well as 0 BS. Figure 1 (left) shows the training errors for the network as a function of the number of remaining weights during pruning by OBS and by OBD. As more weights are pruned the training errors for both OBS and OBD typically increase. Comparing the two graphs for the first pruned weights, the training error for OBD and OBS are roughly equal, after which the training error of OBS is less until the 24th weight 267 268 Hassibi, Stork, Wolff, and Watanabe . 17 . 165 Train E . 22 r E rI .215 .. ',' . , " ' Test .'~ ~.,OBD . . 21 .16 . .... tI _. ? ? ? ? ? ? ._. .205 .155 aBS .2 .15 30 35 40 45 50 55 number of weights 60 65 .195 ' 30 35 40 45 50 55 60 65 number of weights Figure 1: DES and OED training error on a sum of Gaussians prior pattern classification task as a function of the number of weights in the network. (Pruning proceeds right to left.) DES pruning employed 0: = 10- 6 (cf., Eq. 5); OED employed 60 retraining epochs after each pruning. is removed. The reason OED training is initially slightly better is that the network was not at an exact local minimum; indeed in the first few stages the training error for OED actually becomes less than its original value. (Training exhaustively to the true local minimum took prohibitively long.) In contrast, due to the t ---+ 0 approximation DES tries to keep the network response close to where it was, even if that isn't the minimum w*. We think it plausible that if the network were at an exact local minimum DES would have had virtually identical performance. Since OED is using retraining the only reason why OES can outperform after the first steps is that OED has removed an incorrect weight, due to its diagonal approximation. (The reason DES behaves poorly after removing 24 weights - a radically pruned net - may be that the second-order approximation breaks down at this point.) We can see that the minimum on test error occurs before this breakdown, meaning that the failed approximation (Fig. 2) does not affect our choice of the optimal network, at least for this problem. The most important and interesting result is the test error for these pruned networks (Figure 1, right). The test error for OED does not show any consistent behaviour, other than the fact that on the average it generally goes up. This is contrary to what one would expect of a pruning algorithm. It seems that the retraining phase works against the pruning process, by tending to reinforce overfitting, and to reinject the training set noise. For DES, however, the test error consistently decreases until after removing 22 weights a minimum is reached, because the t ---+ 0 approximation avoids reinjecting the training set noise. 4.2 OBS/OBD PRUNING AND "STOPPED" NETWORKS A popular method of avoiding overfitting is to stop training a large net when the validation error reaches a minimum. In order to explore whether pruning could improve the performance on such a "stopped" network (Le., not at w*), we monitored the test error for the above problem and recorded the weights for which a minimum on the test set occured. We then applied OES and OED to this network. Optimal Brain Surgeon: Extensions and Performance Comparisons E ?204 .202 .,, ,, ,, .200 '- , " aBO . 198 ,,'" , -' .. .... " .196 ,\ -, "'--, .194 35 40 45 50 55 60 number of weights Figure 2: A 64-weight network was trained to minimum validation error on the Gaussian problem - not w* - and then pruned by OBD and by OBS. The test error on the resulting network is shown. (Pruning proceeds from right to left.) Note es,pecially that even though the network is far from w*, OBS leads lower test error over a wide range of prunings, even through OBD employs retraining. The results shown in Figure 2 indicate that with OBS we were able to reduce the test error, and this reached a minimum after removing 17 weights. OBD was not able to consistently reduce the test error. This last result and those from Fig. 2 have important consequences. There are no universal stopping criteria based on theory (for the reasons mentioned above), but it is a typical practice to use validation error as such a criterion. As can be seen in Figure 2, the test error (which we here consider a validation error) consistantly decreases to a unique miniumum for pruning by OBS. For the network pruned (and continuously retrained) by OBD, there is no such structure in the validation curves. There seems to be no reliable clue that would permit the user to know when to stop pruning. 4.3 COPIER CONTROL APPLICATION The quality of an image produced by a copier is dependent upon a wide variety of factors: time since last copy, time since last toner cartridge installed, temperature, humidity, overall graylevel of the source document, etc. These factors interact in a highly non-linear fashion, so that mathematical modelling of their interrelationships is difficult. Morita et al. (1992) used backpropagation to train an 8-4-8 network (65 weights) on real-world data, and managed to achieve an RMS voltage error of 0.0124 on a critical control plate. We pruned his network with both OBD with retraining as well as with OBS. When the network was pruned by OBD with retraining, the test error continually increased (erratically) such that at 34 remaining weights, the RMS error was 0.023. When also we pruned the original net by OBS, and the test error gradually decreased such that at the same number of weights the test error was 0.012 - significantly lower than that of the net pruned by OBD. 269 270 Hassibi, Stork, Wolff, and Watanabe 5 CONCLUSIONS We compared pruning by OES and by OED with retraining on a difficult non-linear statistical pattern recognition problem and found that OES led to lower generalization error. We also considered the widely used technique of training large nets to minimum validation error. To our surprise, we found that subsequent pruning by OES lowered generalization error, thereby demonstrating that such networks still have over fitting problems. We have found that the dominant eigenspace approach to OES leads to poor performance. Our simulations support the claim that the t ---+ 0 approximation used in OBS avoids reinjecting training set noise into the network. In contrast, including such t - 0 terms in OES reinjects training set noise and degrades generalization performance, as does retraining in OBD. Acknowledgements Thanks to T. Kailath for support of B.H. through grants AFOSR 91-0060 and DAAL03-91-C-0010. Address reprint requests to Dr. Stork: stork@crc.ricoh.com. References J. Gorodkin, L. K. Hansen, A. Krogh, C. Svarer and O. Winther. (1993) A quantitative study of pruning by Optimal Brain Damage. International Journal of Neural Systems 4(2) 159-169. B. Hassibi & D. G. Stork. (1993) Second order derivatives for network pruning: Optimal Brain Surgeon. In S. J. Hanson, J. D. Cowan and C. L. Giles (eds.), Advances in Neural Information Processing Systems 5, 164-171. San Mateo, CA: Morgan Kaufmann. B. Hassibi, D. G. Stork & G. Wolff. (1993) Optimal Brain Surgeon and general network pruning. Proceedings of ICNN 93, San Francisco 1 IEEE Press. 293-299. Y. LeCun, J. Denker & S. Solla. (1990) Optimal Brain Damage. In D. Touretzky (ed.), Advances in Neural Information Processing Systems 2, 598-605. San Mateo, CA: Morgan Kaufmann. Y. LeCun, P. Simard & B. Pearlmutter. (1993) Automatic learning rate maximization by on-line estimation of the Hessian's eigenvectors. In S. J. Hanson, J. D. Cowan & C. L. Giles (eds.), Advances in Neural Information Processing Systems 5, 156-163. San Mateo, CA: Morgan Kaufmann. T. Morita, M. Kanaya, T. Inagaki, H. Murayama & S. Kato. (1992) Photo-copier image density control using neural network and fuzzy theory. Second International Workshop on Industrial Fuzzy Control ?3 Intelligent Systems December 2-4, College Station, TX, 10. S. Thrun and 23 co-authors. (1991) The Monk's Problems - A performance comparison of different learning algorithms. CMU-CS-91-197 Carnegie-Mellon University Dept. of Computer Science Technical Report. 

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310 PROBABILISTIC CHARACTERIZATION OF NEURAL MODEL COMPUTATIONS Richard M. Golden t University of Pittsburgh, Pittsburgh, Pa. 15260 ABSTRACT Information retrieval in a neural network is viewed as a procedure in which the network computes a "most probable" or MAP estimate of the unknown information. This viewpoint allows the class of probability distributions, P, the neural network can acquire to be explicitly specified. Learning algorithms for the neural network which search for the "most probable" member of P can then be designed. Statistical tests which decide if the "true" or environmental probability distribution is in P can also be developed. Example applications of the theory to the highly nonlinear back-propagation learning algorithm, and the networks of Hopfield and Anderson are discussed. INTRODUCTION A connectionist system is a network of simple neuron-like computing elements which can store and retrieve information, and most importantly make generalizations. Using terminology suggested by Rumelhart & McClelland 1, the computing elements of a connectionist system are called units, and each unit is associated with a real number indicating its activity level. The activity level of a given unit in the system can also influence the activity level of another unit. The degree of influence between two such units is often characterized by a parameter of the system known as a connection strength. During the information retrieval process some subset of the units in the system are activated, and these units in turn activate neighboring units via the inter-unit connection strengths. The activation levels of the neighboring units are then interpreted as t Correspondence should be addressed to the author at the Department of Psychology, Stanford University, Stanford, California, 94305, USA. ? American Institute of Physics 1988 311 the retrieved information. During the learning process, the values of the interunit connection strengths in the system are slightly modified each time the units in the system become activated by incoming information. DERIV ATION OF TIIE SUBJECITVE PF Smolensky 2 demonstrated how the class of possible probability distributions that could be represented by a Hannony theory neural network model can be derived from basic principles. Using a simple variation of the arguments made by Smolen sky , a procedure for deriving the class of probability distributions associated with any connectionist system whose information retrieval dynamics can be summarized by an additive energy function is briefly sketched. A rigorous presentation of this proof may be found in Golden 3. Let a sample space, Sp, be a subset of the activation pattern state space, Sd, for a particular neural network model. For notational convenience, define the term probability function (pf) to indicate a function that assigns numbers between zero and one to the elements of Sp. For discrete random variables, the pf is a probability mass function. For continuous random variables, the pf is a probability density function. Let a particular stationary stochastic environment be represented by the scalar-valued pf, Pe(X)' where X is a particular activation pattern. The pf, Pe(X), indicates the relative frequency of occurrence of activation pattern X in the network model's environment. A second pf defined with respect to sample space Sp also must be introduced. This probability function, ps(X), is called the network's subjective pf. The pf Ps(X) is interpreted as the network's belief that X will occur in the network's environment. The subjective pf may be derived by making the assumption that the information retrieval dynamical system, Ds' is optimal. That is, it is assumed that D s is an algorithm designed to transform a less probable state X into a more probable state X* where the probability of a state is defined by the subjective pf ps(X;A), and where the elements of A are the connection strengths among the units. Or in traditional engineering terminology, it is assumed that D s is a MAP (maximum a posteriori) estimation algorithm. The second assumption is that an energy function, V(X), that is minimized by the system during the information retrieval process can be found with an additivity property. The additivity property says that if the neural network were partitioned into two physically 312 unconnected subnetworks, then Vex) can be rewritten as VI (Xl) + V2(X2 ) where VIis the energy function minimized by the first subnetwork and V2 is the energy function minimized by the second subnetwork. The third assumption is that Vex) provides a sufficient amount of information to specify the probability of activation pattern X. That is, p (X) = G(V(X? where G is some continuous function. And the final assumpti;n (following Smolen sky 2) is that statistical and physical independence are equivalent. To derive ps(X), it is necessary to characterize G more specifically. Note that if probabilities are assigned to activation patterns such that physically independent substates of the system are also statistically independent, then the additivity property of V(X) forces G to be an exponential function since the continuous function that maps addition into multiplication is the exponential . After normalization and the assignment of unity to an irrelevant free parameter 2, the unique subjective pf for a network model that minimizes V(X) during the information retrieval process is: onz p s(X;A) Z =Z -1 exp [ - =Jexp[ - V (X;A)] V (X;A)]dX (1) (2) provided that Z < C < 00. Note that the integral in (2) is taken over sp. Also note that the pf, Ps' and samfle space, Sp, specify a Markov Random Field since (1) is a Gibbs distribution . Example 1: Subjective pfs for associative back-propagation networks The information retrieval equation for an associative back-propagation 6 network can be written in the form ~[I;A] where the elements of the vector 0 are the activity levels for the output units and the elements of the vector I are the activity levels for the input units. The parameter vector A specifies the values 313 of the "connection strengths" among the units in the system. The function cl> specifies the architecture of the network. A natural additive energy function for the information retrieval dynamics of the least squares associative back-propagation algorithm is: V(O) = I()-.4>(I;A) 12, (3) If Sp is defined to be a real vector space such that 0 esp, then direct substitution of V(O) for ViX;A) into (1) and (2) yields a multivariate Gaussian density function with mean cl>(I;A) and covariance matrix equal to the identity matrix multiplied by 1!2. This multivariate Gaussian density function is ps(OII;A). That is, with respect to ps(OII;A), information retrieval in an associative backpropagation network involves retrieving the "most probable" output vector, 0, for a given input vector, I. Example 2: Subjective pis/or Hopfield and BSB networks. The Hopfield 7 and BSB model 8,9 neural network models minimize the following energy function during information retrieval: T Vex) =-X MX (4) where the elements of X are the activation levels of the units in the system. and the elements of M are the connection strengths among the units. Thus, the subjective pf for these networks is: 314 -l T P s< X)= Z exp [X M X] where Z =l:exp [XTM X] (5) where the summation is taken over Sp. APPLICATIONS OF TIlE TIIEORY If the subjective pf for a given connectionist system is known, then tradi- tional analyses from the theory of statistical inference are immediately applicable. In this section some examples of how these analyses can aid in the design and analysis of neural networks are provided. Evaluating Learning Algorithms Learning in a neural network model involves searching for a set of connection strengths or parameters that obtain a global minimum of a learning energy function. The theory proposed here explicitly shows how an optimal learning energy function can be constructed using the model's subjective pf and the environmental pf. In particular, optimal learning is defined as searching for the most probable connection strengths, given some set of observations (samples) drawn from the environmental pf. Given some mild restrictions upon the fonn of the a priori pf associated with the connection strengths, and for a sufficiently large set of observations, estimating the most probable connection strengths (MAP estimation) is equivalent to maximum likelihood estimation 10 A well-known result 11 is that if the parameters of the subjective pf are represented by the parameter vector A, then the maximum likelihood estimate of A is obtained by finding the A * that minimizes the function : 315 E(A) =- <.LOG [p s(X;A)]> (6) where < > is the expectation operator taken with respect to the environmental pf. Also note that (6) is the Kullback-Leibler 12 distance measure plus an irrelevant constant. Asymptotically, E(A) is the logarithm of the probability of A given some set of observations drawn from the environmental pf. Equation (6) is an important equation since it can aid in the evaluation and design of optimal learning algorithms. Substitution of the multivariate Gaussian associated with (3) into (6) shows that the back-propagation algorithm is doing gradient descent upon the function in (6). On the other hand, substitution of (5) into (6) shows that the Hebbian and Widrow-Hoff learning rules proposed for the Hopfield and BSB model networks are not doing gradient descent upon (6). Evaluating Network Architectures The global minimum of ~6) occurs if and only if the subjective and environmental pfs are equivalent 2. Thus, one crucial issue is whether any set of connection strengths exists such that the neural network's subjective pf can be made equivalent to a given environmental pf. If no such set of connection strengths exists, the subjective pf, p s' is defined to be misspecified. White 11 and Lancaster 13 have introduced a statistical test designed to re~ct the null hypothesis that the subjective pf, Ps' is not misspecified. Golden suggests a version of this test that is suitable for subjective pfs with many parameters. REFERENCES 1. D. E. Rumelhart, J. L. McClelland, and the PDP Research Group, Parallel distributed processing: Explorations in the microstructure of cognition, 1, (MIT Press, Cambridge, 1986). 2. P. Smolensky, In D. E. Rumelhart, J. L. McClelland and the PDP Research Group (Eds.), Parallel distributed processing: Explorations in the microstructure of cognition, 1, (MIT Press, Cambridge, 1986), pp. 194-281. 316 3. R. M. Golden, A unified framework for connectionist systems. Unpublished manuscript. 4. C. Goffman, Introduction to real analysis. (Harper and Row, N. Y., 1966), p. 65. 5. J. L. Marroquin, Probabilistic solution of inverse problems. A.I. Memo 860, MIT Press (1985). 6. D. E. Rumelhart, G. E. Hinton, & R. J. Williams, In D. E. Rumelhart, 1. L. McClelland, and the PDP Research Group (Eds.), Parallel distributed processing: Explorations in the microstructure of cognition, 1, (MIT Press, Cambridge, 1986), pp. 318-362. 7. J. 1. Hopfield, Proceedings of the National Academy of Sciences, USA, 79, 2554-2558 (1982). 8. J. A. Anderson, R. M. Golden, & G. L. Murphy, In H. Szu (Ed.), Optical and Hybrid Computing, SPIE, 634,260-276 (1986). 9. R. M. Golden, Journal of Mathematical Psychology, 30,73-80 (1986). 10. H. L. Van Trees, Detection, estimation, and modulation theory. (Wiley, N. Y.,1968). 11. H. White, Econometrica, 50, 1-25 (1982). 12. S. Kullback & R. A. Leibler, Annals of Mathematical Statistics, 22, 79-86 (1951). 13. T. Lancaster, Econometrica, 52, 1051-1053 (1984). ACKNOWLEDGEMENTS This research was supported in part by the Mellon foundation while the author was an Andrew Mellon Fellow in the Psychology Department at the University of Pittsburgh, and partly by the Office of Naval Research under Contract No. N-OOI4-86-K-OI07 to Walter Schneider. This manuscript was revised while the author was an NIH postdoctoral scholar at Stanford University. This research was also supported in part by grants from the Office of Naval Research (Contract No. NOOOI4-87-K-0671), and the System Development Foundation to David Rumelhart. I am very grateful to Dean C. Mumme for comments, criticisms, and helpful discussions concerning an earlier version of this manuscript. I would also like to thank David B. Cooper of Brown University for his suggestion that many neural network models might be viewed within a unified statistical framework. 

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Mixtures of Controllers for Jump Linear and Non-linear Plants Timothy W. Cacciatore Department of Neurosciences University of California at San Diego La Jolla, CA 92093 Steven J. Nowlan Synaptics, Inc. 2698 Orchard Parkway San Jose, CA 95134 Abstract We describe an extension to the Mixture of Experts architecture for modelling and controlling dynamical systems which exhibit multiple modes of behavior. This extension is based on a Markov process model, and suggests a recurrent network for gating a set of linear or non-linear controllers. The new architecture is demonstrated to be capable of learning effective control strategies for jump linear and non-linear plants with multiple modes of behavior. 1 Introduction Many stationary dynamic systems exhibit significantly different behaviors under different operating conditions. To control such complex systems it is computationally more efficient to decompose the problem into smaller subtasks, with different control strategies for different operating points. When detailed information about the plant is available, gain scheduling has proven a successful method for designing a global control (Shamma and Athans, 1992). The system is partitioned by choosing several operating points and a linear model for each operating point. A controller is designed for each linear model and a method for interpolating or 'scheduling' the gains of the controllers is chosen. The control problem becomes even more challenging when the system to be controlled is non-stationary, and the mode of the system is not explicitly observable. One important, and well studied, class of non-stationary systems are jump linear systems of the form: ~~ = A(i)x + B(i)u. where x represents the system state, 719 720 Cacciatore and Nowlan u the input, and i, the stochastic parameter that determines the mode of the sys- tem, is not explicitly observable. To control such a system, one must estimate the mode of the system from the input-output behavior of the plant and then choose an appropriate control strategy. For many complex plants, an appropriate decomposition is not known a priori. One approach is to learn the decomposition and the piecewise solutions in parallel. The Mixture of Experts architecture (Nowlan 1990, Jacobs et a11991) was proposed as one approach to simultaneously learning a task decomposition and the piecewise solutions in a neural network context. This architecture has been applied to control simple stationary plants, when the operating mode of the plant was explicitly available as an input to the gating network (Jacobs and Jordan 1991). There is a problem with extending this architecture to deal with non-stationary systems such as jump linear systems. The original formulation of this architecture was based on an assumption of statistical independence oftraining pairs appropriate for classification tasks. However, this assumption is inappropriate for modelling the causal dependencies in control tasks. We derive an extension to the original Mixture of Experts architecture which we call the Mixture of Controllers. This extension is based on an nth order Markov model and can be implemented to control nonstationary plants. The new derivation suggests the importance of using recurrence in the gating network, which then learns t.o estimate the conditional state occupancy for sequences of outputs. The power of the architecture is illustrated by learning control and switching strategies for simple jump linear and non-stationary nonlinear plants. The modified recurrent architecture is capable of learning both the control and switching for these plants. while a non-recurrent architecture fails to learn an adequate control. 2 Mixtures of Controllers The architecture of the system is shown in figure 1. Xt denotes the vector of inputs to the controller at time t and Yt is the corresponding overall control output. The architecture is identical to the Mixture of Experts architecture, except that the gating network has become recurrent, receiving its outputs from the previous time step as part of its input. The underlying statistical model, and corresponding training procedure for the Mixture of Controllers, is quite different from that originally proposed for the Mixture of Experts. We assume that the system we are interested in controlling has N different modes or states l and we will have a distinct control l\?h for each mode. In general we are interested in the likelihood of producing a sequence of control outputs Yl, ... , YT given a sequence of inputs Xl, ... , XT. This likelihood can be computed as: I1L.:P(YtI St = k,Xt)P(St = kIYl .. ?Yt-I,Xl?? .xd k (1) IThis is an idealization and if N is unknown it is safest to overestimate it. Mixtures of Controllers for Jump Linear and Non-Linear Plants Yt 1 Yt 2 1---i-iYt 3 L-J-H--"1 Yt Figure 1: The Mixture of Controllers architecture. MI, M2 and M3 are feedforward networks implementing controls appropriate for different modes of the system to be controlled. The gating network (Sel.) is recurrent and uses a softmax non-linearity to compute the weight to be assigned to each of the control out.puts. The weighted sum of the controls is then used as the overall control for the plant. where bf represents the probability of producing the desired control Yt given the input Xt and that the system is in state k. If represents the conditional probability of being in state k given the sequence of inputs and outputs seen so far. In order to make the problem tractable, we assume that this conditional probability is completely determined by the current input to the system and the previous state of the system: I: = fW'Y(Xt, {it-I})' Thus we are assuming that our control can be approximated by a Markov process, and since we are assuming that the mode of the system is not explicitly available, this becomes a hidden Markov model. This Markov assumption leads to the particular recurrent gating architecture used in the Mixture of Controllers. If we make the same gaussian assumptions used in the original Mixture of Experts model, we can define a gradient descent procedure for maximizing the log of the likelihood given in Equation 1. Assume b~ = 1 e-(Yt-y~)2/2(72 y'2iu and define f3f = P(YT,"" Yt\Sk, XT,???, Xt), L t = j3 k k Lk f3f,f and R:=~. Lt Then the derivative of the likelihood with respect to the output of one of the controllers becomes: ologL k( k) aYtk = l\ R t r Yt - Yt . (2) 721 722 Cacciatore and Nowlan The derivative of the likelihood with respect to a weight in one of the control networks is computed by accumulating partial derivatives over the sequence of control outputs: (3) For the gating network, we once again use a softmax non-linearity so: exp gtk k It = Then alog L ak 9t _ '""'(R k - .. Lj eXP9~ ~ t _ t k) I't k It-I' (4) The derivatives for the weights in the gating network are again computed by accumulating partial derivatives over output sequences: (5) Equations (2) and (4) turn out to be quite similar to those derived for the original Mixture of Experts architecture. The primary difference is the appearance of (3; rather than bf in the expression for R:. The appearance of /3 is a direct. result of the recurrence introduced into the gating network. {3 can be computed as part of a modified back propagation through time algorithm for the gating network using the recurrence: (6) /3: = + W kjf3f+l b: L j where O;f+l Wkj = olf Equation (6) is the analog of the backward pass in the forward-backward algorithm for standard hidden Markov models. In the simulations reported in the next section, we used an online gradient descent procedure which employs an approximation for (3f which uses only one step of back propagation through time. This approximation did not appear to significantly affect the final performance of the recurrent architecture. 3 Results The performances of the recurrent Mixture of Controllers and non-recurrent Mixture of Experts were compared on three control tasks: a first order jump linear system, a second order jump linear system, and a tracking task that required two nonlinear controllers. The object of the first two jump-linear tasks was to control a plant which switched randomly between two linear systems. The resulting overa.ll systems were highly non-linear. In both the first. and second order cases it was Mixtures of Controllers for Jump Linear and Non-Linear Plants Arst Order Model Trajectory Arst Order Model Traming Error 200 20c00 N~I'OJmnl 150 '5000 b t W j .OCOO <'3 , 00 5000 ?SO RlClJ1Tlnl 0000 .0000 E_ 20c00 3OJOO 40000 S<X110 -100 00 SOD 1000 .SOO To"" Figure 2: Left: Training convergence of Mixtures of Experts and Mixtures of Controllers on first order jump linear system. The vertical axis is average squared error over training sequences and horizontal axis is the number of training sequences seen. Right: Sample test trajectory of first order jump linear system under control of Mixture of Controllers. The system switches states at times 50 and 100. desired to drive all plant outputs to zero (zero-forcing control). Neither the first or second order systems could be successfully controlled by a single linear controller. For both jump-linear tasks, the architecture of the MixtUre of Controllers and Mixture of Experts consisted of two linear experts, and a one layer gating network. The input to the experts was the plant output at the previous time step, while the input to the gating network was the ratio of the plant outputs at the two preceding time steps. An ideal linear controller was designed for each mode of the system. Training targets were derived from outputs of the appropriate ideal controller, using the known mode of the system for the training trajectories. The parameters of the gating and control networks were updated after each pass through sample trajectories which contained several state transitions. The recurrent Mixture of Controllers could be trained to successfully control the first order jump linear system (figure 2), and once trained generalized successfully to novel test trajectories. The non-recurrent Mixture of Experts failed to learn even the training data for the first order jump linear system (note the high asymptote for the training error without recurrence in figure 2). The recurrent Mixture of Controllers was also able to learn to control the second order jump linear system (figure 3), however, it was necessary to teacher force the system during the first 5000 epochs of training by providing the true mode of the system as an extra input to the gating network. This extra input was removed at epoch 5000 and the error initially increases dramatically but the system is able to eventually learn to control the second order jump linear system autonomously. Note that the Mixture of Experts system is actually able to learn a successful control even more rapidly than the Mixture of Controllers when the additional teacher input is provided, however learning again completely fails once this input is removed at epoch 5000 (figure 3). 723 724 Cacciatore and Nowlan Second Order Model Training Error Second Order Model Trajectory Aecurren I Ccnlroler eoo .----~--~--_, '000 3000 -oulpUtl - - Ideal t ---- oulpU12 - - ldul2 TIme Figure 3: Left: Training convergence of Mixt.ures of Experts and Mixtures of Controllers on second order jump linear system. Right: Sample test trajectory of second order jump linear system under control of MixtUre of Controllers. The system again switches states at times 50 and 100. In both first and second order cases, the trained Mixture of Controllers is able to control the system in both modes of system behavior, and to deted mode changes automatically. The difficulty in designing a control for a jump linear system usually lies in identifying the state of the system. No explicit law describing how to identify and switch between control modes is necessary to train the Mixture of Controllers, as this is learned automatically as a byproduct of learning to successfully control the system. Performance of the Mixture of Controllers and the Mixture of Experts was also compared on a more complex task requiring a non-linear control law in each mode. The task wa:s to control the trajectory of a ship to track an object traveling in a straight line, or flee from an object having a random walk trajectory (figure 4). There is a high degree of task interference between the controls appropriate during the two modes of object behaviors. The ship dynamics were t.aken from Miller and Sutton (1990). For both the Mixture of Controllers and the Mixture of Experts two experts were used. The experts received past and present measurements of the object bearing, distance, velocity, and the ship heading and turn rate. The controllers specified the desired turn rate of the ship. A one layer gating network was used which received the velocity of the object as input. Training targets were produced from ideal controllers designed for each object behavior. The ideal controller for the random walk behavior produced a turn rate that headed directly away from the object. The ideal controller for intercepting the object used future information about object position to determine the turn rate which would lead to the closest possible intercept point. Both ideal controllers made use of information not available to the Mixture of Experts or Mixture of Controllers. The Mixture of Controllers and the Mixture of Experts were trained on sequences of Mixtures of Controllers for Jump Linear and Non-Linear Plants b) a) actual 8 Ilil ~ IDD :: 'f\?/fJ''''r?'(r~!-{,?~~N\''1?'I?hf('~/~"r!?''.l,;''?..t,:?+?i'i'II'?~?y. ~/1 200 ~ 0' target! i ;!l X '0 pOliition 60 80 100 correct o. incorrect I 111) time Figure 4: (a) Actual and desired trajectories of ship under control of Mixture of Experts while attempting to intercept target. (b) Gating unit activities as a function of time for trajectory in (a). trajectories where the object changed behaviors multiple times. The weights of the networks were updated after each pass through the trajectories. The input to the gating net in this ta.sk provided more inst antaneous information about the mode of object behavior than was provided in the jump linear tasks. As a result , the nonrecurrent Mixture of Experts was able to achieve a minimum level of performance on the overall task. The recurrent Mixture of Controllers performed much better . The differences between two architectures are revealed by examining the gating network outputs . Without recurrence , the Mixture of Experts gating network could not determine the state of the object with certainty, and compromised by selecting a combination of the correct and incorrect control (figure 4b) . Since the two controls are incompatible, this uncertainty degrades the performance of the overall controller . With recurrence in the ga.ting network, the Mixture of Controllers is able to determine the target state with greater certainty by integrating information from many observations of object behavior . The sharper decisions about which control to use greatly improve tracking performance (figure 5). We explored the ability of the Mixture of Controllers to learn the dynamics of switching by training on trajectories where the object switched behavior with varying frequency. The gating network trained on an object that switched behaviors infrequently was sluggish to respond to transitions, but more noise tolerant than the gating network trained on a frequently switching object. Thus, the gating network is able to incorporate the frequency of transition into its state model. 4 Discussion \Ve have described an extension to the Mixture of Experts architecture for modelling and controlling dynamical systems which exhibit multiple modes of behavior. The algorithm we have presented for updating the parameters of the model is a simple gradient descent procedure . Application of the technique to large scale problems 725 726 Cacciatore and Nowlan b) a) 250 ,:' 09 ."...-.... _- .- correct 08 ? alii I 0' , ] uo actual . . 1'00 desired :>. i 03 i ~ 02f\ target! O\r ~ i ~ .2t) 2D ~ x position ~ ~ ~ rn ?0 incarect 51! .00 1!I) 200 2Slt :a 3SO tOO time Figure 5: (a) Actual and desired trajectories of ship under control of Mixture of Controllers while attempting to intercept target. (b) Gating unit activities as a function of time for trajectory in (a). Note that these are much less noisy than the activities seen in figure 4(b). may require the development of faster converging update algorithms, perhaps based on the generalized EM (GEM) family of algorithms, or a variant of the iterative reweighted least squares procedure proposed by Jordan and Jacobs (1993) for hierarchies of expert networks. Additional work is also required to establish the stability and convergence rate of the algorithm for use in adaptive control applications. References Jacobs, R.A. and Jordan, M.I. A competitive modular connectionist architecture. Neural Information Processing Systems 3 (1991). Jacobs, R.A., Jordan, M.I ., Nowlan, S.J. and Hinton, G .E. Adaptive Mixtures of Local Experts. Neural Computation, 3, 79-87, (1991). Jordan, M.I. and Jacobs, R.A. Hierarchical Mixtures of Experts and the EM algorithm. Neural Computation, (1994). Miller, W.T., Sutton, R.S. and Werbos, P.J. Neural Networks for Control, MIT Press (1993). Nowlan, S.J. Competing Experts: An Experimental Investigation of Associative Mixture Models. Technical Report CRG- TR-90-5, Department of Computer Science, University of Toronto (1990). Shamma, J.S., and Athans, M. Gain scheduling: potential hazards and possible remedies. IEEE Control Systems Magazine, 12:(3), 101-107 (1992). 

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Inverse Dynamics of Speech Motor Control Makoto Hirayama Eric Vatikiotis-Datesol1 Mitsuo Kawato" ATR Human Information Processing Research Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, Japan Abstract Progress ha.s been made in comput.ational implementation of speech production based on physiological dat.a. An inverse dynamics model of the speech articulator's l1111sculo-skeletal system. which is the mapping from art.iculator t.rajectories to e\ectromyogl'aphic (EMG) signals, was modeled using the acquired forward dynamics model and temporal (smoot.hness of EMG activation) and range constraints. This inverse dynamics model allows the use of a faster speech mot.or control scheme, which can be applied to phoneme-tospeech synthesis via musclo-skeletal system dynamics, or to future use in speech recognition. The forward acoustic model, which is the mapping from articulator trajectories t.o the acoustic parameters, was improved by adding velocity and voicing information inputs to distinguish acollst.ic paramet.er differences caused by changes in source characterist.ics. 1 INTRODUCTION Modeling speech articulator dynamics is important not only for speech science, but also for speech processing. This is because many issues in speech phenomena, such as coarticulation or generat.ion of aperiodic sources, are caused by temporal properties of speech articulat.or behavior due t.o musculo-skelet.al system dynamics and const.raints on neurO-l1lotor command activation . .. Also, Laboratory of Parallel Distributed Processing, Research Institute for Electronic Science, Bokkaido University, Sapporo , Hokkaido 060, Japan 1043 1044 Hirayama, Vatikiotis-Bateson, and Kawato We have proposed using neural networks for a computational implementation of speech production based on physiological activities of speech articulator muscles. In previous works (Hirayama, Vatikiotis-Bateson, Kawato and Jordan 1992; Hirayama, Vatikiotis-Bateson, Honda, Koike and Kawato 1993), a neural network learned the forward dynamics, relating motor commands to muscles and the ensuing articulator behavior. From movement t.rajectories, the forward acoustic network generated the acoustic PARCOR parameters (Itakura and Saito, 1969) that were then used to synthesize the speech acoustics. A cascade neural network containing the forward dynamics model along with a suitable smoothness criterion was used to produce a continuous motor command from a sequence of discrete articulatory targets corresponding to the phoneme input string. Along the same line, we have extended our model of speech motor control. In this paper, WI~ focus on modeling the inverse dynamics of the musculo-skeletal system. Having an inverse dynamics model allows us to use a faster control scheme, which permits phoneme-to-speech synthesis via musculo-skeletal system dynamics, and ultimately may be useful in speech recognition. The final sectioll of this paper reports improvements in the forward acoustic model, which were made by incorporating articulator velocity and voicing information to distinguish the acoustic parameter differences caused by changes in source characteristics. 2 INVERSE DYNAMICS MODELING OF MUSCULO-SKELETAL SYSTEM From the viewpoint of control theory, an inverse dynamics model of a controlled object pla.ys an essential role in fecdfonvard cont.rol. That is, an accurate inverse dynamics model outputs an appropriate control sequence that realizes a given desired trajectory by using only fecdforward cOlltrol wi t.hout any feedback information, so long as there is no perturbation from the environment. For speech a rticulators, the main control scheme cannot rely upon feedback control because of sensory feedback delays. Thus, we believe that the inverse dynamics model is essential for biological motor control of speech and for any efficient speech synthesis algorithm based on physiological data. However, the speech articulator system is an excess-degrees-of-freedom system, thus the mapping from art.iculator t.rajectory (posit.ion, velocit.y, accelerat.ion) to electromyographic (E~fG) activity is one-to-many. That is, different EMG combinations exist for the same articulat.or traject.ory (for example, co-contraction of agonist and antagonist muscle pairs). Consequently, we applied the forward modeling approach to learning an inverse model (Jordan alld Rumelhart, 1992), i.e., constrained supervised leaming, as shown in Figure 1. The inputs of the inverse Desired Trajectory Control p----..., Trajectory Forward t------~~ Model Error Figure 1: Inverse dynamics modeling using a forward dynamics model (Jordan and Rumelhart, 1992). r--~--..., Inverse Model I--_ _~ --- Inverse Dynamics of Speech Motor Control 1.0 - --- Actual EMG "optimal" EMG by 10M 0.8 0.6 0.4 0.2 O.O~----------~~----------r-----------~--~----~ o 1 2 Time (s) 3 4 Figure 2: After learning, the inverse model output "optimal" EMG (anterior belly of the digastric) for jaw lowering is compared with actual EMG for the tf'st trajectory. dynamics model are articulator positions, velocities, and accelerations; the outputs are rectified, integrated, and filtered EIVIG for relevant muscles. The forward dynamics model previously reported (Hirayama et al., 1993) was used for determining the error signals of the inverse dynamics model . To choose a realistic EMG patt.ern from among diverse possible sciutions, we use both temporal and range const.raints. The temporal constraint is related to the smoothnt~ss of EMG activat.ion, i.e., minimizing EI\'1G activation change (Uno, Suzuki, and Kawat.o, 1989). The minimum and maximum values of the range constraint were chosen using valucs obt.ained from t.he experimental data. Direct inverse modeling (Albus, 1975) was uscd to det.ermine weights, which were then supplied as initial weights to t.he constrained supervised learning algorithm of Jordan and Rumelhart's (1992) inverse dynamics modeling met.hod. Figure 2 shows an example of t.he inverse dynnmics model output after learning, when a real articulator trajectory, not. included in the training set, was given as the input. Note that the net.work output cannot be exactly t.he same as the actual EMG, as the network chooses a unique "optimal" EMG from many possible EMG patterns that appear in the actual EI\IG for t.he trajectory. --- Experimental data - - - Direct inverse modeli ng Inverse modeling using FDM -0.3 E -0.4 0 -0.5 c ~ UJ 0 Q.. -0.6 -0.7 0 1 2 3 4 Time (s) Figure 3: Trajectories generated by the forward dynamics net.work for the two methods of inverse dynamics modeling compared with t.he desired trajectory (experimental da t.a). 1045 1046 Hirayama, Vatikiotis-Bateson, and Kawato Since the inverse dynamics model was obtained by learning, when the desired trajectory is given to the inverse dynamics model, an articulator trajectory can be generated with the forward dynamics network previously reported (Hirayama et al., 1993). Figure 3 compares trajectories generated by the forward dynamics network using EMG derived from the direct inverse dynamics method or the constrained supervised learning algorithm (which uses the forward dynamics model to determine the inverse dynamics model's "opt.imal" El\IG). The latter method yielded a 30.0 % average reduction in acceleration prediction error over the direct method, thereby bringing the model output trajectory closer to the experimental data. 3 TRAJECTORY FORMATION USING FORWARD AND INVERSE RELAXATION MODEL Previously, to generate a trajectory from discrete phoneme-specific via-points, we used a cascade neural network (c.f., Hirayama. et. al., 1992). The inverse dynamics model allows us t.o use an alternative network proposed by \\fada and Kawato (1993) (Figure 4). The network uses both the forward and inverse models of the controlled object, and updates a given initial rough trajectory passing through the via-points according to t.he dYllamics of the cont.rolled object and a smoothness constraint on the control input. The computation time of the net.work is much shorter than that of the cascade neural network CWada and Kawa.to, 1993). Figure 5 shows a forward dynamics model output trajectory driven by the modelgenerated motor control signals. Unlike \Vada and Kawato's original model (1993) in which generated trajectories always pass through via-points, our tl'ajectories were generated from smoothed motor control signals (i.e., after applying the smoothness constraint) and, consequently, do not. pass through the exact via-points. In this paper, a typical value for each phoneme from experimental data was chosen as the target via-point. and was given in Cartesian coordinates relative to the maxillary incisor. Alt.hough further investigation is needed to refine the phoneme-specific target specifications (e.g. lip aperture targets), reasonable coarticulated trajectories were obtained from series of discret.e via-point t.argets (Figure 5). For engineering applications such as text-to-speech synthesizers using articulatory synthesis, this kind of technique is necessary because realistic coarticula.ted trajectories must serve as input to the articulatory synthesizer. ~ e ~ lal luI IiI (d 'd lsI It I Articulatory Targets Figure 4: Speech t.rajectory formation scheme modified from the forward and inverse relaxation neural network model (\\'ada and Kawato, 1993). Inverse Dynamics of Speech Motor Control Network output ....... Experimental data . ? . Phoneme specific targets -0.3 ? -0.4 .2 -0.5 ~ -0.6 c: = - . .--'''''~-'.....-~.---" - ~"--..". '...... "" .............-.....?. ... '" \. '. '.", '. -0.7 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Time (s) Figure 5: Jaw trajectory generated by the forward and inverse relaxation model. The output of the forward dynamics model is used for this plot. A furthe!' advantage of this network is that. it can be llsed t.o predict phonemespecific via-point.s from t.he realized t.rajectory (vVada, Koike, Vatikiotis-Bateson and Kawato, 1993). This capability will allow us to use our forward and inverse dynamicb models for speech recognition in future, through acoustic to articulatory mapping (Shirai and Kobayashi, 1991; Papcun, Hochberg, Thomas, Laroche, Zacks and Levy, 1992) and the articulatory to phoneme specific via-points mapping discussed above. Because t.rajectories may be recovered from a small set of phoneme??specific via-points, this approach should be readily applicable to problems of speech data compression. 4 DYNAMIC MODELING OF FORWARD ACOUSTICS The secoild area of progress is t.he improvement. in t.he forward acoustic network. Previously (Hirayama et al., 1993), we demonstrat.ed that acoustic signals can be obtained using a neural network that learns the mapping between articulator positions and acoustic PARCOR coetTIcients (ltakura and Saito, 1969; See also, Markel and Gray, ] 976). However, this modeling was effective only for vowels and a limited number of consonants because the architecture of the model was basically the same as that of static articulatory synthesizers (e.g. Mermelst.ein, 1973). For nat.ural speech, aperiodic sources for plosive and sibilant consonants result. in multiple sets of acoustic parameters for the same articulator configurat.ion (i.e., the mapping is one-to-many) ; hence, learning did not fully converge. One approach t.o solving t his problem is to make source modeling completely separat.e from the vocal tract area modeling. However, for synthesis of natural sentences, t.he vocal tract transfer function model requires anot.her model for t.he non-glottal sources associated wit.h consonant production . Since these sources are locat.ed at. various point.s along t.he vocal tract, their interaction is extremely complex. Our approach to solving this one-to-many mapping is to have the neural network learn the acoustic parameters along with the sound source characteristic specific to each phoneme. Thus, we put articulator positions with their velocities and voiced/voiceless informat.ion (e.g ., Markel and Gray, 1976) into the input (Figure 6) because the sound source characterist.ics are made not only by the articulator posi- 1047 1048 Hirayama, Vatikiotis-Bateson, and Kawato Articulator Positions, Velocities & VoicedNoiceless Acoustic Wave ___ G_lot_ta_1s_o_u_rc_e---'I-----L--'--_ _ _ -'--.J.--~~) ) ) Figure 6: Improved forward acoustic network. Inputs to the network are articulator positions and velocities and voiced/voiceless information. tion but also by the dynamic movement of articulators. For simulations, horizontal and vertical motions of jaw, upper and lower lips, and tongue tip and blade were used for the inputs and 12 dimensional PARCOR parameters were used for the outputs of the network. Figure 7(a) shows positionvelocity-voiced/voiceless network out.put compared with posit.ion-only network and experimentally obtained PARCOR parameters for a natural test sentence. Only the first two coefficients are shown. The first part of the test sentence, "Sam sat on top of the potato cooker and waited for Tommy to cut up a bag of tiny tomatoes and pop the beat tips into the pot," is shown in this plot. Figure 7(b)( c) show a part of the synthesized speech driven by funtlamental frequency pulses for voiced sounds and random noises for voiceless sounds. By using velocity and voiced/voiceless inputs, the performance was improved for natural utterances which include many vowels and consonants. The average values of the LPC-cepstrum distance mea.<.;ure between original and synthesized, were 5.17 (dB) for the position-only network and 4.18 (dB) for the position-velocityvoiced/voiceless network. When listening to the output, the sentence can be understood, and almost all vowels and many of the consonants can be classified. The overall clarity and the classifica.tion of some consonants is about as difficult as experienced in noisy international telephone calls. Although there are other potentia.l means to achieve further improvement (e.g. adding more tongue channels, using more balanced training patterns, incorporating nasality information, implementation of better glottal and non-glottal sources), the network synthesizes quite smooth and reasonable acoustic signals by incorporating aspects of the articulator dynamics. 5 CONCLUSION We are modeling the information transfer from phoneme-specific articulatory targets to acoustic wave via the musculo-skeletal system, using a series of neural networks. Electromyographic (EMG) signals are used as the reflection of motor control commands. In this paper, we have focused on the inverse dynamics modeling of the Inverse Dynamics of Speech Motor Control a 0.4 . j. 10 C\I ~ 00 . ... 0.8 0.6 1.0 Position+Velocity+Voiced/Voiceless Network - - - Position-only Network ~.'''._~ \ . PARCOR ~or rest .............. ' .:, '\ r"", '. ~??I??::~ . ? ?,.I. -~ '.. ' "j~.;.;.: ~ ...... .. ' .... -1 .0 0.0 0.2 0.4 0.6 0.8 1.0 b Original Source (Noise + Pulse) -+--~ Synthesized c 1.0 0.5 0.0 Time (s) ---- -- - II 1: 2UlJCJ rr -.__ .:u l ll. n sOCJ{J -- -,--~~~!I"iI=~:.~,_.t_,.i I t " IJ .L ,1 CJ ill L L1. f:~I: o-{.---------'ij, LJCJ 1C I 1((, ,= 0.2 (seconds) C: " -- Figure 7: (a) Model output PARCOR parameters. Only kl and k2 are shown. (b) Original, source model, and synthesized acoustic signals. (c) \Videband spectrogram for the original and synthesized speech. Utterance shown is "Sam sat on top" from a test sentence. 1049 1050 Hirayama, Vatikiotis-Bateson, and Kawato musculo-skeletal system, its control for the transform from discrete linguistic information to continuous motor control signals, and articulatory speech synthesis using the articulator dynamics. '''Ie believe that. modeling the dynamics of articulat.ory motions is a key issue both for elucidating mechanisms of speech motor control and for synthesis of nat'llr'al utterances. Acknowledgetnellts We thank Yoh'ichi Toh'kura for continuous encouragement. Further support was provided by HFSP grants to M. Kawato. References Albus, J. S. (1975) A new approach to manipulator control: The cerebellar model articulation controller (CMAC). Transactions of the ASME Journal of Dynamic System, Afeasurement, and Control, 220-227. Hirayama., M., E. Vatikiotis-Bateson, M. Kawato, and 1\1. 1. Jordan \1992) Forward dynamics modeling of speech motor control using physiological data. In Moody, J. E., Hanson, S. J., and Lippmann, R. P. (eds.) Advances in Neural Information Processing Systems 4. San Mateo, CA: I\lorgan Kaufmann Publishers, 191-198. Hirayama, M., E. Vatikiotis-Bateson, K. Honda, Y. Koike, and M. Kawato (1993) Physiologically based speech synthesis. In Giles, C. L., Hanson, S. J., and Cowan, J. D. (eds.) Advances in Neural Information Processing Systems 5. San Mateo, CA: Morgan Kaufmann Publishers, 658-665. Itakura, F. and S. Saito (1969) Speech analysis and synthesis by partial correlation parameters. Proceeding of Japan Acoustic Society, 2-2-6 (In Japanese). Jordan, M. I. and D. E. Rumelhart (1992) Forward models: Supervised learning with a di'3tal teacher. Cognitive Science, 16, 307-354. Mermelstein, P. (1973) Articulatory model for the study of speech production. Journal of Acoustical Society of America, 53, 1070-1082. Papcun, J., J. Hochberg, T. R. Thomas, T. Laroche, J. Zacks, and S. Levy (1992) Inferring articulation and recognizing gestures from acoustics with a neural network trained on x-ray microbeam data. Jo'urnal of Acoustical Society of America, 92 (2) Pt. 1. Shirai, K. and T. Kobayashi (1991) Estimation of articulatory motion using neural networks. Journal of Phonetics, 19, 379-385. Uno, Y., R. Suzuki, and M. Kawato (1989) The minimum muscle tension change model which reproduces arm movement t.rajectories. Pr'oceedi7l9 of the 4th Symposium on Biological and Physiological Engineering, 299-302 (In Japanese). Wada, Y. and M. Kawat.o (1993) A nemal network model for arm t.rajectory formation of using fOl'ward and inverse dynamics models. Neural Networks, 6, 919-932. Wada, Y., Y. Koike, E. Vatikiotis-Bateson, and M. Kawato (1993) Movement Pattern Recognition Based on the Minimization Principle. Tech nical RI 'port of IEICE, NC93-23, 85-92 (In Japanese). 

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Dual Mechanisms for Neural Binding and Segmentation Paul Sajda and Leif H. Finkel Department of Bioengineering and Institute of Neurological Science University of Pennsylvania 220 South 33rd Street Philadelphia, PA . 19104-6392 Abstract We propose that the binding and segmentation of visual features is mediated by two complementary mechanisms; a low resolution, spatial-based, resource-free process and a high resolution, temporal-based, resource-limited process. In the visual cortex, the former depends upon the orderly topographic organization in striate and extrastriate areas while the latter may be related to observed temporal relationships between neuronal activities . Computer simulations illustrate the role the two mechanisms play in figure/ ground discrimination, depth-from-occlusion, and the vividness of perceptual completion. 1 COMPLEMENTARY BINDING MECHANISMS The "binding problem" is a classic problem in computational neuroscience which considers how neuronal activities are grouped to create mental representations. For the case of visual processing, the binding of neuronal activities requires a mechanism for selectively grouping fragmented visual features in order to construct the coherent representations (i.e. objects) which we perceive. In this paper we argue for the existence of two complementary mechanisms for neural binding, and we show how such mechanisms may operate in the constructiO:l of intermediate-level visual representations. 993 994 Sajda and Finkel Ordered cortical topography has been found in both striate and extrastriate areas and is believed to be a fundamental organizational principle of visual cortex. One functional role for this topographic mapping may be to facilitate a spatial-based binding system. For example, active neurons or neural populations within a cortical area could be grouped together based on topographic proximity while those in different areas could be grouped if they lie in rough topographic register. An advantage of this scheme is that it can be carried out in parallel across the visual field. However, a spatial-based mechanism will tend to bind overlapping or occluded objects which should otherwise be segmented . An alternative binding mechanism is therefore necessary for binding and segmenting overlapping objects and surfaces. Temporal binding is a second type of neural binding. Temporal binding differs from spatial binding in two essential ways; 1) it operates at high spatial resolutions and 2) it binds and segments in the temporal domain, allowing for the coexistence of multiple objects in the same topographic region. Others have proposed that temporal events, such as phase-locked firing or I oscillations may play a role in neural binding (von der Malsburg, 1981; Gray and Singer, 1989, Crick and Koch, 1990). For purposes of this discussion, we do not consider the specific nature of the temporal events underlying neural binding, only that the binding itself is temporally dependent. The disadvantage of operating in the temporal domain is that the biophysical properties of cortical neurons (e.g. membrane time constants) forces this processing to be resource-limited-only a small number of objects or surfaces can be bound and segmented simultaneously. 2 COMPUTING INTERMEDIATE-LEVEL VISUAL REPRESENTATIONS: DIRECTION OF FIGURE We consider how these two classes of binding can be used to compute contextdependent (non-local) characteristics about the visual scene. An example of a context-dependent scene characteristic is contour ownership or direction of figure. Direction of figure is a useful intermediate-level visual representation since it can be used to organize an image into a perceptual scene (e.g. infer relative depth and link segregated features). Figure lA illustrates the relationship between contours and surfaces implied by direction of figure. We describe a model which utilizes both spatial and temporal binding to compute direction of figure (DOF). Prior to computing the DOF, the surface contours in the image are extracted. These contours are then temporally bound by a process we call "contour binding" (Finkel and Sajda, 1992). In the model, the temporal properties of the units are represented by a temporal binding value. We will not consider the details of this process except to say that units with similar temporal binding values are bound together while those with different values are segmented. In vivo, this temporal binding value may be represented by phase of neural firing, oscillation frequency, or some other specific temporal property of neuronal activity. The DOF is computed by circuitry which is organized in a columnar structure, shown in figure 2A. There are two primary circuits which operate to compute the direction of figure; one being a temporal-dependent/spatial-independent (TDSI) circuit selective to "closure", the other a spatial-dependent/temporal-independent Dual Mechanisms for Neural Binding and Segmentation similarity & proximity closure , f ~\.... / - -- / " 1',,- I ' direction of line endings concavities A B Figure 1: A Direction of figure as a surface representation. At point (1) the contour belongs to the surface contour of region A and therefore A owns the contour. This relationship is represented locally as a "direction of figure" vector pointing toward region A. Additional ownership relationships are shown for points (2) and (3). B Cues used in determining direction of figure . (SDTI) circuit selective to "similarity and proximity". There are also two secondary circuits which playa transient role in determining direction of figure. One is based on the observation that concave segments bounded by discontinuities are a cue for occlusion and ownership, while the other considers the direction of line endings as a potential cue. Figure IB summarizes the cues used to determine direction of figure. In this paper, we focus on the TDSI and SDTI circuits since they best illustrate the nature of the dual binding mechanisms. The perceptual consequences of attributing closure discrimination to temporal binding and similarity/proximity to spatial binding is illustrated in figure 3. 2.1 TDSI CIRCUIT Figure 2B(i) shows the neural architecture of the TDSI mechanism. The activity of the TDSI circuit selective for a direction of figure a is computed by comparing the amount of closure on either side of a contour. Closure is computed by summing the temporal dependent inputs over all directions i; T DSl Oi = [~Sf(ti) - L I I Sf- 180o (td] 1 (1) 0 The brackets ([]) indicate an implicit thresholding (if x < 0 then [xl = 0, otherwise [xl x) and Si(ti) and sf- 180 ? (td are the temporal dependent inputs, computed as; (Sj > ST) if { and (2) snt;) = { : ((ti - ~t) < tj < (ti + ~t)) otherwise = 995 996 Sajda and Finkel son TOSI .... toIfrom ~ other DOFcolumns ex - 180 0 t , to/from contour binding (i) (ii) A B Figure 2: A Divisions, inputs, and outputs for a DOF column. B The two primary circuits operating to compute direction of figure. (i) Top view of temporaldependent/spatial-independent (TDSI) circuit architecture. Filled square represents position of a specific column in the network. Unfilled squares represent other DOF columns serving as input to this column. Bold curve corresponds to a surface contour in the input. Shown is the pattern of long-range horizontal connections converging on the right side of the column (side ex). (ii) Top view of spatialdependent/temporal-independent (SDTI) circuit architecture. Shown is the pattern of connections converging on the right side of the column (side ex). where ex and ex - 180 0 represent the regions on either side of the contour, Sj is the activation of a unit along the direction i (For simulations i varies between 0 0 and 315 0 by increments of 45 0 ), 6.t determines the range of temporal binding values over which the column will integrate input, and ST is the activation threshold. The temporal dependence of this circuit implies that only those DOF columns having the same temporal binding value affect the closure computation. 2.2 SDTI CIRCUIT Figure 2B(ii) illustrates the neural architecture of the SDTI mechanism. The SDTI circuit organizes elements in the scene based on "proximity" and "similarity" of orientation. Unlike the TDSI circuit which depends upon temporal binding, the SDTI circuit uses spatial binding to access information across the network. Activity is integrated from units with similar orientation tuning which lie in a direction orthogonal to the contour (i.e. from parallel line segments). The activity of the SDTI circuit selective for a direction of figure ex is computed by comparing input from similar orientations on either side of a contour; SDTl Oi where = _1_ Smax (2: . t sf(Od - 2: . sr- 180o (Od) (3) I is a constant for normalizing the SDTI activity between 0 and 1 and sf (OJ) and sf- 1800 (Od are spatial dependent inputs selective for an orientation 0, Smax Dual Mechanisms for Neural Binding and Segmentation B A Figure 3: A The model predicts that a closed figure could not be discriminated in parallel search since its detection depends on resource-limited temporal binding. B Conversely, proximal parallel segments are predicted to be discriminated in parallel search due to resource-free spatial binding. computed as; (4) where ex and ex - 180? represent the regions on either side of the contour, () is the orientation of the contour, i is the direction from which the unit receives input, Cij is the connection strength (Cij falls off as a gaussian with distance), and Sj(x, y, (}j) is the activation of a unit along the direction i which is mapped to retinotopic location (x, y) and selective for an orientation (}j (For simulations i varies between the following three angles; 1- (}i,1- ((}i -45?), 1- ((}i +45?)). Since the efficacy of the connections, Cij, decrease with distance, columns which are further apart are less likely to be bound together. Neighboring parallel contours generate the greatest activation and the circuit tends to discriminate the region between the two parallel contours as the figure. 2.3 COMPUTED DOF The activity of a direction of figure unit representing a direction ex is given by the sum of the four components; DOF Ci = C 1 (TDSl Ci ) + C2 (SDTrl:) + C3 (CON Ci ) + C4 (DLE Ci ) (5) where the constants define the contribution of each cue to the computed DOF. Note that in this paper we have not considered the mechanisms for computing the DOF given the two secondary cues (concavities (CO N Ci ) and direction of line endings (DLE Ci )) . The DOF activation is computed for all directions ex (For simulations ex varies between 0? and 315? by increments of 45?) with the direction producing the largest activation representing the direction of figure. 3 SIMULATION RESULTS The following are simulations illustrating the role the dual binding mechanisms play in perceptual organization. All simulations were carried out using the NEXUS Neural Simulation Environment (Sajda and Finkel, 1992). 997 998 Sajda and Finkel ~ 100 >l &. ~ "C u: '" '" 60 X A B 60 0 C Figure 4: A 128x128 pixel grayscale image. B Direction of figure computed by the network. Direction of figure is shown as an oriented arrowhead, where the orientation represents the preferred direction of the DOF unit which is most active. C Depth of surfaces. Direction of figure relationships (such as those in the inset of B) are used to infer relative depth. Plot shows % activity of units in the foreground network- higher activity implies that the surface is closer to the viewer. 3.1 FIGURE/GROUND AND DEPTH-FROM-OCCLUSION Figure 4A is a grayscale image used as input to the network. Figure 4B shows the direction of figure computed by the model. Note that though the surface contours are incomplete, the model is still able to characterize the direction of figure and distinguish figure/ground over most of the contour. This is in contrast to models proposing diffusion-like mechanisms for determining figure/ground relationships which tend to fail if complete contour closure is not realized. The model utilizes direction of figure to determine occlusion relationships and stratify objects in relative depth, results shown in figure 4C. This method of inferring the relative depth of surfaces given occlusion is in contrast to traditional approaches utilizing T-junctions. The obvious advantage of using direction of figure is that it is a context-dependent feature directly linked to the representation of surfaces. 3.2 VIVIDNESS OF PERCEPTUAL COMPLETION Our previous work (Finkel and Sajda, 1992) has shown that direction of figure is important for completion phenomena, such as the construction of illusory contours and surfaces. More interestingly, our model offers an explanation for differences in perceived vividness between different inducing stimuli. For example, subjects tend to rank the vividness of the illusory figures in figure 5 from left to right, with the figure on the left being the most vivid and that on the right the least. Our model accounts for this effect in terms of the magnitude of the direction of figure along the illusory contour. Figure 6 shows the individual components contributing to the direction of figure. For a typical inducer, such as the pacman in figure 6, the TDSI and SDTI circuits tend to force the direction of figure of the L-shaped segment to region 1 while the concavity/convexity transformation tries to force the direction of figure of the segment to be toward region 2. This transformation transiently overwhelms the TDSI and SDTI responses, so that the direction of figure of the L-shaped segment is toward region 2. However, the TDSI and SDTI activation will affect the magnitude of the direction of figure, as shown in figure 7. For example, Dual Mechanisms for Neural Binding and Segmentation r .. .... r-., L..I c B A Figure 5: Illusory contour vividness as a function of inducer shape. Three types of inducers are arranged to generate an illusory square . A pacman inducer , B thick L inducer and C thin L inducer. Subjects rank the vividness of the illusory squares from left to right ((A) > (B) > (C)). C?C?Q TDSI component SDn component concavity/convexity component Figure 6: Processes contributing to the direction of figure of the L-shaped contour segment. The TDSI and SDTI circuits assign the contour to region 1, while the change of the concavity to a convexity assigns the segment to region 2. :, ':::::::::;:::::::1 :~ ,. ii f!. i Itll' -:.: .14 .' ii "ii :: .". . . E Ii +E .5 ?? ...... A ". r r B Figure 7: A Activity of SDTI units for the upper left inducer of each stimulus, where the area of each square is proportional to unit activity. The SDTI units try to assign the L-shaped segment to the region of the pacman. Numerical values indicates the magnitude of the SDTI effect. B Magnitude of direction of figure along the L-shaped segment as a function of inducer shape. The direction of figure in all cases is toward the region of the illusory square. 999 1000 Sajda and Finkel the weaker the activation of the TDSI and SDTI circuits, the stronger the activation of the DOF units assigning the L-shaped segment to region 2. Referring back to the inducer types in figure 5, one can see that though the TDSI component is the same for all three inducers (i.e. all three generate the same amount of closure) the SDTI contribution differs, shown quantitatively in figure 7A. The contribution of the SDTI circuit is greatest for the thin L inducers and least for the pacmen inducers-the L-shaped segments for the pacman stimulus are more strongly owned by the surface of the illusory square than those for the thin L inducer. This is illustrated in figure 7B, a plot of the magnitude of the direction of figure for each inducer configuration. This result can be interpreted as the model's ordering of perceived vividness, which is consistent with that of human observers. 4 CONCLUSION In this paper we have argued for the utility of binding neural activities in both the spatial and temporal domains. We have shown that a scheme consisting of these complementary mechanisms can be used to compute context-dependent scene characteristics, such as direction of figure. Finally, we have illustrated with computer simulations the role these dual binding mechanisms play in accounting for aspects of figure/ ground perception, depth-from-occlusion, and perceptual vividness of illusory contours and surfaces. It is interesting to speculate on the relationship between these complementary binding mechanisms and the traditional distinction between preattentive and attentional perception. Acknowledgements This work is supported by grants from ONR (N00014-90-J-1864, N00014-93-1-0681), The Whitaker Foundation, and The McDonnell-Pew Program in Cognitive NeuroSCIence. References F. Crick and C. Koch. Towards a neurobiological theory of consciousness. Seminars in Neuroscience, 2:263-275, 1990. L.H. Finkel and P. Sajda. Object discrimination based on depth-from-occlusion. Neural Computation, 4(6):901-921,1992. C. M. Gray and W. Singer. Neuronal oscillations in orientation columns of cat visual cortex. Proceedings of the National Academy of Science USA, 86:1698-1702, 1989. P. Sajda and L. Finkel. NEXUS: A simulation environment for large-scale neural systems. Simulation, 59(6):358-364, 1992. C. von der Malsburg. The correlation theory of brain function. Technical Report Internal Rep. No. 81-2, Max-Plank-Institute for Biophysical Chemistry, Department of Neurobiology, Gottingen, Germany, 1981. 

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Neural Network Methods for Optimization Problems Arun Jagota Department of Mathematical Sciences Memphis State University Memphis, TN 38152 E-mail: jagota~nextl.msci.memst.edu In a talk entitled "Trajectory Control of Convergent Networks with applications to TSP", Natan Peterfreund (Computer Science, Technion) dealt with the problem of controlling the trajectories of continuous convergent neural networks models for solving optimization problems, without affecting their equilibria set and their convergence properties. Natan presented a class of feedback control functions which achieve this objective, while also improving the convergence rates. A modified Hopfield and Tank neural network model, developed through the proposed feedback approach, was found to substantially improve the results of the original model in solving the Traveling Salesman Problem. The proposed feedback overcame the 2n symmetric property of the TSP problem. In a talk entitled "Training Feedforward Neural Networks quickly and accurately using Very Fast Simulated Reannealing Methods", Bruce Rosen (Asst. Professor, Computer Science, UT San Antonio) presented the Very Fast Simulated Reannealing (VFSR) algorithm for training feedforward neural networks [2]. VFSR Trained networks avoid getting stuck in local minima and statistically guarantee the finding of an optimal weights set. The method can be used when network activation functions are nondifferentiable, and although often slower than gradient descent, it is faster than other Simulated Annealing methods. The performances of conjugate gradient descent and VFSR trained networks were demonstrated on a set of difficult logic problems. In a talk entitled "A General Method for Finding Solutions of Covering problems by Neural Computation", Tal Grossman (Complex Systems, Los Alamos) presented a neural network algorithm for finding small minimal covers of hypergraphs. The network has two sets of units, the first representing the hyperedges to be covered and the second representing the vertices. The connections between the units are determined by the edges of the incidence graph. The dynamics of these two types of units are different. When the parameters of the units are correctly tuned, the stable states of the system correspond to the possible covers. As an example, he found new large square free subgraphs of the hypercube. In a talk entitled "Algebraic and Grammatical Design of Relaxation Nets", Eric 1184 Neural Network Methods for Optimization Problems Mjolsness (Professor, Computer Science, Yale University) presented useful algebraic notation and computer-algebraic syntax for general "programming" with optimization ideas; and also some optimization methods that can be succinctly stated in the proposed notation. He addressed global versus local optimization, time and space cost, learning, expressiveness and scope, and validation on applications. He discussed the methods of algebraic expression (optimization syntax and transformations, grammar models), quantitative methods (statistics and statistical mechanics, multiscale algorithms, optimization methods), and the systematic design approach. In a talk entitled "Algorithms for Touring Knights", Ian Parberry (Associate Professor, Computer Sciences, University of North Texas) compared Takefuji and Lee's neural network for knight's tours with a random walk and a divide-and-conquer algorithm. The experimental and theoretical evidence indicated that the neural network is the slowest approach, both on a sequential computer and in parallel, and for the problems of generating a single tour, and generating as many tours as possible. In a talk entitled "Report on the DIMACS Combinatorial Optimization Challenge" , Arun Jagota (Asst. Professor, Math Sciences, Memphis State University) presented his work, towards the said challenge, on neural network methods for the fast approximate solution of the Maximum Clique problem. The Mean Field Annealing algorithm was implemented on the Connection Machine CM-5. A fast (twotemperature) annealing schedule was experimentally evaluated on random graphs and on the challenge benchmark graphs, and was shown to work well. Several other algorithms, of the randomized local search kind, including one employing reinforcement learning ideas, were also evaluated on the same graphs. It was concluded that the neural network algorithms were in the middle in the solution quality versus running time trade-off, in comparison with a variety of conventional methods. In a talk entitled "Optimality in Biological and Artificial Networks" , Daniel Levine (Professor, Mathematics, UT Arlington) previewed a book to appear in 1995 [1]. Then he expanded his own view, that human cognitive functioning is sometimes, but not always or even most of the time, optimal. There is a continuum from the most "disintegrated" behavior, associated with frontal lobe damage, to stereotyped or obsessive-compulsive behavior, to entrenched neurotic and bureaucratic habits, to rational maximization of some measurable criteria, and finally to the most "integrated" , self-actualization (Abraham Maslow's term) which includes both reason and intuition. He outlined an alternative to simulated annealing, whereby a network that has reached an energy minimum in some but not all of its variables can move out of it through a "negative affect" signal that responds to a comparison of energy functions between the current state and imagined alternative states. References [1] D.S. Levine & W. Elsberry, editors. Optimality in Biological and Artificial Networks? Lawrence Erlbaum Associates, 1995. [2] B. E. Rosen & J. M. Goodwin. Training hard to learn networks using advanced simulated annealing methods. In Proc. of A CM Symp. on Applied Comp .. 1185 

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Fast Pruning Using Principal Components Asriel U. Levin, Todd K. Leen and John E. Moody Department of Computer Science and Engineering Oregon Graduate Institute P.O. Box 91000 Portland, OR 97291-1000 Abstract We present a new algorithm for eliminating excess parameters and improving network generalization after supervised training. The method, "Principal Components Pruning (PCP)", is based on principal component analysis of the node activations of successive layers of the network. It is simple, cheap to implement, and effective. It requires no network retraining, and does not involve calculating the full Hessian of the cost function. Only the weight and the node activity correlation matrices for each layer of nodes are required. We demonstrate the efficacy of the method on a regression problem using polynomial basis functions, and on an economic time series prediction problem using a two-layer, feedforward network. 1 Introduction In supervised learning, a network is presented with a set of training exemplars [u(k), y(k)), k = 1 ... N where u(k) is the kth input and y(k) is the corresponding output. The assumption is that there exists an underlying (possibly noisy) functional relationship relating the outputs to the inputs y=/(u,e) where e denotes the noise. The aim of the learning process is to approximate this relationship based on the the training set. The success of the learned approximation 35 36 Levin, Leen, and Moody is judged by the ability of the network to approximate the outputs corresponding to inputs it was not trained on. Large networks have more functional flexibility than small networks, so are better able to fit the training data. However large networks can have higher parameter variance than small networks, resulting in poor generalization. The number of parameters in a network is a crucial factor in it's ability to generalize. No practical method exists for determining, a priori, the proper network size and connectivity. A promising approach is to start with a large, fully-connected network and through pruning or regularization, increase model bias in order to reduce model variance and improve generalization. Review of existing algorithms In recent years, several methods have been proposed. Skeletonization (Mozer and Smolensky, 1989) removes the neurons that have the least effect on the output error. This is costly and does not take into account correlations between the neuron activities. Eliminating small weights does not properly account for a weight's effect on the output error. Optimal Brain Damage (OBD) (Le Cun et al., 1990) removes those weights that least affect the training error based on a diagonal approximation of the Hessian. The diagonal assumption is inaccurate and can lead to the removal of the wrong weights. The method also requires retraining the pruned network, which is computationally expensive. Optimal Brain Surgeon (OBS) (Hassibi et al., 1992) removes the "diagonal" assumption but is impractical for large nets. Early stopping monitors the error on a validation set and halts learning when this error starts to increase. There is no guarantee that the learning curve passes through the optimal point, and the final weight is sensitive to the learning dynamics. Weight decay (ridge regression) adds a term to the objective function that penalizes large weights. The proper coefficient for this term is not known a priori, so one must perform several optimizations with different values, a cumbersome process. We propose a new method for eliminating excess parameters and improving network generalization. The method, "Principal Components Pruning (PCP)", is based on principal component analysis (PCA) and is simple, cheap and effective. 2 Background and Motivation PCA (Jolliffe, 1986) is a basic tool to reduce dimension by eliminating redundant variables. In this procedure one transforms variables to a basis in which the covariance is diagonal and then projects out the low variance directions. While application of PCA to remove input variables is useful in some cases (Leen et al., 1990), there is no guarantee that low variance variables have little effect on error. We propose a saliency measure, based on PCA, that identifies those variables that have the least effect on error. Our proposed Principal Components Pruning algorithm applies this measure to obtain a simple and cheap pruning technique in the context of supervised learning. Fast Pruning Using Principal Components Special Case: PCP in Linear Regression In unbiased linear models, one can bound the bias introduced from pruning the principal degrees of freedom in the model. We assume that the observed system is described by a signal-plus-noise model with the signal generated by a function linear in the weights: y = Wou + e where u E ~P, Y E ~m, W E ~mxp, and e is a zero mean additive noise. The regression model is Y=Wu. The input correlation matrix is ~ = ~ L:k u(k)uT(k). It is convenient to define coordinates in which ~ is diagonal A = C T ~ C where C is the matrix whose columns are the orthonormal eigenvectors of~. The transformed input variables and weights are u = CT u and W = W C respectively, and the model output can be rewritten as Y = W u . It is straightforward to bound the increase in training set error resulting from re- moving subsets of the transformed input variable. The sum squared error is I = ~ L[y(k) - y(k)f[y(k) - y(k)] k Let Yl(k) denote the model's output when the last p -l components of u(k) are set to zero. By the triangle inequality ~ L[y(k) - h Yl(k)f[y(k) - Yl(k)] k < 1+ ~ L[Y(k) - Yl(k)f[Y(k) - Yl(k)] (1) k The second term in (1) bounds the increase in the training set errorl. This term can be rewritten as ~ p L w; WiAi L[y(k) - Yl(k)f[Y(k) - lh(k)] i=l+l k where Wi denotes the ith column of Wand Ai is the ith eigenvalue. The quantity Wi Ai measures the effect of the ith eigen-coordinate on the output error; it serves as our saliency measure for the weight Wi. w; Relying on Akaike's Final Prediction error (FPE) (Akaike, 1970), the average test set error for the original model is given by J[W] = ~ +-pm pm I(W) where pm is the number of parameters in the model. If p -l principal components are removed, then the expected test set is given by Jl[W] 1 For = N + lm Il(W) N-lm y E Rl, the inequality is replaced by an equality. . 37 38 Levin, Leen, and Moody If we assume that N? l * m, the last equation implies that the optimal generaliza- tion will be achieved if all principal components for which -T _ Wi WiAi 2m! <N are removed. For these eigen-coordinates the reduction in model variance will more then compensate for the increase in training error, leaving a lower expected test set error. 3 Proposed algorithm The pruning algorithm for linear regression described in the previous section can be extended to multilayer neural networks. A complete analysis of the effects on generalization performance of removing eigen-nodes in a nonlinear network is beyond the scope of this short paper. However, it can be shown that removing eigen-nodes with low saliency reduces the effective number of parameters (Moody, 1992) and should usually improve generalization. Also, as will be discussed in the next section, our PCP algorithm is related to the OBD and OBS pruning methods. As with all pruning techniques and analyses of generalization, one must assume that the data are drawn from a stationary distribution, so that the training set fairly represents the distribution of data one can expect in the future. Consider now a feedforward neural network, where each layer is of the form yi = r[WiU i ] = r[Xi] . Here, u i is the input, Xi is the weighted sum of the input, r is a diagonal operator consisting of the activation function of the neurons at the layer, and yi is the output of the layer. 1. A network is trained using a supervised (e.g. backpropagation) training procedure. 2. Starting at the first layer, the correlation matrix :E for the input vector to the layer is calculated. 3. Principal components are ranked by their effect on the linear output of the layer. 2 4. The effect of removing an eigennode is evaluated using a validation set. Those that do not increase the validation error are deleted. 5. The weights of the layer are projected onto the l dimensional subspace spanned by the significant eigenvectors W -+ WClCr where the columns of C are the eigenvectors of the correlation matrix. 6. The procedure continues until all layers are pruned. 2If we assume that -r is the sigmoidal operator, relying on its contraction property, we have that the resulting output error is bounded by Ilell <= IIWlllle",lll where e",l IS error observed at Xi and IIWII is the norm of the matrices connecting it to the output. Fast Pruning Using Principal Components As seen, the algorithm proposed is easy and fast to implement. The matrix dimensions are determined by the number of neurons in a layer and hence are manageable even for very large networks. No retraining is required after pruning and the speed of running the network after pruning is not affected. Note: A finer scale approach to pruning should be used ifthere is a large variation between Wij for different j. In this case, rather than examine w[ WiAi in one piece, the contribution of each wtj Ai could be examined individually and those weights for which the contribution is small can be deleted. 4 Relation to Hessian-Based Methods The effect of our PCP method is to reduce the rank of each layer of weights in a network by the removal of the least salient eigen-nodes, which reduces the effective number of parameters (Moody, 1992). This is in contrast to the OBD and OBS methods which reduce the rank by eliminating actual weights. PCP differs further from OBD and OBS in that it does not require that the network be trained to a local minimum of the error. In spite of these basic differences, the PCP method can be viewed as intermediate between OBD and OBS in terms of how it approximates the Hessian of the error function. OBD uses a diagonal approximation, while OBS uses a linearized approximation of the full Hessian. In contrast, PCP effectively prunes based upon a block-diagonal approximation of the Hessian. A brief discussion follows. In the special case of linear regression, the correlation matrix ~ is the full Hessian of the squared error. 3 For a multilayer network with Q layers, let us denote the numbers of units per layer as {Pq : q = 0 . . . Q}.4 The number of weights (including biases) in each layer is bq = Pq(Pq-l + 1), and the total number of weights in the network is B = L:~=l bq . The Hessian of the error function is a B x B matrix, while the input correlation matrix for each of the units in layer q is a much simpler (Pq-l + 1) X (Pq-l + 1) matrix. Each layer has associated with it Pq identical correlation matrices. The combined set of these correlation matrices for all units in layers q = 1 .. . Q of the network serves as a linear, block-diagonal approximation to the full Hessian of the nonlinear network. 5 This block-diagonal approximation has E~=l Pq(Pq-l + 1)2 non-zero elements, compared to the [E~=l Pq(Pq-l + 1)]2 elements of the full Hessian (used by OBS) and the L:~=l Pq(Pq-l + 1) diagonal elements (used by OBD). Due to its greater richness in approximating the Hessian, we expect that PCP is likely to yield better generalization performance than OBD. 3The correlation matrix and Hessian may differ by a numerical factor depending on the normalization of the squared error. If the error function is defined as one half the average squared error (ASE), then the equality holds. 4The inputs to the network constitute layer O. 5The derivation of this approximation will be presented elsewhere. However, the correspondence can be understood in analogy with the special case of linear regression. 39 40 Levin, Leen, and Moody 0.75 0.75 0.5 0.5 0.25 0.25 0.25 o.~ 0?. 75 -1 -0.25 a) -1 .' -. ~# .. ........- b) -1 Figure 1: a) Underlying function (solid), training data (points), and 10 th order polynomial fit (dashed). b) Underlying function, training data, and pruned regression fit (dotted). The computational complexities of the OBS, OBD, and PCP methods are respectively, where we assume that N 2: B. The computational cost of PCP is therefore significantly less than that of OBS and is similar to that of OBD. 5 Simulation Results Regression With Polynomial Basis Functions The analysis in section 2 is directly applicable to regression using a linear combination of basis functions y = W f (11,) ? One simply replaces 11, with the vector of basis functions f(11,). We exercised our pruning technique on a univariate regression problem using monomial basis functions f(11,) = (1,u,u 2 , ... ,un f with n = 10. The underlying function was a sum of four sigmoids. Training and test data were generated by evaluating the underlying function at 20 uniformly spaced points in the range -1 ~ u ~ + 1 and adding gaussian noise. The underlying function, training data and the polynomial fit are shown in figure 1a. The mean squared error on the training set was 0.00648. The test set mean squared error, averaged over 9 test sets, was 0.0183 for the unpruned model. We removed the eigenfunctions with the smallest saliencies w2 >.. The lowest average test set error of 0.0126 was reached when the trailing four eigenfunctions were removed. 6 . Figure 1b shows the pruned regression fit. 6The FPE criterion suggested pruning the trailing three eigenfunctions. We note that our example does not satisfy the assumption of an unbiased model, nor are the sample sizes large enough for the FPE to be completely reliable. Fast Pruning Using Principal Components 0.9 Figure 2: Prediction of the IP index 1980 - 1990. The solid line shows the performance before pruning and the dotted line the performance after the application of the PCP algorithm. The results shown represent averages over 11 runs with the error bars representing the standard deviation of the spread. 0.85 0.8 '" 0 ''"" r.l al ..N. . 0 . 75 0 .7 ..... ?J ~ 1 0 . 65 ?????????????????????t .................... ...... 0 z 0.6 .......................... 0 . 55 0.50 2 4 6 8 10 Prediction Horizon (month) 12 Time Series Prediction with a Sigmoidal Network We have applied the proposed algorithm to the task of predicting the Index of Industrial Production (IP), which is one of the main gauges of U.S. economic activity. We predict the rate of change in IP over a set of future horizons based on lagged monthly observations of various macroeconomic and financial indicators (altogether 45 inputs). 7 Our standard benchmark is the rate of change in IP for January 1980 to January 1990 for models trained on January 1960 to December 1979. In all runs, we used two layer networks with 10 tanh hidden nodes and 6 linear output nodes corresponding to the various prediction horizons (1, 2, 3, 6, 9, and 12 months). The networks were trained using stochastic backprop (which with this very noisy data set outperformed more sophisticated gradient descent techniques). The test set results with and without the PCP algorithm are shown in Figure 2. Due to the significant noise and nonstationarity in the data, we found it beneficial to employ both weight decay and early stopping during training. In the above runs, the PCP algorithm was applied on top of these other regularization methods. 6 Conclusions and Extensions Our "Principal Components Pruning (PCP)" algorithm is an efficient tool for reducing the effective number of parameters of a network. It is likely to be useful when there are correlations of signal activities. The method is substantially cheaper to implement than OBS and is likely to yield better network performance than OBD.8 7Preliminary results on this problem have been described briefly in (Moody et al., 1993), and a detailed account of this work will be presented elsewhere. 8See section 4 for a discussion of the block-diagonal Hessian interpretation of our method. A systematic empirical comparison of computational cost and resulting network performance of PCP to other methods like OBD and OBS would be a worthwhile undertaking. 41 42 Levin, Leen, and Moody Furthermore, PCP can be used on top of any other regularization method, including early stopping or weight decay.9 Unlike OBD and OBS, PCP does not require that the network be trained to a local minimum. We are currently exploring nonlinear extensions of our linearized approach. These involve computing a block-diagonal Hessian in which the block corresponding to each unit differs from the correlation matrix for that layer by a nonlinear factor. The analysis makes use of GPE (Moody, 1992) rather than FPE. Acknowledgements One of us (TKL) thanks Andreas Weigend for stimulating discussions that provided some of the motivation for this work. AUL and JEM gratefully acknowledge the support of the Advanced Research Projects Agency and the Office of Naval Research under grant ONR NOOOI4-92-J-4062. TKL acknowledges the support of the Electric Power Research Institute under grant RP8015-2 and the Air Force Office of Scientific Research under grant F49620-93-1-0253. References Akaike, H. (1970). Statistical predictor identification. Ann. Inst. Stat. Math., 22:203. Hassibi, B., Stork, D., and Wolff, G. (1992). Optimal brain surgeon and general network pruning. Technical Report 9235, RICOH California Research Center, Menlo Park, CA. Jolliffe, I. T. (1986). Principal Component Analysis. Springer-Verlag. Le Cun, Y., Denker, J., and Solla, S. (1990). Optimal brain damage. In Touretzky, D., editor, Advances in Neural Information Processing Systems, volume 2, pages 598-605, Denver 1989. Morgan Kaufmann, San Mateo. Leen, T. K., Rudnick, M., and Hammerstrom, D. (1990). Hebbian feature discovery improves classifier efficiency. In Proceedings of the IEEE/INNS International Joint Conference on Neural Networks, pages I-51 to I-56. Moody, J. (1992). The effective number of parameters: An analysis of generalization and regularization in nonlinear learning systems. In Moody, J., Hanson, S., and Lippman, R., editors, Advances in Neural Information Processing Systems, volume 4, pages 847-854. Morgan Kaufmann. Moody, J., Levin, A., and Rehfuss, S. (1993). Predicting the u.s. index of industrial production . Neural Network World, 3:791-794. in Proceedings of Parallel Applications in Statistics and Economics '93. Mozer, M. and Smolensky, P. (1989). Skeletonization: A technique for trimming the fat from a network via relevance assesment. In Touretzky, D., editor, Advances in Neural Information Processing Systems, volume 1, pages 107-115. Morgan Kaufmann. Weigend, A. S. and Rumelhart, D. E. (1991). Generalization through minimal networks with application to forecasting. In Keramidas, E. M., editor, INTERFACE'91 - 23rd Symposium on the Interface: Computing Science and Statistics, pages 362-370. 9(Weigend and Rumelhart, 1991) called the rank of the covariance matrix of the node activities the "effective dimension of hidden units" and discussed it in the context of early stopping. 

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WATTLE: A Trainable Gain Analogue VLSI Neural Network Richard Coggins and Marwan Jabri Systems Engineering and Design Automation Laboratory Department of Electrical Engineering J03, University of Sydney, 2006. Australia. Email: richardc@sedal.su.oz.au marwan@sedal.su.oz.au Abstract This paper describes a low power analogue VLSI neural network called Wattle. Wattle is a 10:6:4 three layer perceptron with multiplying DAC synapses and on chip switched capacitor neurons fabricated in 1.2um CMOS. The on chip neurons facillitate variable gain per neuron and lower energy/connection than for previous designs. The intended application of this chip is Intra Cardiac Electrogram classification as part of an implantable pacemaker / defibrillator system. Measurements of t.he chip indicate that 10pJ per connection is achievable as part of an integrated system. Wattle has been successfully trained in loop on parity 4 and ICEG morphology classification problems. 1 INTRODUCTION A three layer analogue VLSI perceptron has been previously developed by [Leong and Jabri, 1993]. This chip named Kakadu uses 6 bit digital weight storage, multiplying DACs in the synapses and fixed value off chip resistive neurons. The chip described in this paper called Wattle has the same synapse arrays as Kakadu, however, has the neurons implemented as switched capacitors on chip. For both Kakadu and Wattle, analogue techniques have been favoured as they offer greater opportunity to achieve a low energy and small area design over standard digital 874 WATfLE: A Trainable Gain Analogue VLSI Neural Network ----------------------, ,, &_-.,..- : NEURON CIRCUIT lout+ ODd Jout- 0UlpUt -=~-~ c:oonoct ..... ------------------------------------ -_.-- --------------------,,, ,, I ~ WEIGHT STORAGE --------------------- -1 HII SYNAPSE CIRCUIT mAC 00 I II , ~---------------------- : I I I ______________________________________________________________ L J Figure 1: Wattle Synapse Circuit Diagram techniques since the transistor count for the synapse can be much lower and the circuits may be biased in subthreshold. Some work has been done in the low energy digital area using subthreshold and optimised threshold techniques, however no large scale circuits have been reported so far. [Burr and Peterson, 1991] The cost of using analogue techniques is however, increased design complexity, sensitivity to noise, offsets and component tolerances. In this paper we demonstrate that difficult nonlinear problems and real world problems can be trained despite these effects. At present, commercially available pacemakers and defibrillators use timing decision trees implemented on CMOS microprocessors for cardiac arrythmia detection via peak detection on a single ventricular lead. Even when atrial leads are used, Intra Cardiac Electrogram (ICEG) morphology classification is required to separate some potentially fatal rhythms from harmless ones. [Leong and J abri, 1992] The requirements of such a morphology classifier are: ? Adaptable to differing morphology within and across patients. ? Very low power consumption. ie. minimum energy used per classification. ? Small size and high reliability. This paper demonstrates how this morphology classification may be done using a neural network architecture and thereby meet the constraints of the implantable arrythmia classification system. In addition, in loop training results will also be given for parity 4, another difficult nonlinear training problem. 875 876 Coggins and Jabri Vdd s reset s clock ~ -+__________~______~__________~ ~_c_lk_0__ t------------------t---------'co='-p-lL----> fan outto COM charging clock '" synapse CIP row connects , 1i::)>----_CIU_____________ _______-----' Figure 2: Wattle Neuron Circuit Diagram - Row Addrus 10x6 Synapse Array neuron. Ihi muHlplexor I Indkclemux 6x4Synapse Array '--- Column Addrus ~ ~ neurons I DD DO DO DO D D .----.Dar"'O'ta---=Re,.-g-.I,....., ..:-e-r-""1 Buffers Figure 3: Wattle Floor Plan ; next layer WATTLE: A Trainable Gain Analogue VLSI Neural Network .. 611"',. Figure 4: Photomicrograph of Wattle 2 ARCHITECTURE Switched capacitors were chosen for the neurons on Wattle after a test chip was fabricated to evaluate three neuron designs. [Coggins and Jabri, 1993] The switched capacitor design was chosen as it allowed flexible gain control of each neuron, investigation of gain optimisation during limited precision in loop training and the realisation of very high effective resistances. The wide gain range of the switched capacitor neurons and the fact that they are implemented on chip has allowed Wattle to operate over a very wide range of bias currents from 1pA LSB DAC current to 10nA LSB DAC current. Signalling on Wattle is fully differential to reduce the effect of common mode noise. The synapse is a multiplying digital to analogue convertor with six bit weights. The synapse is shown in figure L This is identical to the synapse used on the Kakadu chip [Leong and Jabri, 1993]. The MDAC synapses use a weighted current source to generate the current references for the weights. The neuron circuit is shown in figure 2. The neuron requires reset and charging clocks. The period of the charging clock determines the gain. Buffers are used to drive the neuron outputs off chip to avoid the effects of stray pad capacitances. Figure 3 shows a floor plan of the wattle chip . The address and data for the weights access is serial and is implemented by the shift registers on the boundary of the chip. The hidden layer multiplexor allows access to the hidden layer neuron outputs. The neuron demultiplexor switches the neuron clocks between the hidden and output layers. Figure 4 shows a photomicrograph of the wattle die. 3 ELECTRICAL CHARACTERISTICS Tests have been performed to verify the operation of the weighted current source for the MDAC synapse arrays, the synapses, the neurons and the buffers driving the neuron voltages off chip. The influences of noise, offsets, crosstalk and bandwidth of these different elements have been measured. In particular, the system level noise measurement showed that the signal to noise ratio was 40dB. A summary of the electrical characteristics appears in table L 877 878 Coggins and Jabri Table 1: Electrical Characteristics and Specifications Parameter Value Comment Area Technology Resolution Energy per connection LSB DAC current Feedforward delay Synapse Offset Gain cross talk delta 2.2 x 2.2mm~ 1.2um Nwell CMOS 2M2P weights 6bit, gains 7bit 43pJ 200pA 1.5ms 5mV 20% standard process weights on chip, gains off all weights maximum typical @200pA, 3V supply typical maximum maxImum A gain cross talk effect between the neurons was discovered during the electrical testing. The mechanism for this cross talk was found to be transients induced on the current source reference lines going to all the synapses as individual neuron gains timed out. The worst case cross talk coupled to a hidden layer neuron was found to be a 20% deviation from the singularly activated value. However, the training results of the chip do not appear to suffer significantly from this effect. A related effect is the length of time for the precharging of the current summation lines feeding each neuron due to the same transients being coupled onto the current source when each neuron is active. The implication of this is an increase in energy per classification for the network due to the transient decay time. However, one of the current reference lines was available on an outside pin, so the operation of the network free of these transients could also be measured. For this design, including the transient conditions, an energy per connection of 43pJ can be achieved. This may be reduced to 10pJ by modifying the current source to reduce transients and neglecting the energy of the buffers. This is to be compared with typical digital lOnJ per connection and analogue of 60pJ per connection appearing in the literature. [Delcorso et. al., 1993], Table 1. 4 TRAINING BOTH GAINS AND WEIGHTS A diagram of the system used to train the chip is shown in figure 5. The training software is part of a package called MUME [J abri et. al., 1992], which is a multi module neural network simulation environment. Wattle is interfaced to the work station by Jiggle, a general purpose analogue and digital chip tester developed by SEDAL. Wattle, along with gain counter circuitry, is mounted on a separate daughter board which plugs into Jiggle. This provides a software configurable testing environment for Wattle. In loop training then proceeds via a hardware specific module in MUME which writes the weights and reads back the analogue output of the chip. Wattle can then be trained by a wide variety of algorithms available in MUME. Wattle has been trained in loop using a variation on the Combined Search Algorithm (CSA) for limited precision training. [Xie and Jabri, 1992] (Combination of weight perturbation and axial random search). The variation consists of training the gains 

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Lower Boundaries of Motoneuron Desynchronization via Renshaw Interneurons Mitchell Gil Maltenfort It Robert E. Druzinsky Dept. of Physiology Northwestern University Chicago, IT.. 60611 Dept. of Biomedical Engineering Northwestern University Evanston, IT.. 60201 c. w. J. Heckman Zev Rymer Dept. of Physiology and Biomedical Engineering Northwestern University Chicago, IT.. 60611 V. A. Research Service Lakeside Hospital and Dept. of Physiology Northwestern University Chicago, IT.. 60611 Abstract Using a quasi-realistic model of the feedback inhibition ofmotoneurons (MNs) by Renshaw cells, we show that weak inhibition is sufficient to maximally desynchronize MNs, with negligible effects on total MN activity. MN synchrony can produce a 20 - 30 Hz peak in the force power spectrum, which may cause instability in feedback loops. 1 INTRODUCTION The structure of the recurrent inhibitory connections from Renshaw cells (RCs) onto motoneurons (MNs) (Figure 1) suggests that the RC forms a simple negative feedback * send mail to: Mitchell G. Maltenfort, SMPP room 1406, Rehabilitation Insitute of Chicago, 345 East Superior Street, Chicago, IT.. 60611. Email address is mgm@nwu.edu 535 536 Maltenfort, Druzinsky, Heckman, and Rymer loop. Past theoretical work has examined possible roles of this feedback in smoothing or gain regulation of motor output (e.g., Bullock and Contreras-Vidal, 1991; Graham and Redman, 1993), but has assumed relatively strong inhibitory effects from the RC. Experimental observations (Granit et al.,1961) show that maximal RC activity can only reduce MN frring rates by a few impulses per second. although this weak inhibition is sufficient to affect the timing of MN fuings, reducing the probability that any two MNs will fire simultaneously (Adam et al., 1978; Windhorst et al., 1978). In this study, simulations were used to examine the impact of RC inhibition on MN frring synchrony and to predict the effects of such synchrony on force output. + Figure 1: Simplified Schematic of Recurrent Inhibition 2 CONSTRUCTION OF THE MODEL 2.1 MODELING OF INDIVIDUAL NEURONS The integrate-and-fIre neuron model of MacGregor (1987) adequately mimics specific frring patterns. Coupled first-order differential equations govern membrane potential and afterhyperpolarization (AHP) based on injected current and synaptic inputs. A spike is frred when the membrane potential crosses a threshold. The model was modified to include a membrane resistance in order to model MNs of varying current thresholds. Membrane resistance and time constants of model MNs were set to match published data (Gustaffson and Pinter, 1984). The parameters governing AHPs were adjusted to agree with observations from single action potentials and steady-state current-rate plOts (Heckman and Binder, 1991). Realistic frring behavior could be generated for MNs with current thresholds of 4 - 40 nA. Although there are no direct measurements of RC membrane properties available, appropriate parameters were estimated by extrapolation from the MN parameter set. The simulated RC has a 30 ms AHP and a current-rate plot matching that reported by Hultborn and Pierrot-Deseilligny (1979). Spontaneous frring of 8 pps is produced in the model by setting the RC firing threshold to 0.01 mV below resting potential; in vivo Lower Boundaries of Motoneuron Desynchronization via Renshaw Intemeurons this fuing is likely due to descending inputs (Hamm et al., 1987a), but there is no quantitative description of such inputs. The RCs are assumed to be homgeneous. 2.2 CONNECTIVITY OF THE POOL Simulated neurons were arranged along a 16 by 16 grid. The network consists of 256 MNs and 64 RCs, with the RCs ordered on even-numbered rows and columns; as a result, the MN - RC connections are inhomogeneous along the pool. For each trial, MN pools were randomly generated following the distribution of MN current thresholds for a model of the cat medial gastrocnemius motor pool (Heckman and Binder, 1991). Communication between neurons is mediated by synaptic conductances which open when a presynaptic cell fues, then decay exponentially. MN excitation of RCs was set to produce RC fuing rates S; 190 pps (Cleveland et aI., 1981) which linearly increase with MN activity (Cleveland and Ross, 1977). MN activation of RCs scales inversely with MN current threshold (Hultbom et al., 1988). Connectivity is based on observations that synapses from RCs to MNs have a longer spatial range than the reverse (reviewed in Windhorst, 1990). The IPSPs produced by single MN fIrings are 4 - 6 times larger than those produced by single RC fIrings (Hamm et al., 1987b; van Kuelen, 1981). In the model, each MN excites RCs within one column or row of itself, and each RC inhibits MNs up to two rows or columns away; thus, each MN excites 1 - 4 RCs (mean 2.25) and receives feedback from 4 - 9 RCs (mean 6.25). 2.3 ACTIVATION OF THE POOL The MNs are activated by applied step currents. Although this is not realistic, it is computationally efficient. An option in the simulation program allows for the addition of bandlimited noise to the activation current, to simulate a synchronizing common synaptic input. This signal has an rms value of 3% of the mean applied current and is lOW-pass mtered with a cutoff of 30 Hz. This allows us look at the effects due purely to RC activity and to establish which effects persist when the MN pool is being actively synchronized. 3 EFFECTS OF RC STRENGTH ON MN SYNCHRONY 3.1 DEFINITION OF SYNCHRONY COEFFICIENT Consider the total number of spikes frred by the MN pool as a time series. During synchronous firing, the MN spikes will clump together and the time series will have regions of very many or very few MN spikes. When the MNs are de synchronized, the range of spike counts in each time bin will contract towards the mean. It follows that a simple measure of MN synchrony is the the coeffIcient of variation (c. v. = s.d. I mean) of the time series formed by the summed MN activity. Figure 2 shows typical MN pool fIring before and after RC feedback inhibition is added; the changes in "clumping" described above are quite visible in the two plots. 537 538 Maltenfort, Druzinsky, Heckman, and Rymer 3.2 "PLATEAU" OF DESYNCHRONIZATION The magnitude of the synaptic conductance from RCs onto MNs was changed from zero to twice physiological in order to compare the effects of 'weak' and 'strong' recurrent inhibition. At activation levels sufficient to recruit at least 70% of MNs in the pool (mean tiling rate ~ 15 pps), a surprising plateau effect was seen. The synchrony coefficient fell off with RC synaptic conductance until the physiological level was reached, and then no further de synchronization was seen. The effect persisted when synchronizing noise was added (Figure 3). At activation levels sufficient to show this plateau, this "comer" inhibition level was always the same. Synchronized Firing (no RC inhibition) 50 O~~~u-~~~~~~~~~~~~~~~~~~ o 50 100 150 200 Desynchronized Firing (RC inhibition added) 20 O~----~--~--------~~~----~u-~~~~ o 50 100 150 200 Time (ms) Figure 2: Comparison of Synchronous and Asynchronous MN Firing At this comer level, the decrease in mean MN firing rate was ~ 1 pps and not statistically significant. There was also no discernible change in the percentage of the MN pool active. The c.v. of the interspike interval of single MN filings during constant activation is ~ 2.5 % even with RCs active - this implies that the RC system finds an optimal arrangement of the MN fuings and then performs few if any further shifts. When synchronizing noise is added, the RC effect on the interspike interval is swamped by the effect of the synchronizing random input. Figure 4 shows the effect of increasing MN activation on the synchrony coefficients before and after RC inhibition is added. The change is statistically significant at all levels, but is only large at higber levels as discussed above. As activation of the MNs Lower Boundaries of Motoneuron Desynchronization via Renshaw Intemeurons increases, the "before" level of synchrony increases while the "after" level seems to move asymptotically towards a minimum level of about 0.35. This minimum level of MN synchrony, as well as the dependence of the effect on the activation level of the pool, suggests that a certain amount of synchrony becomes inevitable as more MNs are activated and fIre at higher rates. 1.4 * 1.2 1 synchronizing noise added 0.8 0.6 0.4 o 0.002 0.004 0.006 0.008 0.01 RC Synaptic Conductance ijJ.Siemens) Figure 3: MN Firing Synchrony vs. RC Strength 4 EFFECTS OF MN SYNCHRONY ON MUSCLE FORCE 4.1 MODELING OF FORCE OUTPUT Single twitches of motor units are modeled with a second-order model, f(t) =Be-tit, t where the amplitude F and time constant t are matched to MN current threshold according to the model of Heckman and Binder (1991). A rate-based gain factor adapted from Fuglevand (1989) produces fused tetanus at high fuing rates. The tenfold difference in current thresholds maps to a fifty-fold difference in twitch forces. Twitch time constants range 30-90 ms. 539 540 Maltenfort, Druzinsky, Heckman, and Rymer 4.1 EFFECTS OF RECURRENT INHmITION ON FORCE The force model sharply low-pass fllters the neural input signal (S 5 Hz). As a result, the c.v. of the force output is much lower than that of the associated MN input (S 0.01). Although the plot of force c.v. vs. RC strength during constant activation follows the curve in Figure 3, adding synchronizing noise removes any correlation between force .c.v. and magnitude of recurrent inhibition. The effect of recurrent inhibition on mean force is similar to that on the mean firing rate: small (S5 % decrease) and generally not statistically significant 1.8 0 1.6 1.4 ~ c ~u ~ ~ 1.2 before recurrent inhibition 1 bO :5 0.8 ~ 0.6 ~ 0.4 0.2 5 10 15 20 25 30 35 Activation Current (nA) Figure 4: Effects of MN Activation on Synchrony Before and After Recurrent Inhibition When the change in synchrony due to RCs is large, a peak appears in the force power spectrum in the range 20 - 30 Hz. This peak is reduced by RCs even when the MN pool is being actively synchronized (Figure 5). Peaks in the force spectrum match peaks in the spectrum of pooled MN activity, suggesting the effect is due to synchronous MN ruing. Although the magnitude of this peak is small (S 0.5% of mean force), its relatively high frequency suggests that in derivative feedback - where spectral components are multiplied by 21t times their frequency - its impact could be substantial. The feedback loop which measures muscle stretch contains such a derivative component (Hook and Rymer, 1981). Lower Boundaries of Motoneuron Desynchronization via Renshaw Intemeurons 5 DISCUSSION The preceding shows that the ostensibly weak recurrent inhibition is sufficient to sharply reduce the maximum number of synchronous Irrings of a neuron population, while having a negligible effect on the total population activity. This has a broad implication for neural networks in that it suggests the existence of a "switching mechanism" which forces the peaks in the output of an ensemble of neurons to remain below a threshold level without significantly suppressing the total ensemble activity. One possible role for such a mechanism would be in the accommodation to a step or ramp increase in a stimulus. The initial increase synchronizes the neural signal from the receptor, which is then desyncbronized by the recurrent inhibition. The synchronized ruing phase would be sufficient to excite a target neuron past its ruing threshold, but after that, the desyncbronized neural signal would remain well below the target's threshold. 0.2 0.15 0.1 0.05 O~--------~--------~----------~------~ o 10 20 30 40 Frequency (Hz) Figure 5: Recurrent Inhibition Reduces Spectral Peak. 95% confidence limit of means plotted, solid lines before recurrent inhibition and dashed lines after. Acknowledgments The authors are indebted to Dr. Tom Buchanan for use of his IBM RS/6000 workstation. This work was supported by NIH grants NS28076-02 and NS30295-01. 541 542 Maltenfort, Druzinsky, Heckman, and Rymer References Adam D, Windhorst U, Inbar GF: The effects of recurrent inhibition on the crosscorrelated flring patterns of motoneurons (and their relation to signal transmission in the spinal cord-muscle channel). Bioi. Cybern., 29: 229-235, 1978. Bullock D, Contreras-Vidal J: How spinal neural networks reduce discrepancies between motor intention and motor realization. Tech.Report CAS/CNS-91-023, Boston U., 1991. Cleveland S, Kuschmierz A, Ross H-G: Static input-output relations in the spinal recurrent inhibitory pathway. Bioi. Cybern., 40: 223-231, 1981. Cleveland S, Ross H-G: Dynamic properties of Renshaw cells: Frequency response characteristics. Bioi. Cybem., 27: 175-184, 1977. Fuglevand AJ: A motor unit pool model: relationship of neural control properties to isometric muscle tension and the electromyogram. Ph.D. Thesis, U. of Waterloo, 1989. Graham BP, Redman SJ: Dynamic behaviour of a model of the muscle stretch reflex. Neural Networks, 6: 947-962, 1993. Granit R, Haase J, Rutledge LT: Recurrent inhibition in relation to frequency of flring and limitation of discharge rate of extensor motoneurons. J. Physiol., 158: 461-475, 1961. Gustaffson B, Pinter MJ: An investigation of threshold properties among cat spinal amotoneurons. J. Physiol., 357: 453-483, 1984. Hamm 1M, Sasaki S, Stuart DG, Windhorst U, Yuan C-S: Distribution of single-axon recurrent inhibitory post-synaptic potentials in the cat. J. Physiol., 388: 631-651,1987a. Hamm 1M, Sasaki S, Stuart 00, Windhorst U, Yuan C-S: The measurement of single motor-axon recurrent inhibitory post-synaptic potentials in a single spinal motor nucleus in the cat. J. Physiol., 388: 653-664, 1987b . Heckman CJ, Binder MD: Computer simulation of the steady-state input-output function of the cat medial gastrocnemius motoneuron pool. J. Neurophysiol., 65: 952-967, 1991. Houk JC, Rymer WZ: Chapter 8: Neural control of muscle length and tension. In Handbook of Physiology: the Nervous System II pt. I, ed.VB Brooks. Am. Physiol. Soc., Bethesda, MD, 1981. Hultborn H, Peirrot-Deseillgny E: Input-output relations in the pathway of recurrent inhibition to motoneurons in the cat. J. Physiol., 297: 267-287, 1979. MacGregor RJ: Neural and Brain Modeling. Academic Press, San Diego, 1987. Van Kuelen LCM: Autogenetic recurrent inhibition of individual spinal motoneurons of the cat. Neurosci. Lett., 21: 297-300, 1981. Windhorst U: Activation of Renshaw cells. Prog. in Neurobiology, 35: 135-179, 1990. Windhorst U, Adam D, Inbar GF: The effects of recurrent inhibitory feedback in shaping discharge patterns of motoneurones excited by phasic muscle stretches. Bioi. Cybem., 29: 221-227, 1978. 

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Fool.s Gold: Extracting Finite State Machines From Recurrent Network Dynamics John F. Kolen Laboratory for Artificial Intelligence Research Department of Computer and Information Science The Ohio State University Columbus,OH 43210 kolen-j @cis.ohio-state.edu Abstract Several recurrent networks have been proposed as representations for the task of formal language learning. After training a recurrent network recognize a formal language or predict the next symbol of a sequence, the next logical step is to understand the information processing carried out by the network. Some researchers have begun to extracting finite state machines from the internal state trajectories of their recurrent networks. This paper describes how sensitivity to initial conditions and discrete measurements can trick these extraction methods to return illusory finite state descriptions. INTRODUCTION Formal language learning (Gold, 1969) has been a topic of concern for cognitive science and artificial intelligence. It is the task of inducing a computational description of a formal language from a sequence of positive and negative examples of strings in the target language. Neural information processing approaches to this problem involve the use of recurrent networks that embody the internal state mechanisms underlying automata models (Cleeremans et aI., 1989; Elman, 1990; Pollack, 1991; Giles et aI, 1992; Watrous & Kuhn, 1992). Unlike traditional automata-based approaches, learning systems relying on recurrent networks have an additional burden: we are still unsure as to what these networks are doing.Some researchers have assumed that the networks are learning to simulate finite state 501 502 Kolen machines (FSMs) in their state dynamics and have begun to extract FSMs from the networks' state transition dynamics (Cleeremans et al., 1989; Giles et al., 1992; Watrous & Kuhn, 1992). These extraction methods employ various clustering techniques to partition the internal state space of the recurrent network into a finite number of regions corresponding to the states of a finite state automaton. This assumption of finite state behavior is dangerous on two accounts. First, these extraction techniques are based on a discretization of the state space which ignores the basic definition of information processing state. Second, discretization can give rise to incomplete computational explanations of systems operating over a continuous state space. SENSITIVITY TO INITIAL CONDITIONS In this section, I will demonstrate how sensitivity to initial conditions can confuse an FSM extraction system. The basis of this claim rests upon the definition of information processing state. Information processing (lP) state is the foundation underlying automata theory. Two IP states are the same if and only if they generate the same output responses for all possible future inputs (Hopcroft & Ullman, 1979). This definition is the fulcrum for many proofs and techniques, including finite state machine minimization. Any FSM extraction technique should embrace this definition, in fact it grounds the standard FSM minimization methods and the physical system modelling of Crutchfield and Young (Crutchfield & Young, 1989). Some dynamical systems exhibit exponential divergence for nearby state vectors, yet remain confined within an attractor. This is known as sensitivity to initial conditions. If this divergent behavior is quantized, it appears as nondeterministic symbol sequences (Crutchfield & Young, 1989) even though the underlying dynamical system is completely deterministic (Figure 1). Consider a recurrent network with one output and three recurrent state units. The output unit performs a threshold at zero activation for state unit one. That is, when the activation of the first state unit of the current state is less than zero then the output is A. Otherwise, the output is B. Equation 1 presents a mathematical description. Set) is the current state of the system 0 (t) is the current output. S (t + 1) = ro -J [ ~ 2 tanh ( 0 -2 0 0 2 1 . S(t)) 0 2 -1 (1) 1 Figure 2 illustrates what happens when you run this network for many iterations. The point in the upper left hand state space is actually a thousand individual points all within a ball of radius 0.01. In one iteration these points migrate down to the lower corner of the state space. Notice that the ball has elongated along one dimension. After ten iterations the original ball shape is no longer visible. After seventeen, the points are beginning to spread along a two dimensional sheet within state space. And by fifty iterations, we see the network reaching the its full extent of in state space. This behavior is known as sensitivity to initial conditions and is one of three conditions which have been used to characterize chaotic dynamical systems (Devaney, 1989). In short, sensitivity to initial conditions implies Fool's Gold: Extracting Finite State Machines from Recurrent Network Dynamics x~4x(l-x) O(x) x~2x ~ : { = 3.68x(l-x) O(x) = @A x<O.5 x>O.5 mod 1 O(x) x = 1 A x<3 B - <x<3 3 C -<x 3 2 1 2 A C x<O.5 C x>O.5 Figure 1: Examples of deterministic dynamical systems whose discretize trajectories appear nondeterministic. that any epsilon ball on the attractor of the dynamical will exponentially diverge, yet still be contained within the locus of the attractor. The rate of this divergence is illustrated in Figure 3 where the maximum distance between two points is plotted with respect to the number of iterations. Note the exponential growth before saturation. Saturation occurs as the point cloud envelops the attractor. No matter how small one partitions the state space, sensitivity to initial conditions will eventually force the extracted state to split into multiple trajectories independent of the future input sequence. This is characteristic of a nondeterministic state transition. Unfortunately, it is very difficult, and probably intractable, to differentiate between a nondeterministic system with a small number of states or a deterministic with large number of states. In certain cases, however, it is possible to analytically ascertain this distinction (Crutchfield & Young, 1989). THE OBSERVERS' PARADOX One response to this problem is to evoke more computationally complex models such as push-down or linear-bounded automata. Unfortunately, the act of quantization can actually introduce both complexion and complexity in the resulting symbol sequence. Pollack and I have focused on a well-hidden problems with the symbol system approach to understanding the computational powers of physical systems. This work (Kolen & Pollack, 1993; 503 S04 Kolen 1 1 1 I I I output=A 1 Start (e<O.Ol) output=B 1 1 iteration output=A 1 1 1 I I output=A,B 1 17 iterations 1 10 iterations output=A,B 1 25 iterations 1 50 iterations Figure 2: The state space of a recurrent network whose next state transitions are sensitive to initial conditions. The initial epsilon ball contains 1000 points. These points first straddle the output decision boundary at iteration seven. Kolen & Pollack, In press) demonstrated that computational complexity, in terms of Chomsky's hierarchy of formal languages (Chomsky, 1957; Chomsky, 1965) and Newell and Simon's physical symbol systems (Newell & Simon, 1976), is not intrinsic to physical systems. The demonstration below shows how apparently trivial changes in the partitioning of state space can produce symbol sequences from varying complexity classes. Consider a point moving in a circular orbit with a fixed rotational velocity, such as the end of a rotating rod spinning around a fixed center, or imagine watching a white dot on a spinning bicycle wheel. We measure the location of the dot by periodically sampling the location with a single decision boundary (Figure 4, left side). If the point is to the left of boundary at the time of the sample, we write down an "1". Likewise, we write down an "r" when the point is on the other side. (The probability of the point landing on the boundary is zero and can arbitrarily be assigned to either category without affecting the results below.) In the limit, we will have recorded an infinite sequence of symbols containing long sequences of r's and l's. The specific ordering of symbols observed in a long sequence of multiple rotations is Fool's Gold: Extracting Finite State Machines from Recurrent Network Dynamics ell ...... ????? 2.5 ....0c:: 0.. c:: 0 0 ~ ...... 0 .0 0 c:: ? ~ ...... .... "1:;) 8::s ....S ~ ::E ? ? ? 1.5 u ell ? ? ? ? 2 ? 1 ? 0.5 ? ? ??? ? ? ? ? ?? 10 20 30 Iteration number 40 50 Figure 3: Spread of initial points across the attractor as measured by maximum distance. 1 1 r r c Figure 4: On the left, two decision regions which induce a context free language. 9 is the current angle of rotation. At the time of sampling, if the point is to the left (right) of the dividing line, an 1 (r) is generated. On the right, three decision regions which induce a context sensitive language. dependent upon the initial rotational angle of the system. However, the sequence does possess a number of recurring structural regularities, which we call sentences: a run of r's followed by a run of l's. For a fixed rotational velocity (rotations per time unit) and sampling rate, the observed system will generate sentences of the form r n1 m (n, m > 0). (The notation rn indicates a sequence of n r's.) For a fixed sampling rate, each rotational velocity specifies up to three sentences whose number of r's and l's differ by at most one. These sentences repeat in an arbitrary manner. Thus, a typical subsequence of a rotator which produces sentences r n1 n, r n1 n+l ,rn+ 11 n would look like 505 506 Kolen rnln+lrnlnrnln+lrn+l1nrnlnrnln+l. A language of sentences may be constructed by examining the families of sentences generated by a large collection of individuals, much like a natural language is induced from the abilities of its individual speakers. In this context, a language could be induced from a population of rotators with different rotational velocities where individuals generate sentences of the form {r"l n, r"l "+1 ,r"+ll"}, n > O. The reSUlting language can be described by a context free grammar and has unbounded dependencies; the number of 1 's is a function of the number of preceding r's. These two constraints on the language imply that the induced language is context free. To show that this complexity class assignment is an artifact of the observational mechanism, consider the mechanism which reports three disjoint regions: 1, c, and r (Figure 4, right side). Now the same rotating point will generate sequences ofthe form ... rr... rrcc ... ccll... llrr... rrcc ... ccll... ll .... For a fixed sampling rate, each rotational velocity specifies up to seven sentences, r nc ffi l k, when n, m, and k can differ no by no more than one. Again, a language of sentences may be constructed containing all sentences in which the number ofr's, c's, and l's differs by no more than one. The resulting language is context sensitive since it can be described by a context sensitive grammar and cannot be context free as it is the finite union of several context sensitive languages related to r"c"l n. CONCLUSION Using recurrent neural networks as the representation underlying the language learning task has revealed some inherent problems with the concept of this task. While formal languages have mathematical validity, looking for language induction in physical systems is questionable, especially if that system operates with continuous internal states. As I have shown, there are two major problems with the extraction of a learned automata from our models. First, sensitivity to initial conditions produces nondeterministic machines whose trajectories are specified by both the initial state of the network and the dynamics of the state transformation. The dynamics provide the shape of the eventual attractor. The initial conditions specify the allowable trajectories toward that attractor. While clustering methods work in the analysis of feed-forward networks because of neighborhood preservation (as each layer is a homeomorphism), they may fail when applied to recurrent network state space transformations. FSM construction methods which look for single transitions between regions will not help in this case because the network eventually separates initially nearby states across several FSM state regions. The second problem with the extraction of a learned automata from recurrent network is that trivial changes in observation strategies can cause one to induce behavioral descriptions from a wide range of computational complexity classes for a single system. It is the researcher's bias which determines that a dynamical system is equivalent to a finite state automata. Fool's Gold: Extracting Finite State Machines from Recurrent Network Dynamics One response to the first problem described above has been to remove and eliminate the sources of nondeterminism from the mechanisms. Zeng et. a1 (1993) corrected the secondorder recurrent network model by replacing the continuous internal state transformation with a discrete step function. (The continuous activation remained for training purposes.) This move was justified by their focus on regular language learning, as these languages can be recognized by finite state machines. This work is questionable on two points, however. First, tractable algorithms already exist for solving this problem (e.g. Angluin, 1987). Second, they claim that the network is self-clustering the internal states. Self-clustering occurs only at the comers of the state space hypercube because of the discrete activation function, in the same manner as a digital sequential circuit "clusters" its states. Das and Mozer (1994), on the other hand, have relocated the clustering algorithm. Their work focused on recurrent networks that perform internal clustering during training. These networks operate much like competitive learning in feed-forward networks (e.g. Rumelhart and Zipser, 1986) as the dynamics of the learning rules constrain the state representations such that stable clusters emerge. The shortcomings of finite state machine extraction must be understood with respect to the task at hand. The actual dynamics of the network may be inconsequential to the final product if one is using the recurrent network as a pathway for designing a finite state machine. In this engineering situation, the network is thrown away once the FSM is extracted. Neural network training can be viewed as an "interior" method to finding discrete solutions. It is interior in the same sense as linear programming algorithms can be classified as either edge or interior methods. The former follows the edges of the simplex, much like traditional FSM learning algorithms search the space of FSMs. Internal methods, on the other hand, explore search spaces which can embed the target spaces. Linear programming algorithms employing internal methods move through the interior of the defined simplex. Likewise, recurrent neural network learning methods swim through mechanisms with mUltiple finite state interpretations. Some researchers, specifically those discussed above, have begun to bias recurrent network learning to walk the edges (Zeng et al, 1993) or to internally cluster states (Das & Mozer, 1994). In order to understand the behavior of recurrent networks, these devices should be regarded as dynamical systems (Kolen, 1994). In particular, most common recurrent networks are actually iterated mappings, nonlinear versions of Barnsley's iterated function systems (Barnsley, 1988). While automata also fall into this class, they are a specialization of dynamical systems, namely discrete time and state systems. Unfortunately, information processing abstractions are only applicable within this domain and do not make any sense in the broader domains of continuous time or continuous space dynamical systems. Acknowledgments The research reported in this paper has been supported by Office of Naval Research grant number NOOOI4-92-J-1195. I thank all those who have made comments and suggestions for improvement of this paper, especially Greg Saunders and Lee Giles. References Angluin, D. (1987). Learning Regular Sets from Queries and Counterexamples. Information 507 508 Kolen and Computation, 75,87-106. Barnsley, M. (1988). Fractals Everywhere. Academic Press: San Diego, CA. Chomsky, N. (1957). Syntactic Structures. The Hague: Mounton & Co. Chomsky, N. (1965). Aspects of the Theory of Syntax. Cambridge, Mass.: MIT Press. Cleeremans, A, Servan-Schreiber, D. & McClelland, J. L. (1989). Finite state automata and simple recurrent networks. Neural Computation, 1,372-381. Crutchfield, J. & Young, K. (1989). Computation at the Onset of Chaos. In W. Zurek, (Ed.), Entropy, Complexity, and the Physics of Information. Reading: Addison-Wesely. Das, R. & Mozer, M. (1994) A Hybrid Gradient-Descent/Clustering Technique for Finite State Machine Induction. In Jack D. Cowan, Gerald Tesauro, and Joshua Alspector, (Eds.), Advances in Neural Information Processing Systems 6. Morgan Kaufman: San Francisco. Devaney, R. L. (1989). An Introduction to Chaotic Dynamical Systems. Addison-Wesley. Elman, J. (1990). Finding structure in time. Cognitive Science, 14, 179-211. Giles, C. L., Miller, C. B., Chen, D., Sun, G. Z., Chen, H. H. & C.Lee, Y. (1992). Extracting and Learning an Unknown Grammar with Recurrent Neural Networks. In John E. Moody, Steven J. Hanson & Richard P. Lippman, (Eds.), Advances in Neural Information Processing Systems 4. Morgan Kaufman. Gold, E. M. (1969). Language identification in the limit. Information and Control, 10,372381. Hopcroft, J. E. & Ullman, J. D. (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesely. Kolen, J. F. (1994) Recurrent Networks: State Machines or Iterated Function Systems? In M. C. Mozer, P. Smolensky, D. S. Touretzky, J. L. Elman, & AS. Weigend (Eds.), Proceedings of the 1993 Connectionist Models Summer School. (pp. 203-210) Hillsdale, NJ: Erlbaum Associates. Kolen, J. F. & Pollack, J. B. (1993). The Apparent Computational Complexity of Physical Systems. In Proceedings ofthe Fifteenth Annual Conference of the Cognitive Science Society. Laurence Earlbaum. Kolen, J. F. & Pollack, J. B. (In press) The Observers' Paradox: The Apparent Computational Complexity of Physical Systems. Journal of Experimental and Theoretical Artificial Intelligence. Pollack, J. B. (1991). The Induction Of Dynamical Recognizers. Machine Learning, 7.227252. Newell, A. & Simon, H. A (1976). Computer science as empirical inquiry: symbols and search. Communications of the Associationfor Computing Machinery, 19, 113-126. Rumelhart, D. E., and Zipser, D. (1986). Feature Discovery by Competitive Learning. In D. E. Rumelhart, J. L. McClelland, and the PDP Research Group, (Eds.), Parallel Distributed Processing. Volume 1. 151-193. MIT Press: Cambridge, MA Watrous, R. L. & Kuhn, G. M. (1992). Induction of Finite-State Automata Using SecondOrder Recurrent Networks. In John E. Moody, Steven J. Hanson & Richard P. Lippman, (Eds.), Advances in Neural Information Processing Systems 4. Morgan Kaufman. Zeng, Z., Goodman, R. M., Smyth, P. (1993). Learning Finite State Machines With Self-Clustering Recurrent Networks. Neural Computation, 5, 976-990 PART IV NEUROSCIENCE 

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Neural Network Definitions of Highly Predictable Protein Secondary Structure Classes Alan Lapedes Complex Systems Group (TI3) LANL, MS B213 Los Alamos N.M. 87545 and The Santa Fe Institute, Santa Fe, New Mexico Evan Steeg Department of Computer Science University of Toronto, Toronto, Canada Robert Farber Complex Systems Group (TI3) LANL, MS B213 Los Alamos N.M. 87545 Abstract We use two co-evolving neural networks to determine new classes of protein secondary structure which are significantly more predictable from local amino sequence than the conventional secondary structure classification. Accurate prediction of the conventional secondary structure classes: alpha helix, beta strand, and coil, from primary sequence has long been an important problem in computational molecular biology. Neural networks have been a popular method to attempt to predict these conventional secondary structure classes. Accuracy has been disappointingly low. The algorithm presented here uses neural networks to similtaneously examine both sequence and structure data, and to evolve new classes of secondary structure that can be predicted from sequence with significantly higher accuracy than the conventional classes. These new classes have both similarities to, and differences with the conventional alpha helix, beta strand and coil. 809 810 Lapedes, Steeg, and Farber The conventional classes of protein secondary structure, alpha helix and beta sheet, were first introduced in 1951 by Linus Pauling and Robert Corey [Pauling, 1951] on the basis of molecular modeling. Prediction of secondary structure from the amino acid sequence has long been an important problem in computational molecular biology. There have been numerous attempts to predict locally defined secondary structure classes using only a local window of sequence information. The prediction methodology ranges from a combination of statistical and rule-based methods [Chou, 1978] to neural net methods [Qian, 1988], [Maclin, 1992], [Kneller, 1990], [Stolorz, 1992]. Despite a variety of intense efforts, the accuracy of prediction of conventional secondary structure is still distressingly low. In this paper we will use neural networks to generalize the notion of protein secondary structure and to find new classes of structure that are significantly more predictable. We define protein "secondary structure" to be any classification of protein structure that can be defined using only local "windows" of structural information about the protein. Such structural information could be, e.g., the classic cI>'lf angles [Schulz, 1979] that describe the relative orientation of peptide units along the protein backbone, or any other representation of local backbone structure. A classification of local structure into "secondary structure classes", is defined to be the result of any algorithm that uses a representation of local structure as Input, and which produces discrete classification labels as Output. This is a very general definition of local secondary structure that subsumes all previous definitions. We develop classifications that are more predictable than the standard classifications [Pauling, 1951] [Kabsch, 1983] which were used in previous machine learning projects, as well as in other analyses of protein shape. We show that these new, predictable classes of secondary structure bear some relation to the conventional category of "helix", but also display significant differences. We consider the definition, and prediction from sequence, of just two classes of structure. The extension to multiple classes is not difficult, but will not be made explicit here for reasons of clarity. We won't discuss details concerning construction of a representative training set, or details of conventional neural network training algorithms, such as backpropagation. These are well studied subjects that are addressed in e.g., [Stolorz, 1992] in the context of protein secondary structure prediction. We note in passing that one can employ complicated network architectures containing many output neurons (e.g. three output neurons for predicting alpha helix, beta chain, random coil), or many hidden units etc. (c.f. [Stolorz, 1992], [Qian, 1988], [Kneller, 1990]). However, explanatory figures presented in the next section employ only one output unit per net, and no hidden units, for clarity. Neural Network Definitions of Highly Predictable Protein Secondary Structure Classes A widely adopted definition of protein secondary structure classes is due to Kabsch and Sander [Kabsch, 1983]. It has become conventional to use the Kabsch and Sander definition to define, via local structural information, three classes of secondary structure: alpha helix, beta strand, and a default class called random coil. The Kabsch and Sander alpha helix and beta strand classification captures in large part the classification first introduced by Pauling and Corey [Pauling, 1951]. Software implementing the Kabsch and Sander definitions, which take a local window of structural information as Input, and produce the Kabsch and Sander secondary structure classification of the window as Output, is widely available. The key ideas of this paper are contained in Fig. (1). F = I,-C-o-r :e- Ia-h-'o-n-( (-)-l -P)- O-(-P-) ----. [ _ J _P_~ _ _L _ 'R_~ Left Net Maps AA sequence to "secondary structure" . Right Net Maps <l>.\f' to "secondary structure". In this figure the Kabsch and Sander rules are represented by a second neural network. The Kabsch and Sander rules are just an Input/Output mapping (from a local window of structure to a classification of that structure) and may in principle be replaced with an equivalent neural net representing the same Input/Output mapping. We explicitly demonstrated that a simple neural net is capable of representing rules of the complexity of the Kabsch and Sander rules by training a network to perform the same structure classification as the Kabsch and Sander rules, and obtained high accuracy. The representation of the structure data in the right-hand network uses cI>\i' angles. The right-hand net sees a window of cI>\i' angles corresponding to the window of amino acids in the left-hand network. Problems due to the angular periodicity of the cI>\i' angles (i.e ., 360 degrees and 0 degrees are different numbers, but represent the same angle) are eliminated by utilizing both the sin and cos of each angle. 811 812 Lapedes, Steeg, and Farber The representation of the amino acids in the left-hand network is the usual unary representation employing twenty bits per amino acid . Results quoted in this paper do not use a special twenty-first bit to represent positions in a window extending past the ends of a protein. Note that the right-hand neural network could implement extremely general definitions of secondary structure by changing the weights. We next show how to change the weights in a fashion so that new classifications of secondary structure are derived under the important restriction that they be predictable from amino acid sequence. In other words, we require that the synaptic weights be chosen so that the output of the left-hand network and the output of the right-hand network agree for each sequence-structure pair that is input to the two networks. To achieve this, both networks are trained simultaneously, starting from random initial weights in each net, under the sole constraint that the outputs of the two networks agree for each pattern in the training set. The mathematical implementation of this constraint is described in various versions below. This procedure is a general, effective method of evolving predictable secondary structure classifications of experimental data. The goal of this research is to use two mutually self-supervised networks to define new classes of protein secondary structure which are more predictable from sequence than the standard classes of alpha helix, beta sheet or coil. 3 CONSTRAINING THE TWO NETS TO AGREE One way to impose agreement between the outputs of the two networks is to require that they covary when viewed as a stream of real numbers. Note that it is not sufficient to merely require that the outputs of the left-hand and right-hand nets agree by, e.g., minimizing the following objective function (1) P Here, LeftO(p) and RightO(p) represent the outputs of the left-hand and righthand networks, respectively, for the pth pair of input windows: (sequence window -left net) and (structure window -right net). It is necessary to avoid the trivial minimum of E obtained where the weights and thresholds are set so that each net presents a constant Output regardless of the input data. This is easily accomplished in Eqn (1) by merely setting all the weights and thresholds to 0.0. Demanding that the outputs vary, or more explicitly co-vary, is a viable solution to avoiding trivial local minima. Therefore, one can maximize the correlation, P, between the left-hand and right-hand network outputs. The standard correlation measure between two objects, LeftO(p) and RightO(p) is: p = '2:)LeftO(p) - LeftO)(RightO(p) - RightO) (3) P where LeftO denotes the mean of the left net's outputs over the training set, and respectively for the right net. p is zero if there is no variation, and is maximized Neural Network Definitions of Highly Predictable Protein Secondary Structure Classes if there is simultaneously both individual variation and joint agreement. In our situation it is equally desirable to have the networks maximally anti-correlated as it is for them to be correlated. (Whether the networks choose correlation, or anti-correlation, is evident from the behavior on the training set). Hence the minimization of E _p2 would ensure that the outputs are maximally correlated (or anti-correlated). While this work was in progress we received a preprint by Schmidhuber [Schmidhuber, 1992] who essentially implemented Eqn. (1) with an additional variance term (in a totally different context). Our results using this measure seem quite susceptible to local minima and we prefer alternative measures to enforce agreement. = One alternative to enforce agreement, since one ultimately measures predictive performance on the basis of the Mathews correlation coefficient (see, e.g., [Stolorz, 1992]), is to simultaneously train the two networks to maximize this measure . The Mathews coefficient, Gi, for the ith state is defined as: c. _ I - Pini - UiOi [(ni + ui)(ni + Oi)(Pi + Ud(pi + Oi?)1/2 where Pi is the number of examples where the left-hand net and right-hand net both predict class i, ni is the number of examples where neither net predicts ;, Ui counts the examples where the left net predicts i and the right net does not, and 0i counts the reverse. Minimizing E = -Gi 2 optimizes Gi. Other training measures forcing agreement of the left and right networks may be used. Particularly suitable for the situation of many outputs (i .e., more than twoclass discrimination) is "mutual information". Use of mutual information in this context is related to the IMAX algorithm for unsupervised detection of regularities across spatial or temporal data [Becker, 1992]. The mutual information is defined as M " Pi; log ....!2L. p" = 'LJ .. Pi .P.; (4) I,J where Pij is the joint probability of occurrence of the states of the left and right networks. (In previous work [Stolorz, 1992] we showed how Pij may be defined in terms of neural networks) . Minimizing E -M maximizes M. While M has many desirable properties as a measure of agreement between two or more variables [Stolorz, 1992] [Farber, 1992] [Lapedes, 1989] [Korber, 1993], our preliminary simulations show that maximizing M is often prone to poor local maxima. = Finally, an alternative to using mutual information for multi-class, as opposed to dichotomous classification, is the Pearson correlation coefficient, X 2 ? This is defined in terms of Pi; as (5) Our simulations indicate that X 2 , Gi and p are all less susceptible to local minima 813 814 Lapedes, Steeg, and Farber than M. However, these other objective functions suffer the defect that predictability is emphasized at the expense of utility. In other words, they can be maximal for the peculiar situation where a structural class is defined that occurs very rarely in the data, but when it occurs, it is predicted perfectly by the other network. The utility of this classification is therefore degraded by the fact that the predictable class only occurs rarely. Fortunately, this effect did not cause difficulties in the simulations we performed. Our best results to date have been obtained using the Mathews objective function (see Results). 4 RESULTS The database we used consisted of 105 proteins and is identical to that used in previous investigations [Kneller, 1990] [Stolorz, 1992]. The proteins were divided into two groups: a set of 91 "training" proteins, and a distinct "prediction" set of 14 proteins. The resulting database is similar to the database used by Qian & Sejnowski [Qian, 1988] in their neural network studies of conventional secondary structure prediction. When comparison to predictability of conventional secondary structure classes was needed, we defined the conventional alpha, beta and coil states using the Kabsch and Sander definitions and therefore these states are identical to those used in previous work [Kneller, 1990] [Stolorz, 1992]. A window size of 13 residues resulted in 16028 train set examples and 3005 predict set examples. Effects of other windows sizes have not yet been extensively tested. All results, including conventional backpropagation training of Kabsch and Sander classifications, as well as two-net training of our new secondary structure classifications, did not employ an extra symbol denoting positions in a window that extended past the ends of a protein. Use of such a symbol could further increase accuracy. We found that random initial conditions are necessary for the development of interesting new classes. However, random initial conditions also suffer to a certain extent from local minima. The mutual information function, in particular, often gets trapped quickly in uninteresting local minima when evolved from random initial conditions. More success was obtained with the other objective functions discussed above. We have not exhaustively investigated strategies to avoid local minima, and usually just chose new initial conditions if an uninteresting local minimum was encountered. Results were best for two class discrimination using the Mathews objective function and a layer of five hidden units in each net. If one assigns the name "Xclass" to the newly defined structural class, then the Mathews coefficient on the prediction set for the Xclass dichotomy is -0.425 . The Mathews coefficient on the train set for the Xclass dichotomy is -0.508. For comparison, the Mathews coefficient on the same predict set data for dichotomization (using standard backpropagation training with no hidden units), into the standard secondary structure classes Alpha/NotAlpha, Beta/NotBeta, and CoilJNotCoil is 0.33, 0.26, and 0.39, respectively. Adding hidden units gives negligible accuracy increase in predicting the conventional classes, but is important for improved prediction of the new classes. The negative sign of the two-net result indicates anti-correlation - a feature allowed by our objective function. The sign of the correlation is easily assessed on the train set and then can be trivially compensated for in prediction. Neural Network Definitions of Highly Predictable Protein Secondary Structure Classes A natural question to ask is whether the new classes are simply related to the more conventional classes of alpha helix, beta, coil. A simple answer is to compute the Mathews correlation coefficient of the new secondary structure classes with each of the three Kabsch and Sander classes, for those examples in which the sequence network agreed with the structure network's classification. The correlation with Kabsch and Sander's alpha helix is highest: a Mathews coefficient of 0.248 was obtained on the train set, while a Mathews coefficient of 0.247 was obtained on the predict set. There is therefore a significant degree of correlation with the conventional classification of alpha helix, but significant differences exist as well. The new classes are a mixture of the conventional classes, and are not solely dominated by either alpha, beta or coil. Conventional alpha-helices comprise roughly 25% of the data (for both train and predict sets), while the new Xclass comprises 10%. It is quite interesting that an evolution of secondary structure classifications starting from random initial conditions, and hence completely unbiased towards the conventional classifications, results in a classification that has significant relationship to conventional helices but is more predictable from amino acid sequence than conventional helices. Graphical analysis (not shown here) of the new Xclass shows that the Xclass that is most closely related to helix typically extends the definition of helix past the standard boundaries of an alpha-helix. 5 CONCLUSIONS A primary goal of this investigation is to evolve highly predictable secondary structure classes. Ultimately, such classes could be used, e.g., to provide constraints on tertiary structure calculations. Further work remains to derive even more predictable classes and to analyze their physical meaning. However, it is now clear that the use of two, co-evolving, adaptive networks defines a novel and useful machine learning paradigm that allows the evolution of new definitions of secondary structure that are significantly more predictable from primary amino acid sequence than the conventional definitions. Related work is that of [Hunter, 1992], [Hunter, 1992], [Zhang, 1992], [Zhang, 1993] in which clustering either only in sequence space, or only in structure space, is attempted. However, no condition on the compatibility of the clustering is required, so new classes of structure are not guaranteed to be predictable from sequence. Finally, we note that the methods described here might be usefully applied to other cognitive/perceptual or engineering tasks in which correlation of two or more different representations of the same data is required. In this regard the relation of our work to that of independent work of Becker [Becker, 1992], and of Schmidhuber [Schmidhuber, 1992], should be noted. Acknowledgements We are grateful for useful discussions with Geoff Hinton, Sue Becker, and Joe Bryngelson. Sue Becker's contribution of software that was used in the early stages of this project is much appreciated. The research of Alan Lapedes and Robert Farber was supported in part by the U.S. Department of Energy. The authors would like to acknowledge the hospitality of the Santa Fe Institute, where much of this work 815 816 Lapedes, Steeg, and Farber was performed. References [Becker, 1992] [Becker, 1992] [Chou, 1978] [Farber, 1992] [Hunter, 1992] [Hunter, 1992] [Kabsch, 1983] [Kneller, 1990] [Korber, 1993] S. Becker. An Information-theoretic Unsupervised Learning Algorithm for Neural Networks. PhD thesis, University of Toronto (1992) S. Becker, G. Hinton, Nature 355, 161-163 (1992) P. Chou, G. Fasman Adv. Enzymol. 47, 45 (1978) R. Farber, A. Lapedes J. Mol. Bioi. 226 , 471, (1992) 1. Hunter, N. Harris, D. States Proceedings of the Ninth International Conference on Machine Learning, San Mateo, California, Morgan Kaufmann Associates (1992) L. Hunter, D. States, IEEE Expert, 7(4) 67-75 (1992) W. Kabsch, C. Sander Biopolymers 22, 2577 (1983) D. Kneller, F. Cohen, R. Langridge J. Mol. Bioi. 214, 171 (1990) B. Korber, R. Farber, D. Wolpert, A. Lapedes P.N.A.S. - in press (1993) [Lapedes, 1989] A. Lapedes, C.Barnes, C. Burks, R.Farber, K. Sirotkin in Computers and DNA editors: G.Bell, T. Marr, (1989) R. Maclin, J. W. Shavlik Proceedings of the Tenth National [Maclin, 1992] Conference on Artificial Intelligence, San Jose, California, Morgan Kauffman Associates (1992) L. Pauling, R. Corey Proc. Nat. Acad. Sci. 37,205 (1951) [Pauling, 1951] N. Qian, T. Sejnowski J. Mol. Bioi. 202, 865 (1988) [Qian, 1988] [Schmidhuber, 1992] J. Schmidhuber Discovering Predictable Classifications, Technical report CU-CS-626-92, Department of Computer Science, University of Colorado (1992) G. Schulz, R. Schirmer Principles of Protein Structure [Schulz, 1979] Springer Verlag, New York, (1979) P. Stolorz, A. Lapedes, X. Yuan J. Mol. Bioi. 225, 363 (1992) [Stolorz, 1992] X. Zhang, D. Waltz in Artificial Intelligence and Molecular [Zhang, 1992] Biology, editor: L. Hunter, AAAI Press (MIT Press) (1992) X. Zhang, J. Fetrow, W. Rennie, D. Waltz, G. Berg, in Pro[Zhang, 1993] ceedings: First International Conference on Intelligent Systems For Molecular Biology, p. 438, editors: L. Hunter, D. Searls, J. Shavlik, AAAI Press, Menlo Park, CA. (1993) 

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Tonal Music as a Componential Code: Learning Temporal Relationships Between and Within Pitch and Timing Components Catherine Stevens Department of Psychology University of Queensland QLD 4072 Australia kates@psych.psy.uq.oz.au Janet Wiles Depts of Psychology & Computer Science University of Queensland QLD 4072 Australia janetw@cs.uq.oz.au Abstract This study explores the extent to which a network that learns the temporal relationships within and between the component features of Western tonal music can account for music theoretic and psychological phenomena such as the tonal hierarchy and rhythmic expectancies. Predicted and generated sequences were recorded as the representation of a 153-note waltz melody was learnt by a predictive, recurrent network. The network learned transitions and relations between and within pitch and timing components: accent and duration values interacted in the development of rhythmic and metric structures and, with training, the network developed chordal expectancies in response to the activation of individual tones. Analysis of the hidden unit representation revealed that musical sequences are represented as transitions between states in hidden unit space. 1 INTRODUCTION The fundamental features of music, derivable from frequency, time and amplitude dimensions of the physical signal, can be described in terms of two systems - pitch and timing. The two systems are frequently disjoined and modeled independently of one another (e.g. Bharucha & Todd, 1989; Rosenthal, 1992). However, psychological evidence suggests that pitch and timing factors interact (Jones, 1992; Monahan, Kendall 1085 1086 Stevens and Wiles & Carterette, 1987). The pitch and timing components can be further divided into tone, octave, duration and accent which can be regarded as a quasi-componential code. The important features of a componential code are that each component feature can be viewed as systematic in its own right (Fodor & Pylyshyn, 1988). The significance of componential codes for learning devices lies in the productivity of the system - a small (polynomial) number of training examples can generalise to an exponential test set (Brousse & Smolensky, 1989; Phillips & Wiles, 1993). We call music a quasicomponential, code as there are significant interactions between, as well as within, the component features (we adopt this term from its use in describing other cognitive phenomena, such as reading, Plaut & McClelland, 1993). Connectionist models have been developed to investigate various aspects of musical behaviour including composition (Hild, Feulner & Menzel, 1992), performance (Sayegh, 1989), and perception (Bharucha & Todd, 1989). The models have had success in generating novel sequences (Mozer, 1991), or developing properties characteristic of a listener, such as tonal expectancies (Todd, 1988), or reflecting properties characteristic of musical structure, such as hierarchical organisation of notes, chords and keys (Bharucha, 1992). Clearly, the models have been designed with a specific application in mind and, although some attention has been given to the representation of musical information (e.g. Mozer, 1991; Bharucha, 1992), the models rarely explain the way in which musical representations are constructed and learned. These models typically process notes which vary in pitch but are of constant duration and, as music is inherently temporal, the temporal properties of music must be reflected in both representation and processing. There is an assumption implicit in cognitive modeling in the in/ormation processing framework that the representations used in any cognitive process are specified a priori (often assumed to be the output of a perceptual process). In the neural network framework, this view of representation has been challenged by the specification of a dual mechanism which is capable of learning representations and the processes which act upon them. Since the systematic properties of music are inherent in Western tonal music as an environment, they must also be reflected in its representation in, and processes of, a cognitive system. Neural networks provide a mechanism for learning such representations and processes, particularly with respect to temporal effects. In this paper we study how representations can be learned in the domain of Western tonal music. Specifically, we use a recurrent network trained on a musical prediction task to construct representations of context in a musical sequence (a well-known waltz), and then test the extent to which the learned representation can account for the classic phenomena of music cognition. We see the representation construction as one aspect of learning and memory in music cognition, and anticipate that additional mechanisms of music cognition would involve memory processes that utilise these representations (e.g. Stevens & Latimer, 1992), although they are beyond the scope of the present paper. For example, Mozer (1992) discusses the development of higher-order, global representations at the level of relations between phrases in musical patterns. By contrast, the present study focusses on the development of representations at the level of relations between individual musical events. We expect that the additional mechanisms would, in part, develop from the behavioural aspects of music cognition that are made explicit at this early, representation construction stage. Tonal Music as a Componential Code 2 METHOD & RESULTS A simple recurrent network (Elman. 1989), consisting of 25 input and output units, and 20 hidden and context units. was trained according to Elman's prediction paradigm and used Backprop Through Time (BPlT) for one time step. The training data comprised the nrst 2 sections of The Blue Danube by Johann Strauss wherein each training pattern represented one event, or note. in the piece, coded as components of tone (12 values), rest (1 value), octave (3 values), duration (6 values) and accent (3 values). In the early stages of training, the network learned to predict the prior probability of events (see Figure 1). This type of information could be encoded in the bias of the output units alone. as it is independent of temporal changes in the input patterns. After further training, the network learned to modify its predictions based on the input event (purely feed forward information) and, later still, on the context in which the event occurred (see Figure 2). Note that an important aspect of this type of recurrent network is that the representation of context is created by the network itself (Elman, 1989) and is not specified a priori by the network designer or the environment (Jordan, 1986). In this way, the context can encode the information in the envirorunent which is relevant to the prediction task. Consequently, the network could, in principle, be adapted to other styles of music without modification of the design, or input and output representations. a. Output vector __ ~--'"'-_---'"~_~~_----' n, L- _JLl ~ _~ 1-' _ _ b. Target histogram A A# B C C# D D# E F F# G G# 4 5 6 1 2 3 4 6 8 S W V A tones octaves durations accent rest Figure 1: Comparison of output vector for the first event at Epoch 4 (a) and a histogram of the target vector averaged over all events (b). The upper grapb is the predicted output of Event 1 at Epoch 4. The lower grapb is a histogram of all the events in the piece, created by averaging all the target vectors. The comparison shows that the net learned initially to predict the mean target before learning the variations specific to each event. 1087 Stevens and Wiles _ _ _ _ _ _ _ _ _ _ _'"'--- __________ ~ 1 _________________ ___________----JI"L- 11..- _ ---.J1 ~ -11_ _ __ --11- --.J1 ~ _ _ _ _ _ _ _--J,...'-___ ~ 1088 -r-I1 --11,....._ _ __ ---I'L-.n ---I1--....,. _______ ---'~~._J1 ________ ______ --...lL-.-Jl_______ __ .....___________ --11..- --..11'--_ __ ~n~ ~n =E~~~h~.~ _____~_____~ ---------------~ ---------------'"'---------------------~ ---------------~~ ~ ___.11 _______ ____ ----~-----~-----''"'--~ ~n__ _ _ _ _ _ _ _ _ _ _ _ _ _ _~n__ -----------~~ -------------~~ .. T..:..;~=-=_ _ _ _~n,--____ _---n --I"\-.J"1 I"L..- _ -..I1_ _ __ ~ --..11_ _ __ ---J1 -1l'--_ __ --..J'l --1'1'--_ __ -..I1'--_ __ __ J'~_n --n --11'---____ ~ --"''----- ---.J1 -11'--_ __ ~ ~n'_ 1 3 2 ---11_ _ _ __ ---l1 ----11'--_ __ ---.J1 -11'--_ __ ---l1 ~ ? 3 2 ? ~ -..--J"L..r, 5 8 7 6 5 --.fl'--____ ---1L----Il -Il_______ _ _ _ _ _ _ _ _ _ _fL.- ---1L- - - - 1 L - _ _ _ _ _ _ _ _ _ _fL.----I"L--Il_ _ __ _______ _ _ __ ---I"L- --..11_______ _ _--In'___________ ~"'L- -11_______ _ _ _ _ _ _ _ _ _ _11- -11_ _ -1L- _ ---fl~_~_ -...J1'-_ _ __ _______--lnl.-____ ---.J1 -11______ __________n.---. il- _ -.Jl~ ~ =E~~=h~2~ ______~____~'"'___ _ _ _ _ _ _ _ _ _ _ _ _ _ _~n__ rL- _ -.11___ _ 8 7 6 1 M-.-.!'L- .. -1L- .. ~ -...1'l- .. .. ~ .. .. -..I1- .. --J'L..- .. ~ 8 7 6 5 ? 3 2 1 8 7 6 5 ? 3 2 1 A M B C CI 0 Of E F F. Gat. 5 6 1 2 3 ? 6 8 S W V R event tones octaves durations accent rest Figure 2: Evolution of the flrst eight events (predicted). The first block (targets) shows the correct sequence of events for the four components. In the second block (Epoch 2), the net is beginning to predict activation of strong and weak accents. In the third block (Epoch 4), the transition from one octave to another is evident. By the fourth block (Epoch 64), all four components are substantially correct. The pattern of activation across the output vector can be interpreted as a statistical description of each musical event. In psychological terms, the pattern of activation reflects the harmonic or chordal expectancies induced by each note (Bharucha, 1987) and characteristic of the tonal hierarchy (Krumhansl & Shepard, 1979). 3.1 NETWORK PERFORMANCE The performance of the network as it learned to master The Blue Danube was recorded at log steps up to 4096 epochs. One recording comprised the output predicted by the network as the correct event was recycled as input to the network (similar to a teacherforcing paradigm). The second recording was generated by feeding the best guess of the Tonal Music as a Componential Code output back as input. The accent - the lilt of the waltz - was incorporated very early in the training. Despite numerous errors in the individual events, the sequences were clearly identifiable as phrases from The Blue Danube and, for the most part, errors in the tone component were consistent with the tonality of the piece. The errors are of psychological importance and the overall performance indicated that the network learnt the typical features of The Blue Danube and the waltz genre. 3.2 INTERACTIONS BETWEEN ACCENT & DURATION Western tonal music is characterised by regularities in pitch and timing components. For example, the occurrence of particular tones and durations in a single composition is structured and regular given that only a limited number of the possible combinations occur. Therefore, one way to gauge performance of the network is to compare the regularities extracted and represented in the model with the statistical properties of components in the training composition. The expected frequencies of accent-duration pairs, such as a quarter-note coupled with a strong accent, were compared with the actual frequency of occurrence in the composition: the accent and duration couplings with the highest expected frequencies were strong quarter-note (35.7), strong half-note (10.2), weak quarter-note (59.0), and weak half-note (16.9). Scrutiny of the predicted outputs of the network over the time course of learning showed that during the initial training epochs there was a strong bias toward the event with the highest expected frequency - weak quarter-note. The output of this accent-duration combination by the network decreased gradually. Prediction of a strong quarter-note by the network reached a value close to the expected frequency of 35.7 by Epoch 2 (33) and then decreased gradually and approximated the actual composition frequency of 19 at Epoch 64. Similarly, by Epoch 64, output of the most common accent and duration pairs was very close to the actual frequency of occurrence of those pairs in the composition. 3. 3 ANALYSIS OF HIDDEN UNIT ~EPRESENT A TIONS An analysis of hidden unit space most often reveals structures such as regions, hierarchies and intersecting regions (Wiles & Bloesch, 1992). In the present network, four subspaces would be expected (tone, octave, duration, accent), with events lying at the intersection of these suo-spaces. A two-dimensional projection of hidden unit space produced from a canonical discriminant analysis (CDA) of duration-accent pairs reveals these divisions (see Figure 3). In essence, there is considerable structure in the way events are represented into clusters of regions with events located at the intersection of these regions. In Figure 3 the groups used in the CDA relate to the output values which are observable groups. An additional CDA using groups based on position in bar showed that the hidden unit space is structured around inferred variables as well as observable ones. 1089 1090 Stevens and Wiles 2ft 21 2ft ? I, .. .. .. ... 2w .. ~ 1 J I I I I Figure 3: Two-dimensional projection of hidden unit space generated by canonical discriminant analysis of duration-accent pairs. Each note in the composition is depicted as a labelled point, and the flrst and third canonical components are represented along the abscissa and ordinate. respectively. The first canonical component divides strong from weak accents (denoted s and w) . In the strong (s) accent region of the third canonical component, quarter- and half-notes are separated (denoted by 2 and 4, respectively), and the remaining right area separates rests, weak quarter- and weak half-notes. The superimposed line shows the first five notes of the opening two bars of the composition as a trajectory through hidden unit space: there is movement along the flrst canonical component depending on accent (s or w) and the second bar starts in the half-note region. 4 DISCUSSION & CONCLUSIONS The focus of this study has been the extraction of information from the envirorunentthe temporal stream of events representing The Blue Danube - and its incorporation into the static parameters of the weights and biases in the network. Evidence for the stages at which information from the environment is incorporated into the network representation is seen in the predicted output vectors (described above and illustrated in Figure 2). Different musical styles contain different kinds of infonnation in the components. For example. the accent and duration components of a waltz take complementary roles in regulating the rhythm. From the durations of events alone, the position of a note in a Tonal Music as a Componential Code bar could be predicted without error. However, if the performer or listener made a single error of duration, a rhythm system based on durations alone could not recover. By contrast, accent is not a completely reliable predictor of the bar structure, but it is effective for recovery from rhythmic errors. The interaction between these two timing components provides an efficient error correction representation for the rhythmic aspect of the system. Other musical styles are likely to have similar regulatory functions performed by different components. For example, consider the use of ornaments, such as trills and mordents, in Baroque harpsichord music which, in the absence of variations in dynamics, help to signify the beat and metric structure. Alternatively, consider the interaction between pitch and timing components with the placement of harmonicallyimportant tones at accented positions in a bar (Jones, 1992). The network described here has learned transitions and relations between and within pitch and timing musical components and not simply the components per se. The interaction between accent and duration components, for example, demonstrates the manifestation of a componential code in Western tonal music. Patterns of activation across the output vector represented statistical regularities or probabilities characteristic of the composition. Notably, the representation created by the network is reminiscent of the tonal hierarchy which reflects the regularities of tonal music and has been shown to be responsible for a number of performance and memory effects observed in both musically trained and untrained listeners (Krumhansl, 1990). The distribution of activity across the tone output units can also be interpreted as chordal or harmonic expectancies akin to those observed in human behaviour by Bharucha & Stoeckig (1986). The hidden unit activations represent the rules or grammar of the musical environment; an interesting property of the simple recurrent network is that a familiar sequence can be generated by the trained network from the hidden unit activations alone. Moreover, the intersecting regions in hidden unit space represent composite states and the musical sequence is represented by transitions between states. Finally, the course of learning in the network shows an increasing specificity of predicted events to the changing context: during the early stages of training, the default output or bias of the network is towards the average pattern of activation across the entire composition but, over time, predictions are refined and become attuned to the pattern of events in particular contexts. Acknowledgements This research was supported by an Australian Research Council Postdoctoral Fellowship granted to the first author and equipment funds to both authors from the Departments of Psychology and Computer Science, University of Queensland. The authors wish to thank Michael Mozer for providing the musical database which was adapted and used in the present simulation. The modification to McClelland & Rumelhart's (1989) bp program was developed by Paul Bakker, Department of Computer Science, University of Queensland. The comments and suggestions made by members of the Computer Science/Psychology Connectionist Research Group at the University of Queensland are acknowledged. References Bharucha. J. J. (1987). Music cognition and perceptual facilitation: A connectionist framework. Music Perception, 5, 1-30. Bharucha, J. J. (1992). Tonality and learnability. In M. R. Jones & S. Holleran (Eds.), Cognitive bases of musical communication, pp. 213-223. WaShington: American Psychological Association . 1091 1092 Stevens and Wiles Bharucha, J. J., & Stoeckig, K. (1986). Reaction time and musical expectancy: Priming of chords. Journal of Experimental Psychology: Human Perception & Peiformance, 12, 403- 410. Bharucha, J., & Todd, P. M. (1989). Modeling the perception of tonal structure with neural nets. Computer Music Journal, 13, 44-53. Brousse, 0., & Smolensky, P. (1989). Virtual memories and massive generalization in connectionist combinatorial learning. In Proceedings of the 11th Annual Conference of the Cognitive Science Society, pp. 380-387. Hillsdale, NJ: Lawrence Erlbaum. Elman, J . L. (1989). Structured representations and connectionist models. (CRL Tech. Rep. No. 8901). San Diego: University of California, Center for Research in Language. Fodor, J. A., & Pylyshyn, Z. W. (1988). Connectionism and cognitive architecture: A critical analysis. Cognition, 28, 3-71. Hild, H., Feulner, J ., & Menzel, W . (1992). HARMONET: A neural net for harmonizing chorales in the style of J. S. Bach. In J. E. Moody, S. J. Hanson & R. Lippmann (Eds.), Advances in Neural Information Processing Systems 4, pp. 267-274. San Mateo, CA.: Morgan Kaufmann. Jones, M. R. (1992). Attending to musical events. In M. R. Jones & S. Holleran (Eds.), Cognitive bases of musical communication, pp. 91-110. Washington: American Psychological Association. Jordan, M. I. (1986). Serial order: A parallel distributed processing approach (Tech. Rep. No. 8604). San Diego: University of California, Institute for Cognitive Science . Krumhansl, C., & Shepard, R. N. (1979) . Quantification of the hierarchy of tonal functions within a diatonic context. Journal of Experimental Psychology: Human Perception & Performance, 5, 579-594. McClelland, 1. L., & Rumelhart, D. E. (1989). Explorations in parallel distributed processing: A handbook of models, programs and exercises. Cambridge, Mass.: MIT Press. Monahan, C. B., Kendall, R. A ., & Carterette, E. C. (1987). The effect of melodic and temporal contour on recognition memory for pitch change. Perception & Psychophysics, 41, 576-600. Mozer, M. C. (1991). Connectionist music composition based on melodic, stylistic, and psychophysical constraints. In P. M. Todd & D. G. Loy (Eds.), Music and connectionism, pp. 195-211. Cambridge, Mass.: MIT Press. Mozer, M. C. (1992). Induction of multiscale temporal structure. In J. E. Moody, S. J. Hanson, & R. P. Lippmann (Eds.), Advances in Neural Information Processing Systems 4, pp. 275-282. San Mateo, CA.: Morgan Kaufmann. Phillips, S., & Wiles, J. (1993). Exponential generalizations from a polynomial number of examples in a combinatorial domain . Submitted to IlCNN, Japan, 1993. Plaut, D. c., & McClelland, J. L. (1993). Generalization with componential attractors: Word and nonword reading in an attractor network. To appear in Proceedings of the 15th Annual Conference of the Cognitive Science Society. Hillsdale, NJ: Erlbaum. Rosenthal, D. (1992). Emulation of human rhythm perception. Computer Music Journal, 16, 64-76. Sayegh, S. (1989). Fingering for string instruments with the optimum path paradigm. Computer Music Journal, 13, 76-84. Stevens, c., & Latimer, C. (1991) . Judgments of complexity and pleasingness in music: The effect of structure, repetition , and training. Australian Journal of Psychology, 43, 17-22. Stevens, C ., & Latimer, C . (1992). A comparison of connectionist models of music recognition and human performance. Minds and Machines, 2, 379-400. Todd, P. M. (1988). A sequential network design for musical applications. In D. Touretzky, G . Hinton & T. Sejnowski (Eds.), Proceedings of the 1988 Connectionist Models Summer School, pp. 76-84. Menlo Park, CA: Morgan Kaufmann. Wiles, J., & Bloesch, A. (1992). Operators and curried functions: Training and analysis of simple recurrent networks. In J. E. Moody, S. J. Hanson, & R . P. Lippmann (Eds.), Advances in Neural Information Processing Systems 4. San Mateo, CA: Morgan Kaufmann. 

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750 A DYNAMICAL APPROACH TO TEMPORAL PATTERN PROCESSING W. Scott Stornetta Stanford University, Physics Department, Stanford, Ca., 94305 Tad Hogg and B. A. Huberman Xerox Palo Alto Research Center, Palo Alto, Ca. 94304 ABSTRACT Recognizing patterns with temporal context is important for such tasks as speech recognition, motion detection and signature verification. We propose an architecture in which time serves as its own representation, and temporal context is encoded in the state of the nodes. We contrast this with the approach of replicating portions of the architecture to represent time. As one example of these ideas, we demonstrate an architecture with capacitive inputs serving as temporal feature detectors in an otherwise standard back propagation model. Experiments involving motion detection and word discrimination serve to illustrate novel features of the system. Finally, we discuss possible extensions of the architecture. INTRODUCTION Recent interest in connectionist, or "neural" networks has emphasized their ability to store, retrieve and process patterns1,2. For most applications, the patterns to be processed are static in the sense that they lack temporal context. Another important class consists of those problems that require the processing of temporal patterns. In these the information to be learned or processed is not a particular pattern but a sequence of patterns. Such problems include speech processing, signature verification, motion detection, and predictive signal processin,r-8. More precisely, temporal pattern processing means that the desired output depends not only on the current input but also on those preceding or following it as well. This implies that two identical inputs at different time steps might yield different desired outputs depending on what patterns precede or follow them. There is another feature characteristic of much temporal pattern processing. Here an entire sequence of patterns is recognized as a single distinct category, ? American Institute of Physics 1988 751 generating a single output. A typical example of this would be the need to recognize words from a rapidly sampled acoustic signal. One should respond only once to the appearance of each word, even though the word consists of many samples. Thus, each input may not produce an output. With these features in mind, there are at least three additional issues which networks that process temporal patterns must address, above and beyond those that work with static patterns. The first is how to represent temporal context in the state of the network. The second is how to train at intermediate time steps before a temporal pattern is complete. The third issue is how to interpret the outputs during recognition, that is, how to tell when the sequence has been completed. Solutions to each of these issues require the construction of appropriate input and output representations. This paper is an attempt to address these issues, particularly the issue of representing temporal context in the state of the machine . We note in passing that the recognition of temporal sequences is distinct from the related problem of generating a sequence, given its first few members 9 .l O?11 . TEMPORAL CLASSIFICATION With some exceptions 10.12 , in most previous work on temporal problems the systems record the temporal pattern by replicating part of the architecture for each time step. In some instances input nodes and their associated links are replicated 3,4. In other cases only the weights or links are replicated, once for each of several time delays 7,8. In either case, this amounts to mapping the temporal pattern into a spatial one of much higher dimension before processing. These systems have generated significant and encouraging results. However, these approaches also have inherent drawbacks. First, by replicating portions of the architecture for each time step the amount of redundant computation is significantly increased. This problem becomes extreme when the signal is sampled very frequently4. :-.l' ext, by re lying on replications of the architecture for each time step, the system is quite inflexible to variations in the rate at which the data is presented or size of the temporal window. Any variability in the rate of the input signal can generate an input pattern which bears little or no resemblance to the trained pattern. Such variability is an important issue, for example, in speech recognition . Moreover, having a temporal window of any fixed length makes it manifestly impossible to detect contextual effects on time scales longer than the window size. An additional difficulty is that a misaligned signal, in its spatial representation, may have very little resemblance to the correctly aligned training signal. That is, these systems typically suffer from not being translationally invariant in time. ~etworks based on relaxation to equilibrium 11,13,14 also have difficulties for use with temporal problems. Such an approach removes any dependence on initial 752 conditions and hence is difficult to reconcile directly with temporal problems, which by their nature depend on inputs from earlier times. Also, if a temporal problem is to be handled in terms of relaxation to equilibrium, the equilibrium points themselves must be changing in time. A NON?REPLICATED, DYNAMIC ARCHITECTURE We believe that many of the difficulties mentioned above are tied to the attempt to map an inherently dynamical problem into a static problem of higher dimension. As an alternative, we propose to represent the history of the inputs in the state of the nodes of a system, rather than by adding additional units. Such an approach to capturing temporal context shows some very immediate advantages over the systems mentioned above . F'irst, it requires no replication of units for each distinct time step. Second, it does not fix in the architecture itself the window for temporal context or the presentation rate. These advantages are a direct result of the decision to let time serve as its own representation for temporal sequences, rather than creating additional spatial dimensions to represent time. In addition to providing a solution to the above problems, this system lends itself naturally to interpretation as an evolving dynamical system. Our approach allows one to think of the process of mapping an evolving input into a discrete sequence of outputs (such as mapping continuous speech input into a sequence of words) as a dynamical system moving from one attractor to another 15 . As a preliminary example of the application of these ideas, we introduce a system that captures the temporal context of input patterns without replicating units for each time step. We modify the conventional back propagation algorithm by making the input units capacitive. In contrast to the conventional architecture in which the input nodes are used simply to distribute the signal to the next layer, our system performs an additional computation. Specifically, let Xi be the value computed by an input node at time ti ' and Ii be the input signal to this node at the same time. Then the node computes successive values according to (1) where a is an input amplitude and d is a decay rate. Thus, the result computed by an input unit is the sum of the current input value multiplied by a, plus a fractional part, d, of the previously computed value of the input unit. In the absence of further input, this produces an exponential decay in the activation of the input nodes. The value for d is chosen so that this decay reaches lie of its original value in a time t characteristic of the time scale for the particular problem, i.e., d=e'tr, where r is the presentation rate. The value for a is chosen to produce a specified maximum value for X, given by 753 al ma /(1-d) . We note that Eq. (1) is equivalent to having a non-modifiable recurrent link with weight d on the input nodes, as illustrated in Fig. l. o 0 Fig. 1: Schematic architecture with capacitive inputs. The input nodes compute values according to Eq. (1). Hidden and output units are identical to standard back propagation nets. The processing which takes place at the input node can also be thought of in terms of an infinite impulse response (IIR) digital filter. The infinite impulse response of the filter allows input from the arbitrarily distant past to influence the current output of the filter, in contrast to methods which employ fixed windows, which can be viewed in terms of finite impulse response (FIR) filters. The capacitive node of Fig. 1 is equivalent to pre-processing the signal with a filter with transfer function a/(1-dz? 1) . This system has the unique feature that a simple transformation of the parameters a and d allows it to respond in a near-optimal way to a signal which differs from the training signal in its rate. Consider a system initially trained at rate r with decay rate d and amplitude a. To make use of these weights for a different presentation rate, r~ one simply adjusts the values a 'and d'according to d' = d r/r ' (2) 1 - d' a' = a ""[:"d (3) 754 These equations can be derived by the following argument. The general idea is that the values computed by the input nodes at the new rate should be as close as possible to those computed at the original rate. Specifically, suppose one wishes to change the sampling rate from r to nr, where n is an integer. Suppose that at a time to the computed value of the input node is Xo ' If this node receives no additional input, then after m time steps, the computed value of the input node will be Xod m . For the more rapid sampling rate, Xod m should be the value obtained after nm time steps. Thus we require (4) which leads to Eq. (2) because n= r7r. Now suppose that an input I is presented m times in succession to an input node that is initially zero. After the the computed value of the input node is mth presentation, (5) Requiring this value to be equal to the corresponding value for the faster presentation rate after nm time steps leads to Eq. (3). These equations, then, make the computed values of the input nodes identical, independent of the presentation rate . Of course, this statement only holds exactly in the limit that the computed values of the input nodes change only infinitesimally from one time step to the next. Thus, in practice, one must insure that the signal is sampled frequently enough that the computed value of the input nodes is slowly changing. The point in weight space obtained after initial training at the rate r has two desirable properties. First, it can be trained on a signal at one sampling rate and then the values of the weights arrived at can be used as a near-optimal starting point to further train the system on the same signal but at a different sampling rate. Alternatively, the system can respond to temporal patterns which differ in rate from the training signal, without any retraining of the weights. These factors are a result of the choice of input representation, which essentially present the same pattern to the hidden unit and other layers, independent of sampling rate. These features highlight the fact that in this system the weights to some degree represent the temporal pattern independent of the rate of presentation. In contrast, in systems which use temporal windows, the weights obtained after training on a signal at one sampling rate would have little or no relation to the desired values of the weights for a differen.t sampling rate or window size. 755 EXPERIMENTS As an illustration of this architecture and related algorithm, a three-layer, 15-30-2 system was trained to detect the leftward or rightward motion of a gaussian pulse moving across the field of input units with sudden changes in direction. The values of d and a were 0.7788 and 0.4424, respectively. These values were chosen to give a characteristic decay time of 4 time steps with a maximum value computed by the input nodes of 2.0 . The pulse was of unit height with a half-width, 0, of 1.3. Figure 2 shows the input pulse as well as the values computed by the input nodes for leftward or rightward motion. Once trained at a velocity of 0.1 unit per sampling time, the velocity was varied over a wide range, from a factor of2 slower to a factor of2 faster as shown in Fig. 3. For small variations in velocity the system continued to correctly identify the type of motion. More impressive was its performance when the scaling relations given in Eqs. (2) and (3) were used to modify the amplitude and decay rate . In this case, acceptable performance was achieved over the entire range of velocities tested. This was without any additional retraining at the new rates. The difference in performance between the two curves also demonstrates that the excellent performance of the system is not an anomaly of the particular problem chosen, but characteristic of rescaling a and d according to Eqs. (2) and (3). We thus see that a simple use of capacitive links to store temporal context allows for motion detection at variable velocities. A second experiment involving speech data was performed to compare the system's performance to the time-delay-neural-network of Watrous and Shastri 8 . In their work, they trained a system to discriminate between suitably processed acoustic signals of the words "no" and "go." Once trained on a single utterance, the system was able to correctly identify other samples of these words from the same speaker. One drawback of their approach was that the weights did not converge to a fixed point. We were therefore particularly interested in whether our system could converge smoothly and rapidly to a stable solution, using the same data, and yet generalize as well as theirs did. This experiment also provided an opportunity to test a solution to the intermediate step training problem. The architecture was a 16-30-2 network. Each of the input nodes received an input signal corresponding to the energy (sampled every 2.5 milliseconds) as a function of time in one of 16 frequency channels. The input values were normalized to lie in the range 0.0 to 1.0. The values of d and a were 0.9944 and 0.022, respectively. These values were chosen to give a characteristic decay time comparable to the length of each word (they were nearly the same length), and a maximum value computed by the input nodes of 4.0. For an input signal that was part of the word "no", the training signal was (t.O, 0.0), while for the word "go" it was (0.0, 1.0). Thus the outputs that were compared to the training signal can be interpreted as evidence for one word or the other at each time step. The error shown in Fig. 4 is the sum of the squares of the 756 difference between the desired outputs and the computed outputs for each time step, for both words, after training up to the number ofiterations indicated along the x-axis. a) input wavepacket 2 3 4 5 6 7 B 9 10 5 6 7 B 9 10 5 6 7 B 9 10 b) rightward motion 2 3 4 c) leftward motion 2 3 4 Fig. 2: a) Packet presented to input nodes. The x-axis represents the input nodes. b) Computed values from input nodes during rightward motion. c) Computed values during leftward motion . 757 100~______________~~~~__~~~::::~ % 80 __ . : ' "':w - - - ~I!"'.::..:/,j/.:). -. . . . . . . . . . . . . . . . . . . 'i - ;~ c o r ? 60 _ I) ? ? I~ ? 40 +- r e c t '0 20 -I- o I I I I I I I .5 1.0 1.5 2.0 I v'lv Fig. 3: Performance of motion detection experiment for various velocities. Dashed curve is performance without scaling and solid curve is with the scaling given in Eqs. (2) and (3). 125.0 100.0 e r r 75.0 50.0 0 r 25.0 0.0 0 500 1000 1500 2000 2500 iterations Fig. 4: Error in no/go discrimination as a function of the number of training iterations. Evidence for each word was obtained by summing the values of the respective nodes over time. This suggests a mechanism for signaling the completion of a sequence: when this sum crosses a certain threshold value, the sequence (in this case, the word) is considered recognized. Moreover, it may be possible to extend this mechanism to apply to the case of connected speech: after a word is recognized, the sums could be reset to zero, and the input nodes reinitialized. Once we had trained the system on a single utterance, we tested the perfor~ance of the resulting weights on additional utterances of the same speaker. 758 Preliminary results indicate an ability to correctly discriminate between "no" and "go." This suggests that the system has at least a limited ability to generalize in this task domain. DISCUSSION At a more general level, this paper raises and addresses some issues of representation. By choosing input and output representations in a particular way, we are able to make a static optimizer work on a temporal problem while still allowing time to serve as its own representation. In this broader context, one realizes that the choice of capacitive inputs for the input nodes was only one among many possible temporal feature detectors. Other possibilities include refractory units, derivative units and delayed spike units. Refractory units would compute a value which was some fraction of the current input. The fraction would decrease the more frequently and recently the node had been "on" in the recent past. A derivative unit would have a larger output the more rapidly a signal changed from one time step to the next. A delayed spike unit might have a transfer function of the form Itne- at , where t is the time since the presentation of the signal. This is similar to the function used by Tank and Hopfield7 , but here it could serve a different purpose. The maximum value that a given input generated would be delayed by a certain amount of time . By similarly delaying the training signal, the system could be trained to recognize a given input in the context of signals not only preceding but also following it. An important point to note is that the transfer functions of each of these proposed temporal feature detectors could be rescaled in a manner similar to the capacitive nodes. This would preserve the property of the system that the weights contain information about the temporal sequence to some degree independent of the sampling rate. An even more ambitious possibility would be to have the system train the parameters, such as d in the capacitive node case. It may be feasible to do this in the same way that weights are trained, namely by taking the partial of the computed error with respect to the parameter in question. Such a system may be able to determine the relevant time scales of a temporal signal and adapt accordingly. ACKNOWLEDGEMENTS We are grateful for fruitful discllssions with Jeff Kephart and the help of Raymond Watrous in providing data from his own experiments. This work was partially supported by DARPA ISTO Contract # N00140-86-C-8996 and ONR Contract # N00014-82-0699_ 759 1. D. Rumelhart, ed., Parallel Distributed Processing, (:\'lIT Press, Cambridge, 1986). 2. J. Denker, ed., Neural Networks for Computing, AlP Conf. Proc.,151 (1986). 3. T. J. Sejnowski and C. R. Rosenberg, NETtalk: A Parallel Network that Learns to Read Aloud, Johns Hopkins Univ. Report No . JHU/EECS-86101 (1986). 4. J.L. McClelland and J.L. Elman, in Parallel Distributed Processing, vol. II, p. 58. 5. W. Keirstead and B.A. Huberman, Phys . Rev. Lett. 56,1094 (1986). 6. A. Lapedes and R. Farber, Nonlinear Signal Processing Using Neural Networks, Los Alamos preprint LA-uR-87-2662 (1987). 7. D. Tank and J. Hopfield, Proc. Nat. Acad. Sci., 84, 1896 (1987). 8. R. Watrous and L. Shastri, Proc. 9th Ann. Conf Cog. Sci. Soc., (Lawrence Erlbaum, Hillsdale, 1987), p. 518. 9. P. Kanerva, Self-Propagating Search: A Unified Theory of Memory, Stanford Univ. Report No. CSLI-84-7 (1984). 10. M.1. Jordan, Proc. 8th Ann. Conf. Cog. Sci. Soc., (Lawrence Erlbaum, Hillsdale, 1986), p. 531. 11. J. Hopfield,Proc. Nat. Acad. SCi., 79, 2554 (1982). 12. S. Grossberg, The Adaptive Brain, vol. II, ch. 6, (North-Holland, Amsterdam, 1987). 13. G. Hinton and T. J. Sejnowski, in Parallel Distributed Processing, vol. I, p. 282. 14. B. Gold, in Neural Networks for Computing, p. 158. 15. T. Hogg and B.A. Huberman, Phys. Rev. A32, 2338 (1985). 

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Processing of Visual and Auditory Space and Its Modification by Experience Josef P. Rauschecker Laboratory of Neurophysiology National Institute of Mental Health Poolesville, MD 20837 Terrence J. Sejnowski Computational Neurobiology Lab The Salk: Institute San Diego, CA 92138 Visual spatial information is projected from the retina to the brain in a highly topographic fashion, so that 2-D visual space is represented in a simple retinotopic map. Auditory spatial information, by contrast, has to be computed from binaural time and intensity differences as well as from monaural spectral cues produced by the head and ears. Evaluation of these cues in the central nervous system leads to the generation of neurons that are sensitive to the location of a sound source in space ("spatial tuning") and, in some animal species, to auditory space maps where spatial location is encoded as a 2-D map just like in the visual system. The brain structures thought to be involved in the multimodal integration of visual and auditory spatial integration are the superior colliculus in the midbrain and the inferior parietal lobe in the cerebral cortex. It has been suggested for the owl that the visual system participates in setting up the auditory space map in the superior. Rearing owls with displacing prisms, for example, shifts the map by a fixed amount. These behavioral and neurobiological findings have been successfully incorporated into a connectionist model of the owl's sound localization system (Rosen, Rumelhart, and Knudsen, 1994). On the other hand, cats that are reared with both eyes sutured shut develop completely normal auditory spatial mechanisms: Precision of sound localization is even improved above normal (Rauschecker and Kniepert, 1994), and a higher number of auditory neurons with sharper spatial tuning is found in parietal cortex of such cats (Rauschecker and Korte, 1993). Non-visual sensory signals and/or motor feedback must be capable, therefore, to calibrate the auditory spatial mechanisms. Activity-dependent Hebbian learning and synaptic competition between inputs to the parietal region from different sensory modalities are sufficient to explain these results. 1186 Processing of Visual and Auditory Space and Its Modification by Experience The question remains how visual and auditory information are kept in spatial register with each other when the animal moves its eyes or head. Experiments in awake behaving monkeys help to solve this problem. Neurons in the lateral intraparietal area of cortex (LIP) respond to visual and auditory stimuli which call for a movement to the same location in space. Neuronal responses in both modalities are modulated by eye position leading to "gain fields", in which the location of a target in headcentered coordinates is encoded via the response strength in a population of neurons (Andersen, Snyder, Li, and Stricanne, 1993). The neurobiological data from owls, cats and monkeys were used to develop a neural network model of multisensory integration (Pouget and Sejnowski, 1993). A set of basis functions was introduced which replace the conventional allocentric representations and produce gain fields similar to monkey parietal cortex. An extension of the model also incorporates the plasticity of this system. Predictive Hebbian learning is used to bring the visual and auditory maps into register. In the network a Hebb rule is gated by a reinforcement term, which is the difference between actual reinforcement and how much reinforcement is expected by the system. It utilizes the activity of diffuse transmitter projection systems, such as noradrenaline (NA), acetylcholine (ACh), and dopamine (DA) , which are known to play an important role for plasticity in the brain of higher mammals. In summary, it appears extremely fruitful to bring together neuroscientists and neural network modelers, because both groups can profit from each other. Neurobiological data are the flesh for realistic network models, and models are helpful to formalize a biological hypothesis and guide the way for further testing. Andersen RA, Snyder LH, Li C-S, Stricanne B (1993) Coordinate transformations in the representation of spatial information. Curr Opinion Neurobiol3: 171-176. Pouget A, Fisher SA, Sejnowski TJ (1993) Egocentric spatial representation in early vision. J Cog Neurosci 5:150-161. Rauschecker JP and Korte M (1993) Auditory compensation for early blindness in cat cerebral cortex. J Neurosci 13:4538-4548. Rauschecker JP and Kniepert U (1994) Enhanced precision of auditory localization behavior in visually deprived cats. Eur J Neurosci 6 (in press). Rosen D, Rumelhart D, Knudsen E (1994) A connectionist model of the owl's sound localization system. In: Advances in Neural Information Processing Systems 6, Cowan J, Tesauro G, Alspector J (eds), San Mateo, CA: Morgan Kaufmann (in press) 1187 

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Hidden Markov Models for Human Genes Pierre Baldi * Jet Propulsion Laboratory California Institute of Technology Pasadena, CA 91109 Yves Chauvin t Net-ID, Inc. 601 Minnesota San Francisco, CA 94107 S0ren Brunak Center for Biological Sequence Analysis The Technical University of Denmark DK-2800 Lyngby, Denmark Jacob Engelbrecht Center for Biological Sequence Analysis The Technical University of Denmark DK-2800 Lyngby, Denmark Anders Krogh Electronics Institute The Technical University of Denmark DK-2800 Lyngby, Denmark .Abstract Human genes are not continuous but rather consist of short coding regions (exons) interspersed with highly variable non-coding regions (introns). We apply HMMs to the problem of modeling exons, introns and detecting splice sites in the human genome. Our most interesting result so far is the detection of particular oscillatory patterns, with a minimal period ofroughly 10 nucleotides, that seem to be characteristic of exon regions and may have significant biological implications. ? and Division of Biology, California Institute of Technology. t and Department of Psychology, Stanford University. 761 762 Baldi, Brunak, Chauvin, Engelbrecht, and Krogh exon intron EXON 3' splice site 5' splice site acceptor site donor site CONSENSUS SEQUENCES I I I I I I I I I I NC AG CCCCCCCC T I G C AG A I GTAGAGT ~------------------~ Figure 1: Structure of eukaryotic genes (not to scale: introns are typically much longer than exons). 1 INTRODUCTION The genes of higher organisms are not continuous. Rather, they consist of relatively short coding regions called exons interspersed with non-coding regions of highly variable length called introns (Fig. 1). A complete gene may comprise as many as fifty exons. Very often, exons encode discrete functional or structural units of proteins. Prior to the translation of genes into proteins, a complex set of biochemical mechanisms is responsible for the precise cutting of genes at the splice junctions, i.e. the boundaries between introns and exons, and the subsequent removal and ligation which results in the production of mature messenger RNA. The translation machinery of the cell operates directly onto the mRNA, converting a primary sequence of nucleotides into the corresponding primary sequence of amino acids, according to the rules of the genetic code. The genetic code converts every three contiguous nucIeotides, or codons, into one of the twenty amino acids (or into a stop signal). Therefore the splicing process must be exceedingly precise since a shift of only one base pair completely upsets the codon reading frame for translation. Many details of the splicing process are not known; in particular it is not clear how acceptor sites (i.e. intron/exon boundaries) and donor sites (i.e. exon/intron boundaries) are recognized with extremely high accuracy. Both acceptor and donor sites are signaled by the existence of consensus sequences, i.e. short sequences of nucleotides which are highly conserved across genes and, to some extent, across species. For instance, Hidden Markov Models for Human Genes most introns start with GT and terminate with AG and additional patterns can be detected in the proximity of the splice sites. The main problem with consensus sequences, in addition to their variability, is that by themselves they are insufficient for reliable splice site detection. Indeed, whereas exons are relatively short with an average length around 150 nucleotides, introns are often much longer, with several thousand of seemingly random nucleotides. Therefore numerous false positive consensus signals are bound to occur inside the introns. The GT dinucleotide constitutes roughly 5% of the dinucleotides in human DNA, but only a very small percentage of these belongs to the splicing donor category, in the order of 1.5%. The dinucleotide AG constitutes roughly 7.5% of all the dinucleotides and only around 1% of these function as splicing acceptor sites. In addition to consensus sequences at the splice sites, there seem to exist a number of other weak signals (Senapathy (1989), Brunak et a1. (1992)) embedded in the 100 intron nucleotides upstream and downstream of an exon. Partial experimental evidence seems also to suggest that the recognition of the acceptor and donor boundaries of an exon may be a concerted process. In connection with the current exponential growth of available DNA sequences and the human genome project, it has become essential to be able to algorithmically detect the boundaries between exons and introns and to parse entire genes. Unfortunately, current available methods are far from performing at the level of accuracy required for a systematic parsing of the entire human genome. Most likely, gene parsing requires the statistical integration of several weak signals, some of which are poorly known, over length scales of a few hundred nucleotides. Furthermore, initial and terminal exons, lacking one of the splice sites, need to be treated separately. 2 HMMs FOR BIOLOGICAL PRIMARY SEQUENCES The parsing problem has been tackled with classical statistical methods and more recently using neural networks (Lapedes (1988), Brunak (1991)), with encouraging results. Conventional neural networks, however, do not seem ideally suited to handle the sort of elastic deformations introduced by evolutionary tinkering in genetic sequences. Another trend in recent years, has been the casting of DNA and protein sequences problems in terms of formal languages using context free grammars, automata and Hidden Markov Models (HMMs). The combination of machine learning techniques which can take advantage of abundant data together with new flexible representations appears particularly promising. HMMs in particular have been used to model protein families and address a number of task such as multiple alignments, classification and data base searches (Baldi et al. (1993) and (1994); Haussler et a1. (1993); Krogh et al. (1994a); and references therein). It is the success obtained with this method on protein sequences and the ease with which it can handle insertions and deletions that naturally suggests its application to the parsing problem. In Krogh et al. (1994b), HMMs are applied to the problem of detecting coding/noncoding regions in bacterial DNA (E. coli), which is characterized by the absence of true introns (like other prokaryotes). Their approach leads to a HMM that integrates both genic and intergenic regions, and can be used to locate genes fairly reliably. A similar approach for human DNA, that is not based on HMMs, but uses dynamic programming and neural networks to combine various gene finding techniques, is described in Snyder and Stormo (1993). In this paper we take a 763 764 Baldi, Brunak, Chauvin, Engelbrecht, and Krogh Main State Entropy Values ci 10 20 30 40 60 60 70 80 90100110120130140160160170 Main State Position 180190200210220230240260260270280290300310320330340360 Main State Position Figure 2: Entropy of emission distribution of main states. first step towards parsing the human genome with HMMs by modeling exons (and flanking intron regions). As in the applications of HMMs to speech or protein modeling, we use left-right architectures to model exon regions, intron regions or their boundaries. The architectures typically consist of a backbone of main states flanked by a sequence of delete states and a sequence of insert states, with the proper interconnections (see Baldi et al. (1994) and Krogh et al. (1994) for more details and Fig. 4 below). The data base used in the experiments to be described consists of roughly 2,000 human internal exons, with the corresponding adjacent introns, extracted from release 78 of the GenBank data base. It is essential to remark that, unlike in the previous experiments on protein families, the exons in the data base are not directly related by evolution. As a result, insertions and deletions in the model should be interpreted in terms of formal operations on the strings rather than evolutionary events. 3 EXPERIMENTS AND RESULTS A number of different HMM training experiments have been carried using different classes of sequences including exons only, flanked exons (with 50 or 100 nucleotides on each side), introns only, flanked acceptor and flanked donor sites (with 100 nucleotides on each side) and slightly different architectures and learning algorithms. Only a few relevant examples will be given here. Hidden Markov Models for Human Genes A g :~ ~..J~~ 00 .20 140 .40 .00 2.0 '00 .20 In ? ? rt at.ta Po ?. tlon C ~ ~ ~ ;: :::t :::: 100 .00 ,.0 ? ?0 320 G ~ :; ~ ;: :::t ~ 40 120 200 '40 T ... 32. Figure 3: Emission distribution from main states. In an early experiment, we trained a model of length 350 using 500 flanked exons, with 100 nucleotides on each side, using gradient descent on the negative loglikelihood (Baldi and Chauvin (1994)). The exons themselves had variable lengths between 50 and 300. The entropy plot (Fig. 2), after 7 gradient descent training cycles, reveals that the HMM has learned the acceptor site quite well but appears to have some difficulties with the donor site. One possible contributing factor is the high variability of the length of the training exons: the model seems to learn two donor sites, one for short exons and one for the other exons. The most striking pattern, however, is the greater smoothness of the entropy in the exon region. In the exon region, the entropy profile is weakly oscillatory, with a period of about 20 base pairs. Discrimination and t-tests conducted on this model show that it is definitely capable of discriminating exon regions, but the confidence level is not sufficient yet to reliably search entire genomes. A slightly different model was subsequently trained using again 500 flanked exons, with the length of the exons between 100 and 200 only. The probability of emitting each one of the four nucleotides, across the main states of the model, are plott.ed in Fig. 3, after the sixt.h gradient descent training cycle. Again the donor site seems harder to learn than the acceptor site. Even more striking are the clear 765 766 Baldi, Brunak, Chauvin, Engelbrecht, and Krogh Figure 4: The repeated segment of the tied model. Note that position 15 is identical to position 5. oscillatory patterns present in the exon region, characterized by a minimal period of 10 nucleotides, with A and G in phase and C and T in anti-phase. The fact that the acceptor site is easier to learn could result from the fact that exons in the training sequences are always flanked by exactly 100 nucleotides upstream. To test this hypothesis, we trained a similar model using the same sequences but in reverse order. Surprisingly, the model still learns the acceptor site (which is now downstream from the donor site) much better than the donor site. The oscillatory pattern in the reversed exon region is still present. The oscillations we observe could also be an artifact of the method: for instance, when presented with random training sequences, oscillatory HMM solutions could appear naturally as local optima of the training procedure. To test this hypothesis, we trained a model using random sequences of similar average composition as the exons and found no distinct oscillatory patterns. We also checked that our data base of exons does not correspond prevalently to a-helical domains of proteins. To further test our findings, we trained a tied exon model with a hard-wired periodicity of 10. The tied model consists of 14 identical segments of length 10 and 5 additional positions in the beginning and end of the model, making a total length of 150. During training the segments are kept identical by tying of the parameters, i.e. the parameters are constrained to be exactly the same throughout learning, as in the weight sharing procedure for neural networks. The model was trained on 800 exon sequences of length between 100 and 200, and it was tested on 262 different sequences. The parameters of the repeated segment, after training, are shown in Fig. 4. Emission probabilities are represented by horizontal bars of corresponding proportional length. There is a lot of structure in this segment. The most prominent feature is the regular expression [AT][AT]G at position 12-14. (The regular expression means "anything but T followed by A or T followed by G".) The same pattern was often found at positions with very low entropy in the "standard models" described above. In order to test the significance, the tied model was compared to a standard model of the same length. The average negative log-likelihood (NNL) they both assign to the exon sequences and to random sequences of similar composition, as well as their number of parameters are shown in the table below. Hidden Markov Models for Human Genes Model Scores Standard model with random seqs Standard model with real seqs Tied model with real seqs # NLL training NLL testing 203.2 200.3 2550 198.8 196.4 2550 198.6 195.6 340 parameters The tied model achieves a level of performance comparable to the standard model but with significantly less free parameters, and therefore a period of 10 in the exons seems to be a strong hypothesis. Note that the period of the pattern is not strictly 10, and we found almost equally good models with a built-in period of 9 or 11. The type of left-to-right architecture we have used is not the ideal model of an exon, because of the large length variations. It would be desirable to have a model with a loop structure such that the segment can be entered as many times as necessary for any given exon (see Krogh et al. (1994b) for a loop structure used for E. coll DNA). This is one of the future lines of research. 4 CONCLUSION In summary, we are applying HMMs and related methods to the problems of exon/intron modeling and human genome parsing. Our preliminary results show that acceptor sites are intrinsically easier to learn than donor sites and that very simple HMM models alone are not sufficient for reliable genome parsing. Most importantly, interesting statistical 10 base oscillatory patterns have been detected in the exon regions. If confirmed, these patterns could have significant biological and algorithmic implications. These patterns could be related to the superimposition of several simultaneous codes (such as triplet code and frame code), and/or to the way DNA is wrapped around histone molecules (Beckmann and Trifonov (1991)). Presently, we are investigating their relationship to reading frame effects by training several HMM models using a data base of exons with the same reading frame. References Beckmann, J.S. and Trifonov, E.N. (1991) Splice Junctions Follow a 205-base Ladder. PNAS USA, 88, 2380-2383. Baldi, P., Chauvin, Y., Hunkapiller, T. and McClure, M. A. (1994) Hidden Markov Models of Biological Primary Sequence Information. PNAS USA, 91, 3, 1059-1063. Baldi, P., Chauvin, Y., Hunkapiller, T. and McClure, M. A. (1993) Hidden Markov Models in Molecular Biology: New Algorithms and Applications. Advances in Neural Information Processing Systems 5, Morgan Kaufmann, 747-754. Baldi, P. and Chauvin, Y. (1994) Smooth On-Line Learning Algorithms for Hidden Markov Models. Neural Computation, 6, 2, 305-316. Brunak, S., Engelbrecht, J. and Knudsen, S. (1991) Prediction of Human mRNA Donor and Acceptor Sites from the DNA Sequence. Journal of Molecular Biology, 220,49-65. 767 768 Baldi, Brunak, Chauvin, Engelbrecht, and Krogh Engelbrecht, J., Knudsen, S. and Brunak S., (1992) GIC rich tract in 5' end of human introns, Journal of Molecular Biology, 221, 108-113. Haussler, D., Krogh, A., Mian, I.S. and Sjolander, K. (1993) Protein Modeling using Hidden Markov Models: Analysis of Globins, Proceedings of the Hawaii International Conference on System Sciences, 1, IEEE Computer Society Press, Los Alamitos, CA, 792-802. Krogh, A., Brown, M., Mian, I. S., Sjolander, K. and Haussler, D. (1994a) Hidden Markov Models in Computational Biology: Applications to Protein Modeling. Journal of Molecular Biology, 235, 1501-153l. Krogh, A., Mian, I. S. and Haussler, D. (1994b) A Hidden Markov Model that Finds Genes in E. Coli DNA, Technical Report UCSC-CRL-93-33, University of California at San ta Cruz. Lapedes, A., Barnes, C., Burks, C., Farber, R. and Sirotkin, K. Application of Neural Networks and Other Machine Learning Algorithms to DNA Sequence Analysis. In G.I. Bell and T.G. Marr, editors. The Procceedings of the Interface Between Computation Science and Nucleic Acid Sequencing Workshop. Proceedings of the Santa Fe Institute, volume VII, pages 157-182. Addison Wesley, Redwood City, CA,1988. Senapathy, P., Shapiro, M.B., and Harris, N.1. (1990) Splice Junctions, Branch Point Sites, and Exons: Sequence Statistics, Identification and Applications to Genome Project. Patterns in Nucleic Acid Sequences, Academic Press, 252-278. Snyder, E.E. and Stormo, G.D. (1993) Identification of coding regions in genomic DNA sequences: an application of dynamic programming and neural networks. Nucleic Acids Research, 21, 607-613. 

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Cross-Validation Estimates IMSE Mark Plutowski t* Shinichi Sakata t Halbert White t* t Department of Computer Science and Engineering t Department of Economics * Institute for Neural Computation University of California, San Diego Abstract Integrated Mean Squared Error (IMSE) is a version of the usual mean squared error criterion, averaged over all possible training sets of a given size. If it could be observed, it could be used to determine optimal network complexity or optimal data subsets for efficient training. We show that two common methods of cross-validating average squared error deliver unbiased estimates of IMSE, converging to IMSE with probability one. These estimates thus make possible approximate IMSE-based choice of network complexity. We also show that two variants of cross validation measure provide unbiased IMSE-based estimates potentially useful for selecting optimal data subsets. 1 Summary To begin, assume we are given a fixed network architecture. (We dispense with this assumption later.) Let zN denote a given set of N training examples. Let QN(zN) denote the expected squared error (the expectation taken over all possible examples) of the network after being trained on zN. This measures the quality of fit afforded by training on a given set of N examples. Let IMSEN denote the Integrated Mean Squared Error for training sets of size N. Given reasonable assumptions, it is straightforward to show that IMSEN = E[Q N(ZN)] - 0"2, where the expectation is now over all training sets of size N, ZN is a random training set of size N, and 0"2 is the noise variance. = CN(zN) denote the "delete-one cross-validation" squared error measure Let CN for a network trained on zN. CN is obtained by training networks on each of the N training sets of size N -1 obtained by deleting a single example; the measure follows 391 392 Plutowski, Sakata, and White by computing squared error for the corresponding deleted example and averaging the results. Let G N,M = G N,M (zN , zM) denote the "generalization" measure obtained by separating the available data of size N + M into a training set zN of size N, and a validation ("test") set zM of size M; the measure follows by training on zN and computing averaged squared error over zM. We show that eN is an unbiased estimator of E[QN_l(ZN)], and hence, estimates 1M SEN-l up to noise variance. Similarly, GN,M is an unbiased estimator of E[QN(ZN, ZM] . Given reasonable conditions on the estimator and on the data generating process we demonstrate convergence with probability 1 of GN,M and eN to E[QN(ZN)] as Nand M grow large. A direct consequence of these results is that when choice is restricted to a set of network architectures whose complexity is bounded above a priori, then choosing the architecture for which either eN (or G N,M) is minimized leads to choice of the network for which 1MSEN is nearly minimized for all N (respectively, N, M) sufficiently large. We also provide results for training sets sampled at particular inputs. Conditional 1M S E is an appealing criterion for evaluating a particular choice of training set in the presence of noise. These results demonstrate that delete-one cross-validation estimates average MSE (the average taken over the given set of inputs,) and that holdout set cross-validation gives an unbiased estimate of E[QN(ZN)IZ N (x N , yN)], given a set of N input values x N for which corresponding (random) output values yN are obtained. Either cross-validation measure can therefore be used to select a representative subset of the entire dataset that can be used for data compaction, or for more efficient training (as training can be faster on smaller datasets) [4]. = 2 2.1 Definitions Learning Task We consider the learning task of determining the relationship between a random vector X and a random scalar y, where X takes values in a subset X of ~r, and Y takes values in a subset Y of~. (e.g. X ~r and Y ~). We refer to X as the input space. The learning task is thus one of training a neural network with r inputs and one output. It is straightforward to extend the following analysis to networks with multiple targets. We make the following assumption on the observations to be used in the training of the networks. = = = Assumption 1 X is a Borel subset of ~r and Y is a Borel subset of~. Let Z Zoo Xi:1Z. Let (n,.1', P) be a probability space with.1' 8(n). X x Y and n = == = The observations on Z (X', Y)' to be used in the training of the network are a realization of an i.i.d. stochastic process {Z, _ (Xi, Yi)' : n ---+ X x V}. = When wEn is fixed, we write z, Z,(w) for each i (Zl, ... ,ZN) and zn (Zl, .. . ,zn). = = 1,2, .... Also write ZN = Assumption 1 allows uncertainty caused by measurement errors of observations as well as a probabilistic relationship between X and Y . It, however, does not prevent a deterministic relation ship between X and Y such that Y g(X) for some measurable mapping g : ~r ---+ ~. = Cross-Validation Estimates IMSE We suppose interest attaches to the conditional expectation of Y given X, written g( x) E(Y IX). The next assumption guarantees the existence of E(Yi IXi) and E(cdXi), Ci Yi - E(YiIXi). Next, for convenience, we assume homoscedasticity of the conditional variance of Yi given Xi. = = Assumption 2 E(y2) < 00. Assumption 3 E(ci IX1 ) = u 2, where u 2 is a strictly positive constant. 2.2 Network Model Let fP(.,.) : X X WP -- Y be a network function with the "weight space" WP, where p denotes the dimension of the "weight space" (the number of weights.) We impose some mild conditions on the network architecture. Assumption 4 For each p E {I, 2, ... ,p}, pEN, WP is a compact subset of ~P, and fP : X x WP -- ~ satisfies the following conditions: 1. fP(., w) : X -- Y is measurable for each wE WP; 2. fP(x,?) : WP -- Y is continuous for all x E X. We further make a joint assumption on the underlying data generating process and the network architecture to assure that the training dataset and the networks behaves appropriately. = Assumption 5 There exists a function D : X -- ~+ [0,00) such that for each x E X and w E WP, IfP(x, w)1 ~ D(x), and E [(D(X?2] < 00. Hence, fP is square integrable for each w P E WP. We will measure network performance using mean squared error, which for weights wP is given by ).(wP;p) E [(Y - fP(X, w P)2]. The optimal weights are the weights that minimize ).(wP;p). The set of all optimal weights are given by WP? {w? E WP : ).( w? ; p) ~ ).( w; p) for any w E Wp}. The index of the best network is p. , given by the smallest p minimizing minwl'Ewl' ).(wP;p), p E {I, 2, ... ,pl. = = 2.3 Least-Squares Estimator When assumptions I and 4 hold, the nonlinear least-squares estimator exists. Formally, we have Lemma 1 Suppose that Assumptions 1 and 4 hold. Then 1. For each N EN, there exists a measurable function INC; p) : ZN -- WP such that IN(ZN; p) solves the following problem with probability one: minwEWI' N- 1 E~l (Yi - J(Xi, w?2 . 2. ).(.; p) : WP __ ~ is continuous on WP, and WP? is not empty. = For convenience, we also define ~ : n -- WP by ~(w) IN(ZN (w);p) for each wEn. Next let i 1 , i 2 , ?.. , iN be distinct natural numbers and let ZN (Zil' ... , ZiN)" Then IN(ZN) given above solves ;; Ef=l (Yi; - f(Xi;, wP?2 with probability one. In particular, we will consider the estimate using the dataset Z~i made by deleting the ith observation from zN. Let Z~i be a random matrix made = 393 394 Plutowski, Sakata, and White by deleting the ith row from ZN. Thus, IN -1 (Z~i; p) is a measurable least squares estimator and we can consider its probabilistic behavior. 3 Integrated Mean Squared Error Integrated Mean Squared Error (IMSE) has been used to regulate network complexity [9]. Another (conditional) version of IMSE is used as a criterion for evaluating training examples [5, 6, 7, 8]. The first version depends only on the sample size, not the particular sample. The second (conditional) version depends additionally upon the observed location of the examples in the input space. 3.1 Unconditional IMSE The (unconditional) mean squared error (MSE) of the network output at a particular input value x is MN(X;p)=E [{g(x)-!(x,IN(ZN;p))}2]. (1) Integrating MSE over all possible inputs gives the unconditional IMSE: IMSEN(p) J (2) E [MN(XjP)], (3) [MN(X, ;p)] J.L(dx) where J.L is the input distribution. 3.2 Conditional IMSE To evaluate exemplars obtained at inputs x N , we modify Equation (1) by conditioning on x N , giving MN(xlx N ;p) E [{g(x) - !(x, IN(ZN))PIX N xN ] . = = The conditional IMSE (given inputs x N ) is then IMSEN(xN;p) 4 J (4) E [MN(XlxN;p)] . (5) MN(xlxN;p)JL(dx) Cross-Validation Cross-validatory measures have been used successfully to assess the performance of a wide range of estimators [10, 11, 12, 13, 14, 15]. Cross-validatory measures have been derived for various performance criteria, including the Kullback-Liebler Information Criterion (KLIC) and the Integrated Squared Error (ISE, asymptotically equivalent to IMSE) [16]. Although provably inappropriate in certain applications [17, 18], optimality and consistency results for the cross-validatory measures have been obtained for several estimators, including linear regression, orthogonal series, splines, histograms, and kernel density estimators [16, 19,20, 21, 22, 23, 24]. The authors are not aware of similar results applicable to neural networks, although two more general, but weaker results do apply [26]. A general result applicable to neural networks shows asymptotic equivalence between cross-validation and Akaike's Criterion for network selection [25,29]' as well as between cross-validation and Moody's Criterion [30, 29]. Cross-Validation Estimates IMSE 4.1 Expected Network Error Given our assumptions, we can relate cross-validation to IMSE. For clarity and notational convenience, we first introduce a measure of network error closely related to IMSE. For each weight w P E WP, we have defined the mean squared error A( wP ; p) in Section 2.2. We define QN to map each dataset to the mean squared error of the estimated network QN(ZN;p) = A(lN(zN;p);p). When Assumption 3 holds, we have A(wP;p) = E [(g(X) - f(X, W P ))2] = E [(g(XN+d - f(XN+l, wP))2] + u2 + u2 as is easily verified. We therefore have QN(zN; p) = E [(g(XN+d - f(XN+l, IN(ZN ;p)))2IZN = zN] + u 2. Thus, by using the law of iterated expectations, we have E[QN(ZN;p)] = IMSEN(p)+u 2. Likewise, given x N E X N , E[QN(ZN; p)IX N = x N] = IMSE(x N ;p) + u 2. (6) 4.2 Cross-Validatory Estimation of Error In practice we work with observable quantities only. In particular, we must estimate the error of network p over novel data ("generalization") from a finite set of examples. Such an estimate is given by the delete-one cross-validation measure: N CN(zN;p) =~ L (Yi - f(Xi,IN_l(zl!i;P)))2 (7) i=l ~ere zl!i, denotes the training set obtained by deleting the ith example. Using z_i insteaa of z avoids a downward bias due to testing upon examples used in training, as we show below (Theorem 3.) Another version of cross-validation is commonly used for evaluating "generalization" when an abundant supply of novel data is available for use as a "hold-out" set: M GN,M(zN,zM;p) =~ L (iii - J(xi,IN(zN;p)))2, (8) i=l where zM 5 = (ZN+l' ... , ZN+M)' Expectation of the Cross-Validation Measures We now consider the relation between cross-validation measure and IMSE. We examine delete-one cross-validation first. Proposition 1 (Unbiasedness of CN) Let Assumptions 1 through 5 hold. Then for given N, CN is an unbiased estimator of 1M SEN-l (p) + u 2: E [CN(ZN;p)] = IMSEN-1(p) +u 2. (9) 395 396 Plutowski, Sakata, and White With hold-out set cross-validation, the validation set ZM gives i.i.d. information regarding points outside of the training set ZN. Proposition 2 (Unbiasedness of GN,M) Let Assumptions 1 through 5 hold. Let = ZM (ZN+l, .. . ,ZM)'. Then for given Nand M, GN,M is an unbiased estimator of IMSEN(p) + u 2 : E [ GN,M(Z N , Z-M ;p) ] = IMSEN(p) + u2 . (10) The latter result is appealing for large M, N. We expect delete-one cross-validation to be more appealing when training data is not abundant. 6 Expectation of Cross-Validation when Sampling at Selected Inputs We obtain analogous results for training sets obtained by sampling at a given set of inputs x N . We first consider the result for delete-one cross-validation. Proposition 3 (Expectation of CN given xN1 Let Assumptions 1 through 5 hold. Then, given a particular set of inputs, x , CN is an unbiased estimator of average MSEN-l + u 2 , the average taken over x N : N N . p) N1 "L.J M N - 1 (I Xi x_i' + u2, i=l where X~i is a matrix made by deleting the ith row of x N ? This essentially gives an estimate of MSEN-l limited to x E x N , losing a degree of freedom while providing no estimate of the M S E off of the training points. For this average to converge to IMSEN-l, it will suffice for the empirical distribution of x N , p,N, to converge to J-lN, i.e ., P,N => J-lN. We obtain a stronger result for hold-out set cross-validation. The hold-out set gives independent information on M SEN off of the training points, resulting in an estimate of IMSEN for given x N . Proposition 4 (Expectation of GN,M given x N ) Let Assumptions 1 through 5 = (ZN+1, .. . , ZN+M )'. Then, given a particular set of inputs, x N , hold. Let ZM GN,M is an unbiased estimator of of IMSEN(x N ; p) + u 2 : E [GN,M(ZN,ZM;p)IX N =xN] IMSEN(xN;p) +u2 ? 7 Strong Convergence of Hold-Out Set Cross-Validation Our conditions deliver not only unbiasedness, but also convergence of hold-out set cross-validation to IMSEN, with probability 1. Theorem 1 (Convergence of Hold-Out Set w.p. 1) Let Assumptions 1 through 5 hold. Also let ZM (ZN+l, ... , ZN+M)'. If for some A > 0 a sequence {MN} of natural numbers satisfies MN > AN for any N = 1,2, ... , then = Cross-Validation Estimates IMSE 8 Strong Convergence of Delete-One Cross-Validation Given an additional condition (uniqueness of optimal weights) we can show strong convergence for delete-one cross-validation. First we establish uniform convergence of the estimators WP(Z~i) to optimal weights (uniformly over 1 < i < N.) Theorem 2 Let Assumptions 1 through 5 hold. Let Z~k be the dataset made by deleting the kth observation from ZN. Then max d (IN-1(Z~i;P), Wp*) l~i~N where d(w, Wp*) --+ 0 a.s.-P as N --+ 00, (11) =infw.Ewp.llw - w*lI. This convergence result leads to the next result that the delete-one cross validation measure does not under-estimate the optimized MSE, namely, infwPEWp ).(wP;p). Theorem 3 Let Assumptions 1 through 5 hold. Then liminfCN(ZN ;p) N-oo > min wEWp ).(w;p) a.s.-P. When the optimum weight is unique, we have a stronger result about convergence of the delete-one cross validation measure. Assumption 6 Wp* is a singleton, i.e., wp* has only one element. Theorem 4 Let Assumptions 1 through 6 hold. Then CN (ZN ;p) - E [QN(ZN ;p)] 9 --+ 0 a.s. as N --+ 00. Conclusion Our results justify the intuition that cross-validation measures unbiasedly and consistently estimate the expected squared error of networks trained on finite training sets, therefore providing means of obtaining 1M S E-approximate methods of selecting appropriate network architectures, or for evaluating particular choice of training set. Use of these cross-validation measures therefore permits us to avoid underfitting the data, asymptotically. Note, however, that although we also thereby avoid overfitting asymptotically, this avoidance is not necessarily accomplished by choosing a minimally complex architecture. The possibility exists that IMSEN-1(p) = 1M S EN -1 (p + 1). Because our cross-validated estimates of these quanti ties are random we may by chance observe CN(ZN;p) > CN(ZN;p+ 1) and therefore select the more complex network, even though the less complex network is equally good. Of course, because the IMSE's are the same, no performance degradation (overfitting) will result in this solution. Acknowledgements This work was supported by NSF grant IRI 92-03532. We thank David Wolpert, J an Larsen, Jeff Racine, Vjachislav Krushkal, and Patrick Fitzsimmons for valuable discussions. 397 398 Plutowski, Sakata, and White References [1] White, H. 1989. "Learning in Artificial Neural Networks: A Statistical Perspective." Neural Computation, 1 i, pp.i25-i6i. MIT Press, Cambridge, MA. (2] Plutowski, Mark E., Shinichi Sakata, and Halbert White. 1993. "Cross-validation delivers strongly consistent unbiased estimates of Integrated Mean Squared Error." To appear. (3] Plutowski, Mark E., and Halbert White. 1993. "Selecting concise training sets from clean data." IEEE Transactions on Neural Networks. 4, 3, pp.305-318. (i) Plutowski, Mark E., Garrison Cottrell, and Halbert White. 1992. "Learning Mackey-Glass from 25 examples, Plus or Minus 2." (Presented at 1992 Neural Information Processing Systems conference.) Jack D. Cowan, Gerald Tesauro, Joshua Aspector (eds.), Advances in neural information processing systems 6, San Mateo, CA: Morgan Kaufmann Publishers. (5] Fedorov, V.V. 1972. Theory of Optimal Experimenta. Academic Prell, New York . (6] Box,G., and N.Draper. 1987. Empirical ModelBuilding and Reaponae Surfacea. Wiley, New York. (7] Khuri, A . I., and J.A . Cornell. 1987. Reaponae Surfacea (Deaigna and Analyaea). Marcel Dekker, Inc., New York . (8] Faraway, Julian J. 1990. "Sequential design for the nonparametric regression of curves and surfaces." Technical Report #177, Department of Statistics, The University of Michigan. (9] Geman, Stuart, Elie Bienenstock, and Rene Doursat. 1992. "Neural network. and the bias/variance dilemma." Neural Computation. 4, I, 1-58. (10] Stone, M. 1959. "Application of a measure of information to the design and comparison of regression experiments." Annals Math. Stat. 30 55-69 (II] "Cross-validatory choice and assessment of statistical predictions." J.R. Statist . Soc. B. 36, 2, Ill-H. (12] Bowman, Adrian W. 198i. "An &1ternative method of cross-validation for the smoothing of density e.timates." Biometrika (198i), 71, 2, pp. 353-60. (13] Bowman, Adrian W., Peter Hall, D.M. Titterington . 198i. "Cross-validation in nonparametric estimation of probabilities and probability den? ities." Biometrika (198i), 71, 2, pp. 3U-51. (Ii] Marron, M. 1987. "A compari.on of crossvalidation techniques in den.ity estimation." The Annals of Stati.tics. 15, I, 152-162. (15] Wahba, Grace. 1990. Spline Modela for Obaervatlonal Data. v. 59 in the CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, March 1990. Softcover, 169 pages, bibliography, author index. ISBN 0-89871-2H-0 (16] Hall, Peter. 1983. "Large sample optimality of lea.t squares cross-validation in density estimation." The Annals of Statistics. 11, i, 1156-117i. (17] Schuster, E .F. , and G.G. Gregory. 1981. "On the nonconsistency of maximum likelihood nonparametric den.ity estimators". Comp . Sci. ok Statistics: Proc. 13th Symp. on the Interface. W.F. Eddy ed. 295-298 . Springer-Verlag. (18] Chow, Y.-S. and S.Geman, L.-D. Wu. 1983. "Consistent cross-validated density estimation." The Annals of Statistics. 11, I, 25-38. (19] Bowman, Adrian W. 1980. "A note on consi.tency of the kernel method for the analysis of categoric&1 data." Biometrika (1980), 67, 3, pp . 682-i. (20] Stone, Charles J. 198i "An asymptotically optimal window selection rule for kernel density estimates. " The Annals of Statistics. 12, i, 12851297. (21] Li, Ker-Chau . 1986. "Asymptotic optimality of C L and generalized cross-validation in ridge regression with application to spline smoothing." The Annals of Statistics . 14, 3, 1101-1112. (22] Li, Ker-Chau. 1987. "Asymptotic optimality for C p , CL, cross-v&1idation, and generalized crossvalidation: di.crete index set." The Annals of Statistics. 15, 3, 958-975. (23] Utreras, Florencio I. 1987. "On generalized crossvalidation for multivariate smoothing spline function .... SIAM J . Sci. Stat. Comput. 8, i, July 1987. (U] Andrews, Don&1d W.K. 1991. "A.ymptotic optimality of generalized C L, cross-validation, and generalized cross-validation in regression with heteroskedastic errors." Journal of Econometrics . 47 (1991) 359-377. North-Holland. (25] Stone, M. 1977. "An asymptotic equivalence of choice of model by cross-v&1idation and Akaike's criterion." J. Roy. Stat. Soc. Ser B, 39, I, ii-i7. (26] Stone, M. 1977. "Asymptotics for and against cross-validation." Biometrika. 64, I, 29-35. (27] Billingsley, Patrick. 1986. Probability Meaaure. Wiley, New York. and (28] Jennrich, R. 1969. "Asymptotic properties of nonlinear least squares estimators." Ann. Math. Stat. 40, 633-6i3. (29] Liu, Yong. 1993. "Neural network model selection using a.ymptotic jackknife estimator and cross-validation method." In Giles, C.L., Hanson, S.J., and Cowan, J.D. (eds.), Advances in neural information processing systems 5, San Mateo, CA: Morgan Kaufmann Publishers. (30] Moody, John E. 1992. "The effective number of parameters, an analysis of generalization and regularization in nonlinear learning system." In Moody, J.E., Hanson, S . J., and Lippmann, R.P., (eds.), Advances in neural information processing systems i, San Mateo, CA: Morgan Kaufmann Publishers . (31] Bailey, Timothy L. and Charles Elk&n. 1993. "Estimating the accuracy of learned concepts." To appear in Proc. International Joint Conference on Artificial Intelligence. (32] White, Halbert. 1993. 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Segmental Neural Net Optimization for Continuous Speech Recognition Ymg Zhao Richard Schwartz John Makhoul George Zavaliagkos BBN System and Technologies 70 Fawcett Street Cambridge MA 02138 Abstract Previously, we had developed the concept of a Segmental Neural Net (SNN) for phonetic modeling in continuous speech recognition (CSR). This kind of neural network technology advanced the state-of-the-art of large-vocabulary CSR, which employs Hidden Marlcov Models (HMM), for the ARPA 1oo0-word Resource Management corpus. More Recently, we started porting the neural net system to a larger, more challenging corpus - the ARPA 20,Ooo-word Wall Street Journal (WSJ) corpus. During the porting, we explored the following research directions to refine the system: i) training context-dependent models with a regularization method; ii) training SNN with projection pursuit; and ii) combining different models into a hybrid system. When tested on both a development set and an independent test set, the resulting neural net system alone yielded a perfonnance at the level of the HMM system, and the hybrid SNN/HMM system achieved a consistent 10-15% word error reduction over the HMM system. This paper describes our hybrid system, with emphasis on the optimization methods employed. 1 INTRODUCTION Hidden Martov Models (HMM) represent the state-of-the-art for large-vocabulary continuous speech recognition (CSR). Recently, neural network technology has been shown to advance the state-of-the-art for CSR by integrating neural nets and HMMs [1,2]. In principle, the advance is based on the fact that neural network modeling can avoid some limitations of the HMM modeling, for example, the conditional-independence assumption of HMMs and the fact that segmental features are hard to incorporate. Our work has been based on the concept of a Segmental Neural Net (SNN) [2]. 1059 1060 Zhao, Schwartz, Makhoul, and Zavaliagkos A segmental neural network is a neural network that attempts to recognize a complete phoneme segment as a single unit. Its basic structure is shown in Figure 1. The input to the network is a fixed length representation of the speech segment, which is obtained from the warping (quasi-linear sampling) of a variable length phoneme segment. If the network is trained to minimize a least squares error or a cross entropy distortion measure, the output of the network can be shown to be an estimate of the posterior probability of the phoneme class given the input segment [3,4]. .core neural network warping phonetic .egment Figure 1: The SNN model samples the frames and produces a single segment score. Our inith1 SNN system comprised a set of one-layer sigmoidal nets. This system is trained to minimize a cross entropy distortion measure by a quasi-Newton error minimization algorithm. A vanable length segment is warped into a fixed length of 5 input frames. Since each frame includes 16 feature~, 14 mel cepstrum, power and difference of power, an input to the neural network forms a 16 x 5 = 80 dimensional vector. Previously, our experimental domain was the ARPA 1000-word Resource Management (RM) Corpus, where we used 53 output phoneme classes. When tested on three independent evaluation sets (Oct 89, Feb 91 and Sep 92), our system achieved a consistent 10-20% word error rate reduction over the state-of-the-art HMM system [2]. 2 THE WALL STREET JOURNAL CORPUS After the final September 92 RM corpus evaluation, we ported our neural network system to a larger corpus - the Wall Street Journal (WSJ) Corpus. The WSJ corpus consists primarily of read speech, with a 5,000- to 20,000-word vocabulary. It is the current ARPA speech recognition research corpus. Compared to the RM corpus, it is a more challenging corpus for the neural net system due to the greater length of WSJ utterances and the higher perplexity of the WSJ task. So we would expect greater difficulty in improving perfOITnClllCe on the WSJ corpus. Segmental Neural Net Optimization for Continuous Speech Recognition 3 TRAINING CONTEXT-DEPENDENT MODELS WITH REGULARIZATION 3.1 WHY REGULARIZATION In contrast to the context-independent modeling for the RM corpus, we are concentrating on context-dependent modeling for the WSJ corpus. In context-dependent modeling, instead of using a single neural net to recognize phonemes in all contexts, different neural networks are used to recognize phonemes in different contexts. Because of the paucity of training data for some context models, we found that we had an overfitting problem. Regularization provides a class of smoothing techniques to ameliorate the overfitting problem [5]. We started using regularization in our initial one-layer sigmoidal neural network system. The regularization tenn added here is to regulate how far the context-dependent parameters can move away from their initial estimates, which are context-independent parameters. TIus is different from the usual weight decay technique in neural net literature, and it is designed specifically for our problem. The objective function is shown below: , ~ ~ [~IOg(1 - J.) + ~ log f} :'IIW d J v ~ .WolI~ (I) Regulanzatton Tenn Distortion measure Er(W) where Ii is the net output for class i, II W II is the Euclidean nonn of all weights in all the networks, IIWol1 is the initial estimate of weights from a context-independent neural network, Nd is the number of data points. ). is the regularization parameter which controls the tradeoff between the "smoothness" of the solution, as measured by IIW - Wo11 2 , and the deviation from the data as measured by the distortion. The optimal )., wllich gives the best generalization to a test set, can be estimated by generalized cross-validation [5]. If the distortion measure as shown in (2) (2) is a qu:.:-ctratic function in tenns of network weights W, the optimal ). is that which gives the minin,um of a generalized cross-validation index V().) [6]: V(),) = IIA()') - Nt d bW I - ~d tr(A()')) (3) where A(>.) = A(AT A+Nd)'I)A T . V()') is an easily calculated function based on singular value decomposition (SVD): (4) where A = U DV T , singular decomposition of A, z = U T b. Figure 2 shows an example plot of V().). A typical optimal ). has an inverse relation to the number of samples in each class, indicating that ). is gradually reduced with the presence of more data. 1061 1062 Zhao, Schwartz, Makhoul, and Zavaliagkos CI: ~ ~------------------------------------------. o 8 o 8 ? ~ I I ~ R I o 2.5*1011 -6 5*10"-7 lambda Figure 2: A typical V(A) Just as the linear least squares method can be generalized to a nonlinear least squares problem by an iterative procedure, so selecting the optimal value of the regularization parameter in a quadratic error criterion can be generalized to a non-quadratic error criterion iteratively. We developed an iterative procedure to apply the cross-validation technique to a non-..]uadratic error function, for example, the cross-entropy criterion Er(W) in (1) as follows: 1. Compute distortion Er(Wn ) for an estimate W n ? 2. Compute gradient gn and Hessian Hn of the distortion Er(Wn ). 3. Compute the singular value decomposition of Hn = V! Dn Vn . Set Zn = v"gn. 4. Evaluate a generalized cross-validation index Vn(A) similar to (2) as follows, for a range of A'S and select the An that gives the minimum Vn ? Segmental Neural Net Optimization for Continuous Speech Recognition N [E r(Hn TifT) d " !dj+Nd.\ 2] - L.Jj (dj+Nd.\)2Zn VnC\) = 2 [Nd - (5) Lj dj:1vd'\] 5. Set Wn+l = Wn - (Hn + NdAn)-lgn. 6. Go to 1 and iterate. Note that A is adjusted at each iteration. The final value of An is taken as the optimal A. Iterative regularization parameter selection shows that A converges, for example, to 1~~' from one of our experiments. 3.2 A TWO-LAYER NEURAL NETWORK SYSTEM WITH REGULARIZATION We then extended our regularization work from the one-layer sigmoidal network system to a two-layer sigmoidal network system. The first layer of the network works as a feature extractor and is shared by all phonetic classes. Theoretically, in order to benefit from its larger capability of representing phonetic segments, the number of hidden units of a two-layer network should be much greater than the number of input dimensions. However, a large number of hidden units can cause serious overfitting problems when the number of training ~amples is less than the number of parameters for some context models. Therefore, regularization is more useful here. Because the second layer can be trained as a one-layer net, the regularization techniques we developed for a one-layer net can be applied here to train the second layer. In our implementation, a weighted least squares error measure was used at the output layer. First, the weights for the two-layer system were initialized with random numbers between -1 and 1. Fixing the weights for the second layer, we trained the first layer by using gradient descent; then fixing the weights for the first layer, we trained the second layer by linear least squares with a re gularization term, without the sigmoidal function at the output. We stopped after one iteration for our initial experiment. 4 TRAINING SNN WITH PROJECTION PURSUIT 4.1 WHY PROJECTION PURSUIT As we described in the previous section, regularization is especially useful in training the second layer of a two-layer network. In order to take advantage of the two-layer layer structure, we want to train the first layer as well. However, once the number of the hidden units is large, the number of weights in the first layer is huge, which makes the first layer very difficult to train. Projection pursuit presents a !lseful technique to use a large hidden layer but still keep the number of weights in the first layer as small as possible. The original pJojection PU13Uit is a nonparametric statistical technique to find interesting low dimensional projections of high dimensional data sets [7]. The parametric version of it, a projection pursuit learning network. (pPLN) has a structure very similar to a two-layer sigmoidal network network [7]. In a traditional two-layer neural network, the weights in the first layer can be viewed as hypetplanes in the input space. It has been proposed that a special function of the first layer is to partition the input space into cells through these hyperplanes [8]. The second layer groups these cells together to form decision regions. 1063 1064 Zhao , Schwartz, Makhoul, and Zavaliagkos The accuracy or resolution of the decision regions is completely specified by the size and density of the cells which is detennined by the number and placement of the first layer hyperplanes in the input space. In a two-layer neural net, since the weights in the first layer can go anywhere, there are no restrictions on the placement of these hyperplanes. In contrast, a projection pursuit learning network. restricts these hyperplanes in some major "interesting" directions. In other words, hidden units are grouped into several distinct directions. Of course, with this grouping, the number of cells in the input space is reduced somewhat. However, the interesting point here is that this resoiction does not reduce the number of cells asymptotically [7]. In other words, grouping hidden units does not affect the number of cells much. Consequently, for a fixed number of hidden units, the number of parameters in the first layer in a projection pursuit learning network. is much less than in a traditional neural network. Therefore, a projection pursuit learning network is easier to train and generalizes better. 4.2 HOW TO TRAIN A PPLN In our implementation, the distinct projection directions were shared by all contextdependent models, and they were trained context-independently. We then trained these direction parameters with back-propagation. The second layer was trained with regularization. Iterations can go back and forth between the two layers. 5 COMBINATIONS OF DIFFERENT MODELS In the last two sections, we talked about using regularization and projection pursuit to optimize our neural network system. In this section, we will discuss another optimization method, combining different models into a hybrid system. The combining method is based on the N-best rescoring paradigm [2]. The N-best rescoring paradigm is a mechanism that allows us to build a hybrid system by combining different knowledge sources. For example, in the RM corpus, we successfully combined the HMM: system, th~ SNN system and word-pair grammar into a single hybrid system which achieved the state-of-the-art. We have been using this N-best rescoring paradigm to combine different models in the WSJ corpus as well. These different models include SNN left context, right context, and diphone models, HMM models, and a language model known as statistical grammar. We will show how to obtain a reasonable combination of different systems from Bayes rule. The goal is to compute P(SIX), the probability of the sentence 5 given the observation sequence X. From Bayes rule, P(SIX)SNN = P(S)P(XIS) P(X) ~ P(S) II P(xIS) P(x) x ~ P(S) II P(xlp, c) x ~ P(S) P(x) II P(Plx, c) :r: P(plc) Segmental Neural Net Optimization for Continuous Speech Recognition where X is a sequence of acoustic features x in each phonetic segment; p and c is the phoneme class and context for the segment, respectively. The following three approximations are used here: ? P(XIS) = Ox P(xIS). ? P(:z:IS) = P(:z:lp, c). ? P(clx) = P(c). Therefol'e, in a SNN system, we use the following approximation from Bayes rule: P(SIX)N N ~ P(S) II P~~~;) x where P(S): Word grammar score. Ox P(plx, c): Neural net score. Ox P(plc): Phone grammar score. These three scores together with HMM scores are combined in the SNN/HMM hybrid system. 6 EXPERIMENTAL RESULTS HMM Baseline SNN RegtIlarization and P!'ojection Pursuit SNN Baseline SNN/HMM Regularization and Projection Pursuit SNN/HMM Development Set 11.0 11. 7 11.2 10.3 9.5 Nov92 Test 8.5 9.1 7.7 7.2 ~~~~~=-~~==~~~------~----------~~--- Table 1: Word Error Rates for 5K, Bigram Grammar Regularization and Projection Pursuit SNN Regularization and Projection Pursuit SNN/HMM Development Set 14.4 14.6 13.0 Nov93 Test 14.0 12.3 Table 2: Word Error Rates for 20K, Trigram Grammar Speaker-independent CSR tests were performed on the 5,000-word (5K) and 20,000-word (20K) ARPA Wall Street Journal corpus. Bigram and trigram statistical grammars were used. The basic neural network structure consists of 80 inputs, 500 hidden units and 46 outputs. There are 125 projection directions in the first layer. Context models consist of 1065 1066 Zhao, Schwartz, Makhoul, and Zavaliagkos right context models and left diphone models. In the right context models, we used 46 different networks to recognize each phoneme in each of the different right contexts. In the left diphone models, a segment input consisted of the first half segment of the current phone plus the second half segment of the previous phone. Word error rates are shown in Tables 1 and 2. Comparing the first two rows of Table 1 and Table 2, we can see that the two-layer neural network system alone is at the level of state-of-the-art HMM systems. Shown in Row 3 and 5 of Table 1, regularization and projection pursuit improve the performance of neural net system. The hybrid SNN/HMM system reduces the word error rate 10%-15% over the HMM system in both tables. 7 CONCLUSIONS Neural net te':hnology is useful in advancing the state-of-the-art in continuous speech recognition system. Optimization methods, like regularization and projection pursuit, improve the performance of the neural net syst?:m. Our hybrid SNN/HMM system reduces the word error rate 10%-15% over the HMM system on 5,000-word and 20,000-word WSJ corpus. Acknowledgments This work was funded by the Advanced Research Projects Agency of the Department of Defense. References [1] M. Cohen, H. Franco, N. Morgan, D. Rumelhart and V. Abrash, "Context-Dependent Multiple Distribution Phonetic Modeling with :MLPS", in em Advances in Neural Information Processing Systems 5, eds. S. J. Hanson, J. D. Cowan and C. L. Giles. Morgan Kaufmann Publishers, San Mateo, 1993. [2] G. Zavaliagkos, Y. Zhao, R. Schwartz and J. Makhoul, " A Hybrid Neural Net System for State-of-the-Art Continuous Speech Recognition", in em Advances in Neural Information Processing Systems 5, eds. S. J. Hanson, J. D. Cowan and C. L. Giles. Morgan Kaufinann Publishers, San Mateo, 1993. [3] A. Barron, "Statistical properties of artificial neural networks," IEEE Cont Decision and Control, Tampa, FL, pp. 280-285, 1989. [4] H. Gish, "A probabilistic approach to the understanding and training of neural network classifiers," IEEE Int. Cont. Acoust.? Speech. Signal Processing, April 1990. [5] G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, 1990. [6] D. M. Bates, M. J. Lindstrom, G. Wahba and B. S. Yandell, "GCVPACK-Routines for Generalized Cross Validation", Comm. Statist.- Simula., 16(4), 1247-1253 (1987). [7] Y. Zhao and C. G. Atkeson, "Implementing Projection Pursuit Learning", to appear in Neural Computation, in preparation. [8] J. Makhoul, A. El-Jaroudi and R. Schwartz, "Partitioning Capabilities of 1\vo-Iayer Neural Networks", IEEE Transactions on Signal Processing, 39, pp. 1435-1440, 1991. PART X COGNITIVE SCIENCE 

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Convergence of Stochastic Iterative Dynamic Programming Algorithms Tommi Jaakkola'" Michael I. Jordan Satinder P. Singh Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract Increasing attention has recently been paid to algorithms based on dynamic programming (DP) due to the suitability of DP for learning problems involving control. In stochastic environments where the system being controlled is only incompletely known, however, a unifying theoretical account of these methods has been missing. In this paper we relate DP-based learning algorithms to the powerful techniques of stochastic approximation via a new convergence theorem, enabling us to establish a class of convergent algorithms to which both TD("\) and Q-Iearning belong. 1 INTRODUCTION Learning to predict the future and to find an optimal way of controlling it are the basic goals of learning systems that interact with their environment. A variety of algorithms are currently being studied for the purposes of prediction and control in incompletely specified, stochastic environments. Here we consider learning algorithms defined in Markov environments. There are actions or controls (u) available for the learner that affect both the state transition probabilities, and the probability distribution for the immediate, state dependent costs (Ci( u)) incurred by the learner. Let Pij (u) denote the probability of a transition to state j when control u is executed in state i. The learning problem is to predict the expected cost of a ... E-mail: tommi@psyche.mit.edu 703 704 Jaakkola, Jordan, and Singh fixed policy p (a function from states to actions), or to obtain the optimal policy (p*) that minimizes the expected cost of interacting with the environment. If the learner were allowed to know the transition probabilities as well as the immediate costs the control problem could be solved directly by Dynamic Programming (see e.g., Bertsekas, 1987). However, when the underlying system is only incompletely known, algorithms such as Q-Iearning (Watkins, 1989) for prediction and control, and TD(>.) (Sutton, 1988) for prediction, are needed. One of the central problems in developing a theoretical understanding of these algorithms is to characterize their convergence; that is, to establish under what conditions they are ultimately able to obtain correct predictions or optimal control policies. The stochastic nature of these algorithms immediately suggests the use of stochastic approximation theory to obtain the convergence results. However, there exists no directly available stochastic approximation techniques for problems involving the maximum norm that plays a crucial role in learning algorithms based on DP. In this paper, we extend Dvoretzky's (1956) formulation of the classical RobbinsMunro (1951) stochastic approximation theory to obtain a class of converging processes involving the maximum norm. In addition, we show that Q-Iearning and both the on-line and batch versions of TD(>.) are realizations of this new class. This approach keeps the convergence proofs simple and does not rely on constructions specific to particular algorithms. Several other authors have recently presented results that are similar to those presented here: Dayan and Sejnowski (1993) for TD(A), Peng and Williams (1993) for TD(A), and Tsitsiklis (1993) for Q-Iearning. Our results appear to be closest to those of Tsitsiklis (1993). 2 Q-LEARNING The Q-Iearning algorithm produces values-"Q-values"-by which an optimal action can be determined at any state. The algorithm is based on DP by rewriting Bellman 's equation such that there is a value assigned to every state-action pair instead of only to a state. Thus the Q-values satisfy Q(s,u) = cs(u) +, L....J ~pssl(u)maxQ(sl,ul) (1) 1.).1 8' where c denotes the mean of c. The solution to this equation can be obtained by updating the Q-values iteratively; an approach known as the vaz'ue iteration method. In the learning problem the values for the mean of c and for the transition probabilities are unknown. However, the observable quantity CSt (Ut) +, maxQ(St+l, u) (2) 1.). where St and Ut are the state of the system and the action taken at time t, respectively, is an unbiased estimate of the update used in value iteration. The Q-Iearning algorithm is a relaxation method that uses this estimate iteratively to update the current Q-values (see below). The Q-Iearning algorithm converges mainly due to the contraction property of the value iteration operator. Convergence of Stochastic Iterative Dynamic Programming Algorithms 2.1 CONVERGENCE OF Q-LEARNING Our proof is based on the observation that the Q-Iearning algorithm can be viewed as a stochastic process to which techniques of stochastic approximation are generally applicable. Due to the lack of a formulation of stochastic approximation for the maximum norm, however, we need to slightly extend the standard results. This is accomplished by the following theorem the proof of which can be found in Jaakkola et al. (1993). Theorem 1 A random iterative process ~n+I(X) = (l-ll:n(X))~n(x)+lin(x)Fn(x) converges to zero w.p.l under the following assumptions: 1) The state space is finite. 2) Ln ll:n(x) = 00, Ln ll:~(x) < 00, Ln lin(x) = E{lin(x)IPn } ~ E{ll:n(x)IPn } uniformly w.p.1. 3) 4) II 00, II ~n IlwI where'Y E (0,1). Var{Fn(x)IPn } ~ C(1+ II ~n Ilw)2, where C is some Ln Ii~(x) < 00, and E{Fn(x)IPn} Ilw~ 'Y constant. Here Pn = {~n, ~n-I, .. ?' Fn - I , ... , ll:n-I,? .. , lin-I, ... } stands for the past at step n. Fn(x), ll:n(x) and lin(x) are allowed to depend on the past insofar as the above conditions remain valid. The notation II . Ilw refers to some weighted maximum norm. In applying the theorem, the ~n process will generally represent the difference between a stochastic process of interest and some optimal value (e.g., the optimal value function). The formulation of the theorem therefore requires knowledge to be available about the optimal solution to the learning problem before it can be applied to any algorithm whose convergence is to be verified. In the case of Q-Iearning the required knowledge is available through the theory of DP and Bellman's equation in particular. The convergence of the Q-Iearning algorithm now follows easily by relating the algorithm to the converging stochastic process defined by Theorem 1.1 Theorem 2 The Q-learning algorithm given by Qt+I(St, Ut) = (1 - ll:t(St, Ut))Qt(St, ut) + ll:t(St, ut}[CSt(ut) + 'Yvt(St+dJ converges to the optimal Q*(s, u) values if 1) The state and action spaces are finite. 2) Lt ll:t(s, u) = 00 and Lt ll:;(s, u) < 00 uniformly w.p.1. 3) Var{cs(u)} is bounded. 1 We note that the theorem is more powerful than is needed to prove the convergence of Q-learning. Its generality, however, allows it to be applied to other algorithms as well (see the following section on TD(>.)). 705 706 Jaakkola, Jordan, and Singh 3) If, = 1, all policies lead to a cost free terminal state w.p.1. Proof. By subtracting Q*(s, u) from both sides of the learning rule and by defining Llt(s, u) = Qt(s, u) - Q*(s, u) together with (3) the Q-learning algorithm can be seen to have the form of the process in Theorem 1 with !3t(s, u) = at(s, u). To verify that Ft(s, u) has the required properties we begin by showing that it is a contraction mapping with respect to some maximum norm. This is done by relating F t to the DP value iteration operator for the same Markov chain. More specifically, maxIE{Ft(i, u)}1 u j < ,max ~Pij(u)maxIQt(j,v) - Q*(j,v)1 u 6 v j ,muax LPij(U)Va(j) = T(Va)(i) j where we have used the notation Va(j) = maXv IQt(j, v)-Q*(j, v)1 and T is the DP value iteration operator for the case where the costs associated with each state are zero. If, < 1 the contraction property of E{ F t (i, u)} can be obtained by bounding I:j Pij(U)Va(j) by maxj Va(j) and then including the, factor. When the future costs are not discounted (, = 1) but the chain is absorbing and all policies lead to the terminal state w.p.1 there still exists a weighted maximum norm with respect to which T is a contraction mapping (see e.g. Bertsekas & Tsitsiklis, 1989) thereby forcing the contraction of E{Ft(i, u)}. The variance of Ft(s, u) given the past is within the bounds of Theorem 1 as it depends on Qt(s, u) at most linearly and the variance of cs(u) is bounded. Note that the proof covers both the on-line and batch versions. 3 o THE TD(-\) ALGORITHM The TD(A) (Sutton, 1988) is also a DP-based learning algorithm that is naturally defined in a Markov environment. Unlike Q-learning, however, TD does not involve decision-making tasks but rather predictions about the future costs of an evolving system. TD(A) converges to the same predictions as a version ofQ-learning in which there is only one action available at each state, but the algorithms are derived from slightly different grounds and their behavioral differences are not well understood. The algorithm is based on the estimates V/\(i) = (1 - 00 A) L An-l~(n)(i) (4) n=l where ~(n)(i) are n step look-ahead predictions. The expected values of the ~>"(i) are strictly better estimates of the correct predictions than the lit (i)s are (see Convergence of Stochastic Iterative Dynamic Programming Algorithms Jaakkola et al., 1993) and the update equation of the algorithm Vt+l(it) = vt(it) + adV/(it) - (5) Vt(it)J can be written in a practical recursive form as is seen below. The convergence of the algorithm is mainly due to the statistical properties of the V? (i) estimates. 3.1 CONVERGENCE OF TDP) As we are interested in strong forms of convergence we need to impose some new constraints, but due to the generality of the approach we can dispense with some others. Specifically, the learning rate parameters an are replaced by a n ( i) which 00 and Ln a~(i) < 00 uniformly w.p.1. These parameters satisfy Ln an(i) allow asynchronous updating and they can, in general, be random variables. The convergence of the algorithm is guaranteed by the following theorem which is an application of Theorem 1. = Theorem 3 For any finite absorbing Markov chain, for any distribution of starting states with no inaccessible states, and for any distributions of the costs with finite variances the TD(A) algorithm given by 1) m Vn+1(i) t = Vn(i) + an(i) L)Ci + ,Vn(it+d t t=l Ln an(i) = 00 and Ln a~(i) < 00 Vn(it)] LbA)t-kXi(k) k=l uniformly w.p.i. 2) t Vt+l(i) = Vt(i) + at(i)[ci + ,Vt(it+d t Vt(id] LbA)t-kXi(k) k=l Lt at(i) = 00 and Ln a;(i) < 00 uniformly w.p.i and within sequences at(i)/maXtESat(i) ----;. 1 uniformly w.p.i. converges to the optimal predictions w.p.i provided" A E [0,1] with ,A < 1. Proof for (1): We use here a slightly different form for the learning rule (cf. the previous section). Vn(i) 1 + an (i)[Gn (i) - E~~~)} Vn(i)] m(i) E{m(i)} {; Vn"(i; k) where Vn"( i; k) is an estimate calculated at the ph occurrence of state i in a sequence and for mathematical convenience we have made the transformation an(i) ----;. E{m(i)}an(i), where m(i) is the number of times state i was visited during the sequence. 707 708 Jaakkola, Jordan, and Singh To apply Theorem 1 we subtract V* (i), the optimal predictions, from both sides of the learning equation. By identifying an(i) := an(i)m(i)/E{m(i)}, f3n(i) := an(i), and Fn(i) := Gn(i) - V*(i)m(i)/E{m(i)} we need to show that these satisfy the conditions of Theorem 1. For an(i) and f3n(i) this is obvious. We begin here by showing that Fn(i) indeed is a contraction mapping. To this end, m?xIE{Fn(i) 1Vn}1 I = miaxIE{~(i)} E{(VnA(i; 1) - V*(i? + (VnA(i;2) - V*(i? +???1 Vn}1 which can be bounded above by using the relation IE{VnA(i; k) - V*(i) 1Vn}1 ~ k, Vn}IO(m(i) - k) 1Vn } < E { IE{VnA(i; k) - V*(i) 1m(i) < P{m(i) ~ k}IE{VnA(i) - V*(i) 1 Vn}1 < I P {m( i) > k} m~x 1Vn (i) - V* (i) 1 I where O(x) = 0 if x < 0 and 1 otherwise. Here we have also used the fact that VnA(i) is a contraction mapping independent of possible discounting. As Lk P {m( i) ~ k} E{ m( i)} we finally get = m~x IE{ Fn( i) 1 Vn} 1 ::; I m?x IVn(i) - V*(i)1 I I The variance of Fn (i) can be seen to be bounded by E{ m4} m~xIVn(i)12 I For any absorbing Markov chain the convergence to the terminal state is geometric and thus for every finite k, E{mk}::; C(k), implying that the variance of Fn(i) is within the bounds of Theorem 1. As Theorem 1 is now applicable we can conclude that the batch version of TD(>.) converges to the optimal predictions w.p.l. 0 Proof for (2) The proof for the on-line version is achieved by showing that the effect of the on-line updating vanishes in the limit thereby forcing the two versions to be equal asymptotically. We view the on-line version as a batch algorithm in which the updates are made after each complete sequence but are made in such a manner so as to be equal to those made on-line. Define G~ (i) = G n (i) + G~ (i) to be a new batch estimate taking into account the on-line updating within sequences. Here G n (i) is the batch estimate with the desired properties (see the proof for (1? and G~ (i) is the difference between the two. We take the new batch learning parameters to be the maxima over a sequence, that is an(i) = maxtES at(i). As all the at(i) satisfy the required conditions uniformly w.p.1 these new learning parameters satisfy them as well. To analyze the new batch algorithm we divide it into three parallel processes: the batch TD( >.) with an (i) as learning rate parameters, the difference between this and the new batch estimate, and the change in the value function due to the updates made on-line. Under the conditions of the TD(>.) convergence theorem rigorous Convergence of Stochastic Iterative Dynamic Programming Algorithms upper bounds can be derived for the latter two processes (see Jaakkola, et al., 1993). These results enable us to write II E{G~ - V*} II < II E{Gn - V*} II + II G~ II < (-y' + C~) II Vn - V* II +C~ where C~ and C~ go to zero with w.p.I. This implies that for any c II Vn - V* II~ c there exists I < 1 such that I II E{G n - V*} II::; I II Vn - V* > 0 and II for n large enough. This is the required contraction property of Theorem 1. In addition, it can readily be checked that the variance of the new estimate falls under the conditions of Theorem 1. Theorem 1 now guarantees that for any c the value function in the on-line algorithm converges w.p.1 into some t-bounded region of V* and therefore the algorithm itself converges to V* w.p.I. 0 4 CONCLUSIONS In this paper we have extended results from stochastic approximation theory to cover asynchronous relaxation processes which have a contraction property with respect to some maximum norm (Theorem 1). This new class of converging iterative processes is shown to include both the Q-Iearning and TD(A) algorithms in either their on-line or batch versions. We note that the convergence of the on-line version of TD(A) has not been shown previously. We also wish to emphasize the simplicity of our results. The convergence proofs for Q-Iearning and TD(A) utilize only highlevel statistical properties of the estimates used in these algorithms and do not rely on constructions specific to the algorithms. Our approach also sheds additional light on the similarities between Q-Iearning and TD(A). Although Theorem 1 is readily applicable to DP-based learning schemes, the theory of Dynamic Programming is important only for its characterization of the optimal solution and for a contraction property needed in applying the theorem. The theorem can be applied to iterative algorithms of different types as well. Finally we note that Theorem 1 can be extended to cover processes that do not show the usual contraction property thereby increasing its applicability to algorithms of possibly more practical importance. References Bertsekas, D. P . (1987). Dynamic Programming: Deterministic and Stochastic Models. Englewood Cliffs, NJ: Prentice-Hall. Bertsekas, D. P ., & Tsitsiklis, J. N. (1989). Parallel and Distributed Computation: Numerical Methods. Englewood Cliffs, NJ: Prentice-Hall. Dayan, P. (1992). The convergence of TD(A) for general A. Machine Learning, 8, 341-362. 709 710 Jaakkola, Jordan, and Singh Dayan, P., & Sejnowski, T. J. (1993). TD(>.) converges with probability 1. CNL, The Salk Institute, San Diego, CA. Dvoretzky, A. (1956). On stochastic approximation. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. University of California Press. Jaakkola, T., Jordan, M. I., & Singh, S. P. (1993). On the convergence of stochastic iterative dynamic programming algorithms. Submitted to Neural Computation. Peng J., & Williams R. J. (1993). TD(>.) converges with probability 1. Department of Computer Science preprint, Northeastern University. Robbins, H., & Monro, S. (1951). A stochastic approximation model. Annals of Mathematical Statistics, 22, 400-407. Sutton, R. S. (1988). Learning to predict by the methods of temporal differences. Machine Learning, 3, 9-44. Tsitsiklis J. N. (1993). Asynchronous stochastic approximation and Q-learning. Submitted to: Machine Learning. Watkins, C.J .C.H. (1989). Learning from delayed rewards. PhD Thesis, University of Cambridge, England. Watkins, C.J .C.H, & Dayan, P. (1992). Q-learning. Machine Learning, 8, 279-292. 

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Neural Network Exploration Using Optimal Experiment Design David A. Cohn Dept. of Brain and Cognitive Sciences Massachusetts Inst. of Technology Cambridge, MA 02139 Abstract Consider the problem of learning input/output mappings through exploration, e.g. learning the kinematics or dynamics of a robotic manipulator. If actions are expensive and computation is cheap, then we should explore by selecting a trajectory through the input space which gives us the most amount of information in the fewest number of steps. I discuss how results from the field of optimal experiment design may be used to guide such exploration, and demonstrate its use on a simple kinematics problem. 1 Introduction Most machine learning research treats the learner as a passive receptacle for data to be processed. This approach ignores the fact that, in many situations, a learner is able, and sometimes required, to act on its environment to gather data. Learning control inherently involves being active; the controller must act in order to learn the result of its action. When training a neural network to control a robotic arm, one may explore by allowing the controller to "flail" for a length of time, moving the arm at random through coordinate space while it builds up data from which to build a model [Kuperstein, 1988]. This is not feasible, however, if actions are expensive and must be conserved. In these situations, we should choose a training trajectory that will get the most information out of a limited number of steps. Manually designing such trajectories is a slow process, and intuitively "good" trajectories often fail to sufficiently explore the state space [Armstrong, 1989]. In 679 680 Cohn this paper I discuss another alternative for exploration: automatic, incremental generation of training trajectories using results from "optimal experiment design." The study of optimal experiment design (OED) [Fedorov, 1972] is concerned with the design of experiments that are expected to minimize variances of a parameterized model. Viewing actions as experiments that move us through the state space, we can use the techniques of OED to design training trajectories. The intent of optimal experiment design is usually to maximize confidence in a given model, minimize parameter variances for system identification, or minimize the model's output variance. Armstrong [1989] used a form of OED to identify link masses and inertial moments of a robot arm, and found that automatically generated training trajectories provided a significant improvement over human-designed trajectories. Automatic exploration strategies have been tried for neural networks (e.g. [Thrun and Moller, 1992]' [Moore, 1994]), but use of OED in the neural network community has been limited. Plutowski and White [1993] successfully used it to filter a data set for maximally informative points, but its application to selecting new data has only been proposed [MacKay, 1992], not demonstrated. The following section gives a brief description of the relevant results from optimal experiment design. Section 3 describes how these results may be adapted to guide neural network exploration and Section 4 presents experimental results of implementing this adaptation. Finally, Section 5 discusses implications of the results, and logical extensions of the current experiments. 2 Optimal experiment design Optimal experiment design draws heavily on the technique of Maximum Likelihood Estimation (MLE) [Thisted, 1988]. Given a set of assumptions about the learner's architecture and sources of noise in the output, MLE provides a statistical basis for learning. Although the specific MLE techniques we use hold exactly only for linear models, making certain computational approximations allows them to be used with nonlinear systems such as neural networks. We begin with a training set of input-output pairs (Xi, Yi)i=l and a learner fw O? We define fw(x) to be the learner's output given input X and weight vector w. Under an assumption of additive Gaussian noise, the maximum likelihood estimate for the weight vector, W, is that which minimizes the sum squared error Esse = 2:7=1(JW(Xi) - Yi)2. The estimate W gives us an estimate of the output at a novel input: if = fw(x) (see e.g. Figure 1a). MLE allows us to compute the variances of our weight and output estimates. Writing the output sensitivity asgw(x) = 8fw(x)/8w, the covariances of ware where the last approximation assumes local linearity of gw(x). (For brevity, the output sensitivity will be abbreviated to g( x) in the rest of the paper.) Neural Network Exploration Using Optimal Experiment Design y Figure 1: a) A set of training examples for a classification problem, and the network's best fit to the data. b) Maximum likelihood estimate of the network's output variance for the same problem. For a given reference input Xr , the estimated output variance is var(x r ) = g(Xr? A- 1g(x r ). (1) Output variance corresponds to the model's estimate of the expected squared distance between its output fw(x) and the unknown "true" output y. Output variance then, corresponds to the model's estimate of its mean squared error (MSE) (see Figure 1b). If the estimates are accurate then minimizing the output variance would correspond to minimizing the network's MSE. In optimal experiment design, we estimate how adding a new training example is expected to change the computed variances. Given a novel X n +1, we can use OED to predict the effect of adding Xn+1 and its as-yet-unknown Yn+1 to the training set. We make the assumption that -1 ( An+1 ~ An + g(xn+dg(xn+d T)-1 , which corresponds to assuming that our current model is already fairly good. Based on this assumption, the new parameter variances will be A~~1 = A~l - A~1g(xn+d(1 + g(Xn+1? A~1g(xn+d)g(xn+t)T A~1. Combined with Equation 1, this predicts that if we take a new example at the change in output variance at reference input Xr will be ~var(Xr ) (g(xrf A~lg(xn+l?2(1 cov(xr, Xn+l)2(1 Xn +1, + g(X n+1)T A;;lg(xn+d) + var(xn+d) (2) To minimize the expected value of var(x r ), we should select Xn+l so as to maximize the right side of Equation 2. For other interesting OED measures, see MacKay [1992] . 681 682 Cohn 3 Adapting OED to Exploration When building a world model, the learner is trying to build a mapping, e.g. from joint angles to cartesian coordinates (or from state-action pairs to next states). If it is allowed to select arbitrary joint angles (inputs) in successive time steps, then the problem is one of selecting the next "query" to make ([Cohn, 1990], [Baum and Lang, 1991]). In exploration, however, one's choices for a next input are constrained by the current input. We cannot instantaneously "teleport" to remote parts of the state space, but must choose among inputs that are available in the next time step. One approach to selecting a next input is to use selective sampling: evaluate a number of possible random inputs, choose the one with the highest expected gain. In a high-dimensional action space, this is inefficient. The approach followed here is that of gradient search, differentiating Equation 2 and hillclimbing on 8jj,var( x r )/ 8X n +l. Note that Equation 2 gives the expected change in variance only at a single point X r , while we wish to minimize the average variance over the entire domain. Explicitly integrating over the domain is intractable, so we must make do with an approximation. MacKay [1992] proposed using a fixed set of reference points and measuring the expected change in variance over them. This produces spurious local maxima at the reference points, and has the undesirable effect of arbitrarily quantizing the input space. Our approach is to iteratively draw reference points at random (either uniformly or according to a distribution of interest), and compute a stochastic approximation of jj, var. By climbing the stochastically approximated gradient, either to convergence or to the horizon of available next inputs, we will settle on an input/action with a (locally) optimal decrease in expected variance. 4 Experimental Results In this section, I describe two sets of experiments. The first attempts to confirm that the gains predicted by optimal experiment design may actually be realized in practice, and the second studies the application of OED to a simple learning task. 4.1 Expected versus actual gain It must be emphasized that the gains predicted by OED are expected gains. These expectations are based on the relatively strong assumptions of MLE, which may not strictly hold. In order for the expected gains to materialize, two "bridges" must be crossed. First, the expected decrease in model variance must be realized as an actual decrease in variance. Second, the actual decrease in model variance must translate into an actual decrease in model MSE. 4.1.1 Expected decreases in variance --+ actual decreases in variance The translation from expected to actual changes in variance requires coordination between the exploration strategy and the learning algorithm: to predict how the variance of a weight will change with a new piece of data, the predictor must know how the weight itself (and its neighboring weights) will change. Using a black Neural Network Exploration Using Optimal Experiment Design 0 . 012 0.0 1 l xx I 2.4 ;If ,, )( ~ ~ > x 2 .8 x x x x X x ""; , 0 . 008 ;:: ... ~ ~ x ~ x x 'tl x XX 0 .00 6 [oJ 1. 6 x Vl i? :E x ..." x x 1. 2 ~ 'tl 0.8 0 . 004 x x x 0.4 0 . 00 2 - - - - - a ct ual =e x pec ted - 0. 4 0 . 002 0. 00 4 0 .006 0 . 0 08 0 . 01 0 . 01 2 0 .002 exp e c t e d d e lta var 0.004 0.00 6 0 .00 8 0 . 01 0 . 012 actua l d e lta va r Figure 2: a) Correlations between expected change in output variance and actual change output variance b) Correlations between actual change in output variance and change in mean squared error. Correlations are plotted for a network trained on 50 examples from the arm kinematics task. box routine like backpropagation to update the weights virtually guarantees that there will be some mismatch between expected and actual decreases in variance. Experiments indicate that, in spite of this, the correlation between predicted and actual changes in variance are relatively good (Figure 2a) . 4.1.2 Decreases in variance -- decreases in MSE A more troubling translation is the one from model variance to model correctness. Given the highly nonlinear nature of a neural network, local minima may leave us in situations where the model is very confident but entirely wrong. Due to high confidence, the learner may reject actions that would reduce its mean squared error and explore areas where the model is correct, but has low confidence. Evidence of this behavior is seen in the lower right corner of Figure 2b, where some actions which produce a large decrease in variance have little effect on the network's MSE. While this decreases the utility of OED, it is not crippling . We discuss one possible solution to this problem at the end of this paper . 4.2 Learning kinematics We have used the the stochastic approximation of ~var to guide exploration on several simple tasks involving classification and regression. Below , I detail the experiments involving exploration of the kinematics of a simple two-dimensional, two-joint arm . The task was to learn a forward model 8 1 x 8 2 - - X X Y through exploration, which could then be used to build a controller following Jordan [1992]. 683 684 Cohn The model was to be learned by a feedforward network with a sigmoid transfer function using a single hidden layer of 8 or 20 hidden units. Figure 3: Learning 2D arm kinematics with 8 hidden units. a) Geometry of the 2D, two-joint arm. b) Sample trajectory using OED-based greedy exploration. On each time step, the learner was allowed to select inputs 8 1 and 8 2 and was then given tip position x and y to incorporate into its training set. It then hillclimbed to find the next 8 1 and 8 2 within its limits of movement that would maximize the stochastic approximation of ~var . On each time step 8 1 and 8 2 were limited to change by no more than ?36? and ?18? respectively. Simulations were performed on the Xerion simulator (made available by the University of Toronto), approximating the variance gradient on each step with 100 randomly drawn points. A sample tip trajectory is illustrated in Figure 3b. We compared the performance of this one-step optimal (greedy) learner, in terms of mean squared error, with that of an identical learner which explored randomly by "flailing." Not surprisingly, the improvement of greedy exploration over random exploration is significant (Figure 4b). The asymptotic performance of the greedy learner was better than that of the random learner, and it reached its asymptote in much few steps. 5 Discussion The experiments described in this paper indicate that optimal experiment design is a promising tool for guiding neural network exploration. It requires no arbitrary discretization of state or action spaces, and is amenable to gradient search techniques. It does, however, have high computational costs and, as discussed in Section 4.1.2, may be led astray if the model settles in a local minimum. 5.1 Alternatives to greedy OED The greedy approach is prone to "boxing itself into a corner" while leaving important parts of the domain unexplored. One heuristic for avoiding local minima is to Neural Network Exploration Using Optimal Experiment Design I O. 28 ~ I 0.24 w ~\ I I o. 201 ~ Ul :0: 0 . 16 : : ::~ 0.21~ \\ I + I . \ O. 18~ \ ~ 0 . 15~ V 0.12-1 I I 0 . 08i 0 .0 4J 0.00 , .. . :::1\( \ 0 . 06 I ~ _ --T \ O. 0 3~ n I o. 001-----,-, e 20 40 80 60 Number of steps 100 120 o 20 ~~-=::::-./ , -,---,-, I , , 40 60 80 100120140160180200 Number of steps Figure 4: Learning 2D arm kinematics. a) MSE for a single exploration trajectory (20 hidden units). b) Plot of MSE for random and greedy exploration vs. number of training examples, averaged over 12 runs (8 hidden units). occasionally check the expected gain in other parts of the input space and move towards them if they promise much greater gain than a greedy step. The theoretically correct but computationally expensive approach is to optimize over an entire trajectory. Trajectory optimization entails starting with an initial trajectory, computing the expected gain over it, and iteratively perturbing points on the trajectory towards towards optimal expected gain (subject to other points along the trajectory being explored). Experiments are currently underway to determine how much of an improvement may be realized with trajectory optimization; it is unclear whether the improvement over the greedy approach will be worth the added computational cost. 5.2 Computational Costs The computational costs of even greedy OED are great . Selecting a next action requires computation and inversion of the hessian {)2 Eue/ ow 2 . Each time an action is selected and taken, the new data must be incorporated into the training set, and the learner retrained . In comparison, when using a flailing strategy or a fixed trajectory, the data may be gathered with little computation, and the learner trained only once on the batch. In this light, the cost of data must be much greater than the cost of computation for optimal experiment design to be a preferable strategy. There are many approximations one can make which significantly bring down the cost of OED. By only considering covariances of weights leading to the same neuron, the hessian may be reduced to a block diagonal form, with each neuron computing its own (simpler) covariances in parallel. As an extreme, one can do away with covariances entirely and rely only on individual weight variances, whose computation is simple. By the same token, one can incorporate the new examples in small batches, only retraining every 5 or so steps. While suboptimal from a data gathering perspective, they appear to still outperform random exploration, and are much cheaper than "full-blown" optimization. 685 686 Cohn 5.3 Alternative architectures We may be able to bring down computational costs and improve performance by using a different architecture for the learner. With a standard feedforward neural network, not only is the repeated compution of variances expensive, it sometimes fails to yield estimates suitable for use as confidence intervals (as we saw in Section 4.1.2). A solution to both of these problems may lie in selection of a more amenable architecture and learning algorithm . One such architecture, in which output variances have a direct role in estimation, is a mixture of Gaussians, which may be efficiently trained using an EM algorithm [Ghahramani and Jordan, 1994]. We expect that it is along these lines that our future research will be most fruitful. Acknowledgements I am indebted to Michael I. Jordan and David J .C. MacKay for their help in making this research possible. This work was funded by ATR Human Information Processing Laboratories, Siemens Corporate Research and NSF grant CDA-9309300. Bibliography B. Armstrong. (1989) On finding exciting trajectories for identification experiments. Int. J. of Robotics Research, 8(6):28-48. E. Baum and K. Lang. (1991) Constructing hidden units using examples and queries. In R . Lippmann et al., eds., Advances in Neural Information Processing Systems 3, Morgan Kaufmann, San Francisco, CA. D. Cohn, L. Atlas and R. Ladner. (1990) Training connectionist networks with queries and selective sampling. In D. Touretzky, editor, Advances in Neural Information Processing Systems 2, Morgan Kaufmann, San Francisco. V. Fedorov. (1972) Theory of Optimal Experiments. Academic Press, New York. Z. Ghahramani and M. Jordan. (1994) Supervised learning from incomplete data via an EM approach. In this volume. M. Jordan and D. Rumelhart. (1992) Forward models: Supervised learning with a distal teacher. Cognitive Science, 16(3):307-354. D. MacKay. (1992) Information-based objective functions for active data selection, Neural Computation 4(4): 590-604. A. Moore. (1994) The parti-game algorithm for variable resolution reinforcement learning in multidimensional state-spaces. In this volume. M. Plutowski and H. White. (1993) Selecting concise training sets from clean data. IEEE Trans. on Neural Networks, 4(2):305-318. R. Thisted. (1988) Elements of Statistical Computing. Chapman and Hall, NY. S. Thrun and K. Moller. (1992) Active Exploration in Dynamic Environments. In J. Moody et aI., editors, Advances in Neural Information Processing Systems 4. Morgan Kaufmann, San Francisco, CA. 

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Odor Processing in the Bee: a Preliminary Study of the Role of Central Input to the Antennal Lobe. Christiane Linster David Marsan ESPeI, Laboratoire d'Electronique 10, Rue Vauquelin, 75005 Paris linster@neurones.espci.fr Claudine Masson Michel Kerszberg Laboratoire de Neurobiologie Comparee des Invertebrees INRNCNRS (URA 1190) 91140 Bures sur Yvette, France masson@jouy.inra.fr Institut Pasteur CNRS (URA 1284) Neurobiologie Moleculaire 25, Rue du Dr. Roux 75015 Paris, France Abstract Based on precise anatomical data of the bee's olfactory system, we propose an investigation of the possible mechanisms of modulation and control between the two levels of olfactory information processing: the antennallobe glomeruli and the mushroom bodies. We use simplified neurons, but realistic architecture. As a first conclusion, we postulate that the feature extraction performed by the antennallobe (glomeruli and interneurons) necessitates central input from the mushroom bodies for fine tuning. The central input thus facilitates the evolution from fuzzy olfactory images in the glomerular layer towards more focussed images upon odor presentation. 1. Introduction Honeybee foraging behavior is based on discrimination among complex odors which is the result of a memory process involving extraction and recall of "key-features" representative of the plant aroma (for a review see Masson et al. 1993). The study of the neural correlates of such mechanisms requires a determination of how the olfactory system successively analyses odors at each stage (namely: receptor cells, antennal lobe interneurons and glomeruli, mushroom bodies). Thus far, all experimental studies suggest the implication of both antennallobe and mushroom bodies in these processes. The signal transmitted by the receptor cells is essentially unstable and fluctuating. The antennallobe appears as the location of noise reduction and feature extraction. The specific associative components operating on the olfactory memory trace would be essentially located in the mushroom bodies. The results of neuroethological experiments indicate furthermore that both the 527 528 Linster, Marsan, Masson, and Kerszberg feed-forward connections from the antennal lobe projection neurons to the mushroom bodies and the feedback connections from the mushroom bodies to the antennal lobe neurons are crucial for the storage and the recall of odor signals (Masson 1977; Erber et al. 1980; Erber 1981). Interestingly, the antennallobe compares to the mammalian olfactory bulb. Computational models of the insect antennal lobe (Kerszberg and Masson 1993; Linster et aI. 1993) and the mammalian olfactory bulb (Anton et a1. 1991; Li and Hopfield 1989; Schild 1988) have demonstrated that feature extraction can be performed in the glomerular layer, but the possible role of central input to the glomerular layer has not been investigated (although it has been included, as a uniform signal, in the Li and Hopfield model). On the other hand, several models of the mammalian olfactory cortex (Hasselmo 1993; Wilson and Bower 1989; LiljenstrOm 1991) have investigated its associative memory function, but have ignored the nature of the input from the olfactory bulb to this system. Based on anatomical and electrophysiological data obtained for the bee's olfactory system (Fonta et aI. 1993; Sun et al. 1993), we propose in this paper to investigate of the possible mechanisms of modulation and control between the two levels of olfactory information processing in a formal neural model. In the model, the presentation of an "odor" (a mixture of several molecules) differentially activates several populations of glomeruli. Due to coupling by local interneurons, competition is triggered between the activated glomeruli, in agreement with a recent proposal (Kerszberg and Masson 1993). We investigate the role of the different types of neurons implicated in the circuitry, and study the modulation of the glomerular states by reentrant input from the upper centers in the brain (i.e. mushroom bodies). 2. Olfactory circuitry in the bee's antennal lobe and mushroom bodies 95% of sensory cells located on the bee's antenna are olfactory (Esslen and Kaissling 1976), and convey signals to the antennal lobes. In the honeybee, due to some overlap of receptor cell responses, the peripheral representation of an odor stimulus is represented in an across fiber code (Fonta et al. 1993). Sensory axons project on two categories of antennal lobe neurons, namely local interneurons (LIN) and output neurons (ON). The synaptic contacts between sensory neurons and antennal lobe neurons, as well as the synaptic contacts between antennallobe neurons are localized in areas of high synaptic density, the antennal lobe glomeruli; each glomerulus represents an identifiable morphological neuropilar sub-unit (of which there are 165 for the worker honeybee) (Arnold et aI. 1985). Local interneurons constitute the majority of antennallobe neurons, and there is evidence that a majority of the LINs are inhibitory. As receptor cells are supposed to synapse mainly with LINs, the high level of excitation observed in the responses of ONs suggests that local excitation also exists (Malun 1991), in the form of spiking or non-spiking LINs, or as a modulation of local excitatbility. All LINs are pluriglomerular, but the majority of them, heterogeneous local interneurons (or HeteroLINs), have a high density of dendrite branches in one particular glomerulus, and sparser branches distributed across other glomeruli. A second category, homogeneous local interneurons (or Homo LINs), distribute their branches more homogeneously over the whole antennal lobe. Similarly, some of the ONs have dendrites invading only one glomerulus (Uniglomerular, or Uni ON), whereas the others (PI uri ON) are pluriglomerular. The axons of both types of ON project to different areas of the protocerebrum, including the mushroom bodies (Fonta et aI. 1993). Odor Processing in the Bee 3. Olfactory processing in the bee's antennal lobe glomeruli Responses of antennal lobe neurons to various odor stimuli are characterized by complex temporal patterns of activation and inactivation (Sun et al. 1993). Intracellularly recorded responses to odor mixtures are in general very complex and difficult to interpret from the responses to single odor components. A tendency to select particular odor related information is expressed by the category of "localized" antennallobe neurons, both Hetero LlNs and Uni ONs. In contrast, "global" neurons, both Homo LINs and Pluri ONs are often more responsive to mixtures than to single components. This might indicate that the related localized glomeruli represent functional sub units which are particularly involved in the discrimination of some key features. An adaptation of the 2DG method to the honeybee antennallobe has permitted to study the spatial distribution of odor related activity in the antennal lobe glomeruli (Nicolas et al. 1993; Masson et al. 1993). Results obtained with several individuals indicate that a correspondence can be established between two different odors and the activity maps they induce. This suggests that in the antennal lobe, different odor qualities with different biological meaning might be decoded according to separate spatial maps sharing a number of common processing areas. 4. Model of olfactory circuitry In the model, we introduce the different categories of neurons described above (Figure 1). Glomeruli are grouped into several regions and each receptor cell projects onto all local interneurons with arborizations in one region. Interneurons corresponding to heterogeneous LlNs can be (i) excitatory, these have a dendritic arborization (input and output synapses) restricted to one glomerulus; they provide "local" excitation, or, (ii) inhibitory, these have a dense arborization (mainly input synapses) in one glomerulus and sparse arborizations (mainly output synapses) in all others; they provide "local inhibition" and "lateral inhibition" between glomeruli. Interneurons corresponding to homogeneous LINs are inhibitory and have sparse arborizations (input and output synapses) in all glomeruli; they provide "uniform inhibition" over the glomerular layer. Output neurons are postsynaptic only to interneurons, they do not receive direct input from receptor cells. Each output neuron collects information from all interneurons in one glomerulus: thus modeling uniglomerular ONs. Implementation: The different neuron populations associated with one glomerulus are represented in the program as one unit (each unit is governed by one differential equation); the output of one unit represents the average firing probability of all neurons in this population (assuming that on the average, all neurons in one population receive the same input and have the same intrinsic properties). All units have membrane constants and a non-linear output function. Connection delays and connection strengths between units are chosen randomly around an average value: this assures a "realistic spatial averaging" over populations. The differential equations associated with the units are translated into difference equations and simulated by synchronous updating (sampling step Sms). 529 530 Linster, Marsan, Masson, and Kerszberg Molecule spectra Receptor cell types Receptor input ~ Global inhibition Glomerular region ___ Local inhibition and lateral inhibition "'-. Local modulation . :. :;:: '.:',' Modulation of global inhibition " Global inhibitory interneuron Localized output neuron Localized excitatory interneuron Localized inhibitory interneuron Figure 1: Organization of the model olfactory circuitry. In the model, we introduce receptor cells with overlapping molecule spectra; each receptor cell has its maximal spiking probability P for the presence of a particular molecule i. The axons of the receptor cells project into distinct regions of the glomerular layer. All allowed connections exist with the same probability, but with different connection strengths. The activity of each glomerulus is represented by its associated output neurons. Central input projects onto the global inhibitory interneurons (modulation of global inhibition) or on all interneurons in one glomerulus (local modulation). o ? o 5. Olfactory processing by the model circuitry In the model, odors are represented as one-dimensional arrays of molecules; each molecule can be present in varying amounts. Due to the gaussian distributions of receptor cell sensitivities, an active molecule activates more than one receptor cell (with varying degrees of activation). As each receptor cell projects into all glomeruli belonging to its target region, thus, a molecular bouquet differentially activates a number of glomeruli in different glomerular regions. This triggers several phenomena: (i) due to the excitatory elements local to each glomerulus, and activated glomerulus tends to enhance the activation it receives from the receptor cells, (ii) the local inhibitory elements are activated (with a certain delay) by the receptor cell activity and by the self-activation of the local excitatory elements, and, (iii) trend to inhibit neighboring glomeruli. These phenomena result in a competition between active glomeruli: during a number of sampling steps, the output activity of each glomerulus (represented by the firing probability of the associated output neuron), oscillates from high activity to low activity. Due to the competition provided by Odor Processing in the Bee the lateral inhibition, the spatial oscillatory activity pattern changes over time, and a stable activity map is reached eventually. A number of glomeruli "win" and stay active, whereas others "loose" and are inhibited (Figure 2). The activities of individual output neurons follow the general pattern described above: oscillation of the activity during a number of sampling steps until the activity "settles" down to a stable value. A stable activity can either be a constant firing probability, or a "stable" oscillation of the firing probability. An output neuron associated to a particular glomerulus may be active for a particular odor input, and silent for others. Complex temporal patterns of excitation and inhibition may occur after stimulus presentation. Thus, the model predicts that odor representation is performed through spatial maps of activity spanning the whole glomerular layer. Individual output neurons, representing the activity of their associated glomeruli may be either excited or inhibited by a particular odor pattern. Glomeruli After stabilization 1 - 15 Figure 2: Behavior of the model after stimulation of the receptor cells with the molecule array indicated in the figure. For several sampling steps (of 5 ms), the activity (firing probability) of the ON associated to each glomerulus is shown. At step I, all glomeruli are differentially activated by the receptor cell input. Lateral inhibition silences all glomeruli during the next sampling step. At step 3, some glomeruli are highly activated (due to their local excitation), whereas others are almost silenced. Then, t spatial activation pattern oscillates for a number of sampling steps (which depends on the strength of the lateral inhibitory connections and on the number of active molecules in the odor array), and finally stabilizes in a spatial activity map. 6. Comparison of odor processing in the Bee's antennal lobe and in the model Antennallobe neurons in the bee show various response patterns to stimulation with pure components and mixtures. Most LINs and ONs respond with simple excitation or inhibition to stimulation, often followed by a hyperpolarized (resp. depolarized) phase. Interestingly, most LINs respond with various degrees of excitation to stimulation with binary odors and mixtures, whereas ONs respond equally often by excitation than by inhibition (Sun et a1. 1993). In the model, LINs receive direct afferent input from receptor cells, and are therefore differentially activated by odor stimulation; they respond with varying degrees of excitation to stimulation with pure components and their mixtures. Output neurons in the model receive indirect input from receptor cells via local interneurons. Output neurons in the model are either activated (if their associated 531 532 Linster, Marsan, Masson, and Kerszberg glomerulus wins the competition) or inhibited (if their associated glomerulus looses the competition) by odor stimulation. In the simulations, output neurons which are excited for a particular odor stimulation belong to an active glomerulus in the spatial activity map associated to that odor. For each odor, a particular activity map is established. An output neuron is either excited or inhibited by a particular odor stimulation, indicating that it takes part in the representation of an activity map across glomeruli, which might be compared to the antennallobe 2DG maps. 7. Modulation of the model dynamics Odor detection by modulation of spontaneous activity At high spontaneous activity, all glomeruli in the model oscillate spontaneously (Figure 3). Odor stimulation tends to synchronize these oscillations, but no feature detection is perfonned. In the model, the underlying activity map which corresponds to the odor signal can only emerge if the spontaneous activity is decreased (Figure 3). Decreasing of the spontaneous activity can be achieved by 5i) activation of the global inhibitory interneurons by central input, or, (ii) decreasing of the spiking threshold of all antennallobe neurons. These data fit well with experimental data (see Sun et al. 1993, Figures 7 and 8). _: ; '; "'j ;' .. ; TT?t?1"Ttrrt" .:&.&&~ &.~ . Stimulus annlication , .II Ii. 500ms Reduction of spontaneous activity Figure 3 Figure 3: Modulation of the spontaneous activity. We show the spiking probabilities of output neurons associated to different glomeruli. Arrows indicate stimulus onset. Stimulus presentation synchronizes the oscillations. A decreasing of the spontaneous activity results in the emergence of the underlying activity map: several output neurons exhibit high activities, whereas the others are silent. Contrast enhancement by modulation of lateral inhibition Presentation of an odor in the model differentially activates many or all glomeruli, which, due to the local excitation, try to enhance the activation due to the odor stimulus. Due to the competition between glomeruli, feature detection is performed in the glomerular layer, which enhances some elements of the stimulus and suppresses others. In the model, for a given odor stimulation, the number of winning glomeruli depends on the strength of the lateral inhibition between glomeruli (Figure 5). At low lateral inhibition, most glomeruli stay active for any odor; no feature extraction is perfonned. Odor Processing in the Bee Increasing of the lateral inhibition focuses the odor maps, which can now differentiate different odor inputs. annat'\(')(\(')('\. ()~ . . . .( ) n e ( ) ( ) . . . .. . . . . . . U II1J D [l] [[] m DDLJ[]D .00.00000. 0 ??000.00. ? ???00.00. ? ????????? .ooeooooo. 0 ??000.00. ? ???00.00. ? ????????? eooeooooo. 0 ?? 000.00. ? ?? eooeoo. ? ......... Figure 5: Stabilized activity maps for different odor stimuli with increasing lateral inhibition strength. At low competition, all glomeruli tend to be active due to their local excitation. Increasing of lateral inhibition permits to enhance the important features of each odor, and leads to uncorrelated activity maps for the different stimulations. Increasing of the lateral inhibition permits to focus a fuzzy olfactory image in the glomerular layer, or to "smell closer". A fuzzy sampling of an odor may be useful at first approach, whereas a more precise analysis of its important components is facilitated by increasing the competition between glomeruli increases contrast enhancement. 8. Discussion We have presented the computational abilities of the neural circuitry in the antennallobe model, based on what is known of the bee's circuitry. Single cell responses and global activity patterns are comparable to the odor processing mechanisms proposed in the insect (Linster et al. 1993; Masson et al. 1993; Kerszberg and Masson 1993) and in the vertebrate (Kauer et al. 1991; Li and Hopfield 1989; Freeman 1991) literature. As suggested by Kerszberg and Masson (1993), we show that odor preprocessing is based on spontaneous dynamics of the antennal lobe glomeruli, and that, in addition, feature detection needs competition between activated glomeruli due to global and lateral inhibition. The model is able to predict the role of the four types of neurons morphologically identified in the bee an ten nal lobe. It also predicts how intracellular recordings and 2DG data can be explained by the odor processing mechanism. Furthermore, modulation of the models dynamics opens up a number of new ideas about the respective role of the two main categories ("localized" and "global") of antennallobe neurons, and the possible role of central input to these neurons. Acknowledgements The authors are grateful to G. Dreyfus and L. Personnaz for fruitful discussions. 533 534 Linster, Marsan, Masson, and Kerszberg References Arnold, G., Masson, C., Budhargusa, S. 1985. Comparative study of the antennal pathway of the workerbee and the drone (Apis mellifera). Cell Tissue Res. 242: 593-605. Erber, J. 1981 Neural correlates of learning in the honeybee. TINS 4:270-273. Erber, J., Masuhr, T., Menzel, R. 1980. Localisation of shon-term memory in the brain of the bee, Apis melli/era. Physiolo. Entomol. 5: 343-358. EssIen, J., Kaissling, K.E. 1976. Zahl und Verteilung antennaler Sensillen bei der Honigbiene. Zoomorphologie 83: 227-251. Fonta, C., Sun, X., Masson, C. 1193. Morphology and spatial distribution of bee antennallobe interneurons responsive to odours. Chemical Senses, 18 (2): pp. 101119. Hasselmo, M.E. 1993. Acetycholine and Learning in a Conical Associative Memory. Neural Computation, 5: 32-44. Kauer, J.S., Neff, S.R., Hamilton, K.A., Cinelli, A.R. 1991. The Salamander Olfactory Pathway: Visualizing and Modeling Circuit Activity. in Olfaction: A Model System for Computational Neuroscience. Davis, J. and Eichenbaum, H. (eds): 4468. MIT Press. Kerszberg, M., Masson, C., 1993. Signal Induced Selection among Spontaneous Activity Patterns of Bee's Olfactory Glomeruli, submitted. Li, Z., Hopfield, J.1., 1989. Modeling the Olfactory Bulb and its Neural Oscillatory Processings. Biological Cybernetics 61:379-392. LiljenstrOm, H. 1991. Modeling the dynamics of olfactory cortex using simplified network units and realistic architecture. International Journal of Neural Systems, (1&2): 115. Linster, C., Masson, C., Kerszberg, M., Personnaz, L., Dreyfus, G. 1993 Computational Diversity in a Fonnal Model of the Insect Macroglomerulus., Neural Computation, 5:239-252. Malun, D. 1991. Inventory and distribution of synapses of identified uniglomerular projection neurons in the antennallobe of periplaneta americana. J.Comp. Neurol. 305: 348-360. Masson, C. 1977. Central olfactory pathways and plasticity of responses to odor stimuli in insects. in Olfaction and Taste VI. Le Magnen, J., Mac Leod, P. (eds) IRL, London: 305-314. Masson, C., Mustaparta, H. 1990. Chemical Information Processing in the Olfactory System of Insects. Physiol. Reviews 70(1):199-245. Masson, C., Pham-Delegue, MH., Fonta, C., Gascuel, J., Arnold, G., Nicolas, G., Kerszberg, M. 1993. Recent advances in the concept of adaptation to natural odour signals in the honeybee Apis mellifera L. Apidologie 24: 169-194. Menzel, R. 1983. Neurobiology of learning and memory: the honeybee as a model system. Naturwissenschaften 70: 504-511. Nicolas, G., Arnold, G., Patte, F., Masson, C. 1993. Distribution regionale de l'incorporation du 3H2-Desoxyglucose dans Ie lobe antennaire de l'ouvriere d'abeille. CR. Acad. Sc. Paris (Sciences de la Vie), 316: 1245-1249. Schild , D. 1988 Principles of odor coding and a neural network for odor discrimination, Biophys. J. 54:1001-101l. Sun, X., Fonta, C., Masson, C. 1993. Odour quality processing by bee antennal lobe neurons. Chemical Senses 18 (4): 355-377. 

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Supervised learning from incomplete data via an EM approach Zoubin Ghahramani and Michael I. Jordan Department of Brain & Cognitive Sciences Massachusett.s Institute of Technology Cambridge, MA 02139 Abstract Real-world learning tasks may involve high-dimensional data sets with arbitrary patterns of missing data. In this paper we present a framework based on maximum likelihood density estimation for learning from such data set.s. VVe use mixture models for the density estimates and make two distinct appeals to the ExpectationMaximization (EM) principle (Dempster et al., 1977) in deriving a learning algorithm-EM is used both for the estimation of mixture components and for coping wit.h missing dat.a. The resulting algorithm is applicable t.o a wide range of supervised as well as unsupervised learning problems . Result.s from a classification benchmark-t.he iris data set-are presented. 1 Introduction Adaptive systems generally operate in environments t.hat are fraught with imperfections; nonet.heless they must cope with these imperfections and learn to extract as much relevant information as needed for their part.icular goals. One form of imperfection is incomplet.eness in sensing information. Incompleteness can arise extrinsically from the data generation process and intrinsically from failures of the system's sensors. For example, an object recognition system must be able to learn to classify images with occlusions, and a robotic controller must be able to integrate multiple sensors even when only a fraction may operate at any given time. In this paper we present a. fra.mework-derived from parametric statistics-for learn- 120 Supervised Learning from Incomplete Data via an EM Approach ing from data sets with arbitrary patterns of incompleteness. Learning in this framework is a classical estimation problem requiring an explicit probabilistic model and an algorithm for estimating the parameters of the model. A possible disadvantage of parametric methods is their lack of flexibility when compared with nonparametric methods. This problem, however, can be largely circumvented by the use of mixture models (McLachlan and Basford, 1988) . Mixture models combine much of the flexibility of nonparametric methods with certain of the analytic advantages of parametric methods. Mixture models have been utilized recently for supervised learning problems in the form of the "mixtures of experts" architecture (Jacobs et al., 1991; Jordan and Jacobs, 1994). This architecture is a parametric regression model with a modular structure similar to the nonparametric decision tree and adaptive spline models (Breiman et al., 1984; Friedman, 1991). The approach presented here differs from these regression-based approaches in that the goal of learning is to estimate the density of the data. No distinction is made between input and output variables; the joint density is estimated and this estimate is then used to form an input/output map. Similar approaches have been discussed by Specht (1991) and Tresp et al. (1993). To estimate the vector function y = I(x) the joint density P(x, y) is estimated and, given a particular input x, the conditional density P(ylx) is formed. To obtain a single estimate of y rather than the full conditional density one can evaluate y = E(ylx), the expectation of y given x. The density-based approach to learning can be exploited in several ways . First, having an estimate of the joint density allows for the representation of any relation between the variables. From P(x, y), we can estimate y = I(x), the inverse x 1-1 (y), or any other relation between two subsets of the elements of the concatenated vector (x, y). = Second, this density-based approach is applicable both to supervised learning and unsupervised learning in exactly the same way. The only distinction between supervised and unsupervised learning in this framework is whether some portion of the data vector is denoted as "input" and another portion as "target". Third, as we discuss in this paper, the density-based approach deals naturally with incomplete data, i.e. missing values in the data set. This is because the problem of estimating mixture densities can itself be viewed as a missing data problem (the "labels" for the component densities are missing) and an Expectation-Maximization (EM) algorithm (Dempster et al., 1977) can be developed to handle both kinds of missing data. 2 Density estimation using EM This section outlines the basic learning algorithm for finding the maximum likelihood parameters of a mixture model (Dempster et al., 1977; Duda and Hart, 1973; Nowlan, 1991). \IVe assume that. t.he data ..:t' = {Xl, ... , XN} are generated independently from a mixture density 1\1 P(Xi) = LP(Xi IWj;(}j)P(Wj), ;=1 (1) 121 122 Ghahramani and Jordan where each component of the mixture is denoted Wj and parametrized by (}j. From equation (1) and the independence assumption we see that the log likelihood of the parameters given the data set is N l((}IX) M = LlogLP(xilwj;Oj)P(Wj). i=1 (2) j=1 By the maximum likelihood principle the best model of the data has parameters that maximize l(OIX). This function, however, is not easily maximized numerically because it involves the log of a sum. Intuitively, there is a "credit-assignment" problem: it is not clear which component of the mixture generated a given data point and thus which parameters to adjust to fit that data point. The EM algorithm for mixture models is an iterative method for solving this credit-assignment problem. The intuition is that if one had access to a "hidden" random variable z that indicated which data point was genera.ted by which component, then the maximization problem would decouple into a set of simple maximizations. Using the indicator variable z, a "complete-data" log likelihood function can be written N lc((}IX, Z) = L M L Zij log P(XdZi; O)P(Zi; (}), (3) ;=1 j=1 which does not involve a log of a summation. Since Z is unknown lc cannot be utilized directly, so we instead work with its expectation, denoted by Q(OI(}k)' As shown by (Dempster et aI., 1977), l(OIX) can be maximized by iterating the following two steps: Estep: Q(OI(}k) E[lc(OIX,Z)IX,(}k] M step: (}k+l argmax Q((}IOk)' (4) o The E (Expectation) step computes the expected complete data log likelihood and the M (Maximization) step finds the parameters that maximize this likelihood. These two steps form the basis of the EM algorithm; in the next two sections we will outline how they can be used for real and discrete density estimation. 2.1 Real-valued data: Inixture of Gaussians Real-valued data can be modeled as a mixture of Gaussians. For this model the E-step simplifies to computing hij E[Zijlxi,Ok], the probability that Gaussian j, as defined by the parameters estimated at time step k, generated data point i. Itj 1- 1 / 2 exp{ -~ (Xi - itj)Tt;l,k(Xi - itj)} h .. = (5) I} L~1 IEfl-l/2exp{-~(Xi - it7)TE,I,k(Xi - it7)}' The M-step re-estimates the means and covariances of the Gaussians 1 using the data set weighted by the hii= = a ~ k+l _ L~l hijXi ) I-Lj N ' Li=1 hij 1 Though this derivation assumes equal priors for the Gaussians, if the priors arc viewed as mixing parameters they can also be learned in the maximization step. Supervised Learning from Incomplete Data via an EM Approach 2.2 Discrete-valued data: Inixture of Bernoullis D-dimensional binary data x = (Xl, . .. ,Xd, . a mixture of !II Bernoulli densities. That is, M P(xIO) = .. XD), Xd E {O, 1}, can be modeled as D L P(Wj) IT /-ljd(1 - /-ljd)(l-Xd). (7) For this model the E-step involves computing h .. I) - n D pX,ld (1 d=l}d 'Ef'!l nf=l P7J _ d p. )(1-Xld) }d (1 - Pld)(1-xld) , (8) and the M-step again re-estimates the parameters by ~ k+l ttj _ 'E~lN hijXi . - (9) 'Ei=l hij More generally, discrete or categorical data can be modeled as generated by a mixture of multinomial densities and similar derivations for the learning algorithm can be applied. Finally, the extension to data with mixed real, binary. and categorical dimensions can be readily derived by assuming a joint density with mixed components of the three types . 3 Learning from inco111plete data In the previous section we presented one aspect of the EM algorithm: learning mixture models. Another important application of EM is to learning from data sets with missing values (Little and Rubin, 1987; Dempster et aI., 1977). This application has been pursued in the statistics literature for non-mixture density estimation problems; in this paper we combine this application of EM with that of learning mixture parameters. We assume that. the data set ,l:' = {Xl ?.. . , XN} is divided into an observed component ,yo and a missing component ;t'm. Similarly, each data vector Xi is divided into (xi, xi) where each data vector can have different missing components-this would be denoted by superscript Dli and OJ. but we have simplified the notation for the sake of clarity. To handle missing data we rewrite the EM algorithm as follows Estep: M step: E[ic( fJl,t'?, ;t'm , Z) I;t'?. Ok] argmax Q(fJlfJk). o (10) Comparing to equation (4) we see that aside from t.he indicator variables Z we have added a second form of incomplete data, ;t'm , corresponding to the missing values in the data set. The E-step of the algorithm estimates both these forms of missing information; in essence it uses the current estimate of the data density to complete the missing values. 123 124 Ghahramani and Jordan 3.1 Real-valued data: mixture of Gaussians We start by writing the log likelihood of the complete data, N =L ic(OIXO, xm, Z) M N M L Zij log P(xdzj, 0) + L L Zij log P(zd O). j (11) j We can ignore the second term since we will only be estimating the parameters of the P(XdZi, 0). Using equation (11) for the mixture of Gaussians we not.e that if only the indicator variables Zi are missing, the E step can be reduced to estimating E[ Zij lXi, 0]. For the case we are interested in, with two types of missing data Zi and xi, we expand equation (11) using m and 0 superscripts to denote subvectors and submatrices of the parameters matching the missing and observed components of the data, Ic(OIXO, xm, Z) = - N M I J L.L.Zij[n log27r + ! log IEj 1- !(xi -l1-jf E;l,OO(xi -l1-j) 22 2 ( 0 Xi - o)T~-l,Om( m m) 1( m m)T~-l,mm( m m)] L...j Xi - I1-j - 2 Xi - I1-j L...j Xi - I1-j ? I1-j Note that after taking the expectation, the sufficient statistics for the parameters involve three unknown terms, Zij, ZijXi, and zijxixiT. Thus we must compute: E[Zijlx?,Ok]' E[Zijxilx?,Ok], and E[ZijxixinTlx?,Ok]. One intuitive approach to dealing with missing data is to use the current estimate of the data density to compute the expectat.ion of the missing data in an E-step, complete the data with these expectations, and then use this completed data to reestimate parameters in an M-step. However, this intuition fails even when dealing with a single two-dimensional Gaussian; the expectation of the missing data always lies along a line, which biases the estimate of the covariance. On the other hand, the approach arising from application of the EM algorithm specifies that one should use the current density estimate to compute the expectation of whatever incomplete terms appear in the likelihood maximization. For the mixture of Gaussians these incomplete terms involve interactions between the indicator variable :;ij and the first and second moments of xi. Thus, simply computing the expectation of the missing data Zi and xi from our model and substituting those values into the M step is not sufficient to guarantee an increase in the likelihood of the parameters. The above terms can be computed as follows: E[ Zij lxi, Ok] is again hij, the probability as defined in (5) measured only on the observed dimensions of Xi, and = = = E[Zijxilxi, Ok] hijE[xilzij 1, xi, Od hij(l1-j + EjOEjO-l (xi -Il.'}?, (12) Defining xi] = E[xi IZij = 1, xi, Ok], the regression of xi on xi using Gaussian j, .. m XimTI xi' ? 0k] -_ h'J..(~mm ~mo~oo-l ~moT ~ m ~ mT) E[ Z'Jxi (13) L...j - L...j ~j L...j + XijXij . The M-step uses these expectations substituted into equations (6)a and (6)b to re-estimate the means and covariances. To re-estimate the mean vector, I1-j' we substitute the values E[xilzij = 1, xi, Ok] for the missing components of Xi in equation (6)a. To re-estimate the covariance matrix we substitute t.he values E[xixiTlzij = 1, xi, Ok] for the outer product matrices involving the missing components of Xi in equation (6)b. Supervised Learning from Incomplete Data via an EM Approach 3.2 Discrete-valued data: Inixture of Bernoullis For the Bernoulli mixture the sufficient statistics for the M-step involve t he incomplete terms E[Zij Ix?, Ok] and E[ Zij xi Ix~, Ok]. The first is equal to hij calculated over the observed subvector of Xi. The second, since we assume that within a class the individual dimensions of the Bernoulli variable are independent., is simply hijl-Lj. The M-step uses these expectations substituted into equation (9). 4 Supervised learning If each vector Xi in the data set is composed of an "input" subvector, x}, and a "target" or output subvector, x?, then learning the joint density of the input and target is a form of supervised learning. In supervised learning we generally wish to predict the output variables from the input variables. In this section we will outline how this is achieved using the estimated density. 4.1 Function approximation For real-valued function approximation we have assumed that the densit.y is estimated using a mixture of Gaussians. Given an input vector x~ we ext ract all the relevant information from the density p(xi, XO) by conditionalizing t.o p(xOlxD. For a single Gaussian this conditional densit.y is normal, and, since P(x 1 , XO) is a mixture of Gaussians so is P(xolx i ). In principle, this conditional density is the final output of the density estimator. That is, given a particular input the network returns the complete conditional density of t.he output. However, since many applications require a single estimate of the output, we note three ways to obtain estimates x of XO = f(x~): the least squares estimate (LSE), which takes XO(xi) = E(xOlxi); stochastic sampling (STOCH), which samples according to the distribution xO(xD "" P(xOlxi); single component LSE (SLSE), which takes xO(xD = E(xOlxLwj) where j = argmaxk P(zklx~). For a given input, SLSE picks the Gaussian with highest posterior and approximates the out.put with the LSE estimator given by that Gaussian alone. The conditional expectation or LSE estimator for a Gaussian mixt.ure is (14) which is a convex sum of linear approximations, where the weights h ij vary nonlinearly according to equation (14) over the input space. The LSE estimator on a Gaussian mixture has interesting relations to algorithms such as CART (Breiman et al., 1984), MARS (Friedman, 1991), and mixtures of experts (Jacobs <.'t al., 1991; Jordan and Jacobs, 1994), in that the mixture of Gaussians competit.ively partitions the input space, and learns a linear regression surface on each part-it.ion. This similarity has also been noted by Tresp et al. (1993) . The stochastic estimator (STOCH) and the single component estimator (SLSE) are better suited than any least squares method for learning non-convex ill verse maps, where the mean of several solutions to an inverse might not be a solut ion. These 125 126 Ghahramani and Jordan Classification with missing inputs Figure 1: Classification of the iris data set. 100 data points were used for training and 50 for testing. Each data point consisted of 4 real-valued attributes and one of three class labels. The figure shows classification performance ? 1 standard error (11 = 5) as a function of proportion missing features for the EM algorithm and for mean imputation (MI), a common heuristic where the missing values are replaced with their unconditional means. 100 ~" 0-1---~, U ;.;:: '"'" !! ! -t EM , \, 60 o.. ,, \ ... U , 'l,_ U II) 40 -'-'tI U MI ~ 20 o 20 40 60 80 100 % missing features estimators take advantage of the explicit representat.ion of the input/output density by selecting one of the several solutions to the inverse. 4.2 Classification Classification problems involve learning a mapping from an input space into a set of discrete class labels. The density estimat.ion framework presented in this paper lends itself to solving classification problems by estimating the joint density of the input and class label using a mixture model. For example, if the inputs have realvalued attributes and there are D class labels, a mixture model with Gaussian and multinomial components will be used: AI 1 ~ ~jd P(x, e = dlO) = ~ P(Wj) (27r)n/2IEj 11/2 exp{ -"2 (x - I-tj fEj1 (x - I-'j (15) n, denoting the joint probability that the data point. is x and belongs to class d, where the ~j d are the parameters for the multinomial. Once this density has been estimated, the maximum likelihood label for a particular input x may be obtained by computing P(C = dlx, 0). Similarly, the class conditional densities can be derived by evaluating P( x Ie = d, 0). Condi tionalizing over classes in this way yields class conditional densities which are in turn mixtures of Gaussians. Figure 1 shows the performance of the EM algorithm on an example classification problem with varying proportions of missing features. We have also applied these algorithms to the problems of clustering 35-dimensional greyscale images and approximating the kinematics of a three-joint planar arm from incomplete data. 5 Discussion Densit.y estimation in high dimensions is generally considered to be more difficultrequiring more parameters-than function approximation. The density-estimationbased approach to learning, however, has two advantages. First, it permits ready incorporation of results from the statistical literature on missing data to yield flexible supervised and unsupervised learning architectures. This is achieved by combining two branches of application of the EM algorithm yielding a set of learning rules for mixtures under incomplete sampling. Supervised Learning from Incomplete Data via an EM Approach Second, estimating the density explicitly enables us to represent any relation between the variables. Density estimation is fundamentally more general than function approximation and this generality is needed for a large class of learning problems arising from inverting causal systems (Ghahramani, 1994). These problems cannot be solved easily by traditional function approximation techniques since the data is not generated from noisy samples of a function, but rather of a relation. Acknow ledgmuents Thanks to D. M. Titterington and David Cohn for helpful comments. This project was supported in part by grants from the McDonnell-Pew Foundation, ATR Auditory and Visual Perception Research Laboratories, Siemens Corporation, the National Science Foundation, and the Office of Naval Research. The iris data set was obtained from the VCI Repository of Machine Learning Databases. References Breiman, L., Friedman, J. H., Olshen, R. A., and Stone, C. J. (1984). Classification and Regression Trees. Wadsworth International Group, Belmont, CA . Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood fwm incomplete data via the EM algorithm. J. Royal Statistical Society Series B, 39:1-38. Duda, R. O. and Hart, P. E. (1973). Pattern Classification and Scene Analysis. Wiley, New York. Friedman, J. H. (1991). Multivariate adaptive regression splines. The Annols of Statistics, 19:1-141. Ghahramani, Z. (1994). Solving inverse problems using an EM approach to density estimation. In Proceedings of the 1993 Connectionist Models Summer School. Erlbaum, Hillsdale, NJ. Jacobs, R., Jordan, M., Nowlan, S., and Hinton, G. (1991). Adaptive mixture of local experts. Neural Computation, 3:79-87. Jordan, M. and Jacobs, R. (1994). Hierarchical mixtures of experts ano the EM algorithm. Neural Computation, 6:181-214. Little, R. J. A. and Rubin, D. B. (1987). Statistical Analysis with Mis.'ling Data. Wiley, New York. McLachlan, G. and Basford, K. (1988). Mixture models: Inference and applications to clustering. Marcel Dekkel'. Nowlan, S. J. (1991). Soft Competitive Adaptation: Neural Network Learning Algorithms based on Fitting Statistical Mixtures. CMV-CS-91-126, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA. Specht, D. F. (1991). A general I'egression neural network. IEEE Trans. Neural Networks, 2(6):568-576. Tresp, V., Hollatz, J., and Ahmad, S. (1993). Network structuring and training using rule-based knowledge. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems 5. Morgan Kaufman Publishers, San Mateo, CA. 127 

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Dopaminergic Neuromodulation Brings a Dynamical Plasticity to the Retina Eric Boussard Jean-Fran~ois Vibert B3E, INSERM U263 Faculte de medecine Saint-Antoine 27 rue Chaligny 75571 Paris cedex 12 Abstract The fovea of a mammal retina was simulated with its detailed biological properties to study the local preprocessing of images. The direct visual pathway (photoreceptors, bipolar and ganglion cells) and the horizontal units, as well as the D-amacrine cells were simulated. The computer program simulated the analog non-spiking transmission between photoreceptor and bipolar cells, and between bipolar and ganglion cells, as well as the gap-junctions between horizontal cells, and the release of dopamine by D-amacrine cells and its diffusion in the extra-cellular space. A 64 x 64 photoreceptors retina, containing 16,448 units, was carried out. This retina displayed contour extraction with a Mach effect, and adaptation to brightness. The simulation showed that the dopaminergic amacrine cells were necessary to ensure adaptation to local brightness. 1 INTRODUCTION The retina is the first stage in visual information processing. One of its functions is to compress the information received from the environment by removing spatial and temporal redundancies that occur in the light input signal. Modelling and computer simulations present an efficient means to investigate and characterize the physiological mechanisms that underlie such a complex process. In fact, filtering depends on the quality of the input image (van Hateren, 1992): 559 560 Boussard and Vibert l.High mean light intensity (high signal to noise ratio). A high-pass filter enhances the edges (contour extraction) and the temporal changes of the input. 2.Low mean light intensity (low signal to noise ratio). The sensitivity of highpass filters to noise makes them inefficient in this case. A low-pass filter, averaging the signal over several receptors, is required to extract the relevant information. There are three aspects in the filtering adaptivity displayed by the retina: adaptivity to i) the global spatial changes in the image, ii) the local spatial changes in the image, iii) the temporal changes in the image. We will focus on the second feature. A biologically plausible mammalian retina was modelled and simulated to explore the local preprocessing of the images. A first model (Bedfer & Vibert, 1992), that did not take into account the dopamine neuromodulation, reproduced some of the behaviors found in the living retina, like a progressive decrease of ganglion cells' firing rate in response to a constant image presented to photoreceptors, reversed post-image, and optic illusion (Hermann grid). The model, however, displayed a poor local adaptivity. It could not give both a good contrast rendering and a Mach effect. The Mach effect is a psychophysical law that is characterized by an edge enhancement (Ratliff, 1965). The retina network produces a double lighter and darker contour from the frontier line between two areas of different brightness in the stimulus. This phenomenon is indispensable for contour extraction. This paper will first present the conditions in which high-pass filtering and low-pass filtering occur exclusively in the retina model. These results are then compared to those obtained with a model that includes dopamine neuromodulation, thus illustrating the role played by dopamine in local adaptivity (Besharse & Iuvone, 1992). 2 METHODS The retina is an unusual neural structure: i) the photoreceptors respond to light by an hyperpolarization, ii) signal transmission from photoreceptors to bipolar units does not involve spikes, neurotransmitter release at these synapses is a continuous function of the membrane potential (Buser & Imbert, 1987). Only ganglion cells generate spikes. Furthermore, horizontal cells are connected by dopamine dependent gap-junctions. Dopamine is an ubiquitous neurotransmitter and neuromodulator in the central nervous system. In the visual pathway, dopamine affects several types of retinal neurons (Witkovsky & Dearry, 1992). Dopamine is released by stimulated D-amacrine and interplexiform cells. It diffuses in the extra-cellular space, and produces: cone shortening and rod elongation, reduced permeability of gap-junctions, increased conductance of glutamate-induced current among horizontal cells, increased conductance of the cone-to- horizontal cell synapse, and retroinhibition on D-amacrine cells (Djamgoz & Wagner, 1992). Our model focused on the adaptive filtering mechanism in the fovea that enables the retina to simultaneously perform both high-pass and low-pass filtering. Therefore, dopamine action on gap-junction between horizontal cells and the retro-inhibition on D-amacrine cells was the only dopamine effect implemented (fig. 1). Our model included the three neuron types of the direct pathway - photoreceptors, bipolar and ganglion units - as well as two types of the indirect pathway - the horizontal and dopaminergic amacrine cells. Only the On pathway of a mammal fovea was studied here. Each neuron type has been modelled with its own anatomical and electrophysiolog- Dopaminergic Neuromodulation Brings a Dynamical Plasticity to the Retina ~ Excitation . . lnhibition .11 II' Gap-junction ~ ~fea:!ne Figure 1: The dopaminergic amacrine units in the modelled retina. The connections of an On center pathway in the simulated retina. Photo: Photoreceptors. Horiz: Horizontal units. Bip: Bipolar units. Gang: Ganglion Units. DA: Dopaminergic Amacrine unit. DA units are stimulated by many bipolar units. With an enough excitation, they can release dopamine in the extracellular space. This released dopamine goes to modulates the conductance value of horizontal gapjunctions. 561 562 Boussard and Vibert ical properties (Wiissle & Boycott, 1991)(Lewick & Dvorak, 1986). The temporal evolution of the membrane potential of each unit can be recorded. 3 RESULTS A 64x64 photoreceptors retina was constructed as a noisy hexagonal frame where photoreceptors, bipolar and ganglion units were connected to their nearest neighbours. Horizontal units were connected to their 18 nearest photoreceptors and bipolar units, with a number of synaptic boutons decreasing as a function of distance. They did not retroact on the nearest photoreceptor. This horizontal layer architecture produces lateral inhibition. Each modelled D-amacrine unit was connected to about fifty bipolar units. The diffusion of released dopamine in the extra-cellular space was simulated. The modelled retina consisted of 16,448 units and 862,720 synapses. At each simulation, the photoreceptors layer was stimulated by an input image. Stimulations were given as a 256x256 pixel image presented to the simulated 64x64 photoreceptor retina. Since the localization of photoreceptors was not regular, each receptor received the input from 16 pixels on the average. The output image was reconstructed using the ganglion units response. For each of the 4096 ganglion units the spike frequency was measured during a given time (according to the experiment) and coded in a grey level for the given unit retinotopic position. Thus, each simulation produced an image of the retina output. This output image was compared to the input image. The input image (stimulus) consisted here of one white disk on a dark background. The results presented, in fig. 2, were obtained after 750 ms of stationary stimulations. The stimuli were here a white disk on a black background. The inputs were stationary to avoid temporal effects owing to evolving inputs. Output images of stationary inputs, however, vanished after 1000 ms. The time was limited to 750 ms to optimize the quality of the output image. Biological datas available on the conductance value suggest that in the mammalian retina the conductance does not remain constant and undergoes a dynamical tuning depending on the local brightness [?]. This provides a range of possible values for the conductance. The behavior of the model was tested for values within this range. Different values lead to different network behaviors. Three types of results were obtained from the simulations : l.Without dopamine action, the conductance values were fixed for all gap-junctions to 1O- 6 S (fig. 2-A). The output image rendered well the contrast in the input image, but did not display the Mach effect (low-pass filtering). 2.Without dopamine action, the conductance values were fixed for all gap-junctions to 1O- 10 S (fig. 2-B). The low conductance value allowed a pronounced Mach effect, but the contrast in the output image was strongly diminished (high-pass filtering). This contrast appears like an average of the two brightness. Only the contour delimited by Mach effect allows the disk to be distinguished. 3.With dopamine, the conductance values were initially set to 1O- 7 S (fig. 2-C) . The output image displayed both the contrast rendering and the Mach effect (locally Dopaminergic Neuromodulation Brings a Dynamical Plasticity to the Retina A 40 32 24 16 8 ?0~---1-1--~22~-3~4---4~5--~56 ' 22 23 24 26 27 28 29 30 32 33 34 B 40 32 ..-----,. 24 16 8 0~~1~2--~24~~3~6--~4=8--~60 c 65 52 LJV'\....I'-_I 39 26 13 0~~12~~2~4--~3~7--4~9~~61 Figure 2: tances. ~ontour extraction (Mach effect) according to gap-junctions conduc- On the left, results obtained after 750 ms of stimulation for an zmage of a white disk on a black background. On the right, sections through the corresponding image. A bscissa: spike count; Ordinates: geographic position of the unit, from the left side to the middle of the left panel. A: without dopamine (fixed Ggap = 10- 6 S). B: without dopamine (fixed Ggap = lO-lOS). C: with dopamine release (starting Ggap 1O- 7 S) . A gives a good contrast rendering, but no Mach effect. B gives a Mach effect, but there is an averaging between darker and lighter areas. C, with dopa minergic neuromodulation, gives both a Mach effect and a good contrast rendering. = 563 564 Boussard and Vibert adaptive filtering). 4 DISCUSSION These results show that the conductance cannot be fixed at a single value for all the gap-junctions. If the conductance value is high (fig. 2-A), the model acts like a low-pass filter. A good contrast rendering was obtained, but there was no Mach effect. If the conductance value is low (fig. 2-B), the model becomes a high-pass filter. A Mach effect was obtained, but the contrast in the post-retinal image was dramatically deteriorated: an undesirable averaging of the brightness between the darker and the more illuminated areas appeared. Therefore in this model the Mach effect was only obtained at the expense of the contrast. A mammalian retina is able to perform both contrast rendering and contour extraction functions together. It works like an adaptive filter. To obtain a similar result, it is necessary to have a variable communication between horizontal units. The simulated retina needs low gap-junctions conductance in the high light intensity areas and high conductance in the low light intensity areas. The conductance of each gap-junction must be tuned according to the local stimulation. The model used to obtain the fig. 2-C takes into account the dopamine release by the D-amacrine cells. Here, the network performs the two antagonist functions of filtering. Dopamine provides our model with the capacity to have a biological behaviour. What is the action of dopamine on network? Dopamine is released by D-amacrine units. Then, it diffuses from its release point into the extra-cellular space among the neurons, reaches gap-junctions and decreases their conductance value. Thus the conductance modulation depends in time and in intensity on the distance between gap-junction and D-amacrine unit. In addition, this action is transient. 5 CONCLUSION Thanks to dopamine neuromodulation, the network is able to subdivise itself into several subnetworks, each having the appropriate gap-junction conductance. Each subnetwork is thus adapted for a better processing of the external stimulus. Dopamine neuromodulation is a chemically addressed system, it acts more diffusely and more slowly than transmission through the axo-synaptic connection system. Therefore neuromodulation adds a dynamical plasticity to the network. References G. Bedfer & J .-F. Vibert. (1992) Image preprocessing in simulated biologicalretina. Proc. 14th Ann. Conf. IEEE EMBS 1570-157l. J. Besharse & P. Iuvone. (1992). Is dopamine a light-adaptive or a dark-adaptive modulator in retina? NeuroChemistry International 20:193-199. P. Buser & M. Imbert. (1987) Vision. Paris: Hermann. M. Djamgoz & H.-J. Wagner. (1992) Localization and function of dopamine in the adult vertebrate retina. NeuroChemistry InternationaI20:139-19l. L. Dowling. (1986) Dopamine: a retinal neuromodulator? Trends In NeuroSciences Dopaminergic Neuromodulation Brings a Dynamical Plasticity to the Retina 9:236-240. W . Levick & D. Dvorak. (1986) The retina - from molecule to network . Trends In NeuroSciences 9:181-185. F. Ratliff. (1965) Mach bands: quantitative studies on neural network in the retina. Holden-Day. J . H. van Hateren. (1992) Real and optimal images in early vision. Nature 360:6870. H. Wassle & B. B. Boycott. (1991) Functional architecture of the mammalian retina. Physiological Reviews 71(2):447-479. P . Witkovsky & A. Dearry. (1992) Functional roles of dopamine in the vertebrate retina. Retinal Research 11:247-292. 565 

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Signature Verification using a "Siamese" Time Delay Neural Network Jane Bromley, Isabelle Guyon, Yann LeCun, Eduard Sickinger and Roopak Shah AT&T Bell Laboratories Holmdel, NJ 07733 jbromley@big.att.com Copyrighte, 1994, American Telephone and Telegraph Company used by permission. Abstract This paper describes an algorithm for verification of signatures written on a pen-input tablet. The algorithm is based on a novel, artificial neural network, called a "Siamese" neural network. This network consists of two identical sub-networks joined at their outputs. During training the two sub-networks extract features from two signatures, while the joining neuron measures the distance between the two feature vectors. Verification consists of comparing an extracted feature vector ~ith a stored feature vector for the signer. Signatures closer to this stored representation than a chosen threshold are accepted, all other signatures are rejected as forgeries. 1 INTRODUCTION The aim of the project was to make a signature verification system based on the NCR 5990 Signature Capture Device (a pen-input tablet) and to use 80 bytes or less for signature feature storage in order that the features can be stored on the magnetic strip of a credit-card. Verification using a digitizer such as the 5990, which generates spatial coordinates as a function of time, is known as dynamic verification. Much research has been carried out on signature verification. Function-based methods, which fit a function to the pen trajectory, have been found to lead to higher performance while parameter-based methods, which extract some number of parameters from a signa- 737 738 Bromley, Guyon, Le Cun, Sackinger, and Shah ture, make a lower requirement on memory space for signature storage (see Lorette and Plamondon (1990) for comments). We chose to use the complete time extent of the signature, with the preprocessing described below, as input to a neural network, and to allow the network to compress the information. We believe that it is more robust to provide the network with low level features and to allow it to learn higher order features during the training process, rather than making heuristic decisions e.g. such as segmentation into balistic strokes. We have had success with this method previously (Guyon et al., 1990) as have other authors (Yoshimura and Yoshimura, 1992). 2 DATA COLLECTION All signature data was collected using 5990 Signature Capture Devices. They consist of an LCD overlayed with a transparent digitizer. As a guide for signing, a 1 inch by 3 inches box was displayed on the LCD. However all data captured both inside and outside this box, from first pen down to last pen up, was returned by the device. The 5990 provides the trajectory of the signature in Cartesian coordinates as a function of time. Both the trajectory of the pen on the pad and of the pen above the pad (within a certain proximity of the pad) are recorded. It also uses a pen pressure measurement to report whether the pen is touching the writing screen or is in the air. Forgers usually copy the shape of a signature. Using such a tablet for signature entry means that a forger must copy both dynamic information and the trajectory of the pen in the air. Neither of these are easily available to a forger and it is hoped that capturing such information from signatures will make the task of a forger much harder. Strangio (1976), Herbst and Liu (1977b) have reported that pen up trajectory is hard to imitate, but also less repeatable for the signer. The spatial resolution of signatures from the 5990 is about 300 dots per inch, the time resolution 200 samples per second and the pad's surface is 5.5 inches by 3.5 inches. Performance was also measured using the same data treated to have a lower resolution of 100 dots per inch. This had essentially no effect on the results. Data was collected in a university and at Bell Laboratories and NCR cafeterias. Signature donors were asked to sign their signature as consistently as possible or to make forgeries. When producing forgeries, the signer was shown an example of the genuine signature on a computer screen. The amount of effort made in producing forgeries varied. Some people practiced or signed the signature of people they knew, others made little effort. Hence, forgeries varied from undetectable to obviously different. Skilled forgeries are the most difficult to detect, but in real life a range of forgeries occur from skilled ones to the signatures of the forger themselves. Except at Bell Labs., the data collection was not closely monitored so it was no surprise when the data was found to be quite noisy. It was cleaned up according to the following rules: ? Genuine signatures must have between 80% and 120% of the strokes of the first signature signed and, if readable, be of the same name as that typed into the data collection system. (The majority of the signatures were donated by residents of North America, and, typical for such signatures, were readable.) The aim of this was to remove signatures for which only Signature Verification Using a "Siamese" Time Delay Neural Network some part of the signature was present or where people had signed another name e.g. Mickey Mouse. ? Forgeries must be an attempt to copy the genuine signature. The aim of this was to remove examples where people had signed completely different names. They must also have 80% to 120% of the strokes of the signature . ? A person must have signed at least 6 genuine signatures or forgeries. In total, 219 people signed between 10 and 20 signatures each, 145 signed genuines, 74 signed forgeries. 3 PREPROCESSING A signature from the 5990 is typically 800 sets of z, y and pen up-down points. z(t) and y(t) were originally in absolute position coordinates. By calculating the linear estimates for the z and y trajectories as a function of time and subtracting this from the original z and y values, they were converted to a form which is invariant to the position and slope of the signature. Then, dividing by the y standard deviation provided some size normalization (a person may sign their signature in a variety of sizes, this method would normalize them). The next preprocessing step was to resample, using linear interpolation, all signatures to be the same length of 200 points as the neural network requires a fixed input size. Next, further features were computed for input to the network and all input values were scaled so that the majority fell between +1 and -1. Ten different features could be calculated, but a subset of eight were used in different experiments: feature 1 pen up = -1 i pen down = +1, (pud) feature 2 x position, as a difference from the linear estimate for x(t), normalized using feature feature feature feature feature feature feature feature the standard deviation of 1/, (x) 3 y position, as a difference from the linear estimate for y(t), normalized using the standard deviation of 1/, (y) 4 speed at each point, (spd) 5 centripetal acceleration, (ace-c) 6 tangential acceleration, (acc-t) 7 the direction cosine of the tangent to the trajectory at each point, (cosS) 8 the direction sine of the tangent to the trajectory at each point, (sinS) 9 cosine of the local curvature of the trajectory at each point, (cost/J) 10 sine of the local curvature of the trajectory at each point, (sint/J) In contrast to the features chosen for character recognition with a neural network (Guyon et al., 1990), where we wanted to eliminate writer specific information, the features such as speed and acceleration were chosen to carry information that aids the discrimination between genuine signatures and forgeries. At the same time we still needed to have some information about shape to prevent a forger from breaking the system by just imitating the rhythm of a signature, so positional, directional amd curvature features were also used. The resampling of the signatures was such as to preserve the regular spacing in time between points. This method penalizes forgers who do not write at the correct speed. 739 740 Bromley, Guyon, Le Cun, Sackinger, and Shah TARGET - ..... t ? - - - - - - - - - ' :01 ~ fltt .... ~ be- . 11 2OOu .... ? ... beUli Figure 1: Architecture 1 consists of two identical time delay neural networks. Each network has an input of 8 by 200 units, first layer of 12 by 64 units with receptive fields for each unit being 8 by 11 and a second layer of 16 by 19 units with receptive fields 12 by 10. 4 NETWORK ARCHITECTURE AND TRAINING The Siamese network has two input fields to compare two patterns and one output whose state value corresponds to the similarity between the two patterns. Two separate sub-networks based on Time Delay Neural Networks (Lang and Hinton, 1988, Guyon et al. 1990) act on each input pattern to extract features, then the cosine of the angle between two feature vectors is calculated and this represents the distance value. Results for two different subnetworks are reported here. Architecture 1 is shown in Fig 1. Architecture 2 differs in the number and size of layers. The input is 8 by 200 units, the first convolutional layer is 6 by 192 units with each unit's receptive field covering 8 by 9 units of the input. The first averaging layer is 6 by 64 units, the second convolution layer is 4 by 57 with 6 by 8 receptive fields and the second averaging layer is 4 by 19. To achieve compression in the time dimension , architecture 1 uses a sub-sampling step of 3, while architecture 2 uses averaging. A similar Siamese architecture was independently proposed for fingerprint identification by Baldi and Chauvin (1992). Training was carried out using a modified version of back propagation (LeCun, 1989). All weights could be learnt, but the two sub-networks were constrained to have identical weights. The desired output for a pair of genuine signatures was for a small angle (we used cosine=l.O) between the two feature vectors and a large angle Signature Verification Using a "Siamese" Time Delay Neural Network Table 1: Summary of the Training. Note: GA is the percentage of genuine signature pairs with output greater than 0, FR the percentage of genuine:forgery signature pairs for which the output was less than O. The aim of removing all pen up points for Network 2 was to investigate whether the pen up trajectories were too variable to be helpful in verification. For Network 4 the training simulation crashed after the 42nd iteration and was not restarted. Performance may have improved if training had continued past this point. pu acc-c acc-t sp cosH sinS cos'" sin~ 2, arc 1 3, arc 1 4, arc 1 5, arc 2 same as network 3, but a larger training set same as 4, except architecture 2 was used (we used cosine= -0.9 and -1.0) if one of the signatures was a forgery. The training set consisted of 982 genuine signatures from 108 signers and 402 forgeries of about 40 of these signers. We used up to 7,701 signature pairsj 50% genuine:genuine pairs, 40% genuine:forgery pairs and 10% genuine:zero-effort pairs. 1 The validation set consisted of 960 signature pairs in the same proportions as the training set. The network used for verification was that with the lowest error rate on the validation set. See Table 1 for a summary of the experiments. Training took a few days on a SPARe 1+. 5 TESTING When used for verification, only one sub-network is evaluated. The output of this is the feature vector for the signature. The feature vectors for the last six signatures signed by each person were used to make a multivariate normal density model of the person's signature (see pp. 22-27 of Pattern Classification and Scene Analysis by Duda and Hart for a fuller description of this). For simplicity, we assume that the features are statistically independent, and that each feature has the same variance. Verification consists of comparing a feature vector with the model of the signature. The probability density that a test signature is genuine, p-yes, is found by evaluating 1 zero-effort forgeries, also known as random forgeries, are those for which the forger makes no effort to copy the genuine signature, we used genuine signatures from other signers to simulate such forgeries. 741 742 Bromley, Guyon, Le Cun, Sackinger, and Shah 100 10 Ic! 10 J 15 70 10 50 ? 40 f 30 20 10 0 I lie lie a. SI2 90 sa sa ... 82 80 Percentage 01 Genuine Signatures Accepted Figure 2: Results for Networks 4 (open circles) and 5 (closed circles). The training of Network 4 was essentially the same as for Network 3 except that more data was used in training and it had been cleaned of noise. They were both based on architecture 1. Network 5 was based on architecture 2. The signature feature vector from this architecture is just 4 by 19 in size. the normal density function. The probability density that a test signature is a forgery, p-no, is assumed, for simplicity, to be a constant value over the range of interest. An estimate for this value was found by averaging the p-yes values for all forgeries. Then the probability that a test signature is genuine is p-yesj(p-yes + pno). Signatures closer than a chosen threshold to this stored representation are accepted, all other signatures are rejected as forgeries. Networks 1, 2 and 3, all based on architecture I, were tested using a set of 63 genuine signatures and 63 forgeries for 18 different people. There were about 4 genuine test signatures for each of the 18 people, and 10 forgeries for 6 of these people. Networks 1 and 2 had identical training except Network 2 was trained without pen up points. Network 1 gave the better results. However, with such a small test set, this difference may be hardly significant. The training of Network 3 was identical to that of Network I, except that x and y were used as input features, rather than acc-c and acc-t. It had somewhat improved performance. No study was made to find out whether the performance improvement came from using x and y or from leaving out acc-c and acc-t. Plamondon and Parizeau (1988) have shown that acceleration is not as reliable as other functions. Figure 2 shows the results for Networks 4 and 5. They were tested using a set of 532 genuine signatures and 424 forgeries for 43 different people. There were about 12 genuine test signatures for each person, and 30 forgeries for 14 of the people. This graph compares the performance of the two different architectures. It takes 2 to 3 minutes on a Sun SPARC2 workstation to preprocess 6 signatures, Signature Verification Using a "Siamese" Time Delay Neural Network collect the 6 outputs from the sub-network and build the normal density model. 6 RESULTS Best performance was obtained with Network 4. With the threshold set to detect 80% of forgeries, 95.5% of genuine signatures were detected (24 signatures rejected). Performance could be improved to 97.0% genuine signatures detected (13 rejected) by removing all first and second signature from the test set 2. For 9 of the remaining 13 rejected signatures pen up trajectories differed from the person's typical signature. This agrees with other reports (Strangio, 1976 Herbst and Liu, 1977b) that pen up trajectory is hard to imitate, but also a less repeatable signature feature. However, removing pen up trajectories from training and test sets did not lead to any improvement (Networks 1 and 2 had similar performance), leading us to believe that pen up trajectories are useful in some cases. Using an elastic matching method for measuring distance may help. Another cause of error came from a few people who seemed unable to sign consistently and would miss out letters or add new strokes to their signature. The requirement to represent a model of a signature in 80 bytes means that the signature feature vector must be encodable in 80 bytes. Architecture 2 was specifically designed with this requirement in mind. Its signature feature vector has 76 dimensions. When testing Network 5, which was based on this architecture, 50% of the outputs were found (surprisingly) to be redundant and the signature could be represented by a 38 dimensional vector with no loss of performance. One explanation for this redundancy is that, by reducing the dimension of the output (by not using some outputs), it is easier for the neural network to satisfy the constraint that genuine and forgery vectors have a cosine distance of -1 (equivalent to the outputs from two such signatures pointing in opposite directions). These results were gathered on a Sun SPARC2 workstation where the 38 values were each represented with 4 bytes. A test was made representing each value in one byte. This had no detrimental effect on the performance. Using one byte per value allows the signature feature vector to be coded in 38 bytes, which is well within the size constraint. It may be possible to represent a signature feature vector with even less resolution, but this was not investigated. For a model to be updatable (a requirement of this project), the total of all the squares for each component of the signature feature vectors must also be available. This is another 38 dimensional vector. From these two vectors the variance can be calculated and a test signature verified. These two vectors can be stored in 80 bytes. 7 CONCLUSIONS This paper describes an algorithm for signature verification. A model of a person's signature can easily fit in 80 bytes and the model can be updated and become more accurate with each successful use of the credit card (surely an incentive for people to use their credit card as frequently as possible). Other beneficial aspects of this verification algorithm are that it is more resistant to forgeries for people who sign 2people commented that they needed to sign a few time to get accustomed to the pad 743 744 Bromley, Guyon, Le Cun, Sackinger, and Shah consistently, the algorithm is independent of the general direction of signing and is insensitive to changes in size and slope. As a result of this project, a demonstration system incorporating the neural network signature verification algorithm was developed. It has been used in demonstrations at Bell Laboratories where it worked equally well for American, European and Chinese signatures. This has been shown to commercial customers. We hope that a field trial can be run in order to test this technology in the real world. Acknowledgements All the neural network training and testing was carried out using SN2.6, a neural network simulator package developed by Neuristique. We would like to thank Bernhard Boser, John Denker, Donnie Henderson, Vic Nalwa and the members of the Interactive Systems department at AT&T Bell Laboratories, and Cliff Moore at NCR Corporation, for their help and encouragement. Finally, we thank all the people who took time to donate signatures for this project. References P. Baldi and Y. Chauvin, "Neural Networks for Fingerprint Recognition", Neural Computation,5 (1993). R. Duda and P. Hart, Pattern Classification and Scene Analysis, John Wiley and Sons, Inc., 1973. I. Guyon, P. Albrecht, Y. LeCun, J. S. Denker and W. Hubbard, "A Time Delay Neural Network Character Recognizer for a Touch Terminal", Pattern Recognition, (1990). N. M. Herbst and C. N. Liu, "Automatic signature verification based on accelerometry", IBM J. Re,. Develop., 21 (1977)245-253. K. J. Lang and G. E. Hinton, "A Time Delay Neural Network Architecture for Speech Recognition", Technical Report CMU-cs-88-152, Carnegie-Mellon University, Pittsburgh, PA,1988. Y. LeCun, "Generalization and Network Design Strategies", Technical Report CRG-TR89-4 University of Toronto Connectionist Research Group, Canada, 1989. G. Lorette and R. Plamondon, "Dynamic approaches to handwritten signature verification", in Computer processing of handwriting, Eds. R. Plamondon and C. G. Leedham, World Scientific, 1990. R. Plamondon and M. Parizeau, "Signature verification from position, velocity and acceleration signals: a comparative study", in Pro<;. 9th Int. Con. on Pattern Recognition, Rome, Italy, 1988, pp 260-265. C. E. Strangio, "Numerical comparison of similarly structured data perturbed by random variations, as found in handwritten signatures", Technical Report, Dept. of Elect. Eng., 1976. I. Yoshimura and M. Yoshimura, "On-line signature verification incorporating the direction of pen movement - an experimental examination of the effectiveness", in From pixel, to features III: frontiers in Handwriting recognition, Eds. S. Impedova and J. C. Simon, Elsevier, 1992. 

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534 The Performance of Convex Set projection Based Neural Networks Robert J. Marks II, Les E. Atlas, Seho Oh and James A. Ritcey Interactive Systems Design Lab, FT-IO University of Washington, Seattle, Wa 98195. ABSTRACT We donsider a class of neural networks whose performance can be analyzed and geometrically visualized in a signal space environment. Alternating projection neural networks (APNN' s) perform by alternately projecting between two or more constraint sets. Criteria for desired and unique convergence are easily established. The network can be configured in either a homogeneous or layered form. The number of patterns that can be stored in the network is on the order of the number of input and hidden neurons. If the output neurons can take on only one of two states, then the trained layered APNN can be easily configured to converge in one iteration. More generally, convergence is at an exponential rate. Convergence can be improved by the use of sigmoid type nonlinearities, network relaxation and/or increasing the number of neurons in the hidden layer. The manner in which the network responds to data for which it was not specifically trained (i.e. how it generalizes) can be directly evaluated analytically. 1. INTRODUCTION In this paper, we depart from the performance analysis techniques normally applied to neural networks. Instead, a signal space approach is used to gain new insights via ease of analysis and geometrical interpretation. Building on a foundation laid elsewhere l - 3 , we demonstrate that alternating projecting neural network's (APNN's) formulated from such a viewpoint can be configured in layered form or homogeneously. Significiantly, APNN's have advantages over other neural network architectures . For example, (a) APNN's perform by alternatingly projecting between two or more constraint sets. Criteria can be established for proper iterative convergence for both synchronous and asynchronous operation. This is in contrast to the more conventional technique of formulation of an energy metric for the neural networks, establishing a lower energy bound and showing that the energy reduces each iteration 4 - 7 ? Such procedures generally do not address the accuracy of the final solution. In order to assure that such networks arrive at the desired globally minimum energy, computationaly lengthly procedures such as simulated annealing are used B - 10 ? For synchronous networks, steady state oscillation can occur between two states of the same energyll (b) Homogeneous neural networks such as Hopfield's content addressable memory4,12-14 do not scale well, i.e. the capacity ? American Institute of Physics 1988 535 of Hopfield's neural networks less than doubles when the number of neurons is doubled 15-16. Also, the capacity of previously proposed layered neural networks 17 ,18 is not well understood. The capacity of the layered APNN'S, on the other hand, is roughly equal to the number of input and hidden neurons 19 ? (c) The speed of backward error propagation learning 17-18 can be painfully slow. Layered APNN's, on the other hand, can be trained on only one pass through the training data 2 ? If the network memory does not saturate, new data can easily be learned without repeating previous data. Neither is the effectiveness of recall of previous data diminished. Unlike layered back propagation neural networks, the APNN recalls by iteration. Under certain important applications, however, the APNN will recall in one iteration. (d) The manner in which layered APNN's generalizes to data for which it was not trained can be analyzed straightforwardly. The outline of this paper is as follows. After establishing the dynamics of the APNN in the next section, sufficient criteria for proper convergence are given. The convergence dynamics of the APNN are explored. Wise use of nonlinearities, e.g. the sigmoidal type nonlinearities 2 , improve the network's performance. Establishing a hidden layer of neurons whose states are a nonlinear function of the input neurons' states is shown to increase the network's capacity and the network's convergence rate as well. The manner in which the networks respond to data outside of the training set is also addressed. 2. THE ALTERNATING PROJECTION NEURAL NETWORK In this section, we established the notation for the APNN. Nonlinear modificiations to the network made to impose certain performance attributes are considered later. Consider a set of N continuous level linearly independent library vectors (or patterns) of length L> N: {?n I OSnSN}. We form the library matrix !:. = [?1 1?2 I ... I?N ] and the neural network interconnect matrix a T = F (!:.T !:. )-1 FT where the superscript T denotes transposition. We divide the L neurons into two sets: one in which the states are known and the remainder in which the states are unknown. This partition may change from application to application. Let Sk (M) be the state of the kth node at time M. If the kth node falls into the known catego~, its state is clamped to the known value (i.e. Sk (M) = Ik where I is some library vector). The states of the remaining floating neurons are equal to the sum of the inputs into the node. That is, Sk (M) = i k , where L i k = p a r1 tp k sp (1) = The interconnect matrix is better trained iteratively2. To include a new library vector ?, the interconnects are updated as ~T ~T~ ~ ~ ! + (EE ) / (E E) where E = (.!. - !) f . 536 If all neurons change state simultaneously (i.e. sp = sp (M-l) ), then the net is said to operate synchronously. If only one neuron changes state at a time, the network is operating asynchronously. Let P be the number of clamped neurons. We have proven l that the neural states converge strongly to the extrapolated library vector if the first P rows of ! (denoted KP) form a matrix of full column rank. That is, no column of ~ can be expressed as a linear combination of those remainin.,v. 2 By strong convergence b , we mean lim II 1 (M) II == 0 where II x II == t iTi. M~OO Lastly, note that subsumed in the criterion that ~ be full rank is the condition that the number of library vectors not exceed the number of known neural states (P ~ N). Techniques to bypass this restriction by using hidden neurons are discussed in section 5. Without loss of generality, we will assume Partition Notation: that neurons 1 through P are clamped and the remaining neurons are floating. We adopt the vectOr partitioning notation 71 Ip = IIp] ~ io 10 where is the P-tuple of the first P elements of 1. and is a vector of the remaining Q = L-P. We can thus write, for example, ~ [ f~ If~ I ... If: ]. Using this partition notation, we can define the neural clamping operator by: 7 _ !l ~ - IL] 7 10 l Thus, the first P elements of I are clamped to P ? The remaining Q nodes "float". Partition notation for the interconnect matrix will also prove useful. Define T r!2 I !lJ L~ where ~2 is a P by P and !4 a Q by Q matrix. 3. STEADY STATE CONVERGENCE PROOFS For purposes of later reference, we address convergence of the network for synchronous operation. Asynchronous operation is addressed in reference 2. For proper convergence, both cases require that ~ be full rank. For synchronous operation, the network iteration in (1) followed by clamping can be written as: ~ ~ s(M+l) =!l ~ sCM) (2) As is illustrated in l - 3, this operation can easily be visualized in an L dimensional signal space. b The referenced convergence proofs prove strong convergence in an infinite dimensional Hilbert space. In a discrete finite dimensional space, both strong and weak convergence imply uniform convergence l9 ? 2D , i.e. 1(M)~t as M~oo. 537 For a given partition with written in partitioned form as [;'(M+J !l clamped P l*J[ !3!4 neurons, (2) can J ~oI'(M) be (3) The states of the P clamped neurons are not affected by their input sum. Thus, there is no contribution to the iteration by ~1 and ~2. We can equivalently write (3) as -+0 s (M+ 1) -;tp-+o = !3 f +!4 s (M) (4 ) We show in that if fp is full rank, then the spectral radius (magnitude of the maximum eigenvalue) of ~4 is strictly less than one 19 ? It follows that the steady state solution of (4) is: (5 ) where, since fp is full rank, we have made use of our claim that -+0 S (00) = -;to f (6) 4. CONVERGENCE DYNAMICS In this section, we explore different convergence dynamics of the APNN when fp is full column rank. If the library matrix displays certain orthogonality characteristics, or if there is a single output (floating) neuron, convergence can be achieved in a single iteration. More generally, convergence is at an exponential rate. Two techniques are presented to improve convergence. The first is standard relaxation. Use of nonlinear convex constraint at each neuron is discussed elsewhere 2 ,19. One Step Convergence: There are at least two important cases where the APNN converges other than uniformly in one iteration. Both require that the output be bipolar (?1). Convergence is in one step in the sense that -;to ? -+0 f = Slgn s (1) (7) where the vector operation sign takes the sign of each element of the vector on which it operates. ,0 . CASE 1: If there is a single output neuron, then, from (4), (5) and (6), sO (1) (1 t LL ) Since the eigenvalue of the (scalar) matrix, !4 = tL L lies between zero and one 1 9, we conclude that 1t LL > O. Thus, if ,0 is restricted to ?1, (7) follows immediately. A technique to extend this result to an arbitrary number of output neurons in a layered network is discussed in section 7. CASE 2: For certain library matrices, the APNN can also display one step convergence. We showed that if the columns of K are orthogonal and the columns of fp are also orthogonal, then one synchronous iteration results in floating states proportional to the steady 538 state values 19 ? Specifically, for the floating neurons, ~o II (1) t P 2 II 1 0 (8) 111112 An important special case of (8) is when the elements of Fare all ?1 and orthogonal. If each element were chosen by a 50-50 coin flip, for example, we would expect (in the statistical sense) that this would be the case. Exponential Convergence: More generally, the convergence rate of the APNN is exponential and is a function of the eigenstructure of .!4. Let {~r I 1 ~ r ~ Q } denote the eigenvectors of .!4 and {A r } the corresponding eigenvalues. Define ~ = [ ~l 1~2 I ... I~o] and the diagonal matrix A4 such that diag ~ = [AI A2 ... Ao] T ? Then we can . A T ? -+ T-+ -. T ? 1 f Wrl.te :!.4.=~ _4 ~. Defl.ne x (M) =~ s (M). S.;nce ~ ~ = I, \t...,. fol ows T ro~ the--+differe-ace equatJ-on i~ ('Up that x(M+l)=~:!.4 ~ ~ sCM) + ~ .!3 =~4 x (M) + g where g = ~.!3 The solution to this difference equation is 1 t. M "r /\ok 't' 1J r = gk = [ 1 _ "kM + 1 ] /\0 ( ,- 1 1 - , /\ok) gk (9) 0 Since the spectral radius of !4 is less than one 19 , ~: ~ 0 as M ~ Our steady state result is thus x k (~) = (1 - Ak ) gk. Equation . [ " M + l ] (9) can therefore be wrl.tten as x k (M) = 1 - /\ok x k (~). The eCflivalent of a "time constant" in this exponential convergence is 1/ tn (111 Ak I). The speed of convergence is thus dictated by the spectral radius of .!4. As we have shown 19 later, adding neurons in a hidden layer in an APNN can significiantly reduce this spectral radius and thus improve the convergence rate. ~. Relaxation: Both the projection and clamping operations can be relaxed to alter the network's convergence without affecting its steady state 20 - 21 ? For the interconnects, we choose an appropriate value of the relaxation parameter a in the interval (0,2) and 9 redefine the interconnect matrix as T aT + (1 a)I or equivalently, = {a(t nn -l)+1 ; n =m a tnrn TO see the effect of such relaxation on convergence, we need simply exam\ne the resulting ::dgenvalues. If .!4 has eigenvalues {A r I, then .!4 has eigenvalues Ar = 1 + a (Ar - 1). A Wl.se choice of a reduces the spectral radius of .!~ with respect to that of .!4' and thus decreases the time constant of the network's convergence. Any of the operators projecting onto convex sets can be relaxed without affecting steady state convergence 19 - 20 ? These include the ~ operator 2 and the sigmoid-type neural operator that projects onto a box. Choice of stationary relaxation parameters without numerical andlor empirical study of each specific case, however, generally remains more of an art than a science. 539 5. LAYERED APNN' S The networks thus far considered are homogeneous in the sense that any neuron can be clamped or floating. If the partition is such that the same set of neurons always provides the network stimulus and the remainder respond, then the networks can be simplified. Clamped neurons, for example, ignore the states of the other neurons. The corresponding interconnects can then be deleted from the neural network architecture. When the neurons are so partitioned, we will refer the APNN as layered. In this section, we explore various aspects of the layered APNN and in particular, the use of a so called hidden layer of neurons to increase the storage capacity of the network. An alternate architecture for a homogeneous APNN that require only Q neurons has been reported by Marks 2 ? Hidden Layers: In its generic form, the APNN cannot perform a simple exclusive or (XOR). Indeed, failure to perform this same operation was a nail in the coffin of the perceptron 22 . Rumelhart et. al.1 7 -18 revived the percept ron by adding additional layers of neurons. Although doing so allowed nonlinear discrimination, the iterative training of such networks can be painfully slow. With the addition of a hidden layer, the APNN likewise generalizes. In contrast, the APNN can be trained by looking at each data vector only once 1 ? Although neural networks will not likely be used for performing XOR's, their use in explaining the role of hidden neurons is quite instructive. The library matrix for the XOR is f- [~ ~ ~ ~ 1 The first two rOwS of F do not form a matrix of full column rank. Our approach is to augment fp with two more rows such that the resulting matrix is full rank. Most any nonlinear combination of the first two rowS will in general increase the matrix rank. Such a procedure, for example, is used in ~-classifiers23 . possible nonlinear operations include multiplication, a logical "AND" and running a weighted sum of the clamped neural states through a memoryless nonlinearity such as a sigmoid. This latter alteration is particularly well suited to neural architectures. To illustrate with the exclusive or (XOR) , a new hidden neural state is set equal to the exponentiation of the sum of the first two rows. A second hidden neurons will be assigned a value equal to the cosine of the sum of the first two neural states multiplied by Tt/2. (The choice of nonlinearities here is arbitrary. ) The augmented library matrix is !:.+ 0 0 1 1 0 0 1 1 0 1 1 e e e2 0 0 1 -1 1 0 540 In either the training or look-up mode, the states of the hidden neurons are clamped indirectly as a result of clamping the input neurons. The playback architecture for this network is shown in Fig .1. The interconnect values for the dashed lines are unity. The remaining interconnects are from the projection matrix formed from !+. Geometrical Interpretation In lower dimensions, the effects of hidden neurons can be nicely illustrated geometrically. Consider the library matrix F = 1 1/2 ] Clearly IP = (1/2 1) . Let the neurons in the hidden layer be determined by the nonlineariy x 2 where x denotes the elements in the first row of f. Then !+ = [t: I t; ] = [ 1/2 1i4 1;2 J The corresponding geometry is shown in Fig. 2 for x the input neuron, y the output and h the hidden neuron. The augmented library vectors are shown and a portion of the generated subspace is shown lightly shaded. The surface of h = x 2 resembles a cylindrical lens in three dimensions. Note that the linear variety corresponding to f = 1/2 intersects the cylindrical lens and subspace only at Similarly, the x = 1 plane intersects the lens and subspace at 2 ? Thus, in both cases, clamping the input corresponding to the first element of one of the two library vectors uniquely determines the library vector. 1+. 1 Convergence Improvement: Use of additional neurons in the hidden layer will improve the convergence rate of the APNN 19 ? Specifically, the spectral radius of the .!4 matrix is decreased as additional neurons are added. The dominant time constant controlling convergence is thus decreased. Capacity: Under the assumption that nonlinearities are chosen such that the augmented fp matrix is of full rank, the number of vectors which can be stored in the layered APNN is equal to the sum of the number of neurons in the input and hidden layers. Note, then, that interconnects between the input and output neurons are not needed if there are a sufficiently large number of neurons in the hidden layer. 6. GENERALIZATION We are assured that the APNN will converge to the desired result if a portion of a training vector is used to stimulate the network. What, however, will be the response if an initialization is used that is not in the training set or, in other words, how does the network generalize from the training set? To illustrate generalization, we return to the XOR problem. Let S5 (M) denote the state of the output neuron at the Mth (synchronous) 541 loyer : , "" " / input - hidden , , "- "/ / / X 3 exp Figure 1. Illustration of a layered APNN fori performing an XOR. Figure 3. Response of the elementary XOR APNN using an exponential and trignometric nonlinearity in the hidden layer. Note that, at the corners, the function is equal to the XOR of the y l( Figure 2. A geometrical illustration of the use of an x 2 nonlinearity to determine the states of hidden neurons. Figure 4. The generalization of the XOR networks formed by thresholding the function in Fig . 3 at 3/4. Different hidden layer nonlinearities result in different generalizations. 542 iteration. If S1 and S2 denote the input clamped value, then S5 (m+1) =t1 5 Sl + t 25 S2 + t35 S3 + t4 5 S4 + t5 5 S5 (m) where S3 =exp (Sl +S2 ) and S4 =cos [1t (S1 + S2) /2] To reach steady state, we let m tend to infinity and solve for S5 (~) : 1 A plot of S5 (~) versus (S1,S2) is shown in Figure 3. The plot goes through 1 and zero according to the XOR of the corner coordinates. Thresholding Figure 3 at 3/4 results in the generalization perspective plot shown in Figure 4. To analyze the network's generalization when there are more than one output neuron, we use (5) of which (10) is a special case. If conditions are such that there is one step convergence, then generalization plots of the type in Figure 4 can be computed from one network iteration using (7). 7. NOTES (a) There clearly exists a great amount of freedom in the choice of the nonlinearities in the hidden layer. Their effect on the network performance is currently not well understood. One can envision, however, choosing nonlinearities to enhance some network attribute such as interconnect reduction, classification region shaping (generalization) or convergence acceleration. (b) There is a possibility that for a given set of hidden neuron nonlinearities, augmentation of the fp matrix coincidentally will result in a matrix of deficent column rank, proper convergence is then not assured. It may also result in a poorly conditioned matrix, convergence will then be quite slow. A practical solution to these problems is to pad the hidden layer with additional neurons. As we have noted, this will improve the convergence rate. (c) We have shown in section 4 that if an APNN has a single bipolar output neuron, the network converges in one step in the sense of (7). Visualize a layered APNN with a single output neuron. If there are a sufficiently large number of neurons in the hidden layer, then the input layer does not need to be connected to the output layer. Consider a second neural network identical to the first in the input and hidden layers except the hidden to output interconnects are different. Since the two networks are different only in the output interconnects, the two networks can be combined into a singlee network with two output neurons. The interconnects from the hidden layer to the output neurons are identical to those used in the single output neurons architectures. The new network will also converge in one step. This process can clearly be extended to an arbitrary number of output neurons. REFERENCES 1. R.J. Marks II, "A Class of Continuous Level Associative Memory Neural Nets," ~. Opt., vo1.26, no.10, p.200S, 1987. 543 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. K.F. Cheung et. al., "Neural Net Associative Memories Based on Convex Set Projections," Proc. IEEE 1st International Conf. on Neural Networks, San Diego, 1987. R.J. Marks II et. al., "A Class of Continuous Level Neural Nets," Proc. 14th Congress of International Commission for Optics Conf., Quebec, Canada, 1987. J.J. Hopfield, "Neural Networks and Physical Systems with Emergent Collective Computational Abilities," Proceedings Nat. Acad. of Sciences, USA, vol.79, p.2554, 1982. J.J. Hopfield et. al., "Neural Computation of Decisions in Optimization Problem," BioI. Cyber., vol. 52, p.141, 1985. ?D. W. Tank et. al., "Simple Neurel Optimization Networks: an AID Converter, Signal Decision Circuit and a Linear Programming Circuit," IEEE Trans. Cir. ~., vol. CAS-33, p.533, 1986. M. Takeda et. ai, "Neural Networks for Computation: Number Representation and Programming Complexity," ~. Opt., vol. 25, no. 18, p.3033, 1986. S. Geman et. al., "Stochastic Relaxation, Gibb's Distributions, and the Bayesian Restoration of Images," IEEE Trans. Pattern Recog. & Machine Intelligence., vol. PAMI-6, p.721, 1984. S. Kirkpatrick et. al. ,"Optimization by Simulated Annealing," Science, vol. 220, no. 4598, p.671, 1983. D.H. Ackley et. al., "A Learning Algorithm for Boltzmann Machines," Cognitive Science, vol. 9, p.147, 1985. K.F. Cheung et. al., "Synchronous vs. Asynchronous Behaviour of Hopfield's CAM Neural Net," to appear in Applied Optics. R.P. Lippmann, "An Introduction to Computing With Neural nets," IEEE ASSP Magazine, p.7, Apr 1987. N. Farhat et. al .. , "Optical Implementation of the Hopfield Model," ~. Opt., vol. 24, pp.1469, 1985. L.E. Atlas, "Auditory Coding in Higher Centers of the CNS," IEEE Eng. in Medicine and Biology Magazine, p.29, Jun 1987. Y.S. Abu-Mostafa et. al., "Information Capacity of the Hopfield Model, " IEEE Trans. Inf. Theory, vol. IT-31, p.461, 1985. R.J. McEliece et. al.,"The Capacity of the Hopfield Associative Memory, " IEEE Trans. Inf. Theory (submitted), 1986. D.E. Rumelhart et. al., Parallel Distributed Prooessing, vol. I & II, Bradford Books, Cambridge, MA, 1986. D.E. Rumelhart et. al., "Learning Representations by Back-Propagation Errors," Nature. vol. 323, no. 6088, p.533, 1986. R.J. Marks II et. al.,"Alternating Projection Neural Networks," ISDL report *11587, Nov. 1987 (Submitted for publication) . D.C. Youla et. al, "Image Restoration by the Method of Convex Projections: Part I-Theory," IEEE Trans. Med. Imaging, vol. MI-1, p.81, 1982. M. I. Sezan and H. Stark. "Image Restoration by the Method of Convex Projections: Part II-Applications and Numerical Results," IEEE Trans. Med. Imaging, vol. MI-1, p.95, 1985. M. Minsky et. al., Perceptrons, MIT Press, Cambridge, MA, 1969. J. Sklansky et. al., Pattern Classifiers and Trainable Machines, Springer-Verlag, New York, 1981. 

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Packet Routing in Dynamically Changing Networks: A Reinforcement Learning Approach Justin A. Boyan School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Michael L. Littman? Cognitive Science Research Group Bellcore Morristown, NJ 07962 Abstract This paper describes the Q-routing algorithm for packet routing, in which a reinforcement learning module is embedded into each node of a switching network. Only local communication is used by each node to keep accurate statistics on which routing decisions lead to minimal delivery times. In simple experiments involving a 36-node, irregularly connected network, Q-routing proves superior to a nonadaptive algorithm based on precomputed shortest paths and is able to route efficiently even when critical aspects of the simulation, such as the network load, are allowed to vary dynamically. The paper concludes with a discussion of the tradeoff between discovering shortcuts and maintaining stable policies. 1 INTRODUCTION The field of reinforcement learning has grown dramatically over the past several years, but with the exception of backgammon [8, 2], has had few successful applications to large-scale, practical tasks. This paper demonstrates that the practical task of routing packets through a communication network is a natural application for reinforcement learning algorithms. *Now at Brown University, Department of Computer Science 671 672 Boyan and Littman Our "Q-routing" algorithm, related to certain distributed packet routing algorithms [6, 7], learns a routing policy which balances minimizing the number of "hops" a packet will take with the possibility of congestion along popular routes. It does this by experimenting with different routing policies and gathering statistics about which decisions minimize total delivery time. The learning is continual and online, uses only local information, and is robust in the face of irregular and dynamically changing network connection patterns and load. The experiments in this paper were carried out using a discrete event simulator to model the transmission of packets through a local area network and are described in detail in [5]. 2 ROUTING AS A REINFORCEMENT LEARNING TASK A packet routing policy answers the question: to which adjacent node should the current node send its packet to get it as quickly as possible to its eventual destination? Since the policy's performance is measured by the total time taken to deliver a packet, there is no "training signal" for directly evaluating or improving the policy until a packet finally reaches its destination. However, using reinforcement learning, the policy can be updated more quickly and using only local information. Let Qx(d, y) be the time that a node x estimates it takes to deliver a packet P bound for node d by way of x's neighbor node y, including any time that P would have to spend in node x's queue. l Upon sending P to y, x immediately gets back y's estimate for the time remaining in the trip, namely t= . min zEnelghbors of y Qy (d, z) If the packet spent q units of time in x's queue and s units of time in transmission between nodes x and y, then x can revise its estimate as follows: new estimate old estimate ~~ LlQx(d,Y)=17( q+s+t - Qx(d,y)) where 17 is a "learning rate" parameter (usually 0.5 in our experiments). The resulting algorithm can be characterized as a version of the Bellman-Ford shortest paths algorithm [1, 3] that (1) performs its path relaxation steps asynchronously and online; and (2) measures path length not merely by number of hops but rather by total delivery time. We call our algorithm "Q-routing" and represent the Q-function Qx( d, y) by a large table. We also tried approximating Qx with a neural network (as in e.g. [8, 4]), which allowed the learner to incorporate diverse parameters of the system, including local queue size and time of day, into its distance estimates. However, the results of these experiments were inconclusive. 1 We denote the function by Q because it corresponds to the Q function used in the reinforcement learning technique of Q-learning [10]. Packet Routing in Dynamically Changing Networks: A Reinforcement Learning Approach -- - - - .... .... ?? ?? Figure 1: The irregular 6 x 6 grid topology 3 RESULTS We tested the Q-routing algorithm on a variety of network topologies, including the 7-hypercube, a 116-node LATA telephone network, and an irregular 6 x 6 grid. Varying the network load, we measured the average delivery time for packets in the system after learning had settled on a routing policy, and compared these delivery times with those given by a static routing scheme based on shortest paths. The result was that in all cases, Q-routing is able to sustain a higher level of network load than could shortest paths. This section presents detailed results for the irregular grid network pictured in Figure l. Under conditions of low load, the network learns fairly quickly to route packets along shortest paths to their destinations. The performance vs. time curve plotted in the left part of Figure 2 demonstrates that the Q-routing algorithm, after an initial period of inefficiency during which it learns the network topology, performs about as well as the shortest path router, which is optimal under low load. As network load increases, however, the shortest path routing scheme ceases to be optimal: it ignores the rising levels of congestion and soon floods the network with packets. The right part of Figure 2 plots performance vs. time for the two routing schemes under high load conditions: while shortest path is unable to tolerate the packet load, Q-routing learns an efficient routing policy. The reason for the learning algorithm's success is apparent in the "policy summary diagrams" in Figure 3. These diagrams indicate, for each node under a given policy, how many of the 36 x 35 point-to-point routes go through that node. In the left part of Figure 3, which summarizes the shortest path routing policy, two nodes in the center of the network (labeled 570 and 573) are on many shortest paths and thus become congested when network load is high. By contrast, the diagram on the right shows that Q-routing, under conditions of high load, has learned a policy which routes 673 674 Boyan and Littman Q-routing Shortest paths ----. 500 400 500 Q-routing Shortest paths ----. ~ I ! V I ? ! : ,I ,I, ,I l I I :' :1 I I " II 1"1 : I, ~ I~ ,'''" !: ~ " ~ I ~ ",1' ,I 1\I , .,' I " ~,' I I', II ~'~I I II ~ J~,ft. ,I 'I ~'I I l ", f ' 400 ~ II 'I" I f I' I I \I I I I I: 1 ",' ~ ':1 I "' , ? IVI~" II ",~ 1 ,,\II I I I I I I I ) "', I I' ,I :: :~ Q) E F I, " I ~ 300 Q) .~ , ,,, , 300 Q) 0 Q) Cl ~ Q) 200 ?> I ~ ,~/~ ,, I !~ .,'" ,, , ,,, " j" 200 I I 100 100 " ,, : I I 0 ------0 2000 4000 6000 8000 10000 0 0 \ 2000 Simulator Time 4000 6000 8000 10000 Simulator Time Figure 2: Performance under low load and high load 1~4--131-+1-7------------1+6--125--1~5 , , ,, ,, 207 45----5:4 ,, ,, ,, ,,, ,, ,, ,, , 54----4;3 1$9 , , , 364--392--396-----------396--393--367 , , ,,, ,, ,, ,, ,,, ,, 375 102---5.9 , , , ,, , 2rS445'l'{)- ........5fS3$&2'?8 , , .. ? ? " ' . , . 1tB--1~--219 , , , 2$s--1~5--110 ~---1-t3--146 , , , 1~4--1t9--1~8 , , , ,, ,, 45----7-6----58 ,, ,,, 3T7 ,,, , ,, 3~4--2t2--2~8-----------~--2tu--3~3 , , . , , 5~----~2 ,,, ,,, ,,, , ? I I . , ' '" . , ' ? I ? . , , , ,I 2$2--21-8-- t 1 4 , , , 2?7--2<t>1--2-1 7 1rB--1t8---~3 , , , 1$O--1-+1--H~2 ,, ,, ,, 79---105---=15 108--121---=14 ,,, , ,,, , ,, ,, ,, ,, ,, ,, ,, ,, Figure 3: Policy summaries: shortest path and Q-routing under high load Packet Routing in Dynamically Changing Networks: A Reinforcement Learning Approach some traffic over a longer than necessary path (across the top of the network) so as to avoid congestion in the center of the network. The basic result is captured in Figure 4, which compares the performances of the shortest path policy and Q-routing learned policy at various levels of network load. Each point represents the median (over 19 trials) of the mean packet delivery time after learning has settled. When the load is very low, the Q-routing algorithm routes nearly as efficiently as the shortest path policy. As load increases, the shortest path policy leads to exploding levels of network congestion, whereas the learning algorithm continues to route efficiently. Only after a further significant increase in load does the Q-routing algorithm, too, succumb to congestion . 1B W u Q-routing Shortest paths ' ' . 16 c: Ql u III Ql '5 14 0- w ~ 12 Ql E i= ~ 10 Ql .~ Qi 0 Ql 0> B ~ Ql ~ 6 _.. --"-- " ,, 4 0.5 1.5 2 2.5 3 Network Load Level 3.5 4 4.5 Figure 4: Delivery time at various loads for Q-routing and shortest paths 3.1 DYNAMICALLY CHANGING NETWORKS One advantage a learning algorithm has over a static routing policy is the potential for adapting to changes in crucial system parameters during network operation. We tested the Q-routing algorithm, unmodified, on networks whose topology, traffic patterns, and load level were changing dynamically: Topology We manually disconnected links from the network during simulation. Qualitatively, Q-routing reacted quickly to such changes and was able to continue routing traffic efficiently. Traffic patterns We caused the simulation to oscillate periodically between two very different request patterns in the irregular grid: one in which all traffic was directed between the upper and lower halves of the network, and one in which all traffic was directed between the left and right halves. Again, 675 676 Boyan and Littman after only a brief period of inefficient routing each time the request pattern switched, the Q-routing algorithm adapted successfully. Load level When the overall level of network traffic was raised during simulation, Q-routing quickly adapted its policy to route packets around new bottlenecks. However, when network traffic levels were then lowered again, adaptation was much slower, and never converged on the optimal shortest paths. This effect is discussed in the next section . 3.2 EXPLORATION Given the similarity between the Q-routing update equation and the Bellman-Ford recurrence for shortest paths, it seems surprising that there is any difference whatsoever between the performance of Q-routing and shortest paths routing at low load, as is visible in Figure 4. However, a close look at the algorithm reveals that Q-routing cannot fine-tune a policy to discover shortcuts, since only the best neighbor's estimate is ever updated. For instance, if a node learns an overestimate of the delivery time for an optimal route, then it will select a suboptimal route as long as that route's delivery time is less than the erroneous estimate of the optimal route 's delivery time. This drawback of greedy Q-Iearning is widely recognized in the reinforcement learning community, and several exploration techniques have been suggested to overcome it [9]. A common one is to have the algorithm select actions with some amount of randomness during the initial learning period[10]. But this approach has two serious drawbacks in the context of distributed routing: (1) the network is continuously changing, thus the initial period of exploration never ends; and more significantly, (2) random traffic has an extremely negative effect on congestion . Packets sent in a suboptimal direction tend to add to queue delays, slowing down all the packets passing through those queues, which adds further to queue delays, etc. Because the nodes make their policy decisions based on only local information, this increased congestion actually changes the problem the learners are trying to solve. Instead of sending actual packets in a random direction, a node using the "full echo" modification of Q-routing sends requests for information to its immediate neighbors every time it needs to make a decision . Each neighbor returns a single number-using a separate channel so as to not contribute to network congestion in our model-giving that node's current estimate of the total time to the destination. These estimates are used to adjust the Qx(d , y) values for each neighbor y. When shortcuts appear, or if there are inefficiencies in the policy, this information propagates very quickly through the network and the policy adjusts accordingly. Figure 5 compares the performance of Q-routing and shortest paths routing with "full echo" Q-routing. At low loads the performance of "full echo" Q-routing is indistinguishable from that of the shortest path policy, as all inefficiencies are purged. Under high load conditions, "full echo" Q-routing outperforms shortest paths but the basic Q-routing algorithm does better still. Our analysis indicates that "full echo" Q-routing constantly changes policy under high load, oscillating between using the upper bottleneck and using the central bottleneck for the majority of crossnetwork traffic. This behavior is unstable and generally leads to worse routing times under high load. Packet Routing in Dynamically Changing Networks: A Reinforcement Learning Approach Q-routing Shortest paths Full Echo ----- 18 2lc: ----- 16 Q) 0 Ul .!!! 14 '" CT Q; ::: ..!! 12 Q) E i= ~ 10 Q) .~ Qi 0 Q) Cl 8 ~ Q) ?> 6 ~,""-,~~ ??._.".L.~~" ??~"??----'::???? . 4 0.5 1.5 2 2.5 3 Network Load Level 3.5 4 4.5 Figure 5: Delivery time at varIOUS loads for Q-routing, shortest paths and "full echo" Q-routing Ironically, the "drawback" of the basic Q-routing algorithm-that it does no exploration and no fine-tuning after initially learning a viable policy-actually leads to improved performance under high load conditions. We still know of no single algorithm which performs best under all load conditions. 4 CONCLUSION This work considers a straightforward application of Q-Iearning to packet routing. The "Q-routing" algorithm, without having to know in advance the network topology and traffic patterns, and without the need for any centralized routing control system, is able to discover efficient routing policies in a dynamically changing network. Although the simulations described here are not fully realistic from the standpoint of actual telecommunication networks, we believe this paper has shown that adaptive routing is a natural domain for reinforcement learning. Algorithms based on Q-routing but specifically tailored to the packet routing domain will likely perform even better. One of the most interesting directions for future work is to replace the table-based representation of the routing policy with a function approximator. This could allow the algorithm to integrate more system variables into each routing decision and to generalize over network destinations. Potentially, much less routing information would need to be stored at each node, thereby extending the scale at which the algorithm is useful. We plan to explore some of these issues in the context of packet routing or related applications such as auto traffic control and elevator control. 677 678 Boyan and Littman Acknowledgements The authors would like to thank for their support the Bellcore Cognitive Science Research Group, the National Defense Science and Engineering Graduate fellowship program, and National Science Foundation Grant IRI-9214873. References [1] R. Bellman. On a routing problem. 16(1):87-90, 1958. Quarterly of Applied Mathematics, [2] J. Boyan. Modular neural networks for learning context-dependent game strategies. Master's thesis, Computer Speech and Language Processing, Cambridge University, 1992. [3] L. R. Ford, Jr. Flows in Networks. Princeton University Press, 1962 . [4] L.-J . Lin. Reinforcement Learning for Robots Using Neural Networks. PhD thesis, School of Computer Science, Carnegie Mellon University, 1993. [5] M. Littman and J. Boyan. A distributed reinforcement learning scheme for network routing. Technical Report CMU-CS-93-165, School of Computer Science, Carnegie Mellon University, 1993. [6] H. Rudin. On routing and delta routing: A taxonomy and performance comparison of techniques for packet-switched networks. IEEE Transactions on Communications, COM-24(1):43-59, January 1976. [7] A. Tanenbaum . Computer Networks. Prentice-Hall, second edition edition, 1989. [8] G. Tesauro. Practial issues in temporal difference learning. Machine Learning, 8(3/4), May 1992. [9] Sebastian B. Thrun. The role of exploration in learning control. In David A. White and Donald A. Sofge, editors, Handbook of Intelligent Control: Neural, Fuzzy, and Adaptive Approaches. Van Nostrand Reinhold, New York, 1992. [10] C . Watkins. Learning from Delayed Rewards. PhD thesis, King's College, Cambridge, 1989. 

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Resolving motion ambiguities K. I. Diamantaras Siemens Corporate Research 755 College Rd . East Princeton, NJ 08540 D. Geiger* Courant Institute, NYU Mercer Street New York, NY 10012 Abstract We address the problem of optical flow reconstruction and in particular the problem of resolving ambiguities near edges. They occur due to (i) the aperture problem and (ii) the occlusion problem, where pixels on both sides of an intensity edge are assigned the same velocity estimates (and confidence). However, these measurements are correct for just one side of the edge (the non occluded one). Our approach is to introduce an uncertamty field with respect to the estimates and confidence measures. We note that the confidence measures are large at intensity edges and larger at the convex sides of the edges, i.e. inside corners, than at the concave side. We resolve the ambiguities through local interactions via coupled Markov random fields (MRF) . The result is the detection of motion for regions of images with large global convexity. 1 Introduction In this paper we discuss the problem of figure ground separation, via optical flow, for homogeneous images (textured images just provide more information for the disambiguation of figure-ground). We address the problem of optical flow reconstruction and in particular the problem of resolving ambiguities near intensity edges . We concentrate on a two frames problem, where all the motion ambiguities we discuss can be disambiguiated by the human visual system. *work done when the author was at the Isaac Newton Institute and at Siemens Corporate Research 977 978 Diamantaras and Geiger Optical flow is a 2D (two dimensional) field defined as to capture the projection of the 3D (three dimensional) motion field into the view plane (retina). The Horn and Schunk[8] formulation of the problem is to impose (i) the brightness constraint dE(;~y,t) = 0, where E is the intensity image, and (ii) the smoothness of the velocity field. The smoothness can be thought of coming from a rigidity or quasi-rigidity assumption (see Ullman [12]). We utilize two improvements which are important for the optical flow computation , (i) the introduction of the confidence measure (Nagel and Enkelman [10], Anandan [1]) and (ii) the application of smoothness while preserving discontinuities (Geman and Geman [6], Blake and Zisserman [2], Mumford and Shah [9]). It is clear that as an object moves with respect to a background not only optical flow discontinuities occur, but also occlusions occur (and revelations). In stereo , occlusions are related to discontinuities (e.g. Geiger et . al 1992 [5]) , and for motion a similar relation must exist . We study ambiguities ocuring at motion discontinuities and occlusions in images. The paper is organized as follows : Section 2 describes the problem with examples and a brief discussion on possible approaches, section 3 presents our approach, with the formulation of the model and a method to solve it , section 4 gives the results . 2 Motion ambiguities Figure 1 shows two synthetic problems involving a translation and a rotation of simple objects in front of stationary backgrounds. Consider the case of the square translation (see figure Ia.). Humans perceive the square translating , although block-matching (and any other matching technique) gives translation on both sides of the square edges. Moreover , there are other interpretations of the scene, such as the square belonging to the stationary background and the outside being a translating foreground with a square hole. The examples are synthetic, but emphasize the ambiguities. Real images may have more texture , thus many times helping resolve these ambiguities, but not everywhere. (a) (b) Figure 1: Two image sequences of 128 x 128. (a) Square translation of 3 pixels; (b) "Eight" rotation of 10 0 ? Note that the "eight" has concave and convex regions. Resolving Motion Ambiguities 3 A Markov random field model We describe a model capable of solving these ambiguities. It is based on coupled Markov random fields and thus, based on local processes. Our main contribution is to introduce the idea of uncertainty on the estimates and confidence measures. We propose a Markov field that allows the estimates of each pixel to be chosen among a large neighborhood, thus each pixel estimate can be neglected. We show that convex regions of the image do bias the confidence measures such that the final motion solutions are expected to be the ones with global larger convexity Note that locally, one can have concave regions of a shape that give "wrong" bias (see figure 1 b). 3.1 Block Matching Block matching is the process of correlating a block region of one image, say of size (2w M + 1) x (2w M + 1), with a block regIOn of the other image. Block-matching yields a set of matching errors dir, where (i, j) is a pixel in the image and v = [m, n] is a displacement vector in a search window of size (2ws + 1) x (2ws + 1) around the pixel. We define the velocity measurements gij and the covariance matrix Cij as the mean and variance of the-vector v [m, n] averaged according to the distribution e = _kd mn ']: gij = ~ ~ m,n e _kdmn .] Figure 2 shows the block matching data gij for the two problems discussed above and figure 3 shows the correspondent confidence measurse (inverse of the covariance matrix as defined below). 3.2 The aperture problem and confidence The aperture problem [7] occurs where there is a low confidence on the measurements (data) in the direction along an edge; In particular we follow the approach by [1]. The eigenvalues AI, A2, of C ij correspond to the variance of distribution of v along the directions of the corresponding eigenvectors VI, V2. The confidence of the estimate should be inversely proportional to the variance of the distribution, i.e. the confidence along direction VI (V2) is ex 1/ Al (ex 1/ A2)' All this confidence information can be packaged inside the confidence matrzx defined as follows: Rij=f(Cij+f)-1 (1) where ? is a very small constant that guarantees invertibility. Thus the eigenvalues of Rij are values between 0 and 1 corresponding to the confidence along the directions VI and V2, whereas VI and V2 are still eigenvectors of R ij . The confidence measures at straight edges is high perpendincular to the edges and low (zero) along the edges. However, at corners, the confidence is high on both 979 980 Diamantaras and Geiger directions thus through smoothness this result can be propagated through the other parts of the image, then resolving the aperture problem . 3.3 The localization problem and a binary decision field The localization problem arises due to the local symmetry at intensity edges , where both sides of an edge give the same correspondences. These cases occur when occluded regions are homogeneous and so , block matching, pixel matching or any matching technique can not distinguish which side of the edge is being occluded or is occluding. Even if one considers edge based methods, the same problem arises in the reconstruction stage, where the edge velocities have to be propagated to the rest of the image. In this cases a localization uncertainty is introduced . More precisely, pixels whose matching block contains a strong feature (e .g. a corner) will obtain a high-confidence motion estimate along the direction in which this feature moved. Pixels on both sides of this feature , and at distances less than half the matching window size , W M , will receive roughly the same motion estimates associated with high confidences. However, it could have been just one of the two sides that have moved in this direction . In that case this estimate should not be taken into account on the other side. We note however a bias towards inside of corner regions from the confidence measures. Note that in a corner, despite both sides getting roughly the same velocity estimate and high confidence measures, the inside pixel always get a larger confidence. This bias is due to having more pixels outside the edge of a closed contour than outside, and occurs at the convex regions (e .g. a corner) . Thus, in general , the convex regions will have a stronger confidence measure than outside them . Note that at concavities in the "eight" rotation image, the confidence will be higher outside the "eight" and correct at convex regions . Thus, a global optimization will be required to decide which confidences to "pick up" . Our approach to resolve this ambiguity is to allow for the motion estimate at pixel (i , j) to select data from a neighborhood N ij , and its goal is to maximize the total estimates (taking into account the confidence measures). More precisely, let iij be the vector motion field at pixel (i , j) . We introduce a binary field ai'r that indicates which data gi+m,j+n in a neighborhood N ij of (i,j) should correspond to a motion estimate iij . The size of N ij is given by W M + 1 to overcome the localization uncertainty. For a given lattice point (i , j) the boolean parameters ai'r should be mutually exclusive, i.e. only one of them , a~?n? , should be equal to 1 indicating that iij should correspond to gi+m. ,j+n. , While the rest a'?;n , m =f:. m* , n =f:. n*, n ? = 1). The conditional probability reflects should be zero (or 2:m.n.EN'J both an uncertainty due to noise and an uncertainty due to spatial localization of the data a7r Resolving Motion Ambiguities 3.4 The piecewise smooth prior The prior probability of the motion field fij is a piecewise smoothness condition , as in [6]. P(f . a. h, v) = ~1 exp{ - (I: J.L( i-hi) )llitJ - il-1.) 11 2+J-L( I-Vi] )lIiij - ii ,J_1112+~'i) (hlJ +Vi) )) } . I IJ (3) where h ij = 0 (Vij = 0) if there is no motion discontinuity separating pixels (i , j) , (i - l,j) ((i,j), (i,j - 1) , otherwise h ij = 1 (vii = 1). The parameter J.L has to be estimated. We have considered that the cost to create motion discontinuities 1(1 - 6eij), should be lowered at intensity edges (see Poggio et al. [11]) , i.e Iii where eij is the intensity edge and 0 ~ 6 ~ 1 and I have to be estimated . = 3.5 The posterior distribution The posterior distribution is given by Bayes' law P(f a h vlg R) = 1 P(g Rlf a)P(f a h v) = !e-V(f,o ,h,v;g) , ' " P(g, R) " ' " Z (4) where L{ L V(j, a , h, v) ij a~TIIRi+m,j+n(jij - gi+m ,j+n)11 2 mneN'J + J.L(I- hij)llfij - fi_i,jI12 + J.L(I- Vij)llfij - !i,j_1112 lij(hij+Vij)} + (5) is the energy of the system . Ideally, we would like to mInImIZe V under all possible configurations of the fields f , h, v and a , while obeying the constraint EmneN'J ai]n = 1. 3.6 Mean field techniques Introducing the inverse temperature parameter (3(= liT) we can obtain the transformed probability distribution 1 P{3(j , alg , R) = _e-{3V(f,o) Z{3 (6) where Z{3 L exp{ -(3 L J.L?j Ilfij U} Ii-i,j W + J.L~j Ilfij - Ii ,i _1112} ij L x (LexP{-(3L {o} ij mneN'J a~TIIRi+m ,j+n(jij - gi+m,j+n)11 2 }) (7) 981 982 Diamantaras and Geiger where J-lij = J-l(1 - Vij) and J-l?j = J-l(1 - hij ). We have to obey the constraint LmnEN' l o:ir = 1. For the sake of simplicity we have assumed that the neighborhood N ij around site (i, j) is N ij = {( i + m, j + n): -1 ~ m ~ 1, -1 ~ n ~ I}. The second factor in (7) can be explicitly computed. Employing the mean field techniques proposed in [3] and extended in [4] we can average out the variables h, v and 0: (including the constraint) and yield 1 :LII( L {J} ij (exp{-j1"~+m,j+n(fij - 9i+m,j+n)11 2 }) m,n=-1 (elll +el-'lIf'l-f,-1,lI12)(el'l +el-'llf.l-f.o1-dI 2)) (8) which yields the following effective energy Veff(f) since Z{3 = L{f} e-{3 V eff(f) . Using the saddle point approximation, i.e. considering Z{3 ~ e -(3Veff Cf) with! minimizing Veff (/; g). the mean field equations become 0= L cii.t ~+m,j+n(hj - 9i+m,J+n) + J-lij!1v lij + J-l7j !1 h iij mn = with J-lij J-l( 1 - Vij) and (hj - h,j -1), and J-lt (9) The normalization constant Z{3 called the partition function, has the important property that lim - {3-00 ~ In Z{3 = {f,cr,h,v} min {V(f, 0:, h, v)} (10) fJ Then using an annealing method we let j1 - 00 and the minimum of V{3 = - ~ In Z{3 approaches asymptotically the desired minimum. Resolving Motion Ambiguities 4 Results We have applied an iterative method along with an annealing schedule to solvE' the above mean field equations for f3 ~ 00. The method was run on the two examples already described. Figure 4 depicts the results of the experiments . The system chooses a natural interpretation (in agreement with human perception) , namely it interprets the object (e.g. the square in the first example or the eight-shaped region in the second example) moving and the background being stationary. In the beginning of the annealing process the localization field a may produce "erroneous" results, however the neighbor information eventually forces the pixels outside the moving object to coincide with the rest of the background which has zero motion . For the pixels inside the object, on the contrary, the neighbor information eventually reinforces the adoption of the motion of the edges. References [1] P. Anandan, "Measuring Visual Motion from Image Sequences", PhD thesis. COINS Dept., Univ. Massachusetts, Amherst, 1987. [2] A. Blake and A. Zisserman, "Visual Reconstruction", Cambridge , Mass, MIT press, 1987. [3] D. Geiger and F. Girosi, "Parallel and Deterministic Algorithms for MRFs: Surface Reconstruction and Integration", IEEE PAMI: 13(5), May 1991. [4] D. Geiger and A. Yuille, "A Common Framework for Image Segmentation" , Int . J. Comput. Vision, 6(3) , pp . 227-243, 1991. [5] D. Geiger and B. Ladendorf and A. Yuille , "Binocular stereo with occlusion" , Computer Vision- ECCV92, ed . G. Sandini, Springer-Verlag, 588, pp 423-433 , May 1992 . [6] S. Geman and D. Geman , "Stochastic Relaxation , Gibbs Distributions , and the Bayesian Restoration of Images", IEEE PAMI 6, pp . 721-741 , 1984. [7] E. C . Hildreth, "The measurement of visual motion" , MIT press , 1983. [8] B.K.P. Horn and B.G. Schunk , "Determining optical flow", Artificial Intelligence, vol 17, pp . 185-203, August 1981. [9] D. Mumford and J. Shah , "Boundary detection by minimizing functionals, 1" , Proc. IEEE Conf. on Computer Vision & Pattern Recognition , San Francisco, CA,1985. [10] H.-H. Nagel and W. Enkelmann, "An Investigation of Smoothness Constraints for the Estimation of Displacement Vector Fields from Image Sequences" , IEEE PAMI: 8, 1986. [11] T. Poggio and E. B. Gamble and J. J. Little, "Parallel Integration of Vision Module" , Science, vol 242 , pp . 436-440 , 1988. [12] S. Ullman, "The Interpretation of Visual Motion" , Cambridge, Mass, MIT press, 1979. 983 984 Diamantaras and Geiger (b) (a) (c) Figure 2: Block matching data giJ' Both sides of the edges have the same data (and same confidence). White represents motion to the right (x-direction) or up (y-direction). Black is the complement . (a) The x-component of the data for the square translation. (b) The x-component of the data for the rotation and (c) the y-component of the data. (a) (b) Figure 3: The confidence R extracted from the block matching data gij. The display is the sum of both eigenvalues , i.e. the trace of R. Both sides of the edges have the same confidence. White represents high confidence. (a) For the square translation. (b) For the rotation. (a) (b) (c) Figure 4: The final motion estimation , after 20000 iterations , resolved the ambiguities with a natural interpretation of the scene. J.l = 10, 6 = 1" = 100. (a) square translation (b) x component of the motion rotation (c) y component of the motion rotation 

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Optimal Stochastic Search and Adaptive Momentum Todd K. Leen and Genevieve B. Orr Oregon Graduate Institute of Science and Technology Department of Computer Science and Engineering P.O.Box 91000, Portland, Oregon 97291-1000 Abstract Stochastic optimization algorithms typically use learning rate schedules that behave asymptotically as J.t(t) = J.to/t. The ensemble dynamics (Leen and Moody, 1993) for such algorithms provides an easy path to results on mean squared weight error and asymptotic normality. We apply this approach to stochastic gradient algorithms with momentum. We show that at late times, learning is governed by an effective learning rate J.tejJ = J.to/(l - f3) where f3 is the momentum parameter. We describe the behavior of the asymptotic weight error and give conditions on J.tejJ that insure optimal convergence speed. Finally, we use the results to develop an adaptive form of momentum that achieves optimal convergence speed independent of J.to. 1 Introduction The rate of convergence for gradient descent algorithms, both batch and stochastic, can be improved by including in the weight update a "momentum" term proportional to the previous weight update. Several authors (Tugay and Tanik, 1989; Shynk and Roy, 1988) give conditions for convergence of the mean and covariance of the weight vector for momentum LMS with constant learning rate. However stochastic algorithms require that the learning rate decay over time in order to achieve true convergence of the weight (in probability, in mean square, or with probability one). 477 478 Leen and Orr This paper uses our previous work on weight space probabilities (Leen and Moody, 1993; Orr and Leen, 1993) to study the convergence of stochastic gradient algorithms with annealed learning rates of the form Jl = Jlo/t, both with and without momentum. The approach provides simple derivations of previously known results and their extension to stochastic descent with momentum. Specifically, we show that the mean squared weight misadjustment drops off at the maximal rate ex 1/ t only if the effective learning rate JlejJ = Jlo/(1 - (3) is greater than a critical value which is determined by the Hessian. These results suggest a new algorithm that automatically adjusts the momentum coefficient to achieve the optimal convergence rate. This algorithm is simpler than previous approaches that either estimate the curvature directly during the descent (Venter, 1967) or measure an auxilliary statistic not directly involved in the optimization (Darken and Moody, 1992). 2 Density Evolution and Asymptotics We consider stochastic optimization algorithms with weight w E RN. We confine attention to a neighborhood of a local optimum w* and express the dynamics in terms of the weight error v w - w*. For simplicity we treat the continuous time algorithm 1 = d~~t) = Jl(t) H[ v(t), x(t)] (1) where Jl(t) is the learning rate at time t, H is the weight update function and x(t) is the data fed to the algorithm at time t. For stochastic gradient algorithms H = - \7 v ?(v, x(t)), minus the gradient of the instantaneous cost function. Convergence (in mean square) to w* is characterized by the average squared norm of the weight error E [ 1V 12] = Trace C where C - J dNv vvT P(v,t) (2) is the weight error correlation matrix and P(v, t) is the probability density at v and time t. In (Leen and Moody, 1993) we show that the probability density evolves according to the Kramers-Moyal expansion ap(v, t) at 00 ( _l)i 2: ., 2. i=l N 2: il,???j;=l ai aVil aV12 ... aVj; 1 Although algorithms are executed in discrete time, continuous time formulations are often advantagous for analysis. The passage from discrete to continuous time is treated in various ways depending on the needs of the theoretical exposition. Kushner and Clark (1978) define continous time functions that interpolate the discrete time process in order to establish an equivalence between the asymptotic behavior of the discrete time stochastic process, and solutions of an associated deterministic differential equation. Heskes et ai. (1992) draws on the results of Bedeaux et ai. (1971) that link (discrete time) random walk trajectories to the solution of a (continuous time) master equation. Heskes' master equation is equivalent to our Kramers-Moyal expansion (3). Optimal Stochastic Search and Adaptive Momentum where H j " denotes the jlh component of the N-component vector H, and (.. ')x denotes averaging over the density of inputs. Differentiating (2) with respect to time, u~ing (3) and integrating by parts, we obtain the equation of motion for the weight error correlation dd~ J-L(t) J d N V P(v, t) [v (H(v, xf)x J-L(t)2 2.1 + (H(v, x))x vTJ J d N V P(v, t) (H(v, x) H(v, x)T)x + (4) Asymptotics of the Weight Error Correlation Convergence of v can be understood by studying the late time behavior of (4). Since the update function H(v, x) is in general non-linear in v, the time evolution of the correlation matrix Cij is coupled to higher moments E [Vi Vj Vk ??? ] of the weight error. However, the learning rate is assumed to follow a schedule J-L(t) that satisfies the requirements for convergence in mean square to a local optimum. Thus at late times the density becomes sharply peaked about v = 0 2 . This suggests that we expand H(v, x) in a power series about v = 0 and retain the lowest order non-trivial terms in (4) leaving: dC dt =- J-L(t) [ (R C) + (C RT) ] + J-L(t)2 D , (5) where R is the Hessian of the average cost function (E) x' and D = (H(O,x)H(O,xf)x (6) is the diffusion matrix, both evaluated at the local optimum w*. (Note that RT = R.) We use (5) with the understanding that it is valid for large t. The solution to (5) is C(t) = U(t,to)C(to)UT(t,to) + tito d7 J-L(7)2 U(t,7) D UT (t,7) where the evolution operator U(t2' td is U(t2, t1) = exp [ -R 1:' (7) (8) dr tt(r) ] We assume, without loss of generality, that the coordinates are chosen so that R is diagonal (D won't be) with eigenvalues Ai, i = 1 ... N. Then with J-L(t) = J-Lo/t we obtain E[lvI 2 ] = Trace [C(t)] t; N { Cii (to) ( ttO ) 1 [ - t 21-1-0 Ai 21-1-0 A, -1 (to) to t 1} . (9) 2In general the density will have nonzero components outside the basin of w* . We are neglecting these, for the purpose of calculating the second moment of the the local density in the vicinity of w*. 479 480 Leen and Orr We define 1 J.lerit == - - - (10) 2 Amin and identify two regimes for which the behavior of (9) is fundamentally different: 1. J.lo > J.lcri( E [lvI 2 ] drops off asymptotically as lit. 2. J.lo < J.lerit: E [lvI 2 ] drops off asymptotically as ( i.e. more slowly than lit. t) (2 ~o ATnin ) Figure 1 shows results from simulations of an ensemble of 2000 networks trained by LMS, and the prediction from (9). For the simulations, input data were drawn from a gaussian with zero mean and variance R = 1.0. The targets were generated by a noisy teacher neuron (i.e. targets =w*x +~, where (~) = 0 and (e) = (72). The upper two curves in each plot (dotted) depict the behavior for J.lo < J.lerit = 0.5. The remaining curves (solid) show the behavior for J.lo > J.lerit. 0 0 ,... ,... ..... I I C\I C\I i.CII i.CII W CIt? WC') oJ oJ ~' ~ I ClI 0 0 "r 'of U? U? I 100 1000 5000 50000 100 1000 5000 50000 Fig.1: LEFT - Simulation results from an ensemble of 2000 one-dimensional LMS algorithms with R = 1.0, (72 = 1.0 and /-L = /-Lo/t. RIGHT - Theoretical predictions from equation (9). Curves correspond to (top to bottom) /-Lo = 0.2, 0.4, 0.6, 0.8, 1.0, 1.5 . By minimizing the coefficient of lit in (9), the optimal learning rate is found to be J.lopt = 11 Amin. This formalism also yields asymptotic normality rather simply (Orr and Leen, 1994). These conditions for "optimal" (Le. lit) convergence of the weight error correlation and the related results on asymptotic normality have been previously discussed in the stochastic approximation literature (Darken and Moody, 1992; Goldstein, 1987; White, 1989; and references therein) . The present formal structure provides the results with relative ease and facilitates the extension to stochastic gradient descent with momentum. 3 Stochastic Search with Constant Momentum The discrete time algorithm for stochastic optimization with momentum is: v(t + 1) = v(t) + J.l(t) H[v(t), x(t)] + f3 f!(t) (11) Optimal Stochastic Search and Adaptive Momentum v(t + 1) - v(t) n(t) + /1(t) H[v(t), x(t)] n(t + 1) + ((3 - 1) n(t), (12) or in continuous time, dv(t) dt dn(t) dt - /1(t) H[v(t), x(t)] + (3 n(t) (13) /1(t) H[v(t), x(t)] + ((3 - (14) 1) n(t). As before, we are interested in the late time behavior of E [lvI 2 ]. To this end, we define the 2N-dimensional variable Z = (v, nf and, following the arguments of the previous sections, expand H[v(t), x(t)] in a power series about v = 0 retaining the lowest order non-trivial terms. In this approximation the correlation matrix C _ E[ZZT] evolves according to dC dt = - T 2KC + CK + /1(t) D (15) with (16) I is the N x N identity matrix, and Rand D are defined as before. The evolution operator is now U(t2' ttl = [t' dr K(r)] (17) exp and the solution to (15) is C = U(t, to) C(to) U T (t, to) + t dr /1 (r) U(t, r) D U ltD 2 T (t, r) (18) The squared norm of the weight error is the sum of first N diagonal elements of C. In coordinates for which R is diagonal and with /1(t) = /10 It, we find that for t ? to E[lvI2] '" t, {c,,(to) (t;) 'i~~' + This reduces to (9) when (3 = O. Equation (19) defines two regimes of interest: 1. /10/(1 - (3) 2. /10/(1 - (3) > /1cri( E[lvI2] drops off asymptotically as lit. < /1cri( E[lvI2] drops off asymptotically as (~) I.e. more slowly than lit. 21-'Q'xmjn 1 ~ 481 482 Leen and Orr The form of (19) and the conditions following it show that the asymptotics of gradient descent with momentum are governed by the effective learning rate _ MejJ = M 1 - {3 . Figure 2 compares simulations with the predictions of (19) for fixed Mo and various {3. The simulations were performed on an ensemble of 2000 networks trained by LMS as described previously but with an additional momentum term of the form given in (11). The upper three curves (dotted) show the behavior of E[lvI 2 ] for MejJ < Merit? The solid curves show the behavior for MejJ > Merit? The derivation of asymptotic normality proceeds similarly to the case without momentum. Again the reader is referred to (Orr and Leen, 1994) for details . ...., ...., ...... N ...... N (\I' (\I' < ~ "> -C') > -C') ur' ur' C) C) 0'<:1" ...J, 0'<:1" ...J, , III , III 100 1000 5000 100 50000 1000 SOOO 50000 t t Fig.2: LEFT - Simulation results from an ensemble of 2000 one-dimensional LMS algorithms with mome~tum with R = 1.0, (12 = 1.0, and /10 - 0.2. RIGHTTheoretical predictions from equation (19). Curves correspond to (top to bottom) {3 = 0.0, 004, 0.5, 0.6, 0.7, 0.8 . 4 Adaptive Momentum Insures Optimal Convergence The optimal constant momentum parameter is obtained by minimizing the coefficient of lit in (19). Imposing the restriction that this parameter is positive 3 gives (3opt As with Mopt, = max(O, 1 - (20) MOAmin). this result is not of practical use because, in general, Amin is unknown. For I-dimensional linear networks, an alternative is to use the instantaneous estimate of A, :\(t) = x 2 (t) where x(t) is the network input at time t. We thus define the adaptive momentum parameter to be (3adapt = max(O, 1 - MOX 2 ) (I-dimension). (21) An algorithm based on (21) insures that the late time convergence is optimally fast. An alternative route to achieving the same goal is to dispense with the momentum term and adaptively adjust the learning rate. Vetner (1967) proposed an algorithm 3 E[lvI 2 ] diverges for 1{31 > 1. For -1 < {3 < 0, E[lvI 2 ] appears to converge but oscillations are observed. Additional study is required to determine whether {3 in this range might be useful for improving learning. Optimal Stochastic Search and Adaptive Momentum that iteratively estimates A for 1-D algorithms and uses the estimate to adjust J.Lo. Darken and Moody (1992) propose measuring an auxiliary statistic they call "drift" that is used to determine whether or not J.Lo > J.Lcrit. The adaptive momentum scheme generalizes to multiple dimensions more easily than Vetner's algorithm, and, unlike Darken and Moody's scheme, does not involve calculating an auxiliary statistic not directly involved with the minimization. A natural extension to N dimensions is to define a matrix of momentum coefficients, 'Y = I - J.Lo X xT, where I is the N x N identity matrix. By zeroing out the negative eigenvalues of 'Y, we obtain the adaptive momentum matrix (3adapt = I - ex xT, where e = min(J.Lo, 1/(xT x)). (22) =1.5 -1+_~_--::=====-_~~~ L?9(t) 1 2 3 Fig.3: Simulations of 2-D LMS with 1000 networks initialized at Vo = (.2, .3) and with = 1, ).1 = .4, ).2 = 4, and /-Lcrit = 1.25. LEFT- {3 = 0, RIGHT - {3 = (3adapt. Dashed curves correspond to adaptive momentum. (72 Figure 3 shows that our adaptive momentum not only achieves the optimal convergence rate independent of the learning rate parameter J.Lo but that the value of log(E[lvI2]) at late times is nearly independent of J.Lo and smaller than when momentum is not used. The left graph displays simulation results without momentum. Here, convergence rates clearly depend on J.Lo and are optimal for J.Lo > J.Lcrit = 1.25. When J.Lo is large there is initially significant spreading in v so that the increased convergence rate does not result in lower log(E[lvI 2]) until very late times (t ~ 105 ). The graph on the right shows simulations with adaptive momentum. Initially, the spreading is even greater than with no momentum, but log(E[lvI 2]) quickly decreases to reach a much smaller value. In addition, for t ~ 300, the optimal convergence rate (slope=-l) is achieved for all three values of J.Lo and the curves themselves lie almost on top of one another. In other words, at late times (t ;::: 300), the value of log(E[lvI 2]) is independent of J.Lo when adaptive momentum is used. 5 Summary We have used the dynamics of the weight space probabilities to derive the asymptotic behavior of the weight error correlation for annealed stochastic gradient algorithms with momentum. The late time behavior is governed by the effective learning rate J.Lejj J.Lo/(l - (3) . For learning rate schedules J.Lolt, if J.Leff > 1/(2 Arnin) , then the squared norm of the weight error v - w - w* falls off as lit. From these results we have developed a form of momentum that adapts to obtain optimal convergence rates independent of the learning rate parameter. = 483 484 Leen and Orr Acknowledgments This work was supported by grants from the Air Force Office of Scientific Research (F49620-93-1-0253) and the Electric Power Research Institute (RP8015-2). References D. Bedeaux, K. Laktos-Lindenberg, and K. Shuler. (1971) On the Relation Between Master Equations and Random Walks and their Solutions. Journal of Mathematical Physics, 12:2116-2123. Christian Darken and John Moody. (1992) Towards Faster Stochastic Gradient Search. In J.E. Moody, S.J. Hanson, and R.P. Lipmann (eds.) Advances in Neural Information Processing Systems, vol. 4. Morgan Kaufmann Publishers, San Mateo, CA, 1009-1016. Larry Goldstein. (1987) Mean Square Optimality in the Continuous Time Robbins Monro Procedure. Technical Report DRB-306, Dept. of Mathematics, University of Southern California, LA. H.J. Kushner and D.S. Clark. (1978) Stochastic Approximation Methods for Constrained and Unconstrained Systems. Springer-Verlag, New York. Tom M. Heskes, Eddy T.P. Slijpen, and Bert Kappen. (1992) Learning in Neural Networks with Local Minima. Physical Review A, 46(8):5221-5231. Todd K. Leen and John E. Moody. (1993) Weight Space Probability Densities in Stochastic Learning: 1. Dynamics and Equilibria. In Giles, Hanson, and Cowan (eds.), Advances in Neural Information Processing Systems, vol. 5, Morgan Kaufmann Publishers, San Mateo, CA, 451-458. G. B. Orr and T. K. Leen. (1993) Weight Space Probability Densities in Stochastic Learning: II. Transients and Basin Hopping Times. In Giles, Hanson, and Cowan (eds.), Advances in Neural Information Processing Systems, vol. 5, Morgan Kaufmann Publishers, San Mateo, CA, 507-514. G. B. Orr and T. K. Leen. (1994) Momentum and Optimal Stochastic Search. In M. C. Mozer, P. Smolensky, D. S. Touretzky, J. L. Elman, and A. S. Weigend (eds.), Proceedings of the 1993 Connectionist Models Summer School, 351-357. John J. Shynk and Sumit Roy. (1988) The LMS Algorithm with Momentum Updating. Proceedings of the IEEE International Symposium on Circuits and Systems, 2651-2654. Mehmet Ali Tugay and Yal<;in Tanik. (1989) Properties of the Momentum LMS Algorithm. Signal Processing, 18:117-127. J. H. Venter. (1967) An Extension of the Robbins-Monro Procedure. Annals of Mathematical Statistics, 38:181-190. Halbert White. (1989) Learning in Artificial Neural Networks: A Statistical Perspective. Neural Computation, 1:425-464. 

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Convergence of Indirect Adaptive Asynchronous Value Iteration Algorithms Vijaykumar Gullapalli Department of Computer Science University of Massachusetts Amherst, MA 01003 vijay@cs.umass.edu Andrew G. Barto Department of Computer Science University of Massachusetts Amherst, MA 01003 barto@cs.umass.edu Abstract Reinforcement Learning methods based on approximating dynamic programming (DP) are receiving increased attention due to their utility in forming reactive control policies for systems embedded in dynamic environments. Environments are usually modeled as controlled Markov processes, but when the environment model is not known a priori, adaptive methods are necessary. Adaptive control methods are often classified as being direct or indirect. Direct methods directly adapt the control policy from experience, whereas indirect methods adapt a model of the controlled process and compute control policies based on the latest model. Our focus is on indirect adaptive DP-based methods in this paper. We present a convergence result for indirect adaptive asynchronous value iteration algorithms for the case in which a look-up table is used to store the value function. Our result implies convergence of several existing reinforcement learning algorithms such as adaptive real-time dynamic programming (ARTDP) (Barto, Bradtke, & Singh, 1993) and prioritized sweeping (Moore & Atkeson, 1993). Although the emphasis of researchers studying DP-based reinforcement learning has been on direct adaptive methods such as Q-Learning (Watkins, 1989) and methods using TD algorithms (Sutton, 1988), it is not clear that these direct methods are preferable in practice to indirect methods such as those analyzed in this paper. 695 696 Gullapalli and Barto 1 INTRODUCTION Reinforcement learning methods based on approximating dynamic programming (DP) are receiving increased attention due to their utility in forming reactive control policies for systems embedded in dynamic environments. In most of this work, learning tasks are formulated as Markovian Decision Problems (MDPs) in which the environment is modeled as a controlled Markov process. For each observed environmental state, the agent consults a policy to select an action, which when executed causes a probabilistic transition to a successor state. State transitions generate rewards, and the agent's goal is to form a policy that maximizes the expected value of a measure of the long-term reward for operating in the environment. (Equivalent formulations minimize a measure of the long-term cost of operating in the environment.) Artificial neural networks are often used to store value functions produced by these algorithms (e.g., (Tesauro, 1992)). Recent advances in reinforcement learning theory have shown that asynchronous value iteration provides an important link between reinforcement learning algorithms and classical DP methods for value iteration (VI) (Barto, Bradtke, & Singh, 1993). Whereas conventional VI algorithms use repeated exhaustive "sweeps" ofthe MDP's state set to update the value function, asynchronous VI can achieve the same result without proceeding in systematic sweeps (Bertsekas & Tsitsiklis, 1989). If the state ordering of an asynchronous VI computation is determined by state sequences generated during real or simulated interaction of a controller with the Markov process, the result is an algorithm called Real- Time DP (RTDP) (Barto, Bradtke, & Singh, 1993). Its convergence to optimal value functions in several kinds of problems follows from the convergence properties of asynchronous VI (Barto, Bradtke, & Singh, 1993). 2 MDPS WITH INCOMPLETE INFORMATION Because asynchronous VI employs a basic update operation that involves computing the expected value of the next state for all possible actions, it requires a complete and accurate model of the MDP in the form of state-transition probabilities and expected transition rewards. This is also true for the use of asynchronous VI in RTDP. Therefore, when state-transition probabilities and expected transition rewards are not completely known, asynchronous VI is not directly applicable. Problems such as these, which are called MDPs with incomplete information,l require more complex adaptive algorithms for their solution. An indirect adaptive method works by identifying the underlying MDP via estimates of state transition probabilities and expected transition rewards, whereas a direct adaptive method (e.g., Q-Learning (Watkins, 1989)) adapts the policy or the value function without forming an explicit model of the MDP through system identification. In this paper, we prove a convergence theorem for a set of algorithms we call indirect adaptive asynchronous VI algorithms. These are indirect adaptive algorithms that result from simply substituting current estimates of transition probabilities and expected transition rewards (produced by some concurrently executing identification 1 These problems should not be confused with MDPs with incomplete 6tate information, i.e., partially observable MDPs. Convergence of Indirect Adaptive Asynchronous Value Iteration Algorithms algorithm) for their actual values in the asynchronous value iteration computation. We show that under certain conditions, indirect adaptive asynchronous VI algorithms converge with probability one to the optimal value function. Moreover, we use our result to infer convergence of two existing DP-based reinforcement learning algorithms, adaptive real-time dynamic programming (ARTDP) (Barto, Bradtke, & Singh, 1993), and prioritized sweeping (Moore & Atkeson, 1993). 3 CONVERGENCE OF INDIRECT ADAPTIVE ASYNCHRONOUS VI Indirect adaptive asynchronous VI algorithms are produced from non-adaptive algorithms by substituting a current approximate model of the MDP for the true model in the asynchronous value iteration computations. An indirect adaptive algorithm can be expected to converge only if the corresponding non-adaptive algorithm, with the true model used in the place of each approximate model, converges. We therefore restrict attention to indirect adaptive asynchronous VI algorithms that correspond in this way to convergent non-adaptive algorithms. We prove the following theorem: Theorem 1 For any finite 6tate, finite action MDP with an infinite-horizon di6counted performance measure, any indirect adaptive a6ynchronous VI algorithm (for which the corresponding non-adaptive algorithm converges) converges to the optimal value function with probability one if 1) the conditions for convergence of the non-adaptive algorithm are met, 2) in the limit, every action is executed from every 6tate infinitely often, and 3) the e6timate6 of the state-transition probabilities and the expected transition rewards remain bounded and converge in the limit to their true value6 with probability one. Proof The proof is given in Appendix A.2. 4 DISCUSSION Condition 2 of the theorem, which is also required by direct adaptive methods to ensure convergence, is usually unavoidable. It is typically ensured by using a stochastic policy. For example, we can use the Gibbs distribution method for selecting actions used by Watkins (1989) and others. Given condition 2, condition 3 is easily satisfied by most identification methods. In particular, the simple maximumlikelihood identification method (see Appendix A.l, items 6 and 7) converges to the true model with probability one under this condition. Our result is valid only for the special case in which the value function is explicitly stored in a look-up table. The case in which general function approximators such as neural networks are used requires further analysis. Finally, an important issue not addressed in this paper is the trade-off between system identification and control. To ensure convergence of the model, all actions have to be executed infinitely often in every state. On the other hand, on-line control objectives are best served by executing the action in each sta.te that is optimal according to the current value function (i.e., by using the certainty equivalence 697 698 Gullapalli and Barto optimal policy). This issue has received considerable attention from control theorists (see, for example, (Kumar, 1985), and the references therein). Although we do not address this issue in this paper, for a specific estimation method, it may be possible to determine an action selection scheme that makes the best trade-off between identification and control. 5 EXAMPLES OF INDIRECT ADAPTIVE ASYNCHRONOUS VI One example of an indirect adaptive asynchronous VI algorithm is ARTDP (Barto, Bradtke, & Singh, 1993) with maximum-likelihood identification. In this algorithm, a randomized policy is used to ensure that every action has a non-zero probability of being executed in each state. The following theorem for ARDTP follows directly from our result and the corresponding theorem for RTDP in (Barto, Bradtke, & Singh, 1993): Theorem 2 For any discounted MDP and any initial value junction, trial-based 2 ARTDP converges with probability one. As a special case of the above theorem, we can obtain the result that in similar problems the prioritized sweeping algorithm of Moore and Atkeson (Moore & Atkeson, 1993) converges to the optimal value function. This is because prioritized sweeping is a special case of ARTDP in which states are selected for value updates based on their priority and the processing time available. A state's priority reflects the utility of performing an update for that state, and hence prioritized sweeping can improve the efficiency of asynchronous VI. A similar algorithm, Queue-Dyna (Peng & Williams, 1992), can also be shown to converge to the optimal value function using a simple extension of our result. 6 CONCLUSIONS We have shown convergence of indirect adaptive asynchronous value iteration under fairly general conditions. This result implies the convergence of several existing DP-based reinforcement learning algorithms. Moreover, we have discussed possible extensions to our result. Our result is a step toward a better understanding of indirect adaptive DP-based reinforcement learning methods. There are several promising directions for future work. One is to analyze the trade-off between model estimation and control mentioned earlier to determine optimal methods for action selection and to integrate our work with existing results on adaptive methods for MDPs (Kumar, 1985). Second, analysis is needed for the case in which a function approximation method, such as a neural network, is used instead of a look-up table to store the value function. A third possible direction is to analyze indirect adaptive versions of more general DPbased algorithms that combine asynchronous policy iteration with asynchronous 2 As in (Barto, Bradtke, & Singh, 1993), by trial-balled execution of an algorithm we mean its use in an infinite series of trials such that every state is selected infinitely often to be the start state of a trial. Convergence of Indirect Adaptive Asynchronous Value Iteration Algorithms policy evaluation. Several non-adaptive algorithms of this nature have been proposed recently (e.g., (Williams & Baird, 1993; Singh & Gullapalli)). Finally, it will be useful to examine the relative efficacies of direct and indirect adaptive methods for solving MDPs with incomplete information. Although the emphasis of researchers studying DP-based reinforcement learning has been on direct adaptive methods such as Q-Learning and methods using TD algorithms, it is not clear that these direct methods are preferable in practice to indirect methods such as the ones discussed here. For example, Moore and Atkeson (1993) report several experiments in which prioritized sweeping significantly outperforms Q-learning in terms of the computation time and the number of observations required for convergence. More research is needed to characterize circumstances for which the various reinforcement learning methods are best suited. APPENDIX A.1 NOTATION = 1. Time steps are denoted t 1, 2, ..., and Zt denotes the last state observed before time t. Zt belongs to a finite state set S = {I, 2, ... , n}. 2. Actions in a state are selected according to a policy 7r, where 7r(i) E A, a finite set of actions, for 1 :::; i :::; n. 3. The probability of making a transition from state i to state j on executing action a is pa (i, j). 4. The expected reward from executing action a in state i is r(i, a). The reward received at time t is denoted rt(Zt, at). 5. 0 :::; "y < 1 is the discount factor. 6. Let p~(i, j) denote the estimate at time t of the probability of transition from state i to j on executing action a E A. Several different methods can be used for estimating p~( i, j). For example, if n~( i, j) is the observed number of times before time step t that execution of action a when the system was in state i was followed by a transition to state j, and n~(i) = L:jEs nf(i, j) is the number of times action a was executed in state i before time step t, then, for 1 :::; i :::; n and for all a E A, the maximum-likelihood statetransition probability estimates at time t are Aa(' ') Pt ", J a(' ') J = nt~, a (')' nt " 1 < '< _ J _ n. Note that the maximum-likelihood estimates converge to their true values with probability one if nf(i) -+ 00 as t -+ 00, i.e., every action is executed from every state infinitely often. Let pa(i) = [pa(i, 1), ... , pa(i, n)] E [0,1]'\ and similarly, pf(i) = [Pf(i, I), ... , pf(i, n)] E [o,l]n. We will denote the lSI x IAI matrix of transition probabilities associated with state i by P( i) and its estimate at time t by Pt(i). Finally, P denotes the vector of matrices [P(I), ... , P(n)], and Pt denotes the vector [A(I), ... , A(n)]. 699 700 Gullapalli and Barto 7. Let rt(i, a) denote the estimate at time t of the e:Jq>ected reward r(i, a), and let rt denote all the lSI x IAI estimates at time t. Again, if maximumlikelihood estimation is used, " (") rt 'I., G where fill: Gk)h,(Zk, Gk) = L:!=I rk(zk,n II( ") , t 1. S x A -+ {O, 1} is the indicator function for the state-action pair 1.,G. B. ~* denotes the optimal value function for the MDP defined by the estimates and rt of P and r at time t. Thus, Vi E S, A ~*(i) = max{rt(i, a) + "( '" p~(i, i)~*(j)}. ilEA L..-J je S Similarly, V* denotes the optimal value function for the MDP defined by P and r. 9. B t ~ S is the subset of states whose values are updated at time t. Usually, at least Zt E B t ? A.2 PROOF OF THEOREM 1 In indirect adaptive asynchronous VI algorithms, the estimates of the MDP parameters at time step t, Pt and rt, are used in place of the true parameters, P and r, in the asynchronous VI computations at time t. Hence the value function is updated at time t as V. (.) _ { maxaeA{rt(i,a) HI 1. where B(t) ~ - vt(i) + "(L:;Espf(i,i)vt(j)} ifi E B t otherwise, S is the subset of states whose values are updated at time t. A First note that because and rt are assumed to be bounded for all t, Vi is also bounded for all t. Next, because the optimal value function given the model and rt, l't*, is a continuous function of the estimates A and rt, convergence of these estimates w.p. 1 to their true values implies that A 1 V* v.t * 1lI.p. ~ , where V* is the optimal value function for the original MDP. The convergence w.p. 1 of ~* to V* implies that given an ? > 0 there exists an integer T > 0 such that for all t ;:::: T, 11l't* - V*II < Here, II . II (1 - "() ? w.p. 1. 2"( (1) can be any norm on lRn , although we will use the 1/10 or max norm. In algorithms based on asynchronous VI, the values of only the states in B t ~ S are updated at time t, although the value of each state is updated infinitely often. For an arbitrary Z E S, let us define the infinite subsequence {tk}k=O to be the times when the value of state Z gets updated. Further, let us only consider updates at, or after, time T, where T is from equation (1) above, so that t~ ;:::: T for all Z E S. Convergence of Indirect Adaptive Asynchronous Value Iteration Algorithms By the nature of the VI computation we have, for each t ;:::: 1, lVi+l(i) - ~*(i)1 ~ '"Yllvt - ~*II if i E Bt ? (2) Using inequality (2), we can get a bound for Ivt-+l(Z) - ~~(z)1 ,. ,. as Ivt-+dz) - ~!(z)1 < '"Y1I:+lIIVi-0 - ~!II + (1 - '"Y1I:)? ,. ,. 0 w.p.1. (3) We can verify that the bound in (3) is correct through induction. The bound is clearly valid for k = o. Assuming it is valid for k, we show that it is valid for k + 1: Ivt-,.+1 +l(Z) - ~~"+1 (z)1 < '"Yllvt-,.+1 - ~~,.+1 II < '"Y(lIvt?,.+1 - ~~,. II + II~!,. -~!,.+1 II) < '"Ylvt-" + (z) - ~!(z)1 +1' ((1- 1') ?) 1,. l' w.p.l '"Ylvt-+dz) - ~!(z)1 ,. ,. + (1- '"Y)? < '"Yb1l:+1I1vt.o - ~!II + (1 - '"Y1I:)?) + (1 - '"Y)? 0 1'11:+211 Vi- o - w.p.l ~! II + (1 - 1'11:+ 1)?. 0 Taking the limit as k -t 00 in equation (3) and observing that for each z, lim1l:-.00 ~qz) ,. = V*(z) w.p. 1, we obtain lim Ivt-+l(Z) - V*(z)1 11:-.00 ,. < ? w.p.1. Since ? and z are arbitrary, this implies that vt -t V* w.p. 1. 0 Acknowledgements We gratefully acknowledge the significant contribution of Peter Dayan, who pointed out that a restrictive condition for convergence in an earlier version of our result was actually unnecessary. This work has also benefited from several discussions with Satinder Singh. We would also like to thank Chuck Anderson for his timely help in preparing this material for presentation at the conference. This material is based upon work supported by funding provided to A. Barto by the AFOSR, Bolling AFB, under Grant AFOSR-F49620-93-1-0269 and by the NSF under Grant ECS-92-14866. References [1] A.G. Barto, S.J. Bradtke, and S.P. Singh. Learning to act using real-time dynamic programming. Technical Report 93-02, University of Massachusetts, Amherst, MA, 1993. [2] D. P. Bertsekas and J. N. Tsitsiklis. Parallel and Di8tributed Computation: Numerical Method6. Prentice-Hall, Englewood Cliffs, NJ, 1989. [3] P. R. Kumar. A survey of some results in stochastic adaptive control. SIAM Journal of Control and Optimization, 23(3):329-380, May 1985. 701 702 Gullapalli and Barto [4] A. W. Moore and C. G. Atkeson. Memory-based reinforcement learning: Efficient computation with prioritized sweeping. In S. J. Hanson, J. D. Cowan, and C. L. Giles, editors, Advance8 in Neural Information Proceuing Sy8tem8 5, pages 263-270, San Mateo, CA, 1993. Morgan Kaufmann Publishers. [5] J. Peng and R. J. Williams. Efficient learning and planning within the dyna framework. In Proceeding8 of the Second International Conference on Simulation of Adaptive Behavior, Honolulu, HI, 1992. [6] S. P. Singh and V. Gullapalli. Asynchronous modified policy iteration with single-sided updates. (Under review). [7] R. S. Sutton. Learning to predict by the methods of temporal differences. Machine Learning, 3:9-44, 1988. [8] G. J. Tesauro. Practical issues in temporal difference learning. Machine Learning, 8(3/4):257-277, May 1992. [9] C. J. C. H. Watkins. Learning from delayed reward8. PhD thesis, Cambridge University, Cambridge, England, 1989. [10] R. J. Williams and L. C. Baird. Analysis of some incremental variants of policy iteration: First steps toward understanding actor-critic learning systems. Technical Report NU-CCS-93-11, Northeastern University, College of Computer Science, Boston, MA 02115, September 1993. 

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GDS: Gradient Descent Generation of Symbolic Classification Rules Reinhard Blasig Kaiserslautern University, Germany Present address: Siemens AG, ZFE ST SN 41 81730 Miinchen, Germany Abstract Imagine you have designed a neural network that successfully learns a complex classification task. What are the relevant input features the classifier relies on and how are these features combined to produce the classification decisions? There are applications where a deeper insight into the structure of an adaptive system and thus into the underlying classification problem may well be as important as the system's performance characteristics, e.g. in economics or medicine. GDSi is a backpropagation-based training scheme that produces networks transformable into an equivalent and concise set of IF-THEN rules. This is achieved by imposing penalty terms on the network parameters that adapt the network to the expressive power of this class of rules. Thus during training we simultaneously minimize classification and transformation error. Some real-world tasks demonstrate the viability of our approach. 1 Introduction This paper deals with backpropagation networks trained to perform a classification task on Boolean or real-valued data. Given such a classification task in most cases it is not too difficult to devise a network architecture that is capable of learning the input-output relation as represented by a number of training examples. Once training is finished one has a black box which often does a quite good job not 1 Gradient Descent Symbolic Rule Generation 1093 1094 Blasig only on the training patterns but also on some previously unseen test patterns. A good generalization performance indicates that the network has grasped part of the structure inherent in the classification task. The net has figured out which input features are relevant to make a classification decision and which are not. It has also modelled the way the relevant features have to be combined in order to produce the classifying output. In many applications it is important to get an understanding of this information hidden inside the neural network. Not only does this help to create or verify a domain theory, the analysis of this information may also serve human experts to determine, when and in what way the classifier will fail. In order to explicate the network's implicit information, we transform it into a set of rules. This idea is not new, cf. (Saito and Nakano, 1988), (Bochereau and Bourgine, 1990), (Y. Hayashi, 1991) and (Towell and Shavlik, 1992). In contrast to these approaches, which extract rules after BP-training is finished, we apply penalty terms during training to adapt the network's expressive power to that of the rules we want to generate. Consequently the net will be transformable into an equivalent set of rules. Due to their good comprehensibility we restrict the rules to be of the form IF < premise> THEN < conclusion >, where the premise as well as the conclusion are Boolean expressions. To actually make the transformation two problems have to be solved: ? Neural nets are well known for their distributed representation of information; so in order to transform a net into a concise and comprehensible rule set one has to find a way of condensing this information without substantially changing it . ? In the case of backpropagation networks a continuous activation function determines a node's output depending on its activation. However, the dynamic of this function has no counterpart in the context of rule-based descriptions. We address these problems by introducing a penalty function Ep, which we add to the classification error Ec yielding the total back propagation error ET = ED 2 + A* Ep. (1) The Penalty Term The term Ep is intended to have two effects on the network weights. First, by a weight decay component it aims at reducing network complexity by pushing a (hopefully large) fraction of the weights to O. The smaller the net, the more concise the rules describing its behavior will be. As a positive side effect, this component will tend to act as a form of "Occam's razor": simple networks are more likely to exhibit good generalization than complex ones. Secondly, the penalty term should minimize the error caused by transforming the network into a set of rules. Adopting the common approach that each non-input neuron represents one rule, there would be no transformation error if the neurons' activation function were threshold functions; the Boolean node output would then indicate, whether the conclusion is drawn or not. But since backpropagation neurons use continuous activation functions like GDS: Gradient Descent Generation of Symbolic Classification Rules = y tanh (x) to transform their activation value x into the output value y, we are left with the difficulty of interpreting the continuous output of a neuron. Thus our penalty term will be designed to produce a high penalty for those neurons of the backpropagation net, whose behavior cannot be well approximated by threshold neurons, because their activation values are likely to fall into the nonsaturated region of the tanh-function 2 . 1.00 0.00 ,.-------------I I I I I I -1.00 -3.00 0.00 3.00 = Figure 1: We regard Ixl > 3 with Iyl I tanh(x)I > 0.9 as the regions, where a sigmoidal neuron can be approximated by a threshold neuron. The nonsaturated region is marked by the dashed box. For a better understanding of our penalty term one has to be aware of the fact that IF-THEN rules with a Boolean premise and conclusion are essentially Boolean functions. It can easily be shown that any such function can be calculated by a network of threshold neurons provided there is one (sufficiently large) hidden layer. This is still true if we restrict connection weights to the values {-I, 0, I} and node thresholds to be integers (Hertz, Krogh and Palmer, 1991). In order to transfer this scenario to nets with sigmoidal activation functions and having in mind that the activation values of the sigmoidal neurons should always exceed ?3 (see figure 1), we require the nodes' biases to be odd multiples of ?3 and the weights Wji to obey Wji E {-6,0,6}. (2) We shortly comment on the practical problem that sometimes bias values as large as ?6m, (mi being the fan-in of node i) may be necessary to implement certain Boolean functions. This may slow down or even block the learning process. A simple solution to this problem is to use some additional input units with a constant output of +1. If the connections to these units are also subject to the penalty function Ep, it is sufficient to restrict the bias values to (3) hi E {-3, 3}. 2We have to point out that the conversion of sigmoidal neurons to threshold neurons will reduce the net's computational power: there are Boolean functions which can be computed by a net of sigmoidal neurons, but which exceed the capacity of a threshold net of the same topology (Maass, Schnitger and Sontag, 1991). Note that the objective to use threshold units is a consequence of the decision to search for rules of the type IF < premise > THEN < conclusion >. A failure of the net to simultaneously minimize both parts of the error measure may indicate that other rule types are more adequate to handle the given classification task. 1095 1096 Blasig Now we can define penalty functions that push the biases and weights to the desired values. Obviously Eb (the bias penalty) and Ew (the weight penalty) have to be different: (4) Eb(bi ) = 13-lbill E (w .. ) - { w J' - 16 - IWji11 IWjil for for IWjil ~ e IWjil < e (5) The parameter e determines whether a weight should be subject to decay or pushed to attain the value 6 (or -6 respectively). Figure 2 displays the graphs ofthe penalty functions. -3.0 3.0 -6.0 -8 8 6.0 Figure 2: The penalty functions Eb and Ew. The value of e is chosen with the objective that only those weights should exceed this value, which almost certainly have to be nonzero to solve the given classification task. Since we initialize the network with weights uniformly distributed in the interval [-0.5,0.5]' E> = 1.5 works well at the beginning of the training process. The penalty term then has the effect of a pure weight decay. When learning proceeds and the weights converge, we can slowly reduce the value of e, because superfluous weights will already have decayed. So after each sequence of 100 training patterns, say, we decrease e by a factor of 0.995. Observation shows that weights which once exceeded the value of e quickly reach 6 or -6 and that there are relatively few cases where a large weight is reduced again to a value smaller than e. Accordingly, the number of weights in {-6, 6} successively grows in the course of learning, and the criterion to stop training thus influences the number of nonzero weights. The end of training is determined by means of cross validation. However, we do not examine the cross validation performance of the trained net, but that of the corresponding rule set. This is accomplished by calculating the performance of the original net with all weights and biases replaced by their optimal values according to (2) and (3). The weighting factor A of the penalty term (see equation 1) is critical for good learning performance. We pursued the strategy to start learning with A = 0, so that the network parameters first move into a region where the classification error is small. If this error falls below a prespecified tolerance level L, A is incremented by 0.001. The factor A goes down by the same amount, when the error grows larger GDS: Gradient Descent Generation of Symbolic Classification Rules than L3. By adjusting the weighting factor every 100 training patterns we keep the classification error close to the tolerance level. The choice of L of course depends on the learning task. As a heuristic, L should be slightly larger than the classification error attainable by a non-penalized network. 3 Splice-Junction Recognition The DNA, carrying the genetic information of biological cells, can be thought to be composed of two types of subsequences: exons and introns. The task is to classify each DNA position as either an exon-to-intron transition (EI), an intronto-exon transition (IE) or neither (N). The only information available is a sequence of 30 nucleotides (A, C, G or T) before and 30 nucleotides after the position to be classified. Splice-junction recognition is a classification task that has already been investigated by a number of machine learning researchers using various adaptive models. The pattern reservoir contains about 3200 DNA samples, 30% of which were used for training, 10% for cross-validation and 60% for testing. Since we used a grandmothercell coding for the input DNA sequence, the network has an input layer of 4*60 neurons. With a hidden layer of 20 neurons 4 and two output units for the classes EI and IE, this amounts to about 5000 free parameters. The following table compares the classification performance of our penalty term approach and other machine learning algorithms, cf. (Murphy and Aha, 1992). Table 1: Splice-junction recognition: error (in percent) of various machine learning algorithms algorithm KBANN GDS Backprop Perceptron ID3 Nearest Neighbor N 4.62 6.71 5.29 3.99 8.84 31.11 EI 7.56 4.43 5.74 16.32 10.58 11.65 IE 8.47 9.24 10.75 17.41 13.99 9.09 total 6.32 6.75 6.77 10.43 10.56 20.74 Surprisingly, the GDS network turned out to be very small. The weight decay component of our penalty term managed to push all but 61 weights to zero, making use of only three hidden neurons. Thus in addition to performing very well, the network is transformable into a concise rule set, as follows 5 : 3Negative A-values are not allowed. 4. A reasonable size, considering the experiments described in (Shavlik et al., 1991) 5We adopt a. notation commonly used in this domain: @n denotes the position of the first nucleotide in the given sequence being left (negative n) or right (positive n) to the point to be classified. Nucleotide 'V' stands for (,C' or 'T'), 'X' is a.ny of {A, C, G, T}. Consequently, e.g. neuron hidden(2) is active iff at least four of the five nucleotides of the sequence 'GTAXG' are identical to the input pattern at positions 1 to 5 right of the possible splice junction. 1097 1098 Blasig hidden(2): at least 4 nucleotides match sequence 11: 'GTAXG' hidden(11): at least 3 nucleotides match sequence 1-3: 'YAG' hidden(17): at least 1 nucleotides matches sequence 1-1: 'GG' class EI: hidden(2) AID hidden(11) class IE: IOT(hidden(2? AID hidden(17) 4 Prediction of Interest Rates This is an application, where the network input is a vector of real numbers. Since our approach can only handle binary input, we supplement the net with a discretization layer that provides a thermometer code representation (Hancock 1988) of the continuous valued input. In contrast to pure Boolean learning algorithms (Goodman, Miller and Smyth, 1989), (Mezard and Nadal, 1989), which can also be endowed with discretization facilities, here the discretization process is fully integrated into the learning scheme, as the discretization intervals will be adapted by the backpropagation algorithm. The data comprises a total of 226 patterns, which we distribute randomly on three sets: training set (60%), cross-validation set (20%) and test set (20%). The input represents the monthly development of 14 economic time series during the last 19 years. The Boolean target indicates, whether the interest rates will go up or down during the six months succeeding the reference month 6 ? The time series include among others month of the year, income of private households or the amount of German foreign investments. For some time series it is useful not to take the raw feature measurements as input, but the difference between two succeeding measurements; this is advantageous if the underlying time series show only small changes relative to their absolute values. All series were normalized to have values in the range from -1 to +1. We used a network containing a discretization layer of two neurons per input dimension. So there are 28 discretization neurons, which are fully connected to the 10 hidden nodes. The output layer consists of a single neuron. Since our data set is relatively small, the intention to obtain simple rules is not only motivated by the objective of comprehensibility, but also by the notion that we cannot expect a large rule set to be justified by a small amount of training data. In fact, during training 90% of the weights were set to zero and three hidden units proved to be sufficient for this task. Nevertheless the prediction error on the test set could be reduced to 25%. This compares to an error rate of about 20% attainable by a standard backpropagation network with one hidden layer of ten neurons and no input discretization. We thus sacrificed 5% of prediction performance to yield a very compact net, that can be easily transformed into a set of rules. Some of the generated rules are shown below. The first rule e.g. states that interest rates will rise if private income increases AND foreign investments decrease by a certain amount during the reference month. If the rules produce contradicting predictions for a given input, the final decision will be made according to a majority vote. A tie is broken by the bias value of the 61.e. the month where the input data has been measured. GDS: Gradient Descent Generation of Symbolic Classification Rules output unit, which states that by default interest rates will rise. IF (at least 2 ot { increase ot private income < 0.73%, decrease ot toreign investments < 64 MID DM }) THE! (interest rates will rise) ELSE (interest rates will fall). IF (at least 3 ot { increase of business climate estimate < 1.76%, treasury bonds yields (11 month ago) > 7.36%, treasury bonds yields (12 month ago) > 8.2%, increase ot foreign investments < 60 MID DM }) THE! (interest rates will tall) ELSE (interest rates will rise). 5 Conclusion and Future Work G DS is a learning algorithm that utilizes a penalty term in order to prepare a backpropagation network for rule extraction. The term is designed to have two effects on the network's weights: ? By a weight decay component, the number of nonzero weights is reduced: thus we get a net that can hopefully be transformed into a concise and comprehensible rule set . ? The penalty term encourages weight constellations that keep the node activations out of the nonsaturated part of the activation function. This is motivated by the fact that rules of the type IF < premise > THEN < conclusion > can only mimic the behavior of threshold units. The important point is that our penalty function adapts the net to the expressive power of the type of rules we wish to obtain. Consequently, we are able to transform the network into an equivalent rule set. The applicability of GDS was demonstrated on two tasks: splice-junction recognition and the prediction of German interest rates. In both cases the generated rules not only showed a generalization performance close to or even superior to what can be attained by other machine learning approaches such as MLPs or ID3. The rules also prove to be very concise and comprehensible. This is even more remarkable, since both applications represent real-world tasks with a large number of inputs. Clearly the applied penalty terms impose severe restrictions on the network parameters: besides minimizing the number of nonzero weights, the weights are restricted to a small set of distinct values. Last but not least, the simplification of sigmoidal to threshold units also affects the net's computational power. There are applications, where such a strong bias may negatively influence the net's learning capabilities. Furthermore our current approach is only applicable to tasks with binary target patterns. These limitations can be overcome by dealing with more general rules than those of the Boolean IF-THEN type. Future work will go into this direction. 1099 1100 Blasig Acknowledgements I wish to thank Hans-Georg Zimmermann and Ferdinand Hergert for many useful discussions and for providing the data on interest rates, and Patrick Murphy and David Aha for providing the UCI Repository of ML databases. This work was supported by a grant of the Siemens AG, Munich. References L. Bochereau, P. Bourgine. (1990) Extraction of Semantic Features and Logical Rules from a Multilayer Neural Network. Proceedings of the 1990 IJCNN - Washington DC, Vol.II 579-582. R.M. Goodman, J .W. Miller, P. Smyth. (1989) An Information Theoretic Approach to Rule-Based Connectionist Expert Systems. Advances in Neural Information Processing Systems 1, 256-263. San Mateo, CA: Morgan Kaufmann. P.J .B. Hancock. (1988) Data Representation in Neural Nets: an Empirical Study. Proc. Connectionist Summer School. Y. Hayashi. (1991) A Neural Expert System with Automated Extraction of Fuzzy If-Then Rules and its Application to Medical Diagnosis. Advances in Neural Information Processing Systems 3, 578-584. San Mateo, CA: Morgan Kaufmann. J. Hertz, A. Krogh, R.G. Palmer. (1991) Introduction to the Theory of Neural Computation. Addison-Wesley. C.M. Higgins, R.M. Goodman. (1991) Incremental Learning with Rule-Based Neural Networks. Proceedings of the 1991 IEEE INNS International Joint Conference on Neural Networks - Seattle, Vol.1 875-880. M. Mezard, J .-P. Nadal. (1989) Learning in Feedforward Layered Networks: The Tiling Algorithm. J. Phys. A: Math. Gen. 22, 2191-2203. W. Maass, G. Schnitger, E.D. Sontag. (1991) On the Computational Power of Sigmoids versus Boolean Threshold Circuits. Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, 767-776. P.M. Murphy, D.W. Aha. (1992). UCI Repository of machine learning databases [ftp-site: ics.uci.edu: pub/machine-Iearning-databases]. Irvine, CA: University of California, Department of Information and Computer Science. J .R. Quinlan. (1986) Induction of Decision Trees. Machine Learning, 1: 81-106. K. Siato, R. Nakano. (1988) Medical diagnostic expert systems based on PDP model. Proc. IEEE International Conference on Neural Networks Vol. I 255-262. V. Tresp, J. Hollatz, S. Ahmad. (1993) Network Structuring and Training Using Rule-Based Knowledge. Advances in Neural Information Processing Systems 5, 871-878. San Mateo, CA: Morgan Kaufman. G.G. Towell, J.W. Shavlik. (1991) Training Knowledge-Based Neural Networks to Recognize Genes in DNA Sequences. In: Lippmann, Moody, Touretzky (eds.), Advances in Neural Information Processing Systems 3, 530-536. San Mateo, CA: Morgan Kaufmann. 

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Coupled Dynamics of Fast Neurons and Slow Interactions A.C.C. Coolen R.W. Penney D. Sherrington Dept. of Physics - Theoretical Physics University of Oxford 1 Keble Road, Oxford OXI 3NP, U.K. Abstract A simple model of coupled dynamics of fast neurons and slow interactions, modelling self-organization in recurrent neural networks, leads naturally to an effective statistical mechanics characterized by a partition function which is an average over a replicated system. This is reminiscent of the replica trick used to study spin-glasses, but with the difference that the number of replicas has a physical meaning as the ratio of two temperatures and can be varied throughout the whole range of real values. The model has interesting phase consequences as a function of varying this ratio and external stimuli, and can be extended to a range of other models. 1 A SIMPLE MODEL WITH FAST DYNAMIC NEURONS AND SLOW DYNAMIC INTERACTIONS As the basic archetypal model we consider a system of Ising spin neurons (J'i E {-I, I}, i E {I, ... , N}, interacting via continuous-valued symmetric interactions, Iij, which themselves evolve in response to the states of the neurons. The neurons are taken to have a stochastic field-alignment dynamics which is fast compared with the evolution rate of the interactions hj, such that on the time-scale of Iii-dynamics the neurons are effectively in equilibrium according to a Boltzmann distribution, (1) 447 448 Cooien, Penney, and Sherrington where HVoj} ({O"d) =- L JijO"iO"j (2) i<j and the subscript {Jij} indicates that the {Jij} are to be considered as quenched variables. In practice, several specific types of dynamics which obey detailed balance lead to the equilibrium distribution (1), such as a Markov process with single-spin flip Glauber dynamics [1]. The quantity /3 is an inverse temperature characterizing the stochastic gain. For the hj dynamics we choose the form d 1 T' dthj = N(O"iO"j)V,i} - jjJij 1 + viirJij(t) (i < j) (3) where ( .. ')ViJ} refers to a thermodynamic average over the distribution (1) with the effectively instantaneous {Jij}, and TJij (t) is a stochastic Gaussian white noise of zero mean and correlation (TJij(t)TJkl(t')) = 2T'ffi- 1 o(ij),(kl)O(t - t') The first term on the right-hand side of (3) is inspired by the Hebbian process in neural tissue in which synaptic efficacies are believed to grow locally in response to the simultaneous activity of pre- an~ post-synaptic neurons [2]. The second term acts to limit the magnitude of hj; f3 is the characteristic inverse temperature of the interaction system. (A related interaction dynamics without the noise term, equivalent to ffi = 00, was introduced by Shinomoto [3]; the anti-Hebbian version of the above coupled dynamics was studied in layered systems by Jonker et al. [4, 5].) Substituting for (O"iO"j) in terms of the distribution (1) enables us to re-write (3) as d NT' dthj a = - aJij 11. ({Jij}) + VNTJij(t) (4) where the effective Hamiltonian 11. ({ hj}) is given by 11. ({ Jij }) 2 = - /31 In Z {3 ( { Jij }) + 21 jjN ~ ~<. Jij l (5) J where Z{3 ({ hj}) is the partition function associated with (2): 2 COUPLED SYSTEM IN THERMAL EQUILIBRIUM We now recognise (4) as having the form of a Langevin equation, so that the equilibrium distribution of the interaction system is given by a Boltzmann form. Henceforth, we concentrate on this equilibrium state which we can characterize by a partition function Zt3 an d an associated 'free energy' Ft3: Z{3 =JPdJij [Z{3 ({ Jij}) S<J r ~ exp [- ffijjN ~ Ji~] S<J - F{3 = -f3- - 1 In Z{3 (6) Coupled Dynamics of Fast Neurons and Slow Interactions where n _ ~/j3. We may use Z~ as a generating functional to produce thermodynamic averages of state variables <I> ( {O"d; {Jij}) in the combined system by adding suitable infinitesimal source terms to the neuron Hamiltonian (2): HP.j}({O"d) -+ Hp.j} ({O"d) + A<p({ud;{Jij}) oplim A-+O ?:J: UA = (<p({O"d;{Jij})){J } 'J _ IfL<j dhj (<p({O"d; {Jij}))plj}e-~1l(Plj}) IfL<jdhj e-~1l({J?j}) (7) where the bar refers to an average over the asymptotic {hj} dynamics. The form (6) with n -+ 0 is immediately reminiscent of the effective partition function which results from the application of the replica trick to replace In Z by limn-+o ~(zn - 1) in dealing with a quenched average for the infinite-ranged spinglass [6], while n = 1 relates to the corresponding annealed average, although we note that in the present model the time-scales for neuron and interaction dynamics remain completely disparate. These observations correlate with the identification of n with fi / j3, which implies that n -+ 0 corresponds to a situation in which the interaction dynamics is dominated by the stochastic term T)ij (t), rather than by the behaviour of the neurons, while for n = 1 the two characteristic temperatures are the same. For n -+ 00 the influence of the neurons on the interaction dynamics dominates. In fact, any real n is possible by tuning the ratio between the two {3's. In the formulation presented in this paper n is always non-negative, but negative values are possible if the Hebbian rule of (3) is replaced by an anti-Hebbian form with (UiO"j) replaced by - (O"iO"j) (the case of negative n is being studied by Mezard and co-workers [7]). The model discussed above is range-free/infinite-ranged and can therefore be analyzed in the thermodynamic limit N -+ 00 by the replica mean-field theory as devised for the Sherrington-Kirkpatrick spin-glass [6, 8, 9]. This can be developed precisely for integer n [6, 8, 9, 10] and analytically continued. In the usual manner there enters a spin-glass order parameter (, f- b) where the superscripts are replica labels. q"{6 is given by the extremum of ~ F({q1'6})=_LL:[q1'O]2+ ln Tr exp [ L:O"1'q1' OO"O] 2 2 2J-ln 1'<6 {O"1'} J-ln 1'<6 while Z~ is proportional to exp [NextrF ({q1'6})]. In the replica-symmetric region (or ansatz) one assumes q1'O = q. We will first choose as the independent variables nand j3 and briefly discuss the phase picture of our model (full details can be found in [11]). The system exhibits a transition from a paramagnetic state (q = 0) to an ordered state (q > 0) at a critical j3c(n). For n ::; 2 this transition is second order at j3c = 1, down to the SK 449 450 Coolen, Penney, and Sherrington spin-glass limit, n - 0, but for n > 2 the coupled dynamics leads to a qualitative, as well as quantitative, change to first order. Replica symmetry is stable above a critical value n c (!3), at which there is a de Almeida-Thouless (AT) transition (c.f. Kondor [12]). As expected from spin-glass studies, n c (f3) goes to zero as {3 ! 1 but rises for larger /3, having a maximum of order 0.3 at {3 of order 2. Thus, for n > nc(max) ::::: 0.3 there is no instability against small replica-symmetry breaking fluctuations, while for smaller n there is re-entrance in this stability. The transition from a paramagnetic to an ordered state and the onset of local RS instability for various temperatures is shown in Figure 1. 3 EXTERNAL FIELDS Several simple modifications of the above model are possible. One consists of adding external fields to the spin dynamics and/or to the interaction dynamics, by making the substitutions HV,j} ({O"d) ~ HV'J} ({O"d) - LOiO"i i 1? ( {Jii }) ~ 1? ({Jij }) - L hi Kij i<i in (2) and (5) respectively. These external fields may be viewed as generating fields in the sense of (7); for example For neural network models a natural first choice for the external fields would be Oi = hei and Kij = Keiej, E {-I, I}, where the ei are quenched random variables corresponding to an imposed pattern. Without loss of generality all the can be taken as +1, via the gauge transformation O"i ~ O"iei, Jii ~ Jiieiei. Henceforth we shall make this choice. The neuron perturbation field h induces a finite 'magnetization' characterized by a new order parameter ei ei ma = (O"f) which is independent of Q: in the replica-symmetric assumption (which turns out to be stable with respect to variation in this parameter). As in the case of the spin-glass, there is now a critical surface in (h, n, {3) space characterizing the onset of replica symmetry breaking. In introducing the interaction perturbation field K we find that K/ J-l is the analogue of the mean exchange J o in the SK spin-glass model, ]2 ({3nJ-l )-1 being the analogue of the variance. If large enough, this field leads to a spontaneous 'ferromagnetic' order. = Again we find further examples of both second and first order transitions (details 0, q 0) to ferromagnetic can be found in [11]). For the paramagnetic (P; m (F; m I=- 0, q I=- 0) case, the transition is second order at the SK value f3J a 1 so long as ({3])-2 ~ 3n - 2. Only when ({3])-2 < 3n - 2 do the interaction dynamics = = = Coupled Dynamics of Fast Neurons and Slow Interactions 1.2 ~ ll--_P_A_RAM __ A_GN_ET _____ -.-.-------.. .?-?,...? 0.8 T 0.6 WA'M'IS GLASS SPIN GLASS 0.2 1 = n 2 3 1. Dotted line: first order transition, solid line: Figure 1: Phasediagrarn for j second order transition. The separation between Mattis-glass and spin-glass phase is defined by the de Almeida-Thouless instability 451 452 Coolen, Penney, and Sherrington influence the transition, changing it to first order at a lower temperature. Regarding the ferromagnetic to spin-glass (SGj m = 0, q "# 0) transition, this exhibits both second order (lower .70) and first order (higher J o) sections separated by a tricritical point for n less than a critical value of the order of 3.3. This tricritical point exhibits re-entrance as a function of n. 4 COMPARISON BETWEEN COUPLED DYNAMICS AND SK MODEL In order to clarify the differences, we will briefly summarize the two routes that lead to an SK-type replica theory: Coupled Dynamics: Fast Ising spin neurons + slow dynamic interactions, d 1 -J .. = -((J'(J'){J .} dt lJ N I J 'J Free energy: 1 f- - --_-logZ, f3N Define: + -KN - IIJlJ.. r z =fIT dhj + GWN e-1ht({J,j}) i<j io =K/ /-t, Thermodynamics: 1 D f- = - f3n extr G ({q'Y }; {m'Y}) + const. N-+oo: SK spin-glass: Ising spins + fixed random interactions, P(Jij) = [27rJ2]-~e-~[J;j-Jo]2/J2 Free energy: 1 1 . l[n f =--logZ=--hmZ -1 ] f3N f3N n-O n Selt-averaging: Physical scaling: Jo = Jo/N, J = J/Vii Thermodynamics: f = - n_O lim 131 extr G ({q'YD}j {m'Y}) + const. n Coupled Dynamics of Fast Neurons and Slow Interactions 5 DISCUSSION \Ve have obtained a solvable model with which a coupled dynamics of fast stochastic neurons and slow dynamic interactions can be studied analytically. Furthermore it presents the replica method from a novel perspective, provides a direct interpretation of the replica dimension n in terms of parameters controlling dynamical processes and leads to new phase transition characters. As a model for neural learning the specific example analyzed here is however only a first step, with hand K as introduced corresponding to only a single pattern. Its adaptation to treat many patterns is the next challenge. One type of generalization is to consider the whole system as of lower connectivity with only pairs of connected sites being available for interaction upgrade. For example, the system could be on a lattice, in which case the corresponding coupled partition function will have the usual greater complication of a finite-dimensional system, or randomly connected with each bond present with a probability C IN, in which case there results an analogue of the Viana-Bray [13] spin-glass. In each of these cases the explicit factors involving N in the {hj} dynamics (3) should be removed (their presence or absence being determined by the need for statistical relevance and physical scaling). Yet another generalization is to higher order interactions, for example to p-neuron ones: L Hp} ({O"d) = i Ji l , ... ,i 1' O"i 1 0"i 2 ?? . 00i 1' l, ? . . ,i l' with corresponding interaction dynamics d 1 r-J? dt 11 , " .t.1' -- -(0"' N Sl'" 0"'11' ){J} - rIIJ? tl . , .. ?,l1' 1 + -ffi T)' . Sl,? .. ,l1' (t) or to more complex neuron types. If the symmetry-breaking fields Kij in the interaction dynamics are choosen at random, we obtain a curious theory in which we find replicas on top of replicas (the replica trick would be used to deal with the quenched disorder of the K ij , for a model in which replicas are already present. due to the coupled dynamics). Finally, our approach can in fact be generalized to any statistical mechanical system which in equilibrium is described by a Boltzmann distribution in which the Hamiltonian has (adiabatically slowly) evolving parameters. By choosing these parameters to evolve according to an appropriate Langevin process (involving the free energy of the underlying faRt system) one always arrives at a replica theory describing the coupled system in equilibrium. Acknowledgements Financial support from the U.K. Science and Engineering Research Council under grants 9130068X and GR/H26703, from the European Community under grant SISCI *915121, and from Jesus College, Oxford, is gratefully acknowledged. 453 454 Coolen, Penney, and Sherrington References [1] Glauber R.J. (1963) J. Math. Phys. 4 294 [2] Hebb D.O. (1949) 'The Organization of Behaviour' (Wiley, New York) [3] Shinomoto S. (1987) J. Phys. A: Math. Gen. 20 L1305 [4] Jonker II.J.J. and Cool en A.C.C. (1991) J. Phys. A: Math. Gen. 24 4219 [5] Jonker H.J.J., Coolen A.C.C. and Denier van der Gon J.J. (1993) J. Phys. A: Math. Gen. 26 2549 [6] Sherrington D. and Kirkpatrick S. (1975) Phys. Rev. Lett. 35 1792 [7] Mezard M. prit1ate communication [8] Kirkpatrick S. and Sherrington D. (1978) Phys. Rev. B 17 4384 [9] Mezard M., Parisi G. and Virasoro M.A. (1987) 'Spin Glass Theory and Beyond' (World Scientific, Singapore) [10] Sherrington D. (1980) J. Phys. A: Math. Gen. 13 637 [11] Penney R.W., Cool en A.C.C. and Sherrington D. (1993) J. Phys. A: Math. Gen. 26 3681-3695 [12] Kondor I. (198~{) J. Phys. A: Math. Gen. 16 L127 [13] Viana L. and Bray A.J. (1983) J. Phys. C 16 6817 

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An Analog VLSI Saccadic Eye Movement System Timothy K. Horiuchi Brooks Bishofberger and Christof Koch Computation and Neural Systems Program California Institute of Technology MS 139-74 Pasadena, CA 91125 Abstract In an effort to understand saccadic eye movements and their relation to visual attention and other forms of eye movements, we in collaboration with a number of other laboratories - are carrying out a large-scale effort to design and build a complete primate oculomotor system using analog CMOS VLSI technology. Using this technology, a low power, compact, multi-chip system has been built which works in real-time using real-world visual inputs. We describe in this paper the performance of an early version of such a system including a 1-D array of photoreceptors mimicking the retina, a circuit computing the mean location of activity representing the superior colliculus, a saccadic burst generator, and a one degree-of-freedom rotational platform which models the dynamic properties of the primate oculomotor plant. 1 Introduction When we look around our environment, we move our eyes to center and stabilize objects of interest onto our fovea. In order to achieve this, our eyes move in quick jumps with short pauses in between. These quick jumps (up to 750 deg/sec in humans) are known as saccades and are seen in both exploratory eye movements and as reflexive eye movements in response to sudden visual, auditory, or somatosensory stimuli. Since the intent of the saccade is to bring new objects of interest onto the fovea, it can be considered a primitive attentional mechanism. Our interest 582 An Analog VLSI Saccadic Eye Movement System lies in understanding how saccades are directed and how they might interact with higher attentional processes. To pursue this goal, we are designing and building a closed-loop hardware system based on current models of the saccadic system. Using traditional software methods to model neural systems is difficult because neural systems are composed of large numbers of elements with non-linear characteristics and a wide range of time-constants. Their mathematical behavior can rarely be solved analytically and simulations slow dramatically as the number and coupling of elements increases. Thus, real-time behavior, a critical issue for any system evolved for survival in a rapidly changing world, becomes impossible. Our approach to these problems has been to fabricate special purpose hardware that reflects the organization of real neural systems (Mead, 1989; Mahowald and Douglas, 1991; Horiuchi et al., 1992.) Neuromorphic analog VLSI technology has many features in common with nervous tissue such as: processing strategies that are fast and reliable, circuits that are robust against noise and component variability, local parameter storage for the construction of adaptive systems and low-power consumption. Our analog chips and the nervous system both use low-accuracy components and are significantly constrained by wiring. The design of the analog VLSI saccadic system discussed here is part of a long-term effort of a number of laboratories ( Douglas and Mahowald at Oxford University, Clark at Harvard University, Sejnowski at UCSD and the Salk Institute, Mead and Koch at Caltech) to design and build a complete replica of the early mammalian visual system in analog CMOS VLSI. The design and fabrication of all circuits is carried out via the US-government sponsored silicon service MOSIS, using their 2 J.1.m line process. 2 An Analog VLSI Saccadic System Figure 1: Diagram of the current system. The system obtains visual inputs from a photoreceptor array, computes the target location within a model of the superior colliculus and outputs the saccadic burst command to drive the eyeball. While not discussed here, an auditory localization 583 S84 Horiuchi, Bishofberger, and Koch input is being developed to trigger saccades to acoustical stimuli. 2.1 The Oculomotor Plant The oculomotor plant is a one degree-of-freedom turntable which is driven by a pair of antagonistic-pulling motors. In the biological system where the agonist muscle pulls against a passive viscoelastic force, the fixation position is set by balancing these two forces. In our system, the viscoelastic properties of the oculomotor plant are simulated electronically and the fixation point is set by the shifting equilibrium point of these forces. In order maintain fixation off-center, like the biological system, a tonic signal to the motor controller must be maintained. 2.2 Photoreceptors The front-end of the system is an adaptive photoreceptor array (Delbriick, 1992) which amplifies small changes in light intensity yet adapts quickly to gross changes in lighting level. The current system uses a 1-D array of 32 photoreceptors 40 microns apart. This array provides the visual input to the superior colliculus circuitry. The gain control occurs locally at each pixel of the image and thus the maximum sensitivity is maintained everywhere in the image in contrast to traditional imaging arrays which may provide washed out or blacked-out areas of an image when the contrast within an image is too large. In order to trigger reflexive, visually-guided saccades, the output of the photoreceptor array is coupled to the superior colliculus model by a luminance change detection circuit. A change in luminance somewhere in the image sends a pulse of current to the colliculus circuit which computes the center of this activity. This coupling passes a current signal which is proportional to the absolute-value of the temporal derivative of a photoreceptor's voltage output, (i.e. IIdI(x, t)/dtll where I(x,t) is the output of the photreceptor array). While we are initially building a 1-D system, 2-D photoreceptor arrays have been built in anticipation of a two degree-of-freedom system. While these photoreceptor circuits have been successfully constructed, we do not discuss the results here since the performance of these circuits are described in the literature (Delbriick 1992). 2.3 Superior Colliculus Model The superior colliculus, located on the dorsal surface of the midbrain, is a key area in the behavioral orientation system of mammals. The superficial layers have a topographic map of visual space and the deeper layers contain a motor map of saccadic vectors. Microstimulation in this area initiates saccades whose metrics are related to the location stimulated. This type of representation is known as a population coding. Many neurons in the deeper layers of superior colliculus are multisensory and will generate saccades to auditory and somatosensory targets as well as visual targets. While it is clear that the superior colliculus performs a multitude of integrative functions between sensory modalities and attentional processes, our initial model of superior colliculus simply computes the center of activity from the population code seen in the superficial layers (i.e. the photoreceptor array) using the weighted average techniques developed by DeWeerth (1991) for computing the centroid of An Analog VLSI Saccadic Eye Movement System Centroid Circuit Output vs. Target Error 10 ~ 2.. 2.6 / V? V ./ r:r'/ 2.0 I.' V .?J .:J) / ./ ?211 ?10 o 10 20 Figure 2: Output of the centroid circuit for a flashed red LED target at different angles away from the center position. Note that the output of the circuit was sampled 1 msec. after stimulus onset to account for capacitive delays. brightness. The results of the photoreceptor/centroid circuits are shown in Fig. 2. In the case of visually-guided saccades, retinal error translates directly into motor error and thus we can use the photoreceptors directly as our inputs. This simplified retina/superior colliculus model provides the motor error which is then passed on to the burst generator. 2.4 Saccadic Burst Generator The burst generator model (Fig. 3) driving the oculomotor plant receives as its input, desired change in eye position from the superior colliculus model and creates a two-component signal, a pulse and a step (Fig. 4). A pair of these pulse/step signals drive the two muscles of the eye which in turn moves the retinal array, thus closing the loop. The burst generator model is a double integrator model based on the work by Jurgens, et al (1981) and Lisberger et al (1987) which uses initial motor error as the input to the system. This motor error is injected into the "integrating" burst neuron which has negative feedback onto itself. This arrangement has the effect of firing a number of spikes proportional to the initial value of motor error. In the circuit, this integrator is implemented by a 1.9 pF capacitor. This burst of spikes serves to drive the eye rapidly against the viscosity. The burst is also integrated by the "neural integrator" (another 1.9 pF capacitor) which holds the local estimate of the current eye position from which the tonic, or holding signal is generated. Figs. 4 and 5 show output data from the burst generator chip and the response of the physical mechanism to this output. The inputs to the burst generator chip are 1) a voltage indicating desired eye position and 2) a digital trigger signal. The outputs are a pair of asynchronous digital pulse trains which carry the pulse/step signals which drive the left and right motors. 585 S86 Horiuchi, Bishofberger, and Koch 3 Discussion As we are still in the formative stages of our project, our first goal has been to demonstrate a closed-loop system which can fixate a particular stimulus whose image is falling onto its photoreceptor array. The first set of chips represent dramatically simplified circuits in order to capture the first-order behavior of the system while using known representations. Owing to the large number of parameters that must be set, and their sensitivity to variations, we have begun a study to investigate biologically plausible approaches to automatic parameter-setting. In the short term we intend to dramatically refine the models used at each stage, most notably the superior colliculus which is involved in the integration of non-visual sources of saccade targets (e.g. memory or audition), and in the mechanisms used for target selection or fixation. In the longer run, we plan to model the interaction of this system with other oculomotor processes such as smooth pursuit, VOR, OKR, AND vergence eye movements. While the biological microcircuits of the superior colli cui us and brainstem burst generator are not well known, more is understood about the representations found in these areas. By exploring the advantages and disadvantages of various computational models in a working system, it is hoped that a truly robust system will emerge as well as better models to explain the biological data. The construction of a compact hardware system which operates in real-time can often provide a more intuitive understanding of the closed-loop system. In addition, a visually-attentive hardware system which is physically small and low-power has numerous applications in the real world such as in mobile robotics or remote surveillance. 4 Acknowledgements Many thanks go to Prof. Carver Mead and his group for developing the foundations of this research. Our laboratory is partially supported by grants from the Office of Naval Research and the Rockwell International Science Center. Tim Horiuchi is supported by a grant from the Office of Naval Research. 5 References T. Delbriick and C. Mead, (1993) Ph.D. Thesis, California Institute of Technology. S. P. DeWeerth, (1991) Ph.D. Thesis, California Institute of Technology. T. Horiuchi, W. Bair, B. Bishofberger, A. Moore, J. Lazzaro, C. Koch, (1992) Int. J. Computer Vision 8:3,203-216. R. Jiirgens, W. Becker, and H. H. Kornhuber, (1981) BioI. Cybern. 39:87-96. S. G. Lisberger, E. J. Morris, and L. Tychsen, (1987) Ann. Rev. Neurosci. 10:97129. M. Mahowald, and R. Douglas, (1991) Nature 354:515-518. C. Mead, (1989) Analog VLSI and Neural Systems, Addison-Wesley. An Analog VLSI Saccadic Eye Movement System Left Motor Neuron Left Burst Neuron Motor Error < 0 Motor Error> 0 Other inputs: VOR/OKR ---t~ Smooth Pursuit Neural Integrator I~II.IIII Right Burst Neuron III I Right Motor Neuron Figure 3: Schematic diagram of the burst generator. The burst neuron "samples" the motor error when it receives a trigger signal (not shown) and begins firing as a sigmoidal function of the motor error. The spikes feedback and discharge the integrator and the burst is shut down. This "pulse" signal drives the eye against the viscosity. This signal is also integrated by the neural integrator which contributes the "step" portion of the motor command to hold the eye in its final position. The neural integrator has additional velocity inputs for other oculomotor behavior such as smooth pursuit, VOR and OKR. Note that the burst neuron for the other muscle is silent in this direction. S87 588 Horiuchi, Bishoiberger, and Koch -0.25 0..00 0..25 0.50. 0..75 1.00 1.25 1.50. 1.75 2.00 2.25 lime (Iecondl) (10..2 ) Figure 4: Spike signals in the circuit during a small saccade. (7.5 degrees to the right, starting from 4.8 degrees to the right.) Top: Burst neuron, Middle: Neural Integrator, Bottom: Motor neuron. (one of the outputs of the chip) Note that the "neuron" circuit currently used increases both its pulse frequency and pulse duration for large input currents, causing the voltage saturation seen in the bottom trace. Eye Position VB. Time 80. 60. 40. I I!! :ii 20. 0. ~ i -20. -40. -60. ~D+---~----~----+----+----;---~~--~----+---~----~ -0.025 0.,000 0.,025 0..050. 0..075 0..100 0..125 0.,150. 0..175 0..200 0.,225 lime (lecondl) Figure 5: Horizontal position vs . time for 21 different saccades. Peak angular velocity achieved for the 60 degree saccade to the right was approximately 870 degrees per second. The input command was changed uniformly from -60 to +60 degrees. An Analog VLSI Saccadic Eye Movement System Final Eye Position VS. Burst Command Voltage 80 60 40 I I :1 ~ 20 0 ~ i it; ?20 -40 -60 ?80 1.25 1.50 1.75 2 .00 2 .25 2.50 2.75 3.00 3.25 3.50 l,.,ul Voha. to BUI'II Generator (center - 2.5v) Figure 6: Linearity of the system for the position data given in the previous figure. Final eye position was computed as the average eye position during the last 20 msec. of each trace. Average of 10 Saccades from "center" to 30 deg. R 35 - .. .. .'.- .. 30 25 I 20 8 :-e 15 ~ 10 ! 5 0 ?5 -0.025 0.000 0.025 O.OSO 0 .075 0.100 0 .125 0 .150 0.175 0.200 0.225 lime (Iecondl) Figure 7: Repeatability: The solid line shows averaged eye position (relative to center) vs. time for 10 identical saccades. The dashed lines show a standard deviation on each side of the mean. Most of the variability is attributed to problems with friction. 589 

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