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Two Iterative Algorithms for Computing the Singular Value Decomposition from Input / Output Samples Terence D. Sanger Jet Propulsion Laboratory MS 303-310 4800 Oak Grove Drive Pasadena, CA 91109 Abstract The Singular Value Decomposition (SVD) is an important tool for linear algebra and can be used to invert or approximate matrices. Although many authors use "SVD" synonymously with "Eigenvector Decomposition" or "Principal Components Transform", it is important to realize that these other methods apply only to symmetric matrices, while the SVD can be applied to arbitrary nonsquare matrices. This property is important for applications to signal transmission and control. I propose two new algorithms for iterative computation of the SVD given only sample inputs and outputs from a matrix. Although there currently exist many algorithms for Eigenvector Decomposition (Sanger 1989, for example), these are the first true samplebased SVD algorithms. 1 INTRODUCTION The Singular Value Decomposition (SVD) is a method for writing an arbitrary nons quare matrix as the product of two orthogonal matrices and a diagonal matrix. This technique is an important component of methods for approximating nearsingular matrices and computing pseudo-inverses. Several efficient techniques exist for finding the SVD of a known matrix (Golub and Van Loan 1983, for example). 144 Singular Value Decomposition p r- --------------------------- U __--I~ s 1I _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ I Figure 1: Representation of the plant matrix P as a linear system mapping inputs u into outputs y. LT SR is the singular value decomposition of P. However, for certain signal processing or control tasks, we might wish to find the SVD of an unknown matrix for which only input-output samples are available. For example, if we want to model a linear transmission channel with unknown properties, it would be useful to be able to approximate the SVD based on samples of the inputs and outputs of the channel. If the channel is time-varying, an iterative algorithm for approximating the SVD might be able to track slow variations. 2 THE SINGULAR VALUE DECOMPOSITION The SVD of a nonsymmetric matrix P is given by P = LT SR where Land Rare matrices with orthogonal rows containing the left and right "singular vectors" , and S is a diagonal matrix of "singular values". The inverse of P can be computed by inverting S, and approximations to P can be formed by setting the values of the smallest elements of S to zero. For a memoryless linear system with inputs u and outputs y = Pu, we can write y LT SRu which shows that R gives the "input" transformation from inputs to internal "modes", S gives the gain of the modes, and LT gives the "output" transformation which determines the effect of each mode on the output. Figure 1 shows a representation of this arrangement. = The goal of the two algorithms presented below is to train two linear neural networks Nand G to find the SVD of P. In particular, the networks attempt to invert P by finding orthogonal matrices Nand G such that NG ~ p- 1 , or P NG = I. A particular advantage of using the iterative algorithms described below is that it is possible to extract only the singular vectors associated with the largest singular values. Figure 2 depicts this situation, in which the matrix S is shown smaller to indicate a small number of significant singular values. There is a close relationship with algorithms that find the eigenvalues of a symmetric matrix, since any such algorithm can be applied to P pT = LT S2 Land pT P = RT S2 R in order to find the left and right singular vectors. But in a behaving animal or operating robotic system it is generally not possible to compute the product with pT, since the plant is an unknown component of the system. In the following, I will present two new iterative algorithms for finding the singular value decomposition 145 146 Sanger p , - - -- -- -- -- -' I I I I I I__ _ _ _ __ _ _ _ _ _ J y u N G Figure 2: Loop structure of the singular value decomposition for control. The plant is P = LT SR, where R determines the mapping from control variables to system modes, and LT determines the outputs produced by each mode . The optimal sensory network is G L, and the optimal motor network is N RT S-l. Rand L are shown as trapezoids to indicate that the number of nonzero elements of S (the "modes") may be less than the number of sensory variables y or motor variables u. = = of a matrix P given only samples of the inputs u and outputs y. 3 THE DOUBLE GENERALIZED HEBBIAN ALGORITHM The first algorithm is the Double Generalized Hebbian Algorithm (DGHA), and it is described by the two coupled difference equations b-.G = l(zyT - LT[zzT]G) (1) b-.N T = l(zuT - LT[zzT]N T ) (2) where LT[ ] is an operator that sets the above diagonal elements of its matrix argument to zero, y Pu, z Gy, and I is a learning rate constant. = = Equation 1 is the Generalized Hebbian Algorithm (Sanger 1989) which finds the eigenvectors of the autocorrelation matrix of its inputs y. For random uncorrelated inputs u, the autocorrelation of y is E[yyT] LT S2 L, so equation 1 will cause G to converge to the matrix of left singular vectors L . Equation 2 is related to the Widrow-Hoff (1960) LMS rule for approximating u T from z, but it enforces orthogonality of the columns of N. It appears similar in form to equation 1, except that the intermediate variables z are computed from y rather than u. A graphical representation of the algorithm is given in figure 3. Equations 1 and 2 together RT S-l L is an cause N to converge to RT S-l , so that the combination N G approximation to the plant inverse. = = Theorem 1: (Sanger 1993) If y = Pu, z = Gy, and E[uuT ] = I, then equations 1 and 2 converge to the left and right singular vectors of P . Singular Value Decomposition u y p IGHA Figure 3: Graphic representation of the Double Generalized Hebbian Algorithm. G learns according to the usual G HA rule, while N learns using an orthogonalized form of the Widrow- Hoff LMS Rule. Proof: After convergence of equation 1, E[zzT] will be diagonal, so that E[LT[zzT]] E[zzT]. Consider the Widrow-Hoff LMS rule for approximating uT from z: ~NT = 'Y(zu T - zzT NT). = (3) After convergence of G, this will be equivalent to equation 2, and will converge to the same attractor. The stable points of 3 occur when E[uz T - NzzT] 0, for which N = RT 5- 1 = ? The convergence behavior of the Double Generalized Hebbian Algorithm is shown in figure 4. Results are measured by computing B GP N and determining whether B is diagonal using a score "L...I~) ...... b~. I) = ?= L . b?2 1 1 The reduction in ? is shown as a function of the number of (u, y) examples given to the network during training, and the curves in the figure represent the average over 100 training runs with different randomly-selected plant matrices P. Note that the Double Generalized Hebbian Algorithm may perform poorly in the presence of noise or uncontrollable modes. The sensory mapping G depends only on the outputs y, and not directly on the plant inputs u. So if the outputs include noise or autonomously varying uncontrollable modes, then the mapping G will respond to these modes. This is not a problem if most of the variance in the output is due the inputs u, since in that case the most significant output components will reflect the input variance transmitted through P. 4 THE ORTHOGONAL ASYMMETRIC ENCODER The second algorithm is the Orthogonal Asymmetric Encoder (OAE) which is described by the equations (4) 147 148 Sanger 0.7 Double Generalized Hebbian Algorithm 0.& 0.5 j 0.4 j I! is 0.3 0.2 0.1 ... ... .. . . . . . . .. . . . . . . . ~~~-~~~~~~~~~-~~~~~ Exomple Figure 4: Convergence of the Double Generalized Hebbian Algorithm averaged over 100 random choices of 3x3 or 10xlO matrices P. (5) where z = NT u. This algorithm uses a variant of the Backpropagation learning algorithm (Rumelhart et al. 1986). It is named for the "Encoder" problem in which a three-layer network is trained to approximate the identity mapping but is forced to use a narrow bottleneck layer. I define the "Asymmetric Encoder Problem" as the case in which a mapping other than the identity is to be learned while the data is passed through a bottleneck. The "Orthogonal Asymmetric Encoder" (OAE) is the special case in which the hidden units are forced to be uncorrelated over the data set. Figure 5 gives a graphical depiction of the algorithm. Theorem 2: (Sanger 1993) Equations vectors of P. 4 and 5 converge to the left and right singular Proof: Suppose z has dimension m. If P = LT SR where the elements of S are distinct, and E[uu T ] I, then a well-known property of the singular value decomposition (Golub and Van Loan 1983, , for example) shows that = E[IIPu - C T NT ullJ (6) is minimized when = LrnU, NT = V Rm , and U and V are any m x m matrices 1mS/;;". (L~ and Rm signify the matrices of only the first m for which UV columns of LT or rows of R.) If we want E[zzT] to be diagonal, then U and V must be diagonal. OAE accomplishes this by training the first hidden unit as if m = 1, the second as if m = 2, and so on. = = CT For the case m 1, the error 6 is minimized when C is the first left singular vector of P and N is the first right singular vector. Since this is a linear approximation problem, there is a single global minimum to the error surface 6, and gradient descent using the backpropagation algorithm will converge to this solution. Singular Value Decomposition u y p I , ~as!pr.2Pa~ti~ I Figure 5: The Orthogonal Asymmetric Encoder algorithm computes a forward approximation to the plant P through a bottleneck layer of hidden units. After convergence, the remaining error is E[II(P - GT N T plant matrix as )ull1. If we decompose the i=l where Ii and ri are the rows of Land R, and the remaining error is Si are the diagonal elements of S, then n P2 = LlisirT i=2 which is equivalent to the original plant matrix with the first singular value set to zero. If we train the second hidden unit using P2 instead of P, then minimization of E[IIP2 u - GT NT ull1 will yield the second left and right singular vectors. Proceeding in this way we can obtain the first m singular vectors. Combining the update rules for all the singular vectors so that they learn in parallel leads to the governing equations of the OAE algorithm which can be written in matrix form as equations 4 and 5 . ? (Bannour and Azimi-Sadjadi 1993) proposed a similar technique for the symmetric encoder problem in which each eigenvector is learned to convergence and then subtracted from the data before learning the succeeding one. The orthogonal asymmetric encoder is different because all the components learn simultaneously. After convergence, we must multiply the learned N by S-2 in order to compute the plant inverse. Figure 6 shows the performance of the algorithm averaged over 100 random choices of matrix P. Consider the case in which there may be noise in the measured outputs y. Since the Orthogonal Asymmetric Encoder algorithm learns to approximate the forward plant transformation from u to y, it will only be able to predict the components of y which are related to the inputs u. In other words, the best approximation to y based on u is if ~ Pu, and this ignores the noise term. Figure 7 shows the results of additive noise with an SNR of 1.0. 149 150 Sanger 0 .7 Orthogonal Asymmetric Encoder 0.8 " 0.5 I!! c? 0.4 c j ii c 0.3 0 GO .l!I 0 0.2 , 0.1 .. . . . ..... . . . . . ..... 0 0 50 100 150 200 250 300 350 400 450 500 550 800 850 700 750 800 850 900 950 Example Figure 6: Convergence of the Orthogonal Asymmetric Encoder averaged over 100 random choices of 3x3 or 10xlO matrices P. Acknowledgements This report describes research done within the laboratory of Dr. Emilio Bizzi in the department of Brain and Cognitive Sciences at MIT. The author was supported during this work by a National Defense Science and Engineering Graduate Fellowship, and by NIH grants 5R37 AR26710 and 5ROINS09343 to Dr. Bizzi. References Bannour S., Azimi-Sadjadi M. R., 1993, Principal component extraction using recursive least squares learning, submitted to IEEE Transactions on Neural Networks. Golub G. H., Van Loan C. F., 1983, Matrix Computations, North Oxford Academic P., Oxford. Rumelhart D. E., Hinton G. E., Williams R. J., 1986, Learning internal representations by error propagation, In Parallel Distributed Processing, chapter 8, pages 318-362, MIT Press, Cambridge, MA. Sanger T. D., 1989, Optimal unsupervised learning in a single-layer linear feedforward neural network, Neural Networks, 2:459-473. Sanger T. D., 1993, Theoretical Elements of Hierarchical Control in Vertebrate Motor Systems, PhD thesis, MIT. Widrow B., Hoff M. E., 1960, Adaptive switching circuits, In IRE WESCON Conv. Record, Part 4, pages 96-104. Singular Value Decomposition 2 OAE with 50% Added Noise 1? ? 3x3 101110 ! .? j 1 j I CI ... ... . . . . . . ... . . . oL-.---,--,--~~~:;:::::;:::::;~~ o 50 100 150 200 250 300 350 400 450 500 650 &00 860 700 750 &00 860 000 860 ElIIlmple Figure 7: Convergence of the Orthogonal Asymmetric Encoder with 50% additive noise on the outputs, averaged over 100 random choices of 3x3 or 10xlO matrices P. 151 

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233 HIGH ORDER NEURAL NETWORKS FOR EFFICIENT ASSOCIATIVE MEMORY DESIGN I. GUYON?, L. PERSONNAZ?, J. P. NADAL?? and G. DREYFUS? ? Ecole Superieure de Physique et de Chimie Industrielles de la Ville de Paris Laboratoire d'Electronique 10, rue Vauquelin 75005 Paris (France) ?? Ecole Normale Superieure Groupe de Physique des Solides 24, rue Lhomond 75005 Paris (France) ABSTRACT We propose learning rules for recurrent neural networks with high-order interactions between some or all neurons. The designed networks exhibit the desired associative memory function: perfect storage and retrieval of pieces of information and/or sequences of information of any complexity. INTRODUCTION In the field of information processing, an important class of potential applications of neural networks arises from their ability to perform as associative memories. Since the publication of J. Hopfield's seminal paper 1, investigations of the storage and retrieval properties of recurrent networks have led to a deep understanding of their properties. The basic limitations of these networks are the following: - their storage capacity is of the order of the number of neurons; - they are unable to handle structured problems; - they are unable to classify non-linearly separable data. ? American Institute of Physics 1988 234 In order to circumvent these limitations, one has to introduce additional non-linearities. This can be done either by using "hidden", non-linear units, or by considering multi-neuron interactions2 . This paper presents learning rules for networks with multiple interactions, allowing the storage and retrieval, either of static pieces of information (autoassociative memory), or of temporal sequences (associative memory), while preventing an explosive growth of the number of synaptic coefficients. AUTOASSOCIATIVE MEMORY The problem that will be addressed in this paragraph is how to design an autoassociative memory with a recurrent (or feedback) neural network when the number p of prototypes is large as compared to the number n of neurons. We consider a network of n binary neurons, operating in a synchronous mode, with period t. The state of neuron i at time t is denoted by (Ji(t), and the state of the network at time t is represented by a vector ~(t) whose components are the (Ji(t). The dynamics of each neuron is governed by the following relation: (Ji(t+t) = sgn vi(t). (1 ) In networks with two-neuron interactions only, the potential vi(t) is a linear function of the state of the network: For autoassociative memory design, it has been shown 3 that any set of correlated patterns, up to a number of patterns p equal to 2 n, can be made the stable states of the system, provided the synaptic matrix is computed as the orthogonal projection matrix onto the subspace spanned by the stored vectors. However, as p increases, the rank of the family of prototype vectors will increase, and finally reach the value of n. In such a case, the synaptic matrix reduces to the identity matrix, so that all 2 n states are stable and the energy landscape becomes flat. Even if such an extreme case is avoided, the attractivity of the stored states decreases with increasing p, or, in other terms, 235 the number of fixed points which are not the stored patterns increases; this problem can be alleviated to a large extent by making a useful use of these "spurious" fixed points4. Another possible solution consists in "gardening" the state space in order to enlarge the basins of attraction of the fixed points5. Anyway, no dramatic improvements are provided by all these solutions since the storage capacity is always O(n). We now show that the introduction of high-order interactions between neurons, increases the storage capacity proportionally to the number of connections per neuron. The dynamical behaviour of neuron i is still governed by (1). We consider two and three-neuron interactions, extension to higher order are straightforward. The potential vi (t) is now defi ned as It is more convenient, for the derivation of learning rules, to write the potential in the matrix form: ~(t) = C ;t(t), where :?(t) is an m dimensional vector whose components are taken among the set of the (n 2+n)/2 values: a1 , ... , an' a1 a2 , ... , aj al ' ... , a n-1 an. As in the case of the two-neuron interactions model, we want to compute the interaction coefficients so that the prototypes are stable and attractor states. A condition to store a set of states Q:k (k=1 to p) is that y'k= Q:k for all k. Among the solutions, the most convenient solution is given by the (n,m) matrix c=I,r l (2) where I, is the (n,p) matrix whose columns are the Q:k and rl is the (p,m) pseudoinverse of the (m,p) matrix r whose columns are the { . This solution satisfies the above requirements, up to a storage capacity which is related to the dimension m of vectors :?. Thus, in a network with three-neuron 236 interactions, the number of patterns that can be stored is O(n 2). Details on these derivations are published in Ref.6. By using only a subset of the products {aj all. the increase in the number of synaptic coefficients can remain within acceptable limits, while the attractivity of the stored patterns is enhanced, even though their number exceeds the number of neurons ; this will be examplified in the simulations presented below. Finally, it can be noticed that, if vector ~contains all the {ai aj}' i=1, ... n, j=1, ... n, only, the computation of the vector potential ~=C~can be performed after the following expression: where ~ stands for the operation which consists in squaring all the matrix coefficients. Hence, the computation of the synaptic coefficients is avoided, memory and computing time are saved if the simulations are performed on a conventional computer. This formulation is also meaningful for optical implementations, the function ell being easily performed in optics 7. In order to illustrate the capabilities of the learning rule, we have performed numerical simulations which show the increase of the size of the basins of attraction when second-order interactions, in addition to the first-order ones, are used. The simulations were carried out as follows. The number of neurons n being fixed, the amount of second-order interactions was chosen ; p prototype patterns were picked randomly, their components being ?1 with probability 0.5 ; the second-order interactions were chosen randomly. The synaptic matrix was computed from relation (2). The neural network was forced into an initial state lying at an initial Hamming distance Hi from one of the prototypes {!k ; it was subsequently left to evolve until it reached a stable state at a distance Hf from {!k. This procedure was repeated many times for each prototype and the Hf were averaged over all the tests and all the prototypes. Figures 1a. and 1b. are charts of the mean values of Hf as a function of the number of prototypes, for n = 30 and for various values of m (the dimension of 237 vector ':/.). These curves allowed us to determine the maximum number of prototype states which can be stored for a given quality of recall. Perfect recall implies Hf =0 ; when the number of prototypes increases, the error in recall may reach Hf =H i : the associative memory is degenerate. The results obtained for Hi In =10% are plotted on Figure 1a. When no high-order interactions were used, Hf reached Hi for pIn = 1, as expected; conversely, virtually no error in recall occured up to pIn = 2 when all second-order interactions were taken into account (m=465). Figure 1b shows the same quantities for Hi=20 0/0 ; since the initial states were more distant from the prototypes, the errors in recall were more severe. 1.2 1.2 1.0 1.0 0.8 0.8 0.6 :f A 0.6 :f A f f v (8) 0.4 v 0.2 0.2 0.0 0.0 2 0 3 (b) 0.4 2 0 3 pIn pIn Fig. 1. Improvement of the attractivity by addition of three-neuron interactions to the two-neuron interactions. All prototypes are always stored exactly (all curves go through the origin). Each point corresponds to an average over min(p,10) prototypes and 30 tests for each prototype. [] Projection: m = n = 30; ? 1 a: Hi I n =10 % ; m = 120 ; ? m = 180; 0 m = 465 (all interactions) 1 b : Hi In =20%. TEMPORAL SEQUENCES (ASSOCIATIVE MEMORY) The previous section was devoted to the storage and retrieval of items of information considered as fixed points of the dynamics of the network (autoassociative memory design). However, since fully connected neural networks are basically dynamical systems, they are natural candidates for 238 storing and retrieving information which is dynamical in nature, i.e., temporal sequences of patterns8: In this section, we propose a general solution to the problem of storing and retrieving sequences of arbitrary complexity, in recurrent networks with parallel dynamics. Sequences consist in sets of transitions between states {lk_> Q:k+ 1, k=1, ... , p. A sufficient condition to store these sets of transitions is that y..~ Q:k+ 1 for all k. In the case of a linear potential y"=C Q:, the storage prescription proposed in ref.3 can be used: C=r,+r,I, where r, is a matrix whose columns are the Q:k and r,+ is the matrix whose columns are the successors Q:k+ 1 of Q:k. If P is larger than n, one can use high-order interactions, which leads to introduce a non-linear potential Y..=C ';f. , with ';f. as previously defined. We proposed in ref. 10 the following storage prescription : (3) The two above prescriptions are only valid for storing simple sequences, where no patterns occur twice (or more). Suppose that one pattern occurs twice; when the network reaches this bifurcation point, it is unable to make a decision according the deterministic dynamics described in (1), since the knowledge of the present state is not sufficient. Thus, complex sequences require to keep, at each time step of the dynamics, a non-zero memory span. The vector potential Y..=C':J. must involve the states at time t and t-t, which leads to define the vector ';f. as a concatenation of vectors Q:(t), ~(t-t), Q:(t)?Q:(t), Q:(t)?Q:(t-t), or a suitable subset thereof. The subsequent vector Q:(t+t) is still determined by relation (1). In this form, the problem is a generalization of the storage of patterns with high order interactions, as described above. The storage of sequences can be still processed by relation (3). The solution presented above has the following features: i) Sequences with bifurcation points can be stored and retrieved. ii) The dimension of the synaptic matrix is at most (n,2(n 2+n)), and at least (n,2n) in the linear case, so that at most 2n(n2+n) and at least 2n2 synapses are required. 239 iii) The storage capacity is O(m), where m is the dimension of the vector ';t . iv) Retrieval of a sequence requires initializing the network with two states in succession. The example of Figure 2 illustrates the retrieval performances of the latter learning rule. We have limited vector ';t to Q:(t}?Q:(t-t). In a network of n=48 neurons, a large number of poems have been stored, with a total of p=424 elementary transitions. Each state is consists in the 6 bit codes of 8 letters. ALOUETTE JETE PLUMERAI ALOUETTE GENTILLE ALOUETTE ALOUETTE JETE PLUMERAI JE NE OLVMERAI AQFUETTE JEHKILLE SLOUETTE ALOUETTE JETE PLUMERAI Fig. 2. One of the stored poems is shown in the first column. The network is initialized with two states (the first two lines of the second column). After a few steps, the network reaches the nearest stored sequence. LOCAL LEARNING Finally, it should be mentioned that all the synaptic matrices introduced in this paper can be computed by iterative, local learning rules. For autoassociative memory, it has been shown analytically9 that the procedure: with Cij(O) = 0, which is a Widrow-Hoff type learning rule, yields the projection matrix, when 240 the number of presentations of the prototypes {~k} goes to infinity, if the latter are linearly independent. A derivation along the same lines shows that, by repeated presentations of the prototype transitions, the learning rules: lead to the exact solutions (relations (2) and (3) respectively), if the vectors }< are Ii nearly independent. GENERALIZATION TASKS Apart from storing and retrieving static pieces of information or sequences, neural networks can be used to solve problems in which there exists a structure or regularity in the sample patterns (for example presence of clumps, parity, symmetry ... ) that the network must discover. Feed-forward networks with multiple layers of first-order neurons can be trained with back-propagation algorithms for these purposes; however, one-layer feed-forward networks with mUlti-neuron interactions provide an interesting alternative. For instance, a proper choice of vector ':I. (second-order terms only) with the above learning rule yields a perfectly straightforward solution to the exclusive-OR problem. Maxwell et al. have shown that a suitable high-order neuron is able to exhibit the "ad hoc network solution" for the contiguity problem 11. CONCLUSION The use of neural networks with high-order interactions has long been advocated as a natural way to overcome the various limitations of the Hopfield model. However, no procedure guaranteed to store any set of information as fixed points or as temporal sequences had been proposed. The purpose of the present paper is to present briefly such storage prescriptions and show 241 some illustrations of the use of these methods. Full derivations and extensions will be published in more detailed papers. REFERENCES 1. 2. J. J. Hopfield, Proc. Natl. Acad. Sci. (USA) la, 2554 (1982). P. Peretto and J. J. Niez, BioI. Cybern. M, 53 (1986). P. Baldi and S. S. Venkatesh, Phys. Rev. Lett . .5.6 , 913 (1987). For more references see ref.6. 3. L. Personnaz, I. Guyon, G. Dreyfus, J. Phys. Lett. ~ , 359 (1985). L. Personnaz, I. Guyon, G. Dreyfus, Phys. Rev. A ~ , 4217 (1986). 4. I. Guyon, L. Personnaz, G. Dreyfus, in "Neural Computers", R. Eckmiller and C. von der Malsburg eds (Springer, 1988). 5. E. Gardner, Europhys. Lett . .1, 481 (1987). G. Poppel and U.Krey, Europhys. Lett.,.1, 979 (1987). 6. L. Personnaz, I. Guyon, G. Dreyfus, Europhys. Lett. .1,863 (1987). 7. D. Psaltis and C. H. Park, in "Neural Networks for Computing", J. S. Denker ed., (A.I.P. Conference Proceedings 151, 1986). 8. P. Peretto, J. J. Niez, in "Disordered Systems and Biological Organization", E. Bienenstock, F. Fogelman, G. Weisbush eds (Springer, Berlin 1986). S. Dehaene, J. P. Changeux, J. P. Nadal, PNAS (USA)~, 2727 (1987). D. Kleinfeld, H. Sompolinsky, preprint 1987. J. Keeler, to appear in J. Cog. Sci. For more references see ref. 9. 9. I. Guyon, L. Personnaz, J.P. Nadal and G. Dreyfus, submitted for publication. 10. S. Diederich, M. Opper, Phys. Rev. Lett . .5.6, 949 (1987). 11. T. Maxwell, C. Lee Giles, Y. C. Lee, Proceedings of ICNN-87, San Diego, 1987. 

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Robot Learning: Exploration and Continuous Domains David A. Cohn MIT Dept. of Brain and Cognitive Sciences Cambridge, MA 02139 The goal of this workshop was to discuss two major issues: efficient exploration of a learner's state space, and learning in continuous domains. The common themes that emerged in presentations and in discussion were the importance of choosing one's domain assumptions carefully, mixing controllers/strategies, avoidance of catastrophic failure, new approaches with difficulties with reinforcement learning, and the importance of task transfer. 1 Domain assumptions Andrew Moore (CMU) discussed the problem of standardizing and making explicit the set of assumptions that researcher makes about his/her domain. He suggested that neither "fewer assumptions are better" nor "more assumptions are better" is a tenable position, and that we should strive to find and use standard sets of assumptions. With no such commonality, comparison of techniques and results is meaningless. Under Moore's guidance, the group discussed the possibility of designing an algorithm which used a number of well-chosen assumption sets and switched between them according to their empirical validity. Suggestions were made to draw on the AI approach of truth maintenance systems. This theme of detecting failure of an assumption set/strategy was echoed in the discussion on mixing controllers and avoiding failure (described below). 2 Mixing controllers and strategies Consensus appeared to be against using single monolithic approaches, and in favor of mixing controllers. Spatial mixing resulted in local models, as advocated by Stefan Schaal (MIT) using locally weighted regression. Controllers could also be mixed over the entire domain. Jeff Schneider (Rochester) discussed mixing a na'ive feedback controller with a "coaching signal." Combining the coached controller with an un coached one further improved performance. During the main conference, Satinder Singh (MIT) described a controller that learned by reinforcement to mix the strategies of two "safe" but suboptimal controllers, thus avoiding unpleasant surprises and catastrophic failure. 1169 1170 Cohn 3 Avoiding failure The issue of det,ecting impending failure and avoiding catastrophic failure clarified differences in several approaches. A learning strategy that learns in few trials is useless on a real robot if the initial trials break the robot by crashing it into walls. Singh's approach has implicit failure avoidance, but may be hampered by an unnecessarily large margin of safety. Terry Sanger (JPL) discussed a trajectory extension algorithm by which a controller could smoothly "push the limits" of its performance, and detect impending failure of the control strategy. 4 Reinforcement learning Reinforcement learning seems to have come into its own, with people realizing the diverse ways in which it may be applied to problems. Long-Ji Lin (Siemens) described his group's unusual but successful application of reinforcement learning in landmark-based navigation. The Siemens RatBot uses reinforcement to select landmarks on the basis of their recognizability and their value to the eventual precision of position estimation. The reinforcement signal is simply the cost and the robot's final position error after it has used a set of landmarks. Jose del R. Millan (JRC) presented an approach similar to Schneider's, but training a neural controller with reinforcement learning. As the controller's performance improves, it supplants the mobile robot's reactive "instincts," which are designed to prevent catastrophic failure. With new applications, however, come new pitfalls. Leemon Baird (WPAFB) showed how standard reinforcement learning approaches can fail when adapted to exploration in continuous time. He then described the "advantage updating" algorithm which was designed to work in noisy domains with continuous or small time steps. The issue of exploration in continuous space, especially with noise, has not been as easily addressed. Jiirgen Schmidhuber (TUM) described the approach one should take if interested solely in exploration: use prior information gain as a reinforcement signal to decide on an "optimal" action. The ensuing discussion centered on the age-old and intractable tradeoff between exploration and exploitation. Final consensus was that we, as a group, should become more familiar with the literature on dual control, which addresses exactly this issue. 5 Task transfer An unorchestrated theme that emerged from the discussion was the need to address, or even define, task transfer. As with last year's workshop on Robot Learning, it was generally agreed that "one-task learning" is not a suitable goal when designing a learning robot. During the discussion, Long-Ji Lin (Siemens) and Lori Pratt (CSM) described several types of task transfer that are considered in the literature. These included model learning for multiple tasks, hierarchical control and learning, and concept (or bias) sharing across tasks. 

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Comparisoll Training for a Resclleduling Problem ill Neural Networks Didier Keymeulen Artificial Intelligence Laboratory Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussels Belgium Martine de Gerlache Prog Laboratory Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussels Belgium Abstract Airline companies usually schedule their flights and crews well in advance to optimize their crew pools activities. Many events such as flight delays or the absence of a member require the crew pool rescheduling team to change the initial schedule (rescheduling). In this paper, we show that the neural network comparison paradigm applied to the backgammon game by Tesauro (Tesauro and Sejnowski, 1989) can also be applied to the rescheduling problem of an aircrew pool. Indeed both problems correspond to choosing the best solut.ion from a set of possible ones without ranking them (called here best choice problem). The paper explains from a mathematical point of view the architecture and the learning strategy of the backpropagation neural network used for the best choice problem. We also show how the learning phase of the network can be accelerated. Finally we apply the neural network model to some real rescheduling problems for the Belgian Airline (Sabena). 1 Introduction Due to merges, reorganizations and the need for cost reduction, airline companies need to improve the efficiency of their manpower by optimizing the activities of their crew pools as much as possible. A st.andard scheduling of flights and crews is usually made well in advance but many events, such as flight delays or the absence of a crew member make many schedule cha.nges (rescheduling) necessary. 801 802 Keymeulen and de Gerlache Each day, the CPR1 team of an airline company has to deal with these perturbations. The problem is to provide the best answer to these regularly occurring perturbations and to limit their impact on the general schedule. Its solution is hard to find and usually the CPR team calls on full reserve crews. An efficient rescheduling tool taking into account the experiences of the CPR team could substantially reduce the costs involved in rescheduling notably by limit.ing the use of a reserve crew. The paper is organized as follow. In the second section we describe the rescheduling task. In the third section we argue for the use of a neural network for the rescheduling task and we apply an adequate architecture for such a network. Finally in the last section, we present results of experiments with schedules based on actual schedules used by Sabena. 2 Rescheduling for an Airline Crew Pool When a pilot is unavailable for a flight it becomes necessary to replace him, e.g. to reschedule the crew. The rescheduling starts from a list of potential substitute pilots (PSP) given by a scheduling program based generally on operation research or expert syst.em technology (Steels, 1990). The PSP list obtained respects legislation and security rules fixing for example t.he number of flying hours per month, the maximum number of consecutive working hour and the number of training hours per year and t.heir schedule. From the PSP list, the CPR team selects the best candidat.es taking into account t.he schedule stability and equity. The schedule stability requires that possible perturbations of the schedule can be dealt with with only a minimal rescheduling effort. This criterion ensures work stability t.o the crew members and has an important influence on their social behavior. The schedule equity ensures the equal dist.ribution of the work and payment among the crew members during the schedule period. One may think to solve this rescheduling problem in t.he same way as the scheduling problem itself using software t.ools based on operational research or expert system approach. But t.his is inefficient. for t.wo reasons, first., the scheduling issued from a scheduling system and its adapt.at.ion t.o obt.ain an acceptable schedule takes days. Second this system does not t.ake into account the previous schedule. It follows that the updat.ed one may differ significantly from the previous one after each perturbation. This is unaccept.able from a pilot's point of view. Hence a specific procedure for rescheduling is necessary. 3 Neural Network Approach The problem of reassigning a new crew member to replace a missing member can be seen as the problem of finding the best pilot in a pool of potential substitute pilots (PSP), called the best choice problem. To solve the best choice problem, we choose the neural network approach for two reasons. First the rules llsed by the expert. are not well defined: to find the best PSP, lCrew Pool Rescheduler Comparison Training for a Rescheduling Problem in Neural Networks the expert associates implicit.ly a score value to each profile. The learning approach is precisely well suited to integrate, in a short period of time, t.he expert knowledge given in an implicit form. Second, t.he neural network approach was applied with success to board-games e.g. the Backgammon game described by Tesauro (Tesauro and Sejnowski, 1989) and the Nine l\ll en's Morris game described by Braun (Braun and al., 1991). These two games are also exa.mples of best choice problem where the player chooses the best move from a set of possible ones. 3.1 Profile of a Potential Substitute Pilot To be able to use the neural network approach we have to identify the main features of the potential substitute pilot and to codify them in terms of rating values (de Gerlache and Keymeulen, 1993). We based our coding scheme on the way the expert solves a rescheduling problem. He ident.ifies the relevant parameters associated with the PSP and the perturbed schedule. These parameters give three types of information. A first type describes the previous, present and future occupation of the PSP. The second t.ype represents information not in the schedule such as the human relationship fadars. The assocjat.ed values of t.hese two t.ypes of parameters differ for f'ach PSP. The last t.ype of paramet.ers describes the context of the rescheduling, namely t.he characteristics of t.he schedule . This last type of parameters are the same for all the PSP. All t.hese paramet.ers form the profile of a PSP associated to a perturbed schedule. At each rescheduling problem corresponds one perturbed schedule j and a group of 11 PSpi to which we associate a Projile~ (PSpi, PertU1?berLSchedulej) . Implicitly, the expert associates a rating value between 0 and 1 to each parameter of the P1'ojile; based on respectively its little or important impact on the result.ing schedule if the P S pi was chosen. The rating value reflects the relative importance of the parameters on the stability and the equity of the resulting schf'dnle obt.ained after the pilots substitution. = 3.2 Dual Neural Network It would have been possible to get more information from the expert than only the best profile. One of the possibilities is to ask him to score every profile associated with a perturbed planning. From this associat.ion we could immediately construct a scoring function which couples each profile with a specific value, namely its score. Another possibility is to ask the expert to rank all profiles associated with a perturbed schedule. The corresponding ranking function couples each profile with a value such that the values associat.ed with the profiles of the same perturbed schedule order the profiles according t.o t.heir rank. The decision making process used by the rescheduler team for the aircrew rescheduling problem does not consist in the evaluation of a scoring or ranking function . Indeed only the knowledge of the best profile is useful for the rescheduling process. From a neura.l net.work architectural point of view, because the ranking problem is a generalization of the best choice problem, a same neural net.work architecture can be used. But the difference between the best choice problem and t.he scoring problem is such that two different neural network architectures are associated to them. As we show in this section, although a backpropagatian network is sufficient to learn a scoring function, its architecture, its learning and its retrieval procedures must be 803 804 Keymeulen and de Gerlache adapted to learn the best profile. Through a mathematical formulation of the best choice problem, we show that the comparison paradigm of Tesauro (Tesauro, 1989) is suited to the best choice problem and we suggest how to improve the learning convergence. 3.2.1 Comparing Function For the best choice problem the expert gives the best profile Projilefest associated with the perturbed schedule j and that for m pert.urbed schedules. The problem consists then to learn the mapping of the m * n profiles associated with the m perturbed schedules into the m best profiles, one for each pert.urbed schedule. One way to represent this association is through a comparing function. This function has as input a profile, represented by a vector xj, and returns a single value. When a set of profiles associated with a perturbed schedule are evaluated by the function, it returns the lowest value for the best profile. This comparing function integrates the information given by the expert and is sufficient to reschedule any perturbed schedule solved in the past by the expert. Formally it is defined by: Comp(J.1>e) = C(Projile)) (1) C ompare jBest < C ompcl1>C ,j. {V)' Vi=fBest with)' = 1, ... ,111. with i=l, ... ,n The value of Comp(J.1>e) are not known a priori and have only a meaning when they are compared to the value Comp(J.1>ef est of the comparing function for the best profile. 3.2.2 Geometrical Interpretation To illustrate the difference between the neural network learning of a scoring function and a comparing function, we propose a geometrical interpretation in the case of a linear network having as input vect.ors (profiles) XJ, ... ,XJ, ... ,Xp associated with a perturbed schedule j. The learning of a scoring function which associat.es a score Score; with each input vector xj consists in finding a hyperplane in the input vector space which is tangent to the circles of cent.er and radius SC01>e{ (Fig. 1). On the contrary the learning of a comparing function consists t.o obt.ain t.he equation of an hyperplane such that the end-point of the vector Xfest is nearer the hyperplane than the end-points of the other input vectors associated with the same perturbed schedule j (Fig. 1). xf XJ 3.2.3 Learning We use a neural network approach to build the comparing function and the mean squared error as a measure of the quality of t.he approximation. The comparing function is approximated by a non-linear function: C(P1>ojile;) = N?(W,Xj) where W is the weight. vector of the neural network (e.g backpropagat.ion network). The problem of finding C which has the property of (1) is equivalent to finding the function C that minimizes the following error function (Braun and al., 1991) where <I> is the sigmoid function : Comparison Training for a Rescheduling Problem in Neural Networks ,-- --_ .. _... """", , ,, \ ,, , , x,, I I , ,,, ,, , .,' I " -- ,.,1' x..,,, ', ....... --,' W( Wl ,w2'''')' .. ,W L) wlh .w. - 1 Figure 1: Geometrical Interpretation of the learning of a Scoring Function (Rigth) and a Comparing Function (Left) n I: (2) = i 1 i -::f Best To obtain t.he weight vector which mll1UTIlzes the error funct.ion (2), we use the property that t.he -gr~ld?~(W) point.s in the direct.ion in which the error function will decrease at the fastest possible rate. To update t.he weight we have thus to calculate the partial derivative of (2) with each components of the weight vector ltV: it is made of a product of three factors. The evaluation of the first two factors (the sigmoid and the derivative of the sigmoid) is immediate. The third factor is the partial derivative of the non-linear function N ?, which is generally calculated by using the generalized delta rule learning law (Rumelhart. and McClelland, 1986), Unlike the linear associator network, for the backpropagation network, the error e3t function (2) is not equivalent to the error function where the difference is associated with the input vector of the backpropagation network because: Xl X; (3) By consequence to calculate t.he three factors of the partial derivative of (2), we have to introduce separately at the bottom of the network t.he input vector of the best profile X!e3t and the input vector of a less good profile Then we have to memorize theIl' partial contribution at each node of the network and multiply their contributions before updating the weight . Using this way to evaluate the derivative of (2) and to update t.he weight, the simplicity of the generalized delta rule learning law has disappeared . XJ. 805 806 Keymeulen and de Gerlache 3.2.4 Architecture Tesauro (Tesauro and Sejnowski, 1989) proposes an architecture, t.hat we call dual neural network, and a learning procedure such that the simplicity of the generalized delta rule learning law can still be used (Fig. 2). The same kind of architecture, called siamese network, was recently used by Bromley for the signature verification (Bromley and al., 1994). The dual neural network architecture and the learning strategy are justified mathematically at one hand by the decomposition of the partial derivative of the error function (2) in a sum of two terms and at the other hand by the asymmetry property of the sigmoid and its derivative. The architecture of the dual neural network consists to duplicate the multi-layer network approximating the comparing function (1) and to connect the output of both to a unique output node through a positive unit weight for the left network and negative unit weight. for the right network. During the learning a couple of profiles is presented to the dual neural network: a best profile X e3t and a less good profile X!. The desired value at the output node of the dual neural network is 0 when the left network has for input the best profile and the right network has for input a less good profile and 1 when these profiles are permuted. During the recall we work only with one of the two multi-layer networks, suppose the left one (the choice is of no importance because they are exactly the same). The profiles JY~ associated with a perturbed schedule j are presented at the input of the left. multi-layer network. The best profile is the one having the lowest value at the output of the left multi-layer network. f Through this mathematical formulation we can use the suggestion of Braun to improve the learning convergence (Braun and al., 1991). They propose to replace the positive and negative unit weight het.ween the output node of the multi-layer networks and the output. node of the dual neural network by respect.ively a weight value equal to V for the left net.work and - V for the right. network. They modify the value of V by applying the generalized delt.a rule which has no significant impact on the learning convergence. By manually increasing the factor V during the learning procedure, we improve considerably the learning convergence due to its asymmetric impact on the derivative of ?<I>(W) with W: the modification of the weight vector is greater for couples not yet learned than for couples already learned. 4 Results The experiments show the abilit.y of our model to help the CPR team of the Sabena Belgian Airline company to choose the best profile in a group of PSPs based on the learned expertise of the team. To codify the profile we identify 15 relevant parameters. They constitute the input of our neural network. The training data set was obtained by analyzing the CPR team at work during 15 days from which we retain our training and test perturbed schedules. We consider that the network has learned when the comparing value of the best profile is less than the comparing value of t.he other profiles and that for all training perturbed schedules. At that time ?cJ>(W) is less t.han .5 for every couple of profiles. The left graph of Figure 3 shows t.he evolution of t.he mean error over t.he couples Comparison Training for a Rescheduling Problem in Neural Networks 807 Dual Neural Network ?1 C_ .. n C_,.n Belt. .1Ir.(~.l 'J BeltoJ ) .11r.(~. t ZMulI.La,'"" -'-od ..... .) ,~ Nouol NoI",ort.o 10.6lo.~ -+ (X -+ .X But~ toJ 10,910", 10.110.710.11 ) ? wdb XIJ B.... I63IO.2!O-.1 b.61 0.21 0.91 0.21 10.11 0,91 0.71 0.91 0.11 0.11 0,61 Wllb X 2?1 X B.... I =X But.l it.t ) 10.610.710.1/ 0.11 0.711.01 0.21 ,~ XB.... I 10.310.210.11 0.61 0.21 0.91 0.21 X 2?1 10,616316.91 631 0.11 6.71 0.21 XB..... lo. 1 Io.9Io.7! X I .. X I.. 0.910.110.11 0.&1 10.610.710.1( 0.11 0.711.0 I 0.11 XB..... 'S .. = X Be... Figure 2: The training of a dual neural network. during the training . The right graph shows the improvement of the convergence when the weight V is increased regularly during the training process. lnclasilll V OJ or the Dull NaIl'll Ndworll o.s OA 0.? 0.1 OJ M~--~~~---------------------- 02 ~--~~------------------------ 0.1 01 Nwnbcr of'lhilitg liIl) liIO ~~~~~~~~~~~~~~~~ 111(1) IlO) Figure 3: Convergence of the dual neural network architecture. The network does not converge when we introduce contradictory decisions in our training set. It is possible to resolve them by adding new context parameters in the coding scheme of the profile. After learning, our network shows generalization capacity by retrieving the best profile for a new perturbed schedule that is similar to one which has already been learned. The degree of similarity required for the generalization remains a topic for further study. Nunt.crci nW~ 808 Keymeulen and de Gerlache 5 Conclusion In conclusion, we have shown that the rescheduling problem of an airline crew pool can be stated as a decision making problem, namely the identification of the best potential substitute pilot. We have stressed the importance of the codification of the information used by the expert to evaluate the best candidate. We have applied the neural network learning approach to help the rescheduler team in the rescheduling process by using the experience of already solved rescheduling problems. By a mathematical analysis we have proven the efficiency of the dual neural network architecture. The mathematical analysis permits also to improve the convergence of the network. Finally we have illustrated the method on rescheduling problems for the Sabena Belgian Airline company. Acknowledgments We thank the Scheduling and Rescheduling team of Mr. Verworst at Sabena for their valuable information given all along this study; Professors Steels and D'Hondt from the VUB and Professors Pastijn, Leysen and Declerck from the Military Royal Academy who supported this research; Mr. Horner and Mr. Pau from the Digital Europe organization for their funding. We specially thank Mr. Decuyper and Mr. de Gerlache for their advices and attentive reading. References H. Braun, J. Faulner & V. Uilrich. (1991) Learning strategies for solving the problem of planning using backpropagation. In Proceedings of Fourth International Conference on Neural Networks and their Applications, 671-685. Nimes, France. J. Bromley, I. Guyon, Y . Lecun, E. Sackinger, R. Shah . (1994). Signature verification using a siamese delay neural network. In J. Cowan, G. Tesauro & J. Alspector (eds.), Advances in Neural Information Processing Systems 1. San Mateo, CA: Morgan Kaufmann. M. de Gerlache & D. Keymeulen. (1993) A neural network learning strategy adapted for a rescheduling problem. In Proceedings of Fourth International Conference on Neural Networks and their Applications, 33-42 . Nimes, France. D. Rumelhart & J. McClelland. (1986) Parallel Distributed Processing: Explorations in the Microstructure of Cognition I [1 II. Cambridge, MA: MIT Press. L. Steels. (1990) Components of expertise. AI Maga.zine, 11(2):29-49 . G. Tesauro. (1989) Connectionist learning of expert preferences by comparison training. In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 1, 99-106. San Mateo, CA: Morgan Kaufmann. G. Tesauro & T.J. Sejnowski. (1989) A parallel network that learns to play backgammon. Artificial Intelligence, 39:357-390. 

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Exploiting Chaos to Control the Future Gary W. Flake* Guo-Zhen Sun t Yee-Chun Lee t Hsing-Hen Chen t Institute for Advance Computer Studies University of Maryland College Park, MD 20742 Abstract Recently, Ott, Grebogi and Yorke (OGY) [6] found an effective method to control chaotic systems to unstable fixed points by using only small control forces; however, OGY's method is based on and limited to a linear theory and requires considerable knowledge of the dynamics of the system to be controlled. In this paper we use two radial basis function networks: one as a model of an unknown plant and the other as the controller. The controller is trained with a recurrent learning algorithm to minimize a novel objective function such that the controller can locate an unstable fixed point and drive the system into the fixed point with no a priori knowledge of the system dynamics. Our results indicate that the neural controller offers many advantages over OGY's technique. 1 Introduction Recently, Ott, Grebogi and Yorke (OGY) [6] proposed a simple but very good idea. Since any small perturbation can cause a large change in a chaotic trajectory, it is possible to use a very small control force to achieve a large trajectory modification. Moreover, due to the ergodicity of chaotic motion, any state in a chaotic *Department of Computer Science, peyote@umiacs.umd.edu tLaboratory for Plasma Research 647 648 Flake, Sun, Lee, and Chen attractor can be reached by a small control force . Since OGY published their work, several experiments and simulations have proven the usefulness of OGY's method. One prominent application of OGY's method is the prospect of controlling cardiac chaos [1] . We note that there are several unfavorable constraints on OGY's method. First, it requires a priori knowledge of the system dynamics, that is, the location of fixed points. Second, due to the limitation of linear theory, it will not work in the presence of large noise or when the control force is as large as beyond the linear region from which the control law was constructed. Third, although the ergodicity theory guarantees that any state after moving away from the desired fixed point will eventually return to its linear vicinity, it may take a very long time for this to happen, especially for a high dimensional chaotic attractor. In this paper we will demonstrate how a neural network (NN) can control a chaotic system with only a small control force and be trained with only examples from the state-space. To solve this problem, we introduced a novel objective function which measures the distance between the current state and its previous average. By minimizing this objective function, the NN can automatically locate the fixed point. As a preliminary step , a training set is used to train a forward model for the chaotic dynamics. The work of Jordan and Rumelhart [4] has shown that control problems can be mapped into supervised learning problems by coupling the outputs of a controller NN (the control signals) to the inputs of a forward model of a plant to form a multilayer network that is indirectly recurrent. A recurrent learning a.lgorithm is used to train the controller NN. To facilitate learning we use an extended radial basis function (RBF) network for both the forward model and the controller. To benchmark with OGY's result, the Himon map is used as a numerical example. The numerical results have shown the preliminary success of the proposed scheme. Details will be given in the following sections. In the next section we give our methodology and describe the general form of the recurrent learning algorithm used in our experiments. In Section 3, we discuss RBF networks and reintroduce a more powerful version. In Section 4, the numerical results are presented in detail. Finally, in Section 5, we give our conclusions. 2 Recurrent Learning for Control Let kC) denote a NN whose output, tit, is composed through a plant, l(?), with unknown dynamics. The output of the unknown plant (the state), it+l' forms part of the input for the NN a.t the next time step, hence the recurrency. At each time step the state is also passed to an output function , gC), which computes the sensation, Yt+l. The time evolution of this system is more accurately described by fit it+l Yt+l k(it ,!h+l'W) {(it, fit) g(Xt+I), where ii7+1 is the desired sensation for time t + 1 and W represents the trainable weights for the network . Additionally, we define the temporally local and global Exploiting Chaos to Control the Future error functionals J t = ~11Y7 - Ytl1 2 and E = L~I Ji, where N is the final time step for the system. The real-time recurrent learning (RTRL) algorithm [9] for training the network weights to minimize E is based on the fair assumption that minimizing the local error functionals with a small learning rate at each time step will correspond to minimizing the global error. To derive the learning algorithm, we can imagine the system consisting of the plant, controller, and error functionals as being unfolded in time. From this perspective we can view each instance of the controller NN as a separate NN and thus differentiate the error functionals with respect to the network weights at different times. Hence, we now add a time index to Wt to represent this fact. However, when we use W without the time index, the term should be understood to be time invariant. We can now define the matrix f t =~ a~t = ~it a aiI-t- 1 + (axt aiI L.J a....t- 1 + a....ait ) ?) - i=O UWi ?) UUt-l Wt-I ?)- UUt-l Xt-I Xt-l ft-I, (1) which further allows us to define aJi aw aE aw (2) (3) Equation 2 is the gradient equation for the RTRL algorithm while Equation 3 is for the backpropagation through time (BPTT) learning algorithm [7]. The gradients defined by these equations are usually used with gradient descent on a multilayer perceptron (MLP) . We will use them on RBF networks. 3 The CNLS Network The Connectionist Normalized Local Spline (eNLS) network [3] is an extension of the more familiar radial basis function network of Moody and Darken [5]. The forward operation of the network is defined by (4) where (5) All of the equations in this section assume a single output. Generalizing them for multiple outputs merely adds another index to the terms. For all of our simulations, we choose to distribute the centers, iii, based on a sample of the input space. 649 650 Flake, Sun, Lee, and Chen Additionally, the basis widths, f3i' are set to an experimentally determined constant . Because the output, <p, is linear in the terms Ii and d~, training them is very fast. To train the CNLS network on a prediction problem we, can use a quadratic error function of the form E = ~(y(i) - qj(i?2, where y(i) is the target function that we wish to approximate. We use a one-dimensional Newton-like method [8] which yields the update equations If + 7J (y(i) - <P(i?L'~~~i)' ~ + 7J (y(x) - <p(x?~=---=-!J...?...!..lo..::....L-- The right-most update rules form the learning algorithm when using the CNLS network for prediction, where 7J is a learning rate that should be set below 1.0. The left-most update rules describe a more general learning algorithm that can be used when a target output is unknown. When using the CNLS network architecture as part of a recurrent learning algorithm we must be able to differentiate the network outputs with respect to the inputs. Note that in Equations 1 and 2 each of the terms aXt/aUt-l, aUt-daxt-l, ait/Bit- 1 , and Biii/ aii can either be exactly solved or approximated by differentiating a CNLS network. Since the CNLS output is highly nonlinear in its inputs, computing these partial derivatives is not quite as elegant as it would be in a MLP . Nevertheless, it can be done. We skip the details and just show the end result: ann a: = ~ d~Pi(X) + 2~(pj (x) qj f3j l=l 4 (aj - i)) - 2<p(x)::;, (6) J=l Adaptive Control By combining the equations from the last two sections, we can construct a recurrent learning scheme for RBF networks in a similar fashion to what has been done with MLP networks. To demonstrate the utility of our technique, we have chosen a wellstudied nonlinear plant that has been successfully modeled and controlled by using non-neural techniques. Specifically, we will use the Henon map as a plant, which has been the focus of much of the research of OGY [6]. We also adopt some of their notation and experimental constraints. 4.1 The Himon Map The Henon map [2] is described by the equations (7) (8) Exploiting Chaos to Control the Future = = where A Ao + p and p is a control parameter that may be modified at each time step to coerce the plant into a desirable state. For all simulations we set Ao 1.29 and B 0.3 which gives the above equations a chaotic attracter that also contains an unstable fixed point. Our goal is to train a CNLS network that can locate and drive the map into the unstable fixed point and keep it there with only a minimal amount of information about the plant and by using only small values of p. = The unstable fixed point (XF, YF) in Equations 7 and 8 can be easily calculated as XF = YF ~ 0.838486. Forcing the Henon map to the fixed point is trivial if the controller is given unlimited control of the parameter. To make the problem more realistic we define p* as the maximum magnitude that p can take and use the rule below on the left if Ipi < p* _ {p if Ipl < p* if p > p* Pn 0 if Ipl > p* if p < -p* while OGY use the rule on the right. The reason we avoid the second rule is that it cannot be modeled by a CNLS network with any precision since it is step-like. The next task is to define what it means to "control" the Henon map. Having analytical knowledge of the fixed point in the attracter would make the job of the controller much easier, but this is unrealistic in the case where the dynamics of the plant to control are unknown. Instead, we use an error function that simply compares the current state of the plant with an average of previous states: et=21 [(Xt-(x)r) 2+(Yt-(Y)r) 2] , (9) where (.)r is the average of the last T values of its argument. This function approaches zero when the map is in a fixed point for time length greater than T. This function requires no special knowledge about the dynamics of the plant, yet it still enforces our constraint of driving the map into a fixed point. The learning algorithm also requires the partial derivatives of the error function with respect to the plant state variables, which are oet!f)xt = Xt - (x}r and oet!oYt = Yt - (Y)r. These two equations and the objective function are the only special purpose equations used for this problem. All other equations generalize from the derivation of the algorithm. Additionally, since the "output" representation (as discussed earlier) is identical to the state representation, training on a distinct output function is not strictly necessary in this case. Thus, we simplify the problem by only using a single additional model for the unknown next-state function of the Henon map. 4.2 Simulation To facilitate comparison between alternate control techniques, we now introduce the term f6t where 6t is a random variable and f is a small constan~ which specifies the intensity of the noise. We use a Gaussian distribution for bt such that the distribution has a zero mean, is independent, and has a variance of one. In keeping with [6], we discard any values of 6t which are greater in magnitude than 10. For training we set f 0.038. However, for tests on the real controller, we will show results for several values of f. = 651 652 Flake, Sun, Lee, and Chen (a) ? ? ? ? (b) ? ? (c) ? (f) ? ? ? """ 'r (d) ? ? - (e) . ? . Figure 1: Experimental results from training a neural controller to drive the Himon map into a fixed point. From (a) to (f), the values of fare 0.035, 0.036, 0.038, 0.04,0 .05, and 0.06, respectively. The top row corresponds to identical experiments performed in [6]. We add the noise in two places. First, when training the model, we add noise to the target output of the model (the next state). Second, when testing the controller on the real Henon map, we add the noise to the input of the plant (the previous state). In the second case, we consider the noise to be an artifact of our fictional measurements; that is, the plant evolves from the previous noise free state. Training the controller is done in two stages: an off-line portion to tune the model and an on-line stage to tune the controller. To train the model we randomly pick a starting state within a region (-1.5, 1.5) for the two state variables. We then iterate the map for one hundred cycles with p = 0 so that the points will converge onto the chaotic attractor. Next, we randomly pick a value for p in the range of (-p*, p*). The last state from the iteration is combined with this control parameter to compute a target state. We then add the noise to the new state values. Thus, the model input consists of a clean previous state and a control parameter and the target values consist of the noisy next state. We compute 100 training patterns in this manner. Using the prediction learning algorithm for the CNLS network we train the model network on each of the 100 patterns (in random order) for 30 epochs. The model quickly converges to a low average error. In the next stage, we use the model network to train the controller network in two ways. First, the model acts as the plant for the purposes of computing a next state. Additionally, we differentiate the model for values needed for the RTRL algorithm. We train the controller for 30 epochs, where each epoch consists of 50 cycles. At the beginning of each epoch we initialize the plant state to some random values (not necessarily on the chaotic attracter ,) and set the recurrent history matrix, Exploiting Chaos to Control the Future ... - .... _ ... -. _ .._..... ... _-- .. _.... ... _. -.-_ ..... .. (a) (b) . -.- -_... _---.-- . .... _... ...... __ .- -. - ..... . .-.-.. _. _- .... _... .. (c) Figure 2: Experimental results from [6]. From left to right, the values of f. are 0.035, 0.036, and 0.038, respectively. r t, to zero. Then, for each cycle, we feed the previous state into the controller as input. This produces a control parameter which is fed along with the previous state as input into the model network, which in turn produces the next state. This next state is fed into the error function to produce the error signal. At this point we compute all of the necessary values to train the controller for that cycle while maintaining the history matrix. In this way, we train both the model and control networks with only 100 data points, since the controller never sees any of the real values from the Henon map but only estimates from the model. For this experiment both the control and model RBF networks consist of 40 basis functions. 4.3 Summary Our results are summarized by Figure 1. As can be seen, the controller is able to drive the Henon Map into the fixed point very rapidly and it is capable of keeping it there for an extended period of time without transients. As the level of noise is increased, it can be seen that the plant maintains control for quite some time . The first visible spike can be observed when f. = 0.04. These results are an improvement over the results generated from the best nonneural technique available for two reasons: First, the neural controller that we have trained is capable of driving the Henon map into a fixed point with far fewer transients then other techniques. Specifically, alternate techniques , as illustrated in Figure 2, experience numerous spikes in the map for values of f. for which our controller is spike-free (0.035 - 0.038). Second, our training technique has smaller data requirements and uses less special purpose information. For example, the RBF controller was trained with only 100 data points compared to 500 for the nonneural. Additionally, non-neural techniques will typically estimate the location of the fixed point with an initial data set. In the case of [6] it was assumed that the fixed point could be easily discovered by some technique, and as a result all of their experiments rely on the true (hard-coded) fixed point. This, of course, could be discovered by searching the input space on the RBF model, but we have instead allowed the controller to discover this feature on its own. 653 654 Flake, Sun, Lee, and Chen 5 Conclusion and Future Directions A crucial component of the success of our approach is the objective function that measures the distance between the current state and the nearest time average. The reason why this objective function works is that during the control stage the learning algorithm is minimizing only a small distance between the current point and the "moving target." This is in contrast to minimizing the large distance between the current point and the target point, which usually causes unstable long time correlation in chaotic systems and ruins the learning. The carefully designed recurrent learning algorithm and the extended RBF network also contribute to the success of this approach. Our results seem to indicate that RBF networks hold great promise in recurrent systems. However, further study must be done to understand why and how NNs could provide more useful schemes to control real world chaos. Acknowledgements We gratefully acknowledge helpful comments from and discussions with Chris Barnes, Lee Giles, Roger Jones, Ed Ott, and James Reggia. This research was supported in part by AFOSR grant number F49620-92-J-0519. References [1] A. Garfinkel, M.L. Spano, and W.L. Ditto. Controlling cardiac chaos. Science, 257(5074):1230, August 1992. [2] M. HEmon. A two-dimensional mapping with a strange attractor. Communications in Mathematical Physics, 50:69-77, 1976. [3] R.D. Jones, Y.C. Lee, C.W. Barnes, G.W. Flake, K. Lee, P.S. Lewis, and S. Qian. Function approximation and time series prediction with neural network. In Proceedings of the International Joint Conference on Neural Networks, 1990. [4] M.1. Jordan and D.E. Rumelhart. Forward models: Supervised learning with a distal teacher. Technical Report Occasional Paper #40, MIT Center for Cognitive Science, 1990. [5] J. Moody and C. Darken. Fast learning in networks of locally-tuned processing units. Neural Computation, 1:281-294, 1989. [6] E. Ott, C. Grebogi, and J .A. Yorke. Controlling chaotic dynamical systems. In D.K. Campbell, editor, CHAOS: Soviet-American Perspectives on Nonlinear Science, pages 153-172. American Institute of Physics, New York, 1990. [7] F.J. Pineda. Generalization of back-propagation to recurrent neural networks. Physical Review Letters, 59:2229-2232, 1987. [8] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. Numerical Recipes. Cambridge University Press, Cambridge, 1986. [9] R.J. Williams and D. Zipser. Experimental analysis of the real-time recurrent learning algorithm. Connection Science, 1:87-111, 1989. 

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How to Choose an Activation Function H. N. Mhaskar Department of Mathematics California State University Los Angeles, CA 90032 hmhaska@calstatela.edu c. A. Micchelli IBM Watson Research Center P. O. Box 218 Yorktown Heights, NY 10598 cam@watson.ibm.com Abstract We study the complexity problem in artificial feedforward neural networks designed to approximate real valued functions of several real variables; i.e., we estimate the number of neurons in a network required to ensure a given degree of approximation to every function in a given function class. We indicate how to construct networks with the indicated number of neurons evaluating standard activation functions. Our general theorem shows that the smoother the activation function, the better the rate of approximation. 1 INTRODUCTION The approximation capabilities of feedforward neural networks with a single hidden layer has been studied by many authors, e.g., [1, 2, 5]. In [10], we have shown that such a network using practically any nonlinear activation function can approximate any continuous function of any number of real variables on any compact set to any desired degree of accuracy. A central question in this theory is the following. If one needs to approximate a function from a known class of functions to a prescribed accuracy, how many neurons will be necessary to accomplish this approximation for all functions in the class? For example, Barron shows in [1] that it is possible to approximate any function satisfying certain conditions on its Fourier transform within an L2 error of O(1/n) using a feedforward neural network with one hidden layer comprising of n 2 neurons, each with a sigmoidal activation function. On the contrary, if one is interested in a class of functions of s variables with a bounded gradient on [-1, I]S , 319 320 Mhaskar and Micchelli then in order to accomplish this order of approximation, it is necessary to use at least 0(11$) number of neurons, regardless of the activation function (cf. [3]). In this paper, our main interest is to consider the problem of approximating a function which is known only to have a certain number of smooth derivatives. We investigate the question of deciding which activation function will require how many neurons to achieve a given order of approximation for all such functions. We will describe a very general theorem and explain how to construct networks with various activation functions, such as the Gaussian and other radial basis functions advocated by Girosi and Poggio [13] as well as the classical squashing function and other sigmoidal functions. In the next section, we develop some notation and briefly review some known facts about approximation order with a sigmoidal type activation function. In Section 3, we discuss our general theorem. This theorem is applied in Section 4 to yield the approximation bounds for various special functions which are commonly in use. In Section 5, we briefly describe certain dimension independent bounds similar to those due to Barron [1], but applicable with a general activation function. Section 6 summarizes our results. 2 SIGMOIDAL-TYPE ACTIVATION FUNCTIONS In this section, we develop some notation and review certain known facts. For the sake of concreteness, we consider only uniform approximation, but our results are valid also for other LP -norms with minor modifications, if any. Let s 2: 1 be the number of input variables. The class of all continuous functions on [-1, IP will be denoted by C$. The class of all 27r- periodic continuous functions will be denoted by C$*. The uniform norm in either case will be denoted by II . II. Let IIn,I,$,u denote the set of all possible outputs of feedforward neural networks consisting of n neurons arranged in I hidden layers and each neuron evaluating an activation function (j where the inputs to the network are from R$. It is customary to assume more a priori knowledge about the target function than the fact that it belongs to C$ or cn. For example, one may assume that it has continuous derivatives of order r 2: 1 and the sum of the norms of all the partial derivatives up to (and including) order r is bounded. Since we are interested mainly in the relative error in approximation, we may assume that the target function is normalized so that this sum of the norms is bounded above by 1. The class of all the functions satisfying (or if the functions are periodic). In this this condition will be denoted by paper, we are interested in the universal approximation of the classes W: (and their periodic versions). Specifically, we are interested in estimating the quantity W: (2.1) W:'" sup En,l,$,u(f) JEW: where (2.2) En,l,$,u(f) := p E Anf n,l,s,1T III - PII? The quantity En,l,s ,u(l) measures the theoretically possible best order of approximation of an individual function I by networks with 11 neurons. We are interested How to Choose an Activation Function in determining the order that such a network can possibly achieve for all functions in the given class. An equivalent dual formulation is to estimate (2.3) En,l,s,O'(W:) := min{m E Z : sup Em,l,s,O'(f) ~ lin}. fEW: This quantity measures the minimum number of neurons required to obtain accuracy of lin for all functions in the class W:. An analogous definition is assumed for W:* in place of W: . Let IH~ denote the class of all s-variable trigonometric polynomials of order at most n and for a continuous function f, 27r-periodic in each of its s variables, E~(f):= min (2.4) PEIH~ Ilf - PII? We observe that IH~ can be thought of as a subclass of all outputs of networks with a single hidden layer comprising of at most (2n + 1)" neurons, each evaluating the activation function sin X. It is then well known that (2.5) Here and in the sequel, c, Cl, ... will denote positive constants independent of the functions and the number of neurons involved, but generally dependent on the other parameters of the problem such as r, sand (j. Moreover, several constructions for the approximating trigonometric polynomials involved in (2.5) are also well known. In the dual formulation, (2.5) states that if (j(x) := sinx then (2.6) It can be proved [3] that any "reasonable" approximation process that aims to approximate all functions in W:'" up to an order of accuracy lin must necessarily depend upon at least O(ns/r) parameters. Thus, the activation function sin x provides optimal convergence rates for the class W:*. The problem of approximating an r times continuously differentiable function f R s --+ R on [-1, I]S can be reduced to that of approximating another function from the corresponding periodic class as follows. We take an infinitely many times differentiable function 1f; which is equal to 1 on [-2,2]S and 0 outside of [-7r, 7rp. The function f1f; can then be extended as a 27r-periodic function. This function is r times continuously differentiable and its derivatives can be bounded by the derivatives of f using the Leibnitz formula. A function that approximates this 27r-periodic function also approximates f on [-I,I]S with the same order of approximation. In contrast, it is not customary to choose the activation function to be periodic. In [10] we introduced the notion of a higher order sigmoidal function as follows. Let k > O. We say that a function (j : R --+ R is sigmoidal of order k if ( 2.7) lim (j( x) - 1 x-+oo xk -, lim (j(x) - 0 x-+-oo xk - and (2.8) xE R. , 321 322 Mhaskar and Micchelli A sigmoidal function of order 0 is thus the customary bounded sigmoidal function. We proved in [10] that for any integer r ~ 1 and a sigmoidal function r - 1, we have (j of order if s = 1, if s > 2. (2.9) Subsequently, Mhaskar showed in [6] that if (j is a sigmoidal function of order k and r ~ 1 then, with I = O(log r/ log k)), >2 (2.10) Thus, an optimal network can be constructed using a sigmoidal function of higher order. During the course of the proofs in [10] and [6], we actually constructed the networks explicitly. The various features of these constructions from the connectionist point of view are discussed in [7, 8, 9]. In this paper, we take a different viewpoint. We wish to determine which activation function leads to what approximation order. As remarked above, for the approximation of periodic functions, the periodic activation function sin x provides an optimal network. Therefore, we will investigate the degree of approximation by neural net.works first in terms of a general periodic activation function and then apply these results to the case when the activation function is not periodic. 3 A GENERAL THEOREM In this section, we discuss the degree of approximation of periodic functions using periodic activation functions. It is our objective to include the case of radial basis functions as well as the usual "first. order" neural networks in our discussion. To encompass both of these cases, we discuss the following general formuation. Let s ~ d 2: 1 be integers and ?J E Cd?. We will consider the approximation of functions in ca. by linear combinat.ions of quantities of the form ?J(Ax + t) where A is a d x s matrix and t E Rd. (In general, both A and t are parameters ofthe network.) When d = s, A is the identity matrix and ?J is a radial function, then a linear combination of n such quantities represents the output of a radial basis function network with n neurons. When d = 1 then we have the usual neural network with one hidden layer and periodic activation function ?J. We define the Fourier coefficients of ?J by the formula (3.1) , 1 ?J(m) := (2 )d 7r 1 . ?J(t)e- zm .t dt, [-lI',lI']d Let (3.2) and assume that there is a set J co Itaining d x s matrices with integer entries such that (3.3) How to Choose an Activation Function where AT denotes the transpose of A. If d = 1 and ?(l) #- 0 (the neural network case) then we may choose S4> = {I} and J to be Z8 (considered as row vectors). If d = sand ?J is a function with none of its Fourier coefficients equal to zero (the radial basis case) then we may choose S4> zs and J {Is x s}. For m E Z8, we let k m be the multi-integer with minimum magnitude such that m = ATk m for some A = Am E J. Our estimates will need the quantities = (3.4) mn := = min{I?(km)1 : -2n::; m::; 2n} and (3.5) N n := max{lkml : -2n::; m < 2n} where Ikml is the maximum absolute value of the components of km. In the neural network case, we have mn = 1?(1)1 and N n = 1. In the radial basis case, N n = 2n. Our main theorem can be formulated as follows. THEOREM 3.1. Let s ~ d ~ 1, n ~ 1 and N ~ N n be integers, f E Cn , ?J E C d*. It is possible to construct a network (3.6) such that (3.7) In (3.6), the sum contains at most O( n S Nd) terms, Aj E J, tj E R d, and dj are linear functionals of f, depending upon n, N, <p. The estimate (3.7) relates the degree of approximation of f by neural networks explicitly in terms of the degree of approximation of f and ?J by trigonometric polynomials. Well known estimates from approximation theory, such as (2.5), provide close connections between the smoothness of the functions involved and their degree of trigonometric polynomial approximation. In particular, (3.7) indicates that the smoother the function ?J the better will be the degree of approximation. In [11], we have given explicit constructions of the operator Gn, N,4>. The formulas in [11] show that the network can be trained in a very simple manner, given the Fourier coefficients of the target function. The weights and thresholds (or the centers in the case of the radial basis networks) are determined universally for all functions being approximated . Only the coefficients at the output layer depend upon the function . Even these are given explicitly as linear combinations of the Fourier coefficients of the target function. The explicit formulas in [11] show that in the radial basis case, the operator G n ,N,4> actually contains only O( n + N)S summands. 4 APPLICATIONS In Section 3, we had assumed that the activation function ?J is periodic. If the activation function (J is not periodic, but satisfies certain decay conditions near 323 324 Mhaskar and Micchelli it is still possible to construct a periodic function for which Theorem 3.1 can be applied. Suppose that there exists a function 1j; in the linear span of Au,J := {(T( Ax + t) A E J, t E R d}, which is integrable on R d and satisfies the condition that 00, for some (4.1) T> d. Under this assumption, the function ( 4.2) 1j;0 (x):= L 1j;(x - 27rk) kEZ d is a 27r-periodic function integrable on [-7r, 7r]s. We can then apply Theorem 3.1 with 1j;0 instead of ?. In Gn,N,tjJo, we next replace 1j;0 by a function obtained by judiciously truncating the infinite sum in (4.2). The error made in this replacement can be estimated using (4.1). Knowing the number of evaluations of (T in the expression for '1/) as a finite linear combination of elements of Au,J, we then have an estimate on the degree of approximation of I in terms of the number of evaluations of (T. This process was applied on a number of functions (T. The results are summarized in Table 1. 5 DIMENSION INDEPENDENT BOUNDS In this section, we describe certain estimates on the L2 degree of approximation that are independent of the dimension of the input space. In this section, II? II denotes th(' L2 norm on [-1, I]S (respectively [-7r, 7r]S) and we approximate functions in the class S Fs defined by (5.1 ) SFtI := {I E C H : II/l1sF,s:= L li(m)l::; I}. mEZ' Analogous to the degree of approximation from IH~, we define the n-th degree of approximation of a function I E CS* by the formula En s(f) (5.2) := , inf ACZ' ,IAI~n III - L i(m)eimOxlI mEA where we require the norm involved to be the L2 norm. In (5.2), there is no need to assume that n is an integer. Let ? be a square integrable 27r-periodic function of one variable. We define the L2 degree of approximation by networks with a single hidden layer by the formula E~~~)f) := PEj~~l"'~ III - PII (.5.3) where m is the largest integer not exceeding n. Our main theorem in this connection is the following THEOREM 5.1. Let s 2: 1, integers n, N (5.4) 2: 1 be an integer, IE SFs , ? E Li and J(1) f:. O. Then, for How to Choose an Activation Function Table 1: Order of magnitude of En,l,s,o-(W:) for different O"S --- Function En Iso- 0' Remarks Sigmoidal, order r - 1 n 1/ r Sigmoidal, order r - 1 n lJ / r +(s+2r)/r 2 s ~ 2, d n IJ / r+ (2r+s )/2r k k ~ 2, s ~ 2, d xk, if x ~ 0, 0, if x (1 < O. + e-x)-l nlJ/r(log n)2 s=d=I,/=1 = 1, I = 1 = 1, I = 1 s~2,d=I,/=1 = Sigmoidal, order k n lJ / r k ~ 2, s ~ 1, d 1, I (log r/ log k)) exp( -lxl 2/2) n 2s / r s=d>2/=1 , Ixlk(log Ixl)6 n( IJ /r)(2+(3s+2r)/ k) =o = d > 2, k > 0, k + seven, 6 = 0 if s odd, 1 if s even, I = 1 S where {6 n } is a sequence of positive numbers, 0 ::; 6n ::; 2, depending upon f such that 6n --- 0 as n --- 00. Moreover, the coefficients in the network that yields (5.1,) are bounded, independent of nand N. We may apply Theorem 5.1 in the same way as Theorem 3.1. For the squashing activation fUllction, this gives an order of approximation O(n-l/2) with a network consisting of n(lo~ n)2 neurons arranged in one hidden layer. With the truncated power function x + (cf. Table 1, entry 3) as the activation function, the same order of approximation is obtained with a network with a single hidden layer and O(n1+1/(2k?) neurons. 6 CONCLUSIONS. We have obtained estimates on the number of neurons necessary for a network with a single hidden layer to provide a gi ven accuracy of all functions under the only a priori assumption that the derivatives of the function up to a certain order should exist. We have proved a general theorem which enables us to estimate this number 325 326 Mhaskar and Micchelli in terms of the growth and smoothness of the activation function. We have explicitly constructed networks which provide the desired accuracy with the indicated number of neurons. Acknowledgements The research of H. N. Mhaskar was supported in part by AFOSR grant 2-26 113. References 1. BARRON, A. R., Universal approximation bounds for superposition of a sigmoidal function, IEEE Trans. on Information Theory, 39. 2. CYBENKO, G., Approximation by superposition of sigmoidal functions, Mathematics of Control, Signals and Systems, 2, # 4 (1989), 303-314. 3. DEVORE, R., HOWARD , R. AND MICCHELLI, C.A., Optimal nonlinear approximation, Manuscripta Mathematica, 63 (1989), 469-478. 4. HECHT-NILESEN, R., Thoery of the backpropogation neural network, IEEE International Conference on Neural Networks, 1 (1988), 593-605. 5. HORNIK, K., STINCHCOMBE, M. AND WHITE, H ., Multilayer feedforward networks are universal approximators, Neural Networks, 2 (1989),359-366. 6. MHASKAR, H. N., Approximation properties of a multilayered feedforward artificial neural network, Advances in Computational Mathematics 1 (1993), 61-80. 7. MHASKAR, H. N., Neural networks for localized approximation of real functions, in "Neural Networks for Signal Processing, III", (Kamm, Huhn, Yoon, Chellappa and Kung Eds.), IEEE New York, 1993, pp. 190-196. 8. MHASKAR, H. N., Approximation of real functions using neural networks, in Proc. of Int. Conf. on Advances in Comput. Math., New Delhi, India, 1993, World Sci. Publ., H. P. Dikshit, C. A. Micchelli eds., 1994. 9. MHASKAR, H. N., Noniterative training algorithms for neural networks, Manuscript, 1993. 10. MHASKAR, H. N. AND MICCHELLI, C. A. , Approximation by superposition of a sigmoidal function and radial basis functions, Advances in Applied Mathematics, 13 (1992),350-373 . 11. MHASKAR, H. N. AND MICCHELLI, C. A., Degree of approximation by superpositions of a fixed function, in preparation. 12. MHASKAR, H. N. AND MICCHELLI, C. A., Dimension independent bounds on the degree of approximation by neural networks, Manuscript, 1993. 13. POGGIO, T. AND GIROSI, F., Regularization algorithms for learning that are equivalent to multilayer networks, Science, 247 (1990), 978-982. 

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Efficient Computation of Complex Distance Metrics Using Hierarchical Filtering Patrice Y. Simard AT&T Bell Laboratories Holmdel, NJ 07733 Abstract By their very nature, memory based algorithms such as KNN or Parzen windows require a computationally expensive search of a large database of prototypes. In this paper we optimize the searching process for tangent distance (Simard, LeCun and Denker, 1993) to improve speed performance. The closest prototypes are found by recursively searching included subset.s of the database using distances of increasing complexit.y. This is done by using a hierarchy of tangent distances (increasing the Humber of tangent. vectors from o to its maximum) and multiresolution (using wavelets). At each stage, a confidence level of the classification is computed. If the confidence is high enough, the c.omputation of more complex distances is avoided. The resulting algorithm applied to character recognition is close to t.hree orders of magnitude faster than computing the full tangent dist.ance on every prot.ot.ypes . 1 INTRODUCTION Memory based algorithms such as KNN or Parzen windows have been extensively used in pattern recognition. (See (Dasal'athy, 1991) for a survey.) Unfortunately, these algorithms often rely 011 simple distances (such a<; Euclidean distance, Hamming distance, etc.). As a result, t.hey suffer from high sensitivity to simple transformations of the input patterns that should leave the classification unchanged (e.g. translation or scaling for 2D images). To make the problem worse, these algorithms 168 Efficient Computation of Complex Distance Metrics Using Hierarchical Filtering are further limited by extensive computational requirements due to the large number of distance computations. (If no optimization technique is used, the computational cost is given in equation 1.) computational cost ~ number of prototypes x dist.ance complexity (1) Recently, the problem of transformation sensitivity has been addressed by the introduction of a locally transformation-invariant metric, the tangent distance (Simard, LeCun and Denker, 1993). The basic idea is that instead of measuring the distance d(A, B) between two patterns A and B, their respective sets of transformations TA and TB are approximated to the first order, and the distance between these two approximated sets is computed. Unfortunately, the tangent distance becomes computationally more expensive as more transformations are taken into consideration, which results in even stronger speed requirements. The good news is that memory based algorithms are well suited for optimization using hierarchies of prototypes, and that this is even more true when the distance complexity is high. In this paper, we applied these ideas to tangent distance in two ways: 1) Finding the closest prototype can be done by recursively searching included subsets of the database using distances of increasing complexity. This is done by using a hierarchy of tangent distances (increasing the number of tangent vectors from 0 to its maximum) and l11ultiresolution (using wavelets). 2) A confidence level can be computed fm each distance. If the confidence in the classification is above a threshold early on, there is no need to compute the more expensive distances. The two methods are described in the next section. Their application on a real world problem will be shown in the result section. 2 FILTERING USING A HIERARCHY OF DISTANCES Our goal is to compute the distance from one unknown pattern to every prototype in a large database in order to determine which one is the closest. It is fairly obvious that some patterns are so different from each other that a very crude approximation of our distance can tell us so. There is a wide range of variation in computation time (and performance) depending on the choice of the distance. For instance, computing the Euclidean distance on n-pixel images is a factor 11/ k of the computation of computing it on k-pixels images. ? Similarly, at a given resolution, computing the tangent distance with 111 tangent vectors is (m + 1)2 times as expensive as computing the Euclidean distance (m = tangent vectors). This observations provided us wit.h a hierarchy of about a dozen different distances ranging in computation time from 4 multiply/adds (Euclidean distance on a 2 x 2 averaged image) to 20,000 multiply /adds (tangent distance, 7 tangent vectors, 16 x 16 pixel images). The resulting filtering algorithm is very straightforward and is exemplified in Figure 1. The general idea is to store the database of prototypes several times at different resolutions and with different tangent. vectors. Each of these resolutions and groups of tangent vectors defines a distance di . These distances are ordered in increasing 169 170 Simard Unknown Pattern ( Prototypes ~ 10~OOO Euc. Dist 2x2 Cost: 4 ~ Confidence Euc. Dist ~ {soc 4x4 Cost: 16 ~ Confidence ~ soo Tang.Dist Category 14 vectors t----t~ 16x16 Confidence Figure 1: Pattern recognition using a hierarchy of distance. The filter proceed from left (starting with the whole database) to right (where only a few prototypes remain). At each stage distances between prototypes and the unknown pattern are computed, sorted and the best candidate prototypes are selected for the next stage. As the complexity of the distance increases, the number of prototypes decreases, making computation feasible. At each stage a classification is attempted and a confidence score is computed. If the confidence score is high enough, the remaining stages are skipped . accuracy and complexity. The first distance dl is computed on all (1\0) prototypes of the database. The closest J\ 1 pat.terns are then selected and identified to the next stage. This process is repeated for each of the distances; i.e. at each stage i, the distance d i is computed on each J\i-l patterns selected by the previous stage. Of course, the idea is that as the complexity of the distance increases, the number of patterns on which this distance must be computed decreases. At the last stage, the most complex and accurate distance is computed on all remaining patterns to determine the classificat.ion. The only difficult part is to det.ermine the minimum I<i patterns selected at each stage for which the filtering does not decrease t.he overall performance. Note that if the last distance used is the most accurat.e distance, setting all J\j to the number of patterns in the database will give optimal performance (at the most expensive cost). Increasing I<i always improves the performance in the sense that it allows to find patterns that are closer for the next distance measure d j + 1 . The simplest way to determine I<i is by selecting a validation set and plotting t.he performance on this validation set as a function of !\j. The opt.imal !\?i is then determined graphically. An automatic way of computing each 1\; is currently being developed. This method is very useful when the performance is not degraded by choosing small J{j. In this case, the dist.ance evaluation is done using distance metrics which are relatively inexpensive to compute. The computation cost becomes: Efficient Computation of Complex Distance Metrics Using Hierarchical Filtering computational cost ~ L number of prototypes at stage i X distance complexity at stage i Curves showing the performance as a function of the value of the result section. 3 !{i (2) will be shown in PRUNING THE SEARCH USING CONFIDENCE SCORES If a confidence score is computed at each stage of the distance evaluation, it is possible for certain patterns to avoid completely computing the most expensive distances. In the extreme case, if the Euclidean distance between two patterns is 0, there is really no need to compute the tangent distance. A simple (and crude) way to compute a confidence score at a given stage i, is to find the closest prototype (for distance di ) in each of the possible classes. The distance difference between the closest class and the next closest class gives an approximation of a confidence of this classification. A simple algorithm is then to compare at stage i the confidence score Cip of the current unknown patt.ern p to a threshold ()j, and to stop the classification process for this pattern as soon as Cip > ()j. The classification will then be determined by the closest prototype at this stage. The computation time will therefore be different depending on the pattern to be classified. Easy patterns will be recognized very quickly while difficult. patterns will need to be compared to some of the prototypes using the most complex distance . The total computation cost is therefore: computational cost ~ L number of prototypes at stage i X distance complexity at. stage i X probabili ty to reach stage i (3 ) Note that if all ()j are high, the performance is maximized but so is the cost . We therefore wish to find the smallest value of Oi which does not degrade the performance (increasing (Jj a.lways improves the performance). As in the previous section, the simplest way to determine the optimal ()j is graphically with a validation set.. Example of curves representing the perfornlance as a function of ()j will be given in the result section . 4 4.1 CHOSING A GOOD HIERARCHY, OPTIMIZATION k-d tree Several hierarchies of distance are possible for optimizing the search process. An incremental nearest neighbor search algorithm based on k-d tree (Broder, 1990) was implemented . The k-d tree structure was interesting because it can potentially be used with tangent distance. Indeed, since the separating hyperplanes have n-1 dimension, they can be made parallel to many tangent vectors at the same time. As much as 36 images of 256 pixels ,,,ith each 7 t.angent. vectors can be separat.ed into two group of 18 images by Olle hyperplane which is parallel to all tangent 171 172 Simard vectors. The searching algorithm is taking some knowledge of the transformation invariance into account when it computes on which side of each hyperplane the unknown pattern is. Of course, when a leaf is reached, the full tangent. distance must be computed. The problem with the k-d tree algorithm however is that in high dimensional space, the distance from a point to a hyperplane is almost always smallel' than the distance between any pair of points. As a result, the unknown pattern must be compared to many prototypes to have a reasonable accuracy. The speed up factor was comparable to our multiresolution approach in the case of Euclidean distance (about 10), but we have not been able to obtain both good performance and high speedup with the k-d tree algorithm applied to tangent distance. This algorithm was not used in our final experiments. 4.2 Wavelets One of the main advantages of the multiresolution approach is that it is easily implemented with wavelet transforms (i\'1allat, 1989), and that in the wavelet space, the tangent distance is conserved (with orthonormal wavelet bases). Furthermore, the multiresolution decomposition is completely orthogonal to the tangent distance decomposition. In our experiment.s, the Haar transform was used. 4.3 Hierarchy of tangent distance Many increasingly accurate approximations can be made for the tangent distance at a given resolution. For instance, the tangent distance can be computed by an iterative process of alternative projections onto the tangent hyperplanes. A hierarchy of distances results, derived from the number of projections performed. This hierarchy is not very good because the initial projection is already fairly expensive. It is more desirable to have a better efficiency in the first stages since only few patterns will be left for the latter stages. Our most successful hierarchy consisted in adding tangent vectors one by one, on both sides. Even though this implies solving a new linear system at each stage, the computational cost is mainly dominated by computing dot products between tangent vectors. These dot-products are then reused in the subsequent stages to create larger linear systems (invol ving more tangent vectors). This hierarchy has the advantage that the first stage is only twice as expensive, yet much more accurate, than the Euclidean distance . Each subsequent stage brings a lot of accuracy at a reasonable cost. (The cost inCl'eases quicker toward the lat.er stages since solving the linear system grows with the cube of the number of tangent vector .) In addition, the last stage is exactly the full tangent distance. As we will see in section 5 the cost in the final stages is negligible. Obviously, the tangent vectors can be added in different order. \Ve did not try to find the optimal order. For character recognition application adding translations first, followed by hyperbolic deformations, the scalings, the thickness deformations and the rotations yielded good performance . Efficient Computation of Complex Distance Metrics Using Hierarchical Filtering z # of T.V. 0 1 2 3 4 5 6 7 0 0 0 1 2 4 6 8 10 12 14 9 10 8 Reso 4 16 64 64 256 256 256 256 256 256 256 # of prot.o (Ki) # of prod 9709 3500 500 125 50 45 25 15 10 5 5 1 1 1 2 5 7 9 11 13 15 17 Probab 1.00 1.00 1.00 0.90 0.60 0.40 0.20 0.10 0.10 0.05 0.0.5 # of mul/add 40,000 56,000 32,000 14,000 40,000 32,000 11,000 4,000 3,000 1,000 1,000 Table 1: Summary computation for the classification of 1 pattern: The first column is the distance index, the second column indicates the number of tangent vector (0 for the Euclidean distance), and the third column indicates the resolution in pixels, the fourth is J{j or the number of prototypes on which the distance di must be computed, the fifth column indicat.es the number of additional dot products which must be computed to evaluate distance di, the sixth column indicates the probability to not skip that stage after the confidence score has been used, and the last column indicates the total average number of multiply-adds which must be performed (product of column 3 to 6) at each stage. 4.4 Selecting the k closests out of N prototypes in O(N) In the multiresolution filter, at the early stages we must select the k closest prototypes from a large number of protot.ypes. This is problematic because the prototypes cannot be sorted since O( N ZagN) is expensive compared to computing N distances at very low resolution (like 4 pixels). A simple solution consist.s in using a variation of "quicksort" or "finding the k-t.h element" (Aho, Hopcroft and Ullman, 1983), which can select the k closests out of N prototypes in O(N). The generic idea is to compute the mean of the distances (an approximation is actually sufficient) and then to split the distances into two halves (of different sizes) according to whether they are smaller or larger than the mean distance. If they are more dist.ances smaller than the mean than k, the process is reiterat.ed on the upper half, ot.herwise it is reiterated on the lowel' half. The process is recursively executed until there is only one distance in each half. (k is then reached and all the k prototypes in the lower halves are closer to the unknown pattei'll than all the N - ~~ prototypes in the upper halves.) Note that. the elements are not sorted and t.hat only t.he expected t.ime is O(N), but this is sufficient for our problem. 5 RESULTS A simple task of pattern classification was used to test the filtering. The prototype set and the test set consisted l'especti vely of 9709 and 2007 labeled images (16 by 16 pixels) of handwritten digit.s. The prot.otypes were also averaged t.o lower 173 174 Simard 5 8 71 t Error in % Error in % 4 / Resolution 16 pIXels 6 ResolutIOn 64 pixels 5 Reso lutlO n 64 pixels 1 tangent veC10r Resol ution 16 pixels 3 4 3 2~~~~~~ o K (in 1000) 10 20 30 40 __~__~~~~ 50 60 70 80 90 100 % of pat. kept. Figure 2: Left: Raw error performance as a function of Kl and 1\2. The final chosen values were J{ 1 = 3500 and [\'2 = 500. Right: Raw error as a function of the percentage of pattern which have not exceeded the confidence threshold Oi. A 100% means all the pattern were passed to the next stage. resolutions (2 by 2, 4 by 4 and 8 by 8) and copied to separate databases . The 1 by 1 resolution was not useful for anything. Therefore the fastest distance was the Euclidean distance on 2 by 2 images, while the slowest distance was the full tangent distance with 7 tangent vectors for both the prototype and the unknown pattern (Simard, LeCun and Denker, 1993). Table 1 summarizes the results. Several observations can be made. First, simple distance metrics are very useful to eliminate large proportions of Pl"Ototypes at no cost in performances. Indeed the Euclidean distance computed on 2 by 2 images can remove 2 third of the prototypes. Figure 2, left, shows the performance as a function of J{l and 1\2 (2 .5 % raw error was considered optimal performance). It can be noticed that for J{j above a certain value, the performance is optimal alld c.onstant. The most complex distances (6 and 7 tangent vectors on each side) need only be computed for 5% of the prototypes. The second observation is that the use of a confidence score can greatly reduce the number of distance evaluations in later stages. For instance the dominant phases of the computation would be with 2, 4 and 6 tangent vectors at resolution 256 if there were not reduced to 60%, 40% and 20% respectively using the confidence sc.ores. Figure 2, right, shows the raw error performance as a function of the percentage of rejection (confidence lower than OJ) at stage i. It can be noticed that above a certain threshold, the performance are optimal and constant . Less than 10% of the unknown patterns need the most. complex distances (5, 6 and 7 tangent vectors on each side), to be comput.ed. Efficient Computation of Complex Distance Metrics Using Hierarchical Filtering 6 DISCUSSION Even though our method is by no way optimal (the order of the tangent vector can be changed, intermediate resolution can be used, etc ... ), the overall speed up we achieved was about 3 orders of magnitude (compared with computing the full tangent distance on all the patterns). There was no significant decrease in performances. This classification speed is comparable with neural network method, but the performance are better with tangent distance (2.5% versus 3%). Furthermore the above methods require no learning period which makes them very attractive for application were the distribution of the patterns to be classified is changing rapidly. The hierarchical filtering can also be combined with learning the prototypes using algorithms such as learning vector quantization (LVQ). References Aho, A. V., Hopcroft, J. E., and Ullman, J. D. (1983). Data Structure and Algorithms. Addison- \V'esley. Broder, A. J. (1990). Strategies for Efficient Incremental Nearest Neighbor Search. Pattern Recognition, 23: 171-178. Dasarathy, B. V. (1991). Nearest Neighbor (NN) Norms: NN Pattern classification Techniques. IEEE Computer Society Press, Los Alamitos, California. Mallat, S. G. (1989). A Theory for I\,I ultiresolution Signal Decomposition: The Wavelet Representation. IEEE Transactions 011 Pattern Analysis and Machine Intelligence, 11, No. 7:674-693. Simard, P. Y., LeCun, Y., and Denker, J. (1993). Efficient Pattern Recognition Using a New Transformation Distance. In Neural Information Processing Systems, volume 4, pages 50-58, Sa.n Mateo, CA. 175 

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Illumination-Invariant Face Recognition with a Contrast Sensitive Silicon Retina Joachim M. Buhmann Rheinische Friedrich-Wilhelms-U niversitiit Institut fUr Informatik II, RomerstraBe 164 0-53117 Bonn, Germany Martin Lades Ruhr-Universitiit Bochum Institut fiir Neuroinformatik 0-44780 Bochum, Germany Frank Eeckman Lawrence Livermore National Laboratory ISCR, P.D.Box 808, L-426 Livermore, CA 94551 Abstract Changes in lighting conditions strongly effect the performance and reliability of computer vision systems. We report face recognition results under drastically changing lighting conditions for a computer vision system which concurrently uses a contrast sensitive silicon retina and a conventional, gain controlled CCO camera. For both input devices the face recognition system employs an elastic matching algorithm with wavelet based features to classify unknown faces. To assess the effect of analog on -chip preprocessing by the silicon retina the CCO images have been "digitally preprocessed" with a bandpass filter to adjust the power spectrum. The silicon retina with its ability to adjust sensitivity increases the recognition rate up to 50 percent. These comparative experiments demonstrate that preprocessing with an analog VLSI silicon retina generates image data enriched with object-constant features. 1 Introdnction Neural computation as an information processing paradigm promises to enhance artificial pattern recognition systems with the learning capabilities of the cerebral cortex and with the 769 770 Buhmann, Lades, and Eeckman adaptivity of biological sensors. Rebuilding sensory organs in silicon seems to be particularly promising since their neurophysiology and neuroanatomy, including the connections to cortex, are known in great detail. This knowledge might serve as a blueprint for the design of artificial sensors which mimic biological perception. Analog VLSI retinas and cochleas, as designed by Carver Mead (Mead, 1989; Mahowald, Mead, 1991) and his collaborators in a seminal research program, will ultimately be integrated in vision and communication systems for autonomous robots and other intelligent information processing systems. The study reported here explores the influence of analog retinal preprocessing on the recognition performance of a face recognition system. Face recognition is a challenging classification task where object inherent distortions, like facial expressions and perspective changes, have to be separated from other image variations like changing lighting conditions. Preprocessing with a silicon retina is expected to yield an increased recognition rate since the first layers of the retina adjust their local contrast sensitivity and thereby achieve invariance to variations in lighting conditions. Our face recognizer is equipped with a silicon retina as an adaptive camera. For comparison purposes all images are registered simultaneously by a conventional CCD camera with automatic gain control. Galleries with images of 109 different test persons each are taken under three different lighting conditions and two different viewing directions (see Fig. 1). These different galleries provide separate statistics to measure the sensitivity of the system to variations in light levels or contrast and image changes due to perspective distortions. Naturally, the performance of an object recognition system depends critically on the classification strategy pursued to identify unknown objects in an image with the models stored in a database. The matching algorithm selected to measure the performance enhancing effect of retinal preprocessing deforms prototype faces in an elastic fashion (Buhmann et aI., 1989; Buhmann et al., 1990; Lades et al., 1993). Elastic matching has been shown to perform well on the face classification task recognizing up to 80 different faces reliably (Lades et al., 1993) and in a translation, size and rotation invariant fashion (Buhmann et aI., 1990). The face recognition algorithm was initially suggested as a simplified version of the Dynamic Link A rchitecture (von der Malsburg, 1981), an innovative neural classification strategy with fast changes in the neural connectivity during recognition stage. Our recognition results and conclusions are expected to be qualitatively typical for a whole range of face/object recognition systems (Turk, Pentland, 1991; Yuille, 1991; Brunelli, Poggio, 1993), since any image preprocessing with emphasis on object constant features facilitates the search for the correct prototype. 2 The Silicon Retina The silicon retina used in the recognition experiments models the interactions between receptors and horizontal cells taking place in the outer plexiform layer of the vertebrate retina. All cells and their interconnections are explicitly represented in the chip so that the following description simultaneously refers to both biological wetware and silicon hardware. Receptors and horizontal cells are electrically coupled to their neighbors. The weak electrical coupling between the receptors smoothes the image and reduces the influence of voltage offsets between adjacent receptors. The horizontal cells have a strong lateral electrical coupling and compute a local background average. There are reciprocal excitatory-inhibitory synapses between the receptors and the horizontal cells. The horizontal cells use shunting inhibition to adjust the membrane conductance of the receptors and Illumination-Invariant Face Recognition with a Contrast Sensitive Silicon Retina thereby adjust their sensitivity locally. This feedback interaction produces an antagonistic center/surround organization of receptive fields at the output The center is represented by the weakly coupled excitatory receptors and the surround by the more strongly coupled inhibitory horizontal cells. The center/surround organization removes the average intensity and expands the dynamic range without response compression. Furthennore, it enhances edges. In contrast to this architecture, a conventional CCD camera can be viewed as a very primitive retina with only one layer of non-interacting detectors. There is no DC background removal, causing potential over- and underexposure in parts of the image which reduces the useful dynamic range. A mechanical iris has to be provided to adjust the mean luminance level to the appropriate setting. Since cameras are designed for faithful image registration rather than vision, on-chip pixel processing, if provided at all, is used to improve the camera resolution and signal-to-noise ratio. Three adjustable parameters allow us to fine tune the retina chip for an object recognition experiment: (i) the diffusivity of the cones (ii) the diffusivity ofthe horizontal cells (iii) the leak in the horizontal cell membrane. Changes in the diffusivities affect the shape of the receptive fields, e.g., a large diffusivity between cones smoothes out edges and produces a blurred image. The other extreme of large diffusivity between horizontal cells pronounces edges and enhances the contrast gain . The retina chip has a resolution of 90 x 92 pixels, it was designed by (Boahen, Andreou, 1992) and fabricated in 2flm n-well technology by MOSIS. 3 Elastic Matching Algorithm for Face Recognition Elastic matching is a pattern classification strategy which explicitly accounts for local distortions. A prototype template is elastically deformed to measure local deviations from a new, unknown pattern. The amount of deformation and the similarity oflocal image features provide us with a decision criterion for pattern classification. The rubbersheet-like behavior of the prototype transformation makes elastic matching a particularly attractive method for face recognition where ubiquitous local distortions are caused for example by perspective changes and different facial expressions. Originally, the technique was developed for handwritten character recognition (Burr, 1981). The version of elastic matching employed for our face recognition experiments is based on attributed graph matching. A detailed description with a plausible interpretation in neural networks terms is published in (Lades et al., (993). Each prototype face is encoded as a planar graph with feature vectors attached to the vertices of the graph and metric information attached to the edges. The feature vectors extract local image information at pixel Xi in a multiscale fashion, i.e., they are functions of wavelet coefficients. Each feature vector establishes a correspondence between a vertex i of a prototype graph and a pixel Xi in the image. The components of a feature vector are defined as the magnitudes of the convolution of an image with a set of two-dimensional, DC free Gaussian kernels centered at pixel Xi. The kernels with the form 1/!'k (X) fl exp = (72 (flx2 - ) - exp (-(72/2) 1 - 2(72 ) [exp (ikX (I) are parameterized by the wave vector k defining their orientations and their sizes. To construct a self-similar set of filter functions we select eight different orientations and five 771 772 Buhmann, Lades, and Eeckman different scales according to k(v, tt) = ~ Tv/2 (cos( itt), sin( itt)) (2) with v E {O, ... ,4};tt E {O, ... , 7}. The multi-resolution data format represents local distortions in a robust way, i.e., only feature vectors in the vicinity x of an image distortion are altered by the changes. The edge labels encode metric information, in particular we choose the difference vectors AXij == Xi - Xj as edge labels. To generate a new prototype graph for the database, the center of a new face is determined by matching a generic face template to it. A 7 x 10 rectangular grid with 10 pixel spacing between vertices and edges between adjacent vertices is then centered at that point. The saliency of image points is taken into account by deforming that generic grid so that each vertex is moved to the nearest pixel with a local maximum in feature vector length. The classification of an unknown face as one of the models in the database or its rejection as an unclassified object is achieved by computing matching costs and distortion costs. The matching costs are designed to maximize the similarity between feature vector J;M of vertex i in the model graph (M) and feature vector Jl (Xi) associated with pixel Xi in the new image (I). The cosine of the angle between both feature vectors -[...., S(JI(x) jM) = '" -M J (Xi) . Ji (3) M Ilf (Xi)IIIIJ; II 1 is suited as a similarity function for elastic matching since global contrast changes in images only scale feature vectors but do not rotate them. Besides maximizing the similarity between feature vectors the elastic matching algorithm penalizes large distortions. The distortion cost term is weighted by a factor ,\ which can be interpreted as a prior for expected distortions. The combined matching cost function which is used in the face recognition system compromises between feature similarity and distortion, i.e, it minimizes the cost function (4) for the model M in the face database with respectto the correspondence points {xf}. (i, j) in Eq. (4) denotes that index j runs over the neighborhood of vertex i and index i runs in the new image over all vertices. By minimizing Eq. (4) the algorithm assigns pixel I to vertex i in the prototype graph M. Numerous classification experiments revealed that a steepest descent algorithm is sufficient to minimize cost function (4) although it is nonconvex and local minima may cause non-optimal correspondences with reduced recognition rates. x; During a recognition experiment all prototype graphs in the database are matched to the new image. A new face is classified as prototype A if H A is minimal and if the significance criterion (5) is fulfilled. The average costs (Ji) and their standard deviation LH are calculated excluding match A. This heuristic is based on the assumption that a new face image strongly Illumination-Invariant Face Recognition with a Contrast Sensitive Silicon Retina l> gr.tl rr.m.1 l>gr.rr~ l>~"~ l>~.tr. .llo'" ~1Ib.ka-:2to,.. > Workstation Datacube ~~ . .. Figure I: Laboratory setup of the face recognition experiments. correlates with the correct prototype but the matching costs to all the other prototype faces is approximately Gaussian distributed with mean (1l) and standard deviation I.H. The threshold parameter 0 is used to limit the rate of false positive matches, i.e., to exclude significant matches to wrong prototypes. 4 Face Recognition Results To measure the recognition rate of the face recognition system using a silicon retina or a CCD camera as input devices, pictures of 109 different persons are taken under 3 different lighting conditions and 2 different viewing directions. This setup allows us to quantify the influence of changes in lighting conditions on the recognition performance separate from the influence of perspective distortions. Figure 2 shows face images of one person taken under two different lighting setups. The images in Figs. 2a,c with both lights on are used as the prototype images for the respective input devices. To test the influence of changing lighting conditions the left light is switched off. The faces are now strongly illuminated from the right side. The CCD camera images (Figs. 2a,b) document the drastic changes of the light settings. The corresponding responses of the silicon retina shown in Figs. 2c,d clearly demonstrate that the local adaptivity of the silicon retina enables the recognition system to extract object structure from the bright and the dark side of the face. For control purposes all recognition experiments have been repeated with filtered CCD camera images. The filter was adjusted such that the power spectra of the retina chip images and the filtered CCD images are identical. The images (e,f) are filtered versions of the images (a,b). It is evident that information in the dark part of image (b) has been erased due to saturation effects of the CCD camera and cannot be recovered by any local filtering procedure. We first measure the performance of the silicon retina under uniform lighting conditions, n3 b ~ .... .. ~ ~ It ,. ...'. - '\. .. ... C .... .- ?... ~ ?. . ~ .~ ..... ? 1? Figure 2: (a) Conventional CCD camera images (a,b) and silicon retina image (c,d) under different lighting conditions. The images (e,O are filtered CCD camera images with a power spectrum adjusted to the images in (c,d). The images (a,c) are used to generate the Illumination-Invariant Face Recognition with a Contrast Sensitive Silicon Retina Table 1: (a) Face recognition results in a well illuminated environment and (b) in an environment with drastic changes in lighting conditions. a b f. p. rate 100% 100/0 50/0 10/0 1000/0 10% 50/0 10/0 silicon retina 83.5 81.7 76.2 71.6 96.3 96.3 96.3 93.6 cony. CCD 86.2 83.5 82.6 79.8 80.7 76.2 72.5 64.2 filt. CCD 85.3 84.4 80.7 75.2 78.0 75.2 72.5 62.4 i.e., both lamps are on and the person looks 20-30 degrees to the right. The recognition system has to deal with perspective distortions only. A gallery of 109 faces is matched to a face database of the same 109 persons. Table la shows that the recognition rate reaches values between 80 and 90 percent if we accept the best match without checking its significance. Such a decision criterion is unpractical for many applications since it corresponds to a false positive rate (f. p. rate) of 100 percent. If we increase the threshold E> to limit false positive matches to less than 1 percent the face recognizer is able to identify three out of four unknown faces. Filtering the CCD imagery does not hurt the recognition performance as the third column in Table 1a demonstrates. All necessary information for recognition is preserved in the filtered CCD images. The situation changes dramatically when we switch off the lamp on the left side of the test person. We compare a test gallery of persons looking straight ahead, but illuminated only from the right side, to our model gallery. Table 1b summarizes the recognition results for different false positive rates. The advantage of using a silicon retina are 20 to 45 percent higher recognition rates than for a system with a CCD camera. For a false positive rate below one percent a silicon retina based recognition system identifies two third more persons than a conventional system. Filtering does not improve the recognition rate of a system that uses a CCD camera as can be seen in the third column. Our comparative face recognition experiment clearly demonstrates that a face recognizer with a retina chip is performing substantially better than conventional CCD camera based systems in environments with uncontrolled, substantially changing lighting conditions. Retina-like preprocessing yields increased recognition rates and increased significance levels. We expect even larger discrepancies in recognition rates if object without a bilateral symmetry have to be classified. In this sense the face recognition task does not optimally explore the potential of adaptive preprocessing by a silicon retina. Imagine an object recognition task where the most significant features for discrimination are hardly visible or highly ambiguous due to poor illumination. High error rates and very low significance levels are an inevitable consequence of such lighting conditions. The limited resolution and poor signal-to-noise ratio of silicon retina chips are expected to be improved by a new generation of chips fabricated in 0.7 /lm CMOS technology with a 775 776 Buhmann, Lades, and Eeckman potential resolution of256 x 256 pixels. Lighting conditions as simulated in ourrecognition experiment are ubiquitous in natural environments. Autonomous robots and vehicles or surveillance systems are expected to benefit from the silicon retina technology by gaining robustness and reliability. Silicon retinas and more elaborate analog VLSI chips for low level vision are expected to be an important component of an Adaptive Vision System. Acknowledgement: It is a pleasure to thank K. A. Boahen for providing us with the retina chips. We acknowledge stimulating discussions with C. von der Malsburg and C. Mead. This work was supported by the German Ministry of Science and Technology (lTR-8800-H 1) and by the Lawrence Livermore National Laboratory (W-7405-Eng-48). References Boahen, K., Andreou, A. 1992. A Contrast Sensitive Silicon Retina with Reciprocal Synapses. Pages 764-772 of: NIPS91 Proceedings. IEEE. Brunelli, R., Poggio, T. (1993). Face Recognition: Features versus Templates. IEEE Trans. on Pattern Analysis Machine Intelligence, 15, 1042-1052. Buhmann, J., Lange, J., von der Malsburg, C. 1989. Distortion Invariant Object Recognition by Matching Hierarchically Labeled Graphs. Pages I 155-159 of' Proc. llCNN, Washington. IEEE. Buhmann, J., Lades, M., von der Malsburg, C. 1990. Size and Distortion Invariant Object Recognition by Hierarchical Graph Matching. Pages II 411-416 of' Proc. llCNN, SanDiego. IEEE. Burr, D. J. (1981). Elastic Matching of Line Drawings. IEEE Trans. on Pat. An. Mach. Intel., 3, 708-713. Lades, M., Vorbriiggen, J.C., Buhmann, J., Lange, J., von der Malsburg, C., Wurtz, R.P., Konen, W. (1993). Distortion Invariant Object Recognition in the Dynamic Link Architecture. IEEE Transactions on Computers, 42, 300-311. Mahowald, M., Mead, C. (1991). The Silicon Retina. Scientific American, 264(5), 76. Mead, C. (1989). Analog VLSI and Neural Systems. New York: Addison Wesley. Turk, M., Pentland, A. (1991). Eigenfaces for Recognition. J. Cog. Sci., 3, 71-86. von der Malsburg, Christoph. 1981. The Correlation Theory of Brain Function. Internal Report. Max-Planck-Institut, Biophys. Chern., Gottingen, Germany. Yuille, A. (1991). Deformable Templates for Face Recognition. J. Cog. Sci., 3, 60-70. 

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The "Softmax" Nonlinearity: Derivation Using Statistical Mechanics and Useful Properties as a Multiterminal Analog Circuit Element I. M. Elfadel Research Laboratory of Electronics Massachusetts Institute of Technology Cambridge, MA 02139 J. L. Wyatt, Jr. Research Laboratory of Electronics Massachusetts Institute of Technology Cambridge, MA 02139 Abstract We use mean-field theory methods from Statistical Mechanics to derive the "softmax" nonlinearity from the discontinuous winnertake-all (WTA) mapping. We give two simple ways of implementing "soft max" as a multiterminal network element. One of these has a number of important network-theoretic properties. It is a reciprocal, passive, incrementally passive, nonlinear, resistive multiterminal element with a content function having the form of informationtheoretic entropy. These properties should enable one to use this element in nonlinear RC networks with such other reciprocal elements as resistive fuses and constraint boxes to implement very high speed analog optimization algorithms using a minimum of hardware. 1 Introduction In order to efficiently implement nonlinear optimization algorithms in analog VLSI hardware, maximum use should be made of the natural properties of the silicon medium. Reciprocal circuit elements facilitate such an implementation since they 882 The "Softmax" Nonlinearity can be combined with other reciprocal elements to form an analog network having Lyapunov-like functions: the network content or co-content. In this paper, we show a reciprocal implementation of the "softmax" nonlinearity that is usually used to enforce local competition between neurons [Peterson, 1989]. We show that the circuit is passive and incrementally passive, and we explicitly compute its content and co-content functions. This circuit adds a new element to the library of the analog circuit designer that can be combined with reciprocal constraint boxes [Harris, 1988] and nonlinear resistive fuses [Harris, 1989] to form fast, analog VLSI optimization networks. 2 Derivation of the Softmax Nonlinearity To a vector y E ~n of distinct real numbers, the discrete winner-take-all (WTA) mapping W assigns a vector of binary numbers by giving the value 1 to the component of y corresponding to maxl<i<n Yi and the value 0 to the remaining components. Formally, W is defined as - - W(y) = (Wl(y), ... Wn(y?T I where for every 1 ~ j ~ n, 1 if YJ' > Yi, V 1 ~ i ~ n Wj (y) = { 0 otherwise (1) Following [Geiger, 1991], we assign to the vector y the "energy" function n Ey(z) = - L ZkYk = _zT y, z E Vnl (2) k=l = where Vn is the set of vertices of the unit simplex Sn {x E ~n, Zi > 0, 1 < i < n and E~=l Zk = 1}. Every vertex in the simplex encodes one possible winner. It is then easy to show that W(y) is the solution to the linear programming problem n max LZkYk. ZE'V .. k=l Moreover, we can assign to the energy Ey(z) the Gibbs distribution e-Ey(Z)/T Py(z) = Py(Zl' ... , zn) = ZT where T is the temperature of the heat bath and ZT is a normalizing constant. Then one can show that the mean of Zj considered as a random variable is given by [Geiger, 1991] A _ Fj(y/T) ey;/T ey;/T = = -z = Ei=l n/T? T eY ' Zj The mapping F : ~n -+ ~n whose components are the Fj's, 1 ~ j < n, is the generalized sigmoid mapping [Peterson, 1989] or "soft max" . It plays, in WTA networks, a role similar to that of the sigmoidal function in Hopfield and backpropagation 883 884 Elfadel and Wyatt v Figure 1: A circuit implementation of softmax with 5 inputs and 5 outputs. This circuit is operated in subthreshold mode, takes the gates voltages as inputs and gives the drain currents as outputs. This circuit is not a reciprocal multiterminal element. networks [Hopfield, 1984, Rumelhart, 1986] and is usually used for enforcing competitive behavior among the neurons of a single cluster in networks of interacting clusters [Peterson, 1989, Waugh, 1993]. A For y E ~n, we denote by FT(y) = F(y IT). The softmax mapping satisfies the following properties: 1. The mapping FT converges pointwise to W over ~n as T -+ 0 and to the center of mass of Sn, *e = *(1,1, . . . , 1)T E ~n, as T -+ +00. 2. The Jacobian DF of the softmax mapping is a symmetric n x n matrix that satisfies DF(y) = diag (F,,(y? - F(y)F(y)T. (3) It is always singular with the vector e being the only eigenvector corresponding to the zero eigenvalue. Moreover, all its eigenvalues are upper-bounded by maxlS"Sn F,,(y) < 1. 3. The soft max mapping is a gradient map, i.e, there exists a "potential" function 'P : ~n -+ ~ such that F = V'P. Moreover 'P is convex. The symbol 'P was chosen to indicate that it is a potential function. It should be noted that if F is the gradient map of 'P then FT is the gradient map of T'PT where 'PT(Y) 'P(yIT). In a related paper [Elfadel, 1993], we have found that the convexity of'P is essential in the study of the global dynamics of analog WTA networks. Another instance where the convexity of 'P was found important is the one reported in [Kosowsky, 1991] where a mean-field algorithm was proposed to solve the linear assignment problem. = The "Softmax" Nonlinearity v, Vz v" Va Ta Figure 2: Modified circuit implementation of softmax. In this circuit all the transistors are diode-connected, and all the drain currents are well in saturation region. Note that for every transistor, both the voltage input and the current output are on the same wire - the drain. This circuit is a reciprocal multiterminal element. 3 Circuit Implementations and Properties Now we propose two simple CMOS circuit implementations of the generalized sigmoid mapping. See Figures 1 and 2. When the transistors are operated in the subthreshold region the drain currents i l , .. . ,in are the outputs of a softmax mapping whose inputs are the gate voltages Vl, ?.. , V n . The explicit v - i characteristics are given by (4) where K, is a process-dependent parameter and Vo is the thermal voltage ([Mead, 1989],p. 36). These circuits have the interesting properties of being unclocked and parallel. Moreover, the competition constraint is imposed naturally through the KCL equation and the control current source. From a complexity point of view, this circuit is most striking since it computes n exponentials, n ratios, and n - 1 sums in one time constant! A derivation similar to the above was independently carried out in [Waugh, 1993] for the circuit of Figure 1. Although the first circuit implements softmax, it has two shortcomings. The first is practical: the separation between inputs and outputs implies additional wiring. The second is theoretical: this circuit is not a reciprocal multiterminal element, and therefore it can't be combined with other reciprocal elements like resistive fuses or constraint boxes to design analog, reciprocal optimization networks. Therefore, we only consider the circuit of Figure 2 and let v and i be the ndimensional vectors representing the input voltages and the output currents, respectively. 1 The softmax mapping i = F(v) represents a voltage-controlled, nonlinear, lCompare with Lazarro et. al.'s WTA circuit [Lazzaro, 1989] whose inputs are currents and outputs are voltages. 885 886 Elfadel and Wyatt resistive multiterminal element. The main result of our paper is the following: 2 Theorem 1 The softmax multiterminal element F is reciprocal, passive, locally passive and has a co-content function given by 1 n ~(v) K, Ie Vo In exp(K,vm/Vo) L = (5) m=l and a content function given by ..w..*(O) _ IeVo ~ im 1nim I - - - L.J -. 'If K, m=l Ie (6) Ie Thus, with this reciprocal, locally passive implementation of the softmax mapping, we have added a new circuit element to the library of the circuit designer. Note that this circuit element implements in an analog way the constraint L:~=1 y" = 1 defining the unit simplex Sn. Therefore, it can be considered a nonlinear constraint box [Harris, 1988] that can be used in reciprocal networks to implement analog optimization algorithms. The expression of ~* is a strong reminder of the information-theoretic definition of entropy. We suggest the name "entropic resistor" for the circuit of Figure 2. 4 Conclusions In this paper, we have discussed another instance of convergence between the statistical physics paradigm of Gibbs distributions and analog circuit implementation in the context of the winner-take-all function. The problem of using the simple, reciprocal circuit implementation of softmax to design analog networks for finding near optimal solutions of the linear assignment problem [Kosowsky, 1991] or the quadratic assignment problem [Simic, 1991] is still open and should prove a challenging task for analog circuit designers. Acknowledgements I. M. Elfadel would like to thank Alan Yuille for many helpful discussions and Fred Waugh for helpful discussions and for communicating the preprint of [Waugh, 1993]. This work was supported by the National Science Foundation under Grant No. MIP-91-17724. References [Peterson, 1989] C. Peterson and B. Soderberg. A new method for mapping optimization problems onto neural networks. International Journal of Neural Systems, 1(1):3 - 22, 1989. 2The concepts of reciprocity, passivity, content, and co-content are fundamental to nonlinear circuit theory. They are carefully developed in [Wyatt, 1992]. The "Softmax" Nonlinearity J. G. Harris. Solving early vision problems with VLSI constraint networks. In Neural Architectures for Computer Vision Workshop, AAAI-88, Minneapolis, MN, 1988. [Harris,1989] J. G. Harris, C. Koch, J. Luo, and J. Wyatt. Resistive fuses: Analog hardware for detecting discontinuities in early vision. In C. Mead and M. Ismail, editors, Analog VLSIImplemenation of Neural Systems. Kluwer Academic Publishers, 1989. [Geiger, 1991] D. Geiger and A. Yuille. A common framework for image segmentation. Int. J. Computer Vision, 6:227 - 253, 1991. [Hopfield, 1984] J. J. Hopfield. Neurons with graded responses have collective computational properties like those of two-state neurons. Proc. Nat'l Acad. Sci., USA, 81:3088-3092, 1984. [Rumelhart, 1986] D. E. Rumelhart et. al. Parallel Distributed Processing, volume 1. MIT Press, 1986. [Waugh, 1993] F. R. Waugh and R. M. Westervelt. Analog neural networks with local competition. I. dynamics and stability. Physical Review E, 1993. in press. [Elfadel, 1993] I. M. Elfadel. Global dynamics of winner-take-all networks. In [Harris, 1988] SPIE Proceedings, Stochastic and Neural Methods in Image Processing, volume 2032, pages 127 - 137, San Diego, CA, 1993. [Kosowsky, 1991] J. J. Kosowsky and A. L. Yuille. The invisible hand algorithm: Solving the assignment problem with statistical physics. TR # 91-1, Harvard Robotics Laboratory, 1991. [Mead, 1989] Carver Mead. Analog VLSI and Neural Systems. Addison-Wesley, 1989. [Lazzaro,1989] J. Lazarro, S. Ryckebush, M. Mahowald, and C. Mead. Winnertake-all circuits of O(n) complexity. In D. S. Touretsky, editor, Advances in Neural Information Processing Systems I, pages 703 - 711. Morgan Kaufman, 1989. [Wyatt, 1992] J. L. Wyatt. Lectures on Nonlinear Circuit Theory. MIT VLSI memo # 92-685,1992. [Simic, 1991] P. D. Simic. Constrained nets for graph matching and other quadratic assignment problems. Neural Computation, 3:169 - 281, 1991. 887 

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Learning Stochastic Perceptrons Under k-Blocking Distributions Mario Marchand Ottawa-Carleton Institute for Physics University of Ottawa Ottawa, Ont., Canada KIN 6N5 mario@physics.uottawa.ca Saeed Hadjifaradji Ottawa-Carleton Institute for Physics University of Ottawa Ottawa, Ont., Canada KIN 6N5 saeed@physics.uottawa.ca Abstract We present a statistical method that PAC learns the class of stochastic perceptrons with arbitrary monotonic activation function and weights Wi E {-I, 0, +I} when the probability distribution that generates the input examples is member of a family that we call k-blocking distributions. Such distributions represent an important step beyond the case where each input variable is statistically independent since the 2k-blocking family contains all the Markov distributions of order k. By stochastic percept ron we mean a perceptron which, upon presentation of input vector x, outputs 1 with probability fCLJi WiXi - B). Because the same algorithm works for any monotonic (nondecreasing or nonincreasing) activation function f on Boolean domain, it handles the well studied cases of sigmolds and the "usual" radial basis functions. 1 INTRODUCTION Within recent years, the field of computational learning theory has emerged to provide a rigorous framework for the design and analysis of learning algorithms. A central notion in this framework, known as the "Probably Approximatively Correct" (PAC) learning criterion (Valiant, 1984), has recently been extended (Hassler, 1992) to analyze the learn ability of probabilistic concepts (Kearns and Schapire, 1994; Schapire, 1992). Such concepts, which are stochastic rules that give the probability that input example x is classified as being positive, are natural probabilistic 280 Mario Marchand, Saeed Hadjifaradji extensions of the deterministic concepts originally studied by Valiant (1984). Motivated by the stochastic nature of many "real-world" learning problems and by the indisputable fact that biological neurons are probabilistic devices, some preliminary studies about the PAC learnability of simple probabilistic neural concepts have been reported recently (Golea and Marchand, 1993; Golea and Marchand, 1994). However, the probabilistic behaviors considered in these studies are quite specific and clearly need to be extended. Indeed, only classification noise superimposed on a deterministic signum function was considered in Golea and Marchand (1993). The probabilistic network, analyzed in Golea and Marchand (1994), consists of a linear superposition of signum functions and is thus solvable as a (simple) case of linear regression. What is clearly needed is the extension to the non-linear cases of sigmolds and radial basis functions. Another criticism about Golea and Marchand (1993 , 1994) is the fact that their learn ability results was established only for distributions where each input variable is statistically independent from all the others (sometimes called product distributions). In fact, very few positive learning results for non-trivial p-concepts classes are known to hold for larger classes of distributions. Therefore, in an effort to find algorithms that will work in practice, we introduce in this paper a new family of distributions that we call k-blocking. As we will argue, this family has the dual advantage of avoiding malicious and unnatural distributions that are prone to render simple concept classes unlearnable (Lin and Vitter, 1991) and of being likely to contain several distributions found in practice. Our main contribution is to present a simple statistical method that PAC learns (in polynomial time) the class of stochastic perceptrons with monotonic (but otherwise arbitrary) activation functions and weights Wi E { -1,0, +1} when the input examples are generated according to any distribution member of the k-blocking family. Due to space constraints, only a sketch of the proofs is presented here. 2 DEFINITIONS The instance (input) space, In, is the Boolean domain {-1, +1}n. The set of all input variables is denoted by X. Each input example x is generated according to some unknown distribution D on In. We will often use PD(X), or simply p(x), to denote the probability of observing the vector value x under distribution D. If U and V are two disjoint subsets of X, Xu and Xv will denote the restriction (or projection) of x over the variables of U and V respectively and p D (xu Ixv) will denote the probability, under distribution D, of observing the vector value Xu (for the variables in U) given that the variables in V are set to the vector value xv. Following Kearns and Schapire (1994), a probabilistic concept (p-concept) is a map c : In ~ [0, 1] for which c(x) represents the probability that example X is classified as positive. More precisely, upon presentation of input x, an output of a = 1 is generated (by an unknown target p-concept) with probability c(x) and an output of a = 0 is generated with probability 1 - c(x). A stochastic perceptron is a p-concept parameterized by a vector of n weights Wi and a activation function fO such that, the probability that input example X is Learning Stochastic Perceptrons under k-Blocking Distributions 281 classified as positive is given by (1) We consider the case of a non-linear function fO since the linear case can be solved by a standard least square approximation like the one performed by Kearns in Schapire (1994) for linear sums of basis functions. We restrict ourselves to the case where fO is monotonic i.e. either nondecreasing or nonincreasing. But since any nonincreasing f (.) combined with a weight vector w can always be represented by a nondecreasing f(?) combined with a weight vector -w, we can assume without loss of generality that the target stochastic perceptron has a nondecreasing f (. ). Hence, we allow any sigmoid-type of activation function (with arbitrary threshold). Also, since our instance space zn is on an-sphere, eq. 1 also include any nonincreasing radial basis function of the type ?(z2) where z = Ix - wi and w is interpreted as the "center" of ?. The only significant restriction is on the weights where we allow only for Wi E {-I, 0, +1}. As usual, the goal of the learner is to return an hypothesis h which is a good approximation of the target p-concept c. But, in contrast with decision rule learning which attempts to "filter out" the noisy behavior by returning a deterministic hypothesis, the learner will attempt the harder (and more useful) task of modeling the target p-concept by returning a p-concept hypothesis. As a measure of error between the target and the hypothesis p-concepts we adopt the variation distance dv (?,?) defined as: err(h,c) = dv(h,c) ~f LPD(X) Ih(x) - c(x)1 x (2) Where the summation is over all the 2n possible values of x. Hence, the same D is used for both training and testing. The following formulation of the PAC criterion (Valiant, 1984; Hassler, 1992) will be sufficient for our purpose. Definition 1 Algorithm A is said to PAC learn the class C of p-concepts by using the hypothesis class H (of p-concepts) under a family V of distributions on instance space In, iff for any c E C, any D E V, any 0 < t,8 < 1, algorithm A returns in a time polynomial in (l/t, 1/8, n), an hypothesis h E H such that with probability at least 1 - 8, err(h, c) < t. 3 K-BLOCKING DISTRIBUTIONS To learn the class of stochastic perceptrons, the algorithm will try to discover each weight Wi that connects to input variable Xi by estimating how the probability of observing a positive output (0" = 1) is affected by "hard-wiring" variable Xi to some fixed value. This should clearly give some information about Wi when Xi is statistically independent from all the other variables as was the case for Golea and Marchand (1993) and Schapire (1992). However, if the input variables are correlated, then the process of fixing variable Xi will carry over neighboring variables which in turn will affect other variables until all the variables are perturbed (even in the simplest case of a first order Markov chain). The information about Wi will 282 Mario Marchand, Saeed HadjiJaradji then be smeared by all the other weights. Therefore, to obtain information only on Wi, we need to break this "chain reaction" by fixing some other variables. The notion of blocking sets serves this purpose. Loosely speaking, a set of variables is said to be a blocking set 1 for variable Xi if the distribution on all the remaining variables is unaffected by the setting of Xi whenever all the variables of the blocking set are set to a fixed value. More precisely, we have: Definition 2 Let B be a subset of X and let U = X - (B U {Xi}). Let XB and Xu be the restriction of X on Band U respectively and let b be an assignment for XB. Then B is said to be a blocking set for variable Xi (with respect to D), iff: PD(xulxB = b,Xi = +1) = PD(xulxB = b,Xi = -1) for all b and Xu In addition, if B is not anymore a blocking set when we remove anyone of its variables, we then say that B is a minimal blocking set for variable Xi. We thus adopt the following definition for the k-blocking family. Definition 3 Distribution D on rn is said to be k-blocking iff IBil < k 1,2?? . n when each Bi is a minimal blocking set for variable Xi. for i = The k-blocking family is quite a large class of distributions. In fact we have the following property: Property 1 All Markov distributions of kth order are members of the 2k-blocking family. Proof: By kth order Markov distributions, we mean distributions which can be exactly written as a Chow(k) expansion (see Hoeffgen, 1993) for some permutation of the variables. We prove it here (by using standard techniques such as in Abend et. al, 1965) for first order Markov distributions, the generalization for k > 1 is straightforward. Recall that for Markov chain distributions we have: p(XjIXj-b??? xI) = p(XjIXj_l) for 1 < j ~ n. Hence: P(XI ... Xj-2, Xj+2? .. XnlXj-b Xj, Xj+!) = p(Xl)p(X2Ixl)??? p(Xj IXj-l)p(Xj+llxj)??? P(XnIXn-l)!p(Xj-b Xj, Xj+!) = p(xI)p(x2Ixd??? p(xj-llxj-2)P(Xj+2Ixj+l)? .. P(XnIXn-l)!p(Xj-I) = P(Xl?? ?Xj-2,Xj+2? ??XnIXj-bXj,Xj+!) where Xj denotes the negation of Xj. Thus, we see that Markov chain distributions are a special case of 2-blocking distributions: the blocking set of each variable consisting only of the two first-neighbor variables. D. The proposed algorithm for learning stochastic perceptrons needs to be provided with a blocking set (of at most k variables) for each input variable. Hoeffgen (1993) has recently proven that Chow(l) and Chow(k > 1) expansions are efficiently learnable; the latter under some restricted conditions. We can thus use these algorithms IThe wording "blocking set" was also used by Hancock & Mansour (Proc. of COLT'91 , 179-183, Morgan Kaufmann Publ.) to denote a property of the target concept. In contrast, our definition of blocking set denotes a property of the input distribution only. Learning Stochastic Perceptrons under k-Blocking Distributions 283 to discover the blocking sets for such distributions. However , the efficient learnability of unrestricted Chow(k > 1) expansions and larger classes of distributions, such as the k-blocking family, is still unknown. In fact, from the hardness results of Hoeffgen (1993), we can see that it is definitely very hard (perhaps NP-complete) to find the blocking sets if the learner has no information available other than the fact that the distribution is k-blocking. On the other hand, we can argue that the "natural" ordering of the variables present in many "real-world" situations is such that the blocking set of any given variable is among the neighboring variables. In vision for example, we expect that the setting of a pixel will directly affect only those located in it's neighborhood; the other pixels being affected only through this neighborhood. In such cases, the neighborhood of a variable "naturally" provides its blocking set. 4 LEARNING STOCHASTIC PERCEPTRONS We first establish (the intuitive fact) that, without making much error, we can always consider that the target p-concept is defined only over the variables which are not almost always set to the same value. Lemma 1 Let V be a set of v variables Xi for which Pr(xi = ai) > 1 - a. Let c be a p-concept and let c' be the same p-concept as c except that the reading of each variable Xi E V is replaced by the reading of the constant value ai. Then err( c' , c) < v . a. Proof: Let a be the vector obtained from the concatenation of all ais and let Xv be the vector obtained from X by keeping only the components Xi which are in V. Then err(c', c) ~ Pr(xv =I- a) ~ L:iEVPr(Xi =I- ai). D. For a given set of blocking sets {Bi }f=l ' the algorithm will try to discover each weight Wi by estimating the blocked influence of Xi defined as: Binf(xilhi) ~f Pr(O' = 11xBi = hi , Xi = +1) - Pr(O' = 11xBi = hi, Xi = -1) where XB i denotes the restriction of x on the blocking set Bi for variable Xi and hi is an assignment for XB i ? The following lemma ensures the learner that Binf(xilhi) contains enough information about Wi. Lemma 2 Let the taryet p-concept be a stochastic perceptron on In having a nondecreasing activation function and weights taken from {-1, 0, +1}. Then, for any assignment hi for the variables in the blocking set Bi of variable Xi, we have: Binf(xilhi) { ~ 0 if Wi = +1 if Wi = 0 0 if Wi = -1 = 0 ~ (3) Proof sketch: Let U = X - (Bi U {Xi}), s = L:jEUWjXj and ( = L:kEBi wkbk? Let pes) denote the probability of observing s (under D). Then Binf(xilhi) = L:sp(s) [f(s + (+ Wi) - f(s + (- Wi)]; from which we find the desired result for a nondecreasing f(?) . D. 284 Mario Marchand, Saeed Hadjifaradji In principle, lemma 2 enables the learner to discover Wi from Binf(xilbi). The learner, however, has only access to its empirical estimate Bfnf(xilbi) from a finite sample. Hence, we will use Hoeffding's inequality (Hoeffding, 1963) to find the number of examples needed for a probability p to be close to its empirical estimate ft with high probability. Lemma 3 (Hoeffding, 1963) Let YI. ... , Y m be a sequence of m independent Bernoulli trials, each succeeding with probability p. Let ft = L:l ~/m. Then: Pr (1ft - pi > E) ~ 2 exp (-2mE2) Hence, by writing Binf(xilbi) in terms of (unconditional) probabilities that can be estimated from all the training examples, we find from lemma 3 that the number mO(E,8,n) of examples needed to have IBfnf(xilbi) - Binf(xilbi)I < E with probability at least 1 - 8 is given by: mO(E,8,n) ~ ~ (:E) 2 ln (~) where /'i, = a k +1 is the lowest permissible value for PD(b i , Xi) (see lemma 1). So, if the minimal nonzero value for IBinf(xilbi)1 is (3, then the number of examples needed to find, with confidence at least 1 - 8, the exact value for Wi among { -1,0, + I} is such that we need to have: Pr(IBfnf(xilbi) - Binf(xilbi)1 < (3/2) > 1 - 8. Thus, whenever (3 is oH2(e- n ), we will need of O(e2n ) examples to find (with prob > 1-8) the value for Wi. So, in order to be able to PAC learn from a polynomial sample, we must arrange ourselves so that we do not need to worry about such low values for IBinf(xilbi)l. We therefore consider the maximum blocked influence defined as: Binf(xi) ~f Binf(xilbn where b; is the vector value for which IBinf(xilbi)1 is the largest. We now show that the learner can ignore all variables Xi for which IBinf(xi)1 is too small (without making much error). Lemma 4 Let c be a stochastic perceptron with nondecreasing activation function f (.) and weights taken from { -1, 0, + 1}. Let V c X and let cv be the same stochastic perceptron as c except that Wi = 0 for all Xi E V and its activation function is changed to f ( . + e). Then, there always exists a value for e such that: err(cv, c) :::; 2: IBinf(xi)I iEV Proof sketch: By induction on IVI. To first verify the lemma for V = {Xl}, let b be a vector of values for the setting of all Xi E Bl and let Xu be a vector of values for the setting of all Xj E U = X - (Bl U {Xl}). Let s = LjEU WjXj and ( = LjEB l WjXj, then for e = WI, we have: err(cv, c) = 2: 2:PD(xul b )PD(blxl = -l)PD(XI = -1) Xu b x If(s + ( + WI) - f(s + ( - wl)1 ~ IBinf(xdl Learning Stochastic Perceptrons under k-Blocking Distributions 285 We now assume that the lemma holds for V = {Xl. X2?? . Xk} and prove it for W = V U {Xk+1}. Let S = {Xk+1} and let f(? + Ow), f(? + Ov) and f(? + Os) denote respectively the activation function for cw, Cv and cs. By inspecting the expressions for err(cv, c) and err(cw, cs), we can see that there always exist a value for Ow E {Ov + Wk+1,OV - Wk+l} and Os E {Wk+l. -wk+d such that err( cw, cs) ::; err( cv, c). And since dv (-, .) satisfies the triangle inequality, err(cw,c)::; err(cv,c) + IBinf(xk+1)I. D. After discovering the weights, the hypothesis p-concept h returned by the learner will simply be the table look-up of the estimated probabilities of observing a positive classification given that ~~= 1 Wi Xi = s for all s values that are observed with sufficient probability (the hypothesis can output any value for the values of s that are observed very rarely). We thus have the following learning algorithm for stochastic perceptrons. Algorithm LearnSP(n, ?, 6, {Bi}i=l) 1. Call m = 128 e: ) 2kHIn e~n) training examples (where k = maXi IBi I). = +1) for each variable Xi. Neglect Xi whenever we have Pr(xi = +1) < ?/(4n) or Pr(xi = +1) > 1 - ?/(4n). 2. Compute Pr(xi 3. For each variable Xi and for each of its blocking vector value hi, compute Bfnf(xilhi). Let h; be the value of hi for which IBfnf(xilhi)1 is the largest. Let Bfnf(xi) = Bfnf(xilh;). 4. For each variable Xi: (a) Let Wi = +1 whenever Bfnf(xi) > ?/(4n). (b) Let Wi = -1 whenever Bfnf(xi) < ?/(4n). (c) Otherwise let Wi = 0 5. Compute Pr(~~=l Wi Xi = s) for s = -n, . .. + n. 6. Return the hypothesis p-concept h formed by the table look-up: h(x) = h'(s) = Pr (0' = 1 t WiXi = s) ~=l for all s for which Pr(~~=l WiXi = s) > ?/(8n let h'(s) = 0 (or any other value). + 8). For the other s values, Theorem 1 Algorithm LearnSP PAC learns the class of stochastic perceptrons on In with monotonic activation functions and weights Wi E {-1, 0, +1} under any k-blocking distribution (when a blocking set for each variable is known). The number of examples required is m = 128 (2:) 2kHIn (l~n) (and the time needed is O(n x m)) for the returned hypothesis to make error at most ? with confidence at least 1 - 6. Proof sketch: From Hoeffding's inequality (lemma 3) we can show that this sample size is sufficient to ensure that: 286 Mario Marchand, Saeed Hadjifaradji ? IPr(Xi = +1) - Pr(xi = +1)1 < ? IBfnf(xi) - Binf(xi)1 < ? IPr(I:~=l WiXi = s) - ~/(4n) with confidence at least 1 - ~/(4n) with confidence at least 1 - Pr(I:~=l WiXi = s)1 < 6/(4n) 6/(4n) ~2/[64(n + 1)] with confidence at least 1- 6/(4n + 4) ? IPr(O" = llI:~=l WiXi = s) - Pr(O" = dence at least 1 - 6/4 llI:~=l WiXi = s)1 < ~/4 with confi- From this and from lemma 1,2 and 4, it follows that returned hypothesis will make error at most ~ with confidence at least 1 - 6. D. Acknowledgments We thank Mostefa Golea, Klaus-U. Hoeffgen and Stefan Poelt for useful comments and discussions about technical points. M. Marchand is supported by NSERC grant OGP0122405. Saeed Hadjifaradji is supported by the MCHE of Iran. References Abend K., Hartley T.J. & Kanal L.N. (1965) "Classification of Binary Random Patterns", IEEE Trans. Inform. Theory vol. IT-II, 538-544. Golea, M. & Marchand M. (1993) "On Learning Perceptrons with Binary Weights", Neural Computation vol. 5, 765-782. Golea, M. & Marchand M. (1994) "On Learning Simple Deterministic and Probabilistic Neural Concepts", in Shawe-Talor J. , Anthony M. (eds.), Computational Learning Theory: EuroCOLT'93, Oxford University Press, pp. 47-60. Haussler D. (1992) "Decision Theoritic Generalizations of the PAC Model for Neural Net and Other Learning Applications", Information and Computation vol. 100,78150. Hoeffgen K.U. (1993) "On Learning and Robust Learning of Product Distributions", Proceedings of the 6th ACM Conference on Computational Learning Theory, ACM Press, 77-83. Hoeffding W. (1963) "Probability inequalities for sums of bounded random variabIes", Journal of the American Statistical Association, vol. 58(301), 13-30. Kearns M.J. and Schapire R.E. (1994) "Efficient Distribution-free Learning ofProbabilistic Concepts", Journal of Computer and System Sciences, Vol. 48, pp. 464-497. Lin J.H. & Vitter J.S. (1991) "Complexity Results on Learning by Neural Nets", Machine Learning, Vol. 6, 211-230. Schapire R.E. (1992) The Design and Analysis of Efficient Learning Algorithms, Cambridge MA: MIT Press. Valiant L.G. (1984) "A Theory of the Learnable", Comm. ACM, Vol. 27, 11341142. 

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Nonlinear Image Interpolation using Manifold Learning Christoph Bregler Computer Science Division University of California Berkeley, CA 94720 bregler@cs.berkeley.edu Stephen M. Omohundro'" Int . Computer Science Institute 1947 Center Street Suite 600 Berkeley, CA 94704 om@research.nj .nec.com Abstract The problem of interpolating between specified images in an image sequence is a simple, but important task in model-based vision . We describe an approach based on the abstract task of "manifold learning" and present results on both synthetic and real image sequences. This problem arose in the development of a combined lip-reading and speech recognition system. 1 Introduction Perception may be viewed as the task of combining impoverished sensory input with stored world knowledge to predict aspects of the state of the world which are not directly sensed. In this paper we consider the task of image interpolation by which we mean hypothesizing the structure of images which occurred between given images in a temporal sequence. This task arose during the development of a combined lipreading and speech recognition system [3], because the time windows for auditory and visual information are different (30 frames per second for the camera vs. 100 feature vectors per second for the acoustic information). It is an excellent visual test domain in general, however, because it is easy to generate large amounts of test and training data and the performance measure is largely "theory independent" . The test consists of simply presenting two frames from a movie and comparing the "'New address: NEe Research Institute, Inc., 4 Independence Way, Princeton, NJ 08540 974 Christoph Bregler. Stephen M. Omohundro Figure 1: Linear interpolated lips. Figure 2: Desired interpolation. hypothesized intermediate frames to the actual ones. It is easy to use footage of a particular visual domain as training data in the same way. Most current approaches to model-based vision require hand-constructed CADlike models. We are developing an alternative approach in which the vision system builds up visual models automatically by learning from examples. One of the central components of this kind of learning is the abstract problem of inducing a smooth nonlinear constraint manifold from a set of examples from the manifold. We call this "manifold learning" and have developed several approaches closely related to neural networks for doing it [2]. In this paper we apply manifold learning to the image interpolation problem and numerically compare the results of this "nonlinear" process with simple linear interpolation. We find that the approach works well when the underlying model space is low-dimensional. In more complex examples, manifold learning cannot be directly applied to images but still is a central component in a more complex system (not discussed here). We present several approaches to using manifold learning for this task. We compare the performance of these approaches to that of simple linear interpolation. Figure 1 shows the results of linear interpolation of lip images from the lip-reading system. Even in the short period of 33 milliseconds linear interpolation can produce an unnatural lip image. The problem is that linear interpolation of two images just averages the two pictures. The interpolated image in Fig. 1 has two lower lip parts instead of just one. The desired interpolated image is shown in Fig. 2, and consists of a single lower lip positioned at a location between the lower lip positions in the two input pictures. Our interpolation technique is nonlinear, and is constrained to produce only images from an abstract manifold in "lip space" induced by learning. Section 2 describes the procedure, Section 4 introduces the interpolation technique based on the induced manifold, and Sections 5 and 6 describe our experiments on artificial and natural images. 2 Manifold Learning Each n * m gray level image may be thought of as a point in an n * m-dimensional space. A sequence of lip-images produced by a speaker uttering a sentence lie on a Nonlinear Image Interpolation Using Manifold Learning - 975 \ Graylevel Dimensions ------ (l6x16 pixel =256 dim. space) Figure 3: Linear vs nonlinear interpolation. 1-dimensional trajectory in this space (figure 3). If the speaker were to move her lips in all possible ways, the images would define a low-dimensional submanifold (or nonlinear surface) embedded in the high-dimensional space of all possible graylevel images. If we could compute this nonlinear manifold, we could limit any interpolation algorithm to generate only images contained in it. Images not on the manifold cannot be generated by the speaker under normal circumstances. Figure 3 compares a curve of interpolated images lying on this manifold to straight line interpolation which generally leaves the manifold and enters the domain of images which violate the integrity of the model. To represent this kind of nonlinear manifold embedded in a high-dimensional feature space, we use a mixture model of local linear patches. Any smooth nonlinear manifold can be approximated arbitrarily well in each local neighborhood by a linear "patch" . In our representation, local linear patches are "glued" together with smooth "gating" functions to form a globally defined nonlinear manifold [2]. We use the "nearest-point-query" to define the manifold. Given an arbitrary point near the manifold, this returns the closest point on the manifold. We answer such queries with a weighted sum of the linear projections of the point to each local patch. The weights are defined by an "influence function" associated with each linear patch which we usually define by a Gaussian kernel. The weight for each patch is the value of its influence function at the point divided by the sum of all influence functions ("partition of unity"). Figure 4 illustrates the nearest-point-query. Because Gaussian kernels die off quickly, the effect of distant patches may be ignored, improving computational performance. The linear projections themselves consist of a dot product and so are computationally inexpensive. For learning, we must fit such a mixture of local patches to the training data. An initial estimate of the patch centers is obtained from k-means clustering. We fit a patch to each local cluster using a local principal components analysis. Fine tuning 976 Christoph Bregler. Stephen M. Omohundro Influence Function PI P2 \ Linear Patch l:,Gi(X) ' Pi(x) P(x) = . .:. .'-=--l:,Gi(X) , Figure 4: Local linear patches glued together to a nonlinear manifold. of the model is done using the EM (expectation-maximization) procedure. This approach is related to the mixture of expert architecture [4], and to the manifold representation in [6]. Our EM implementation is related to [5] , which uses a hierarchical gating function and local experts that compute linear mappings from one space to another space. In contrast, our approach uses a "one-level" gating function and local patches that project a space into itself. 3 Linear Preprocessing Dealing with very high-dimensional domains (e.g. 256 * 256 gray level images) requires large memory and computational resources. Much of this computation is not relevant to the task, however. Even if the space of images is nonlinear, the nonlinearity does not necessarily appear in all of the dimensions. Earlier experiments in the lip domain [3] have shown that images projected onto a lO-dimensional linear subspace still accurately represents all possible lip configurations. We therefore first project the high-dimensional images into such a linear subspace and then induce the nonlinear manifold within this lower dimensional linear subspace. This preprocessing is similar to purely linear techniques [7, 10, 9] . 4 Constraint Interpolation Geometrically, linear interpolation between two points in n-space may be thought of as moving along the straight line joining the two points. In our non-linear approach to interpolation, the point moves along a curve joining the two points which lies in the manifold of legal images. We have studied several algorithms for estimating the shortest manifold trajectory connecting two given points. For the performance results, we studied the point which is halfway along the shortest trajectory. Nonlinear Image Interpolation Using Manifold Learning 4.1 977 "Free-Fall" The computationally simplest approach is to simply project the linearly interpolated point onto the nonlinear manifold. The projection is accurate when the point is close to the manifold. In cases where the linearly interpolated point is far away (i.e. no weight of the partition of unity dominates all the other weights) the closest-pointquery does not result in a good interpolant. For a worst case, consider a point in the middle of a circle or sphere. All local patches have same weight and the weighted sum of all projections is the center point itself, which is not a manifold point. Furthermore, near such "singular" points, the final result is sensitive to small perturbations in the initial position. 4.2 "Manifold-Walk" A better approach is to "walk" along the manifold itself rather than relying on the linear interpolant. Each step of the walk is linear and in the direction of the target point but the result is immediately projected onto the manifold. This new point is then moved toward the target point and projected onto the manifold, etc. When the target is finally reached, the arc length of the curve is approximated by the accumulated lengths of the individual steps. The point half way along the curve is chosen as the interpolant. This algorithm is far more robust than the first one, because it only uses local projections, even when the two input points are far from each other. Figure 5b illustrates this algorithm. 4.3 "Manifold-Snake" This approach combines aspects of the first two algorithms. It begins with the linearly interpolated points and iteratively moves the points toward the manifold. The Manifold-Snake is a sequence of n points preferentially distributed along a smooth curve with equal distances between them. An energy function is defined on such sequences of points so that the energy minimum tries to satisfy these constraints (smoothness, equidistance, and nearness to the manifold): (1) E has value 0 if all Vi are evenly distributed on a straight line and also lie on the manifold. In general E can never be 0 if the manifold is nonlinear, but a minimum for E represents an optimizing solution. We begin with a straight line between the two input points and perform gradient descent in E to find this optimizing solution. 5 Synthetic Examples To quantify the performance of these approaches to interpolation, we generated a database of 16 * 16 pixel images consisting of rotated bars. The bars were rotated for each image by a specific angle. The images lie on a one-dimensional nonlinear manifold embedded in a 256 dimensional image space. A nonlinear manifold represented by 16 local linear patches was induced from the 256 images. Figure 6a shows 978 Christoph Bregler, Stephen M. Omohundro a) "Free Fall" b) "Surface Walk" c) "Surface Snake" Figure 5: Proposed interpolation algorithms. -~/ Figure 6: a) Linear interpolation, b) nonlinear interpolation. two bars and their linear interpolation. Figure 6b shows the nonlinear interpolation using the Manifold- Walk algorithm. Figure 7 shows the average pixel mean squared error of linear and nonlinear interpolated bars. The x-axis represents the relative angle between the two input points. Figure 8 shows some iterations of a Manifold-Snake interpolating 7 points along a 1 dimensional manifold embedded in a 2 dimensional space . .... pi --- ......... /' ./ ~/ .t. ,__ / VV --- --- -- --- --- - Figure 7: Average pixel mean squared error of linear and nonlinear interpolated bars. Nonlinear Image Interpolation Using Manifold Learning 979 QQOOOO oIter. I Iter. 2 Iter. 5 Iter. to Iter. 30 Iter. Figure 8: Manifold-Snake iterations on an induced 1 dimensional manifold embedded in 2 dimensions. Figure 9: 16x16 images. Top row: linear interpolation. Bottom row : nonlinear "manifold-walk" interpolation. 6 Natural Lip Images We experimented with two databases of natural lip images taken from two different subjects. Figure 9 shows a case of linear interpolated and nonlinear interpolated 16 * 16 pixel lip images using the Manifold- Walk algorithm . The manifold consists of 16 4-dimensional local linear patches. It was induced from a training set of 1931 lip images recorded with a 30 frames per second camera from a subject uttering various sentences. The nonlinear interpolated image is much closer to a realistic lip configuration than the linear interpolated image. Figure 10 shows a case of linear interpolated and nonlinear interpolated 45 * 72 pixel lip images using the Manifold-Snake algorithm. The images were recorded with a high-speed 100 frames per second camera l . Because of the much higher dimensionality of the images, we projected the images into a 16 dimensional linear subspace. Embedded in this subspace we induced a nonlinear manifold consisting of 16 4-dimensionallocallinear patches, using a training set of 2560 images. The linearly interpolated lip image shows upper and lower teeth, but with smaller contrast , because it is the average image of the open mouth and closed mouth. The nonlinearly interpolated lip images show only the upper teeth and the lips half way closed, which is closer to the real lip configuration. 7 Discussion We have shown how induced nonlinear manifolds can be used to constrain the interpolation of gray level images. Several interpolation algorithms were proposed IThe images were recorded in the UCSD Perceptual Science Lab by Michael Cohen Christoph Bregler. Stephen M. Omohundro 980 Figure 10: 45x72 images projected into a 16 dimensional subspace. Top row: linear interpolation. Bottom row: nonlinear "manifold-snake" interpolation. and experimental studies have shown that constrained nonlinear interpolation works well both in artificial domains and natural lip images. Among various other nonlinear image interpolation techniques, the work of [1] using a Gaussian Radial Basis Function network is most closely related to our approach. Their approach is based on feature locations found by pixelwise correspondence, where our approach directly interpolates graylevel images. Another related approach is presented in [8]. Their images are also first projected into a linear subspace and then modelled by a nonlinear surface but they require their training examples to lie on a grid in parameter space so that they can use spline methods. References [1] D. Beymer, A. Shahsua, and T. Poggio Example Based Image Analysis and Synthesis M.I.T. A.1. Memo No. 1431, Nov. 1993. [2] C. Bregler and S. Omohundro, Surface Learning with Applications to Lip-Reading, in Advances in Neural Information Processing Systems 6, Morgan Kaufmann, 1994. [3] C. Bregler and Y. Konig, "Eigenlips" for Robust Speech Recognition in Proc. ofIEEE Int. Conf. on Acoustics, Speech, and Signal Processing, Adelaide, Australia, 1994. [4] R.A. Jacobs, M.1. Jordan, S.J. Nowlan, and G.E. Hinton, Adaptive mixtures of local experts in Neural Compuation, 3, 79-87. [5] M.1. Jordan and R. A. Jacobs, Hierarchical Mixtures of Experts and the EM Algorithm Neural Computation, Vol. 6, Issue 2, March 1994. [6] N. Kambhatla and T.K. Leen, Fast Non-Linear Dimension Reduction in Advances in Neural Information Processing Systems 6, Morgan Kaufmann, 1994. [7] M. Kirby, F. Weisser, and G. DangeImayr, A Model Problem in Represetation of Digital Image Sequences, in Pattern Recgonition, Vol 26, No.1, 1993. [8] H. Murase, and S. K. Nayar Learning and Recognition of 3-D Objects from Brightness Images Proc. AAAI, Washington D.C., 1993. [9] P. Simard, Y. Le Cun, J. Denker Efficient Pattern Recognition Using a New Transformation Distance Advances in Neural Information Processing Systems 5, Morgan Kaufman, 1993. [10] M. Turk and A. Pentland, Eigenfaces for Recognition Journal of Cognitive Neuroscience, Volume 3, Number 1, MIT 1991. 

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144 SPEECH RECOGNITION EXPERIMENTS WITH PERCEPTRONS D. J. Burr Bell Communications Research Morristown, NJ 07960 ABSTRACT Artificial neural networks (ANNs) are capable of accurate recognition of simple speech vocabularies such as isolated digits [1]. This paper looks at two more difficult vocabularies, the alphabetic E-set and a set of polysyllabic words. The E-set is difficult because it contains weak discriminants and polysyllables are difficult because of timing variation. Polysyllabic word recognition is aided by a time pre-alignment technique based on dynamic programming and E-set recognition is improved by focusing attention. Recognition accuracies are better than 98% for both vocabularies when implemented with a single layer perceptron. INTRODUCTION Artificial neural networks perform well on simple pattern recognition tasks. On speaker trained spoken digits a layered network performs as accurately as a conventional nearest neighbor classifier trained on the same tokens [1]. Spoken digits are easy to recognize since they are for the most part monosyllabic and are distinguished by strong vowels. It is reasonable to ask whether artificial neural networks can also solve more difficult speech recognition problems. Polysyllabic recognition is difficult because multi-syllable words exhibit large timing variation. Another difficult vocabulary, the alphabetic E-set, consists of the words B, C, D, E, G, P, T, V, and Z. This vocabulary is hard since the distinguishing sounds are short in duration and low in energy. We show that a simple one-layer perceptron [7] can solve both problems very well if a good input representation is used and sufficient examples are given. We examine two spectral representations - a smoothed FFT (fast Fourier transform) and an LPC (linear prediction coefficient) spectrum. A time stabilization technique is described which pre-aligns speech templates based on peaks in the energy contour. Finally, by focusing attention of the artificial neural network to the beginning of the word, recognition accuracy of the E-set can be consistently increased. A layered neural network, a relative of the earlier percept ron [7], can be trained by a simple gradient descent process [8]. Layered networks have been ? American Institute of Physics 1988 145 applied successflJ.lly to speech recognition [1], handwriting recognition [2], and to speech synthesis [11]. A variation of a layered network [3] uses feedback to model causal constraints, which can be useful in learning speech and language. Hidden neurons within a layered network are the building blocks that are used to form solutions to specific problems. The number of hidden units required is related to the problem [1,2]. Though a single hidden layer can form any mapping [12], no more than two layers are needed for disjunctive normal form [4]. The second layer may be useful in providing more stable learning and representation in the presence of noise. Though neural nets have been shown to perform as well as conventional techniques[I,5], neural nets may do better when classes have outliers [5]. PERCEPTRONS A simple perceptron contains one input layer and one output layer of neurons directly connected to each other (no hidden neurons). This is often called a one-layer system, referring to the single layer of weights connecting input to output. Figure 1. shows a one-layer perceptron configured to sense speech patterns on a two-dimensional grid. The input consists of a 64-point spectrum at each of twenty time slices. Each of the 1280 inputs is connected to each of the output neurons, though only a sampling of connections are shown. There is one output neuron corresponding to each pattern class. Neurons have standard linear-weighted inputs with logistic activation. C(1) C(2) C(N-1) C(N) FR:<lBC'V .... 64 units Figure 1. A single layer perceptron sensing a time-frequency array of sample data. Each output neuron CU) (1 <i<N) corresponds to a pattern class and is full connected to the input array (for clarity only a few connections are shown). An input word is fit to the grid region by applying an automatic endpoint detection algorithm. The algorithm is a variation of one proposed by Rabiner and Sambur [9] which employs a double threshold successive approximation 146 method. Endpoints are determined by first detecting threshold crossings of energy and then of zero crossing rate. In practice a level crossing other than zero is used to prevent endpoints from being triggered by background sounds. INPUT REPRESENTATIONS Two different input representations were used in this study. The first is a Fourier representation smoothed in both time and frequency. Speech is sampled at 10 KHz ap.d Hamming windowed at a number of sample points. A 128-point FFT spectrum is computed to produce a template of 64 spectral samples at each of twenty time frames. The template is smoothed twice with a time window of length three and a frequency window of length eight. For comparison purposes an LPC spectrum is computed using a tenth order model on 300-sample Hamming windows. Analysis is performed using the autocorrelation method with Durbin recursion [6]. The resulting spectrum is smoothed over three time frames. Sample spectra for the utterance "neural-nets" is shown in Figure 2. Notice the relative smoothness of the LPC spectrum which directly models spectral peaks. FFT LPC Figure 2. FFT and LPC time-frequency plots for the utterance "neural nets". Time is toward the left, and frequency, toward the right. DYNAMIC TIME ALIGMv1ENT Conventional speech recognition systems often employ a time normalization technique based on dynamic programming [10]. It is used to warp the time scales of two utterances to obtain optimal alignment between their spectral frames. We employ a variation of dynamic programming which aligns energy contours rather than spectra. A reference energy template is chosen for each pattern class, and incoming patterns are warped onto it. Figure 3 shows five utterances of "neural-nets" both before and after time alignment. Notice the improved alignment of energy peaks. 147 ? I ? ? ? I ~ II ~ ~ !I ! >- \b ~ III z W .. .. (a. ) 10 10 ... TIME (b) Figure 3. (a) Superimposed energy plots of five different utterances of "neural nets". (b). Same utterances after dynamic time alignment. POLYSYLLABLE RECOGNITION Twenty polysyllabic words containing three to five syllables were chosen, and five tokens of each were recorded by a single male speaker. A variable number of tokens were used to train a simple perceptron to study the effect of training set size on performance. Two performance measures were used: classification accuracy, and an RMS error measure. Training tokens were permuted to obtain additional experimental data points. Figure 4. Output responses of a perceptron trained with one token per class (left) and four tokens per class (right). 148 Figure 4 shows two representative perspective plots of the output of a perceptron trained on one and four tokens respectively per class. Plots show network response (z-coordinate) as a function of output node (left axis) and test word index (right axis). Note that more training tokens produce a more ideal map - a map should have ones along the diagonal and zeroes everywhere else. Table 1 shows the results of these experiments for three different representations: (1) FFT, (2) LPC and (3) time aligned LPC. This table lists classification accuracy as a function of number of training tokens and input representation. The perceptron learned to classify the unseen patterns perfectly for all cases except the FFT with a single training pattern. Table 1. Polysyllabic Number Training Tokens FFT LPC Time Aligned LPC Permuted Trials Word Recognition 2 1 98.7% 100% 100% 100% 100% 100% 400 300 AccuraclT 3 100% 100% 100% 200 4 100% 100% 100% 100 A different performance measure, the RMS error, evaluates the degree to which the trained network output responses Rjk approximate the ideal targets T jk ? The measure "is evaluated over the N non-trained tokens and M output nodes of the network. Tik equals 1 for J=k and 0 for J=I=k. Figure 5 shows plots of RMS error as a function of input representation and training patterns. Note that the FFT representation produced the highest error, LPC was about 40% less, and time-aligned LPC only marginally better than non-aligned LPC. In a situation where many choices must be made (i.e. vocabularies much larger than 20 words) LPC is the preferred choice, and time alignment could be useful to disambiguate similar words. Increased number of training tokens results in improved performance in all cases. 149 o ci ,-----------------------------~ '"0 .. FFT i "! l- I- ii 0 5 0 g W tJl ~ LPC a: '"0 0 o o TIme Aligned LPC ~--~----~--~----~__~____~ 1.0 2.0 3.0 4.0 Number Traln'ng Tokens Figure 5. RMS error versus number of training tokens for various input representations. E-SET VOCABULARY The E-Set vocabulary consists of the nine E-words of the English alphabet - B, C, D, E, G, P, T, V, Z. Twenty tokens of each of the nine classes were recorded by a single male speaker. To maximize the sizes of training and test sets, half were used for training and the other half for testing. Ten permutations produced a total of 900 separate recognition trials. Figure 6 shows typical LPC templates for the nine classes. Notice the double formant ridge due to the ''E'' sound, which is common to all tokens. Another characteristic feature is the FO ridge - the upward fold on the left of all tokens which characterizes voicing or pitched sound. 150 Figure 6. LP C time-frequency plots for representative tokens of the E-set words. Figure 7. Time-frequency plots of weight values connected to each output neuron ''E'' through "z" in a trained perceptron. 151 Figure 7 shows similar plots illustrating the weights learned by the network when trained on ten tokens of each class. These are plotted like spectra, since one weight is associated with each spectral sample. Note that the patterns have some formant structure. A recognition accuracy of 91.4% included perfect scores for classes C, E, and G. Notice that weights along the FO contour are mostly small and some are slightly negative. This is a response to the voiced ''E" sound common to all classes. The network has learned to discount "voicing" as a discriminator for this vocabulary. Notice also the strong "hilly" terrain near the beginning of most templates. This shows where the network has decided to focus much of its discriminating power. Note in particular the hill-valley pair at the beginning of ''p'' and "T". These are near to formants F2/F3 and could conceivably be formant onset detectors. Note the complicated detector pattern for the ''V'' sound. The classes that are easy to discriminate (C, E, G) produce relatively fiat and uninter~sting weight spaces. A highly convoluted weight space must therefore be correlated with difficulty in discrimination. It makes little sense however that the network should be working hard in the late time C'E" sound) portion of the utterance. Perhaps additional training might reduce this activity, since the network would eventually find little consistent difference there. A second experiment was conducted to help the network to focus attention. The first k frames of each input token were averaged to produce an average spectrum. These average spectra were then used in a simple nearest neighbor recognizer scheme. Recognition accuracy was measured as a function of k. The highest performance was for k=8, indicating that the first 40% of the word contained most of the "action". Figure 8. word. E C P T 0 0 0 0 0 0 100 0 0 0 0 0 0 0 D 0 0 08 0 0 2 0 0 0 E 0 0 0 100 0 0 0 0 0 c 0 0 0 0 100 0 0 0 0 p 0 0 3 0 0 03 4 0 0 T 0 0 0 0 0 0 100 0 0 V 2 0 0 0 0 2 0 0 Z 0 0 0 0 0 0 0 B C B 08 c D V Z 0 08 09 Confusion matrix of the E-set focused on the first 40% of each 152 All words were resampled to concentrate 20 time frames into the first 40% of the word. LPC spectra were recomputed using a 16th order model and the network was trained on the new templates. Performance increased from 91.4% to 98.2%. There were only 16 classification errors out of the 900 recognition tests. The confusion matrix is shown in Figure 8. Learning times for all experiments consisted of about ten passes through the training set. When weights were primed with average spectral values rather than random values, learning time decreased slightly. CONCLUSIONS Artificial neural networks are capable of high performance in pattern recognition applications, matching or exceeding that of conventional classifiers. We have shown that for difficult speech problems such as time alignment and weak discriminability, artificial neural networks perform at high accuracy exceeding 98%. One-layer perceptrons learn these difficult tasks almost effortlessly - not in spite of their simplicity, but because of it. REFERENCES 1. D. J. Burr, "A Neural Network Digit Recognizer", Proceedings of IEEE Conference on Systems, Man, and Cybernetics, Atlanta, GA, October, 1986, pp. 1621-1625. 2. D. J. Burr, "Experiments with a Connectionist Text Reader," IEEE International Conference on Neural Networks, San Diego, CA, June, 1987. 3. M. I. Jordan, "Serial Order: A Parallel Distributed Processing Approach," ICS Report 8604, UCSD Institute for Cognitive Science, La Jolla, CA, May 1986. 4. S. J. Hanson, and D. J. Burr, 'What Connectionist Models Learn: Toward a Theory of Representation in Multi-Layered Neural Networ.ks," submitted for pu blication. 5. W. Y. Huang and R. P. Lippmann, "Comparisons Between Neural Net and Conventional Classifiers," IEEE International Conference on Neural Networks, San Diego, CA, June 21-23, 1987. 6. J. D. Markel and A. H. Gray, Jr., Linear Prediction of Speech, SpringerVerlag, New York, 1976. 7. M. L. Minsky and S. Papert, Perceptrons, MIT Press, Cambridge, Mass., 1969. 153 8. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, ''Learning Internal Representations by Error Propagation," in Parallel Distributed Processing, Vol. 1, D. E. Rumelhart and J. L. McClelland, eds., MIT Press, 1986, pp. 318362. 9. L. R. Rabiner and M. R. Sambur, "An Algorithm for Determining the Endpoints of Isolated Utterances," BSTJ, Vol. 54,297-315, Feb. 1975. 10. H. Sakoe and S. Chiba, "Dynamic Programming Optimization for Spoken Word Recognition," IEEE Trans. Acoust., Speech, Signal Processing, Vol. ASSP-26, No.1, 43-49, Feb. 1978. 11. T. J. Sejnowski and C. R. Rosenberg, "NETtalk: A Parallel Network that Learns to Read Aloud," Technical Report JHU/EECS-86/01, Johns Hopkins University Electrical Engineering and Computer Science, 1986. 12. A. Wieland and R. Leighton, "Geometric Analysis of Neural Network Capabilities," IEEE International Conference on Neural Networks, San Deigo, CA, June 21-24, 1987. 

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Using a Saliency Map for Active Spatial Selective Attention: Implementation & Initial Results Shumeet Baluja baluja@cs.cmu.edu School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Dean A. Pomerleau pomerleau@cs.cmu.edu School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Abstract In many vision based tasks, the ability to focus attention on the important portions of a scene is crucial for good performance on the tasks. In this paper we present a simple method of achieving spatial selective attention through the use of a saliency map. The saliency map indicates which regions of the input retina are important for performing the task. The saliency map is created through predictive auto-encoding. The performance of this method is demonstrated on two simple tasks which have multiple very strong distracting features in the input retina. Architectural extensions and application directions for this model are presented. 1 MOTIVATION Many real world tasks have the property that only a small fraction of the available input is important at any particular time. On some tasks this extra input can easily be ignored. Nonetheless, often the similarity between the important input features and the irrelevant features is great enough to interfere with task performance. Two examples of this phenomena are the famous "cocktail party effect", otherwise known as speech recognition in a noisy environment, and image processing of a cluttered scene. In both cases, the extraneous information in the input signal can be easily confused with the important features, making the task much more difficult. The concrete real world task which motivates this work is vision-based road following. In this domain, the goal is to control a robot vehicle by analyzing the scene ahead, and choosing a direction to travel based on the location of important features like lane marking and road edges. This is a difficult task, since the scene ahead is often cluttered with extraneous features such as other vehicle, pedestrians, trees, guardrails, crosswalks, road signs and many other objects that can appear on or around a roadway. 1 While we have had significant success on the road following task using simple feed-forward neural networks to transform images of the road ahead into steering commands for the vehicle [pomerleau, 1993b], these methods fail when presented with cluttered environments like those encounI . For the gl:neral task of autonomous navigation, these extra features are extremely important, but for restricted task of road following. which is the focus of this paper. these features are merely distractions. Although we are addressing the more general task using the techniques described here in combination with other methods, a description of these efforts is beyond the scope of this paper. 452 Shumeet Baluja, Dean A. Pomerleall tered when driving in heavy traffic, or on city streets. The obvious solution to this difficulty is to focus the attention of the processing system on only the relevant features by masking out the "noise". Because of the high degree of simi1arity between the relevant features and the noise, this filtering is often extremely difficult. Simultaneously learning to perform a task like road following and filtering out clutter in a scene is doubly difficult because of a chicken-and-egg problem. It is hard to learn which features to attend to before knowing how to perform the task, and it is hard to learn how to perform the task before knowing which features to attend to. This paper describes a technique designed to solve this problem. It involves deriving a "saliency map" of the image from a neural network's internal representation which highlights regions of the scene considered to be important. This saliency map is used as feedback to focus the attention of the network's processing on subsequent images. This technique overcomes the chicken-and-egg problem by simultaneously learning to identify which aspects of the scene are important, and how to use them to perform a task. 2 THE SALIENCY MAP A saliency map is designed to indicate which portions of the image are important for completing the required task. The trained network should be able to accomplish two goals with the presentation of each image. The first is to perform the given task using the inputs and the saliency map derived from the previous image, and the second is to predict the salient portions of the next image. 2.1 Implementation The creation of the saliency map is similar to the technique of Input Reconstruction Reliability Estimation (IRRE) by [Pomerleau, 1993]. IRRE attempts to predict the reliability of a network's output. The prediction is made by reconstructing the input image from linear transformations of the activations in the hidden layer, and comparing it with the actual image. IRRE works on the premise that the greater the similarity between the input image and the reconstructed input image, the more the internal representation has captured the important input features, and therefore the more reliable the network's response. A similar method to IRRE can be used to create a saliency map. The saliency map should be determined by the important features in the current image for the task to be performed. Because compressed representations of the important features in the current image are represented in the activations of the hidden units, the saliency map is derived from these, as shown in Figure 1. It should be noted that the hidden units, from which the saliency map is derived, do not necessarily contain information similar to principal components (as is achieved through auto-encoder networks), as the relevant task may only require information on a small portion of the image. In the simple architecture depicted in Figure 1, the internal representation must contain information which can be transformed by a single layer of adjustable weights into a saliency map for the next image. If such a transformation is not possible, separate hidden layers, with input from the task-specific internal representations could be employed to create the saliency map. The saliency map is trained by using the next image, of a time-sequential data set, as the target image for the prediction, and applying standard error backpropagation on the differences between the next image and the predicted next image. The weights from the hidden Using a Saliency Map for Active Spatial Selective Attention (delayed I time step) Predicted Output Units Hidden Units L------! ~':~e ,....--=--., erived aliency p Input Retina x X ----+--- Relevant Portion of Input 453 Figure 1: A simple architecture for using a saliency map. The dashed "chilled line represents connections", i.e. errors from these connections do not propagate back further to impact the activations of the hidden units. This architecture assumes that the target task contains information which will help determine the salient portions of the next frame. units to the saliency map are adjusted using standard backpropagation, but the error terms are not propagated to the weights from the inputs to the hidden units. This ensures that the hidden representation developed by the network is determined only by the target task, and not by the task of prediction. In the implementation used here, the feedback is to the input layer. The saliency map is created to either be the same size as the input layer, or is scaled to the same size, so that it can influence the representation in a straight-forward manner. The saliency map's values are scaled between 0.0 and 1.0, where 1.0 represents the areas in which the prediction matched the next image exactly. The value of 1.0 does not alter the activation of the input unit, a value of 0.0 turns off the activation. The exact construction of the saliency map is described in the next section, with the description of the experiments. The entire network is trained by standard backpropagation; in the experiments presented, no modifications to the standard training algorithm were needed to account for the feedback connections. The training process for prediction is complicated by the potential presence of noise in the next image. The saliency map cannot "reconstruct" the noise in the next image, because it can only construct the portions of the next image which can be derived from the activation of the hidden units, which are task-specific. There/ore, the noise in the next image will not be constructed, and thereby will be de-emphasized in the next time step by the saliency map. The saliency map serves as a filter, which channels the processing to the important portions of the scene [Mozer, 1988]. One of the key differences between the filtering employed in this study, and that used in other focus of attention systems, is that this filtering is based on expectations from multiple frames, rather than on the retinal activations from a single frame. An alternative neural model of visual attention which was explored by [Olshausen et al., 1992] achieved focus of attention in single frames by using control neurons to dynamically modify synaptic strengths. The saliency map may be used in two ways. It can either be used to highlight portions of the input retina or, when the hidden layer is connected in a retinal fashion using weight sharing, as in [LeCun et al., 1990], it can be used to highlight important spatial locations within the hidden layer itself. The difference is between highlighting individual pixels from which the features are developed or highlighting developed features. Discussion of the psychological evidence for both of these types of highlighting (in single-frame retinal activation based context), is given in [pashler and Badgio, 1985]. This network architecture shares several characteristics with a Jordan-style recurrent network [Jordan, 1986], in which the output from the network is used to influence the pro- 454 Shumeet Baluja, Dean A. Pomerleau cessing of subsequent input patterns in a temporal sequence. One important distinction is that the feedback in this architecture is spatially constrained. The saliency map represents the importance of local patches of the input, and can influence only the network's processing of corresponding regions of the input. The second distinction is that the outputs are not general task outputs, rather they are specially constructed to predict the next image. The third distinction is in the form of this influence. Instead of treating the feedback as additional input units, which contribute to the weighted sum for the network's hidden units, this architecture uses the saliency map as a gating mechanism, suppressing or emphasizing the processing of various regions of the layer to which it is fed-back. In some respects, the influence of the saliency map is comparable to the gating network in the mixture of experts architecture [Jacobs et al., 1991]. Instead of gating between the outputs of multiple expert networks, in this architecture the saliency map is used to gate the activations of the input units within the same network. 3 THE EXPERIMENTS In order to study the feasibility of the saliency map without introducing other extraneous factors, we have conducted experiments with two simple tasks described below. Extensions of these ideas to larger problems are discussed in sections 4 & 5. The first experiment is representative of a typical machine vision task, in which the relevant features move very little in consecutive frames. With the method used here, the relevant features are automatically determined and tracked. However, if the relevant features were known a priori, a more traditional vision feature tracker which begins the search for features within the vicinity of the location of the features in the previous frame, could also perform well. The second task is one in which the feature of interest moves in a discontinuous manner. A traditional feature tracker without exact knowledge the feature's transition rules would be unable to track this feature, in the presence of the noise introduced. The transition rules of the feature of interest are learned automatically through the use of the saliency map. In the first task, there is a slightly tilted vertical line of high activation in a 30x32 input unit grid. The width of the line is approximately 9 units, with the activation decaying with distance from the center of the line. The rest of the image does not have any activation. The task is to center a gaussian of activation around the center of the x-intercept of the line, 5 pixels above the top of the image. The output layer contains 50 units. In consecutive images, the line can have a small translational move and/or a small rotational move. Sample training examples are given in Figure 2. This task can be easily learned in the presence of no noise. The task is made much harder when lines, which have the same visual appearance as the real line (in everything except for location and tilt) randomly appear in the image. In this case, it is vital that the network is able to distinguish between the real line and noise line by using information gathered from previous image(s). In the second task, a cross ("+") of size 5x5 appears in a 20x20 grid. There are 16 positions in which the cross can appear, as shown in Figure 2c. The locations in which the cross appears is set according to the transition rules shown in Figure 2c. The object of this problem is to reproduce the cross in a smaller lOx 10 grid, with the edges of the cross extended to the edges of the grid, as shown in Figure 2b. The task is complicated by the presence of randomly appearing noise. The noise is in the form of another cross which appears exactly similar to the cross of interest. Again, in this task, it is vital for the network to be able to distinguish between the real cross, and crosses which appear as noise. As in the first task, this is only possible with knowledge of the previous image(s). Using a Saliency Map for Active Spatial Selective Attention At 455 12 5 9 2 7 1 II 16 10 15 4 \3 3 14 6 8 A2 c Bt B2 Figure 2: (A) The first task, image (AI) with no distractions, image (A2) with one distracting feature. (8) The second task, image (8 I) with no distractions, image (82) with two distractions. (C) Transition rules for the second task. 3.1 Results The two problems described above were attempted with networks trained both with and without noise. Each of the training sessions were also tested with and without the saliency map. Each type of network was trained for the same number of pattern presentations with the same training examples. The results are shown in Table 1. The results reported in Table I represent the error accrued over 10,000 testing examples. For task 1, errors are reported in terms of the absolute difference between the peak's of the Gaussians produced in the output layer, summed for all of the testing examples (the max error per image is 49). In task 2, the errors are the sum of the absolute difference between the network's output and the target output, summed across all of the outputs and all of the testing examples. When noise was added to the examples, it was added in the following manner (for both training and testing sets): In task 1, '1 noise' guarantees a noise element, similarly, '2 noise' guarantees two noise elements. However, in task 2, '1 noise' means that there is a 50% chance of a noise element occurring in the image, '2 noise' means that there is a 50% chance of another noise element occurring, independently of the appearance of the first noise element. The positions of the noise elements are determined randomly. The best performance. in task 1, came from the cases in which there was no noise in testing or training. and no saliency map was used. This is expected, as this task is not difficult when no noise is present. Surprisingly, in task 2, the best case was found with the saliency map, when training with noise and testing without noise. This performed even better than training without noise. Investigation into this result is currently underway. In task 1, when training and testing without noise. the saliency map can hurt performance. If the predictions made by the saliency map are not correct, the inputs appear slightly distorted; therefore, the task to be learned by the network becomes more difficult. Nevertheless. the benefit of using a saliency map is apparent when the test set contains noise. In task 2. the network without the saliency map, trained with noise, and tested without noise cannot perform well; the performance further suffers when noise is introduced into the testing set. The noise in the training prevents accurate learning. This is not the case when the saliency map is used (Table 1, task 2). When the training set contains noise, the network with the saliency map works better when tested with and without noise. 456 Sizumeet Baluja. Dean A. Pomerleau Table 1: Summed Error of 10,000 Testing Examples Testing Set Training Set Task 1 o Noise 1 Noise Task 2 2 Noise o Noise I Noise 2 Noise o Noise (Saliency) 12672 60926 82282 7174 94333 178883 o Noise (No Saliency) 10241 91812 103605 7104 133496 216884 1 Noise (Saliency) 18696 26178 52668 4843 10427 94422 1 Noise (No Saliency) 14336 80668 97433 31673 150078 227650 When the noise increased beyond the level of training, to 2 noise elements per image, the performances of networks trained both with and without the saliency map declined. It is suspected that training the networks with increased noise will improve performance in the network trained with the saliency map. Nonetheless, due to the amount of noise compared to the small size of the input layer, improvements in results may not be dramatic. In Figure 3, a typical test run of the second task is shown. In the figure, the inputs, the predicted and actual outputs, and the predicted and actual saliency maps, are shown. The actual saliency map is just a smaller version of the unfiltered next input image. The input size is 20x20, the outputs are 10xlO, and the saliency map is 10xlO. The saliency map is scaled to 20x20 when it is applied to the next inputs. Note that in the inputs to the network, one cross appears much brighter than the other; this is due to the suppression of the distracting cross by the saliency map. The particular use of the saliency map which is employed in this study, proceeded as follows: the difference between the saliency map (derived from input imagei) and the input imagei+l was calculated. This difference image was scaled to the range of 0.0 to 1.0. Each pixel was then passed through a sigmoid; alternatively, a hard-step function could have been used. This is the saliency map. The saliency map was multiplied by input imagei+ 1; this was used as the input into the network. If the sigmoid is not used, features, such as incomplete crosses, sometimes appear in the input image. This happens because different portions of the cross may have slightly different saliency values associated with them -due to errors in prediction coupled with the scaling of the saliency map. Although the sigmoid helps to alleviate the problem, it does not eliminate it. This explains why training with no noise with a saliency map sometimes does not perform as well as training without a saliency map. 4 ALTERNATIVE IMPLEMENTATIONS An alternative method of implementing the saliency map is with standard additive connections. However, these connection types have several drawbacks in comparison with the multiplicative ones use in this study. First, the additive connections can drastically change the meaning of the hidden unit's activation by changing the sign of the activation. The saliency map is designed to indicate which regions are important for accomplishing the task based upon the features in the hidden representations; as little alteration of the important features as possible is desired. Second, if the saliency map is incorrect, and suggests an area of which is not actually important, the additive connections will cause 'ghost' images to appear. These are activations which are caused only by the influence of the addi- Using a Saliency Map for Active Spatial Selective Attention 457 predicted saliency target saliency predicted output target output inputs 1 4 5 6 7 Figure 3: A typical sequence of inputs and outputs in the second task. Note that when two crosses are in the inputs. one is much brighter than the other. The "noise" cross is de-emphasized. tive saliency map. The multiplicative saliency map, as is implemented here, does not have either of these problems. A second alternative, which is more closely related to a standard recurrent network [Jordan, 1986], is to use the saliency map as extra inputs into the network. The extra inputs serve to indicate the regions which are expected to be important. Rather than hand-coding the method to represent the importance of the regions to the network, as was done in this study, the network learns to use the extra inputs when necessary. Further, the saliency map serves as the predicted next input. This is especially useful when the features of interest may have momentarily been partially obscured or have entirely disappeared from the image. This implementation is currently being explored by the authors for use in a autonomous road lane-tracking system in which the lane markers are not always present in the input image. 5 CONCLUSIONS & FUTURE DIRECTIONS These experiments have demonstrated that an artificial neural network can learn to both identify the portions of a visual scene which are important, and to use these important features to perform a task. The selective attention architecture we have develop uses two simple mechanisms, predictive auto-encoding to form a saliency map, and a constrained form of feedback to allow this saliency map to focus processing in the subsequent image. There are at least four broad directions in which this research should be extended. The first is, as described here, related to using the saliency map as a method for automatically actively focusing attention to important portions of the scene. Because of the spatial dependence of the task described in this paper, with the appropriate transformations, the output units could be directly fed back to the input layer to indicate saliency. Although this does not weaken the result, in terms of the benefit of using a saliency map, future work should also focus on problems which do not have this property to determine how easily a saliency map can be constructed. Will the use of weight sharing be enough to develop the necessary spatially oriented feature detectors? Harder problems are those with target tasks which does not explicitly contain spatial saliency information. An implementation problem which needs to be resolved is in networks which contain more than a single hidden layer: from which layer should the saliency map be con- 458 Shumeet Baluja, Dean A. Pomerleau structed? The trade-off is that at the higher layers, the information contained is more task specific. However, the higher layers may effectively extract the information required to perform the task, without maintaining the information required for saliency map creation. The opposite case is true in the lower layers; these may contain all of the information required, but may not provide enough discrimination to narrow the focus effectively. The third area for research is an alternative use for the saliency map. ANNs have often been criticized for their uninterpretability, and lack of mechanism to explain performance. The saliency map provides a method for understanding, at a high level, what portions of the inputs the ANN finds the most important. Finally, the fourth direction for research is the incorporation of higher level, or symbolic knowledge. The saliency map provides a very intuitive and direct method for focusing the network's attention to specific portions of the image. The saliency map may prove to be a useful mechanism to allow other processes, including human users, to simply "point at" the portion of the image to which the network should be paying attention. The next step in our research is to test the effectiveness of this technique on the main task of interest, autonomous road following. Fortunately, the first demonstration task employed in this paper shares several characteristics with road following. Both tasks require the network to track features which move over time in a cluttered image. Both tasks also require the network to produce an output that depends on the positions of the important features in the image. Because of these shared characteristics, we believe that similar performance improvements should be possible in the autonomous driving domain. Acknowledgments Shumeet Baluja is supported by a National Science Foundation Graduate Fellowship. Support has come from "Perception for Outdoor Navigation" (contract number DACA76-89-C-0014, monitored by the US Army Topographic Engineering Center) and "Unmanned Ground Vehicle System" (contract number DAAE07-90-C-R059, monitored by TACOM). Support has also come from the National Highway Traffic Safety Administration under contract number DTNH22-93-C-07023. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing official policies, either expressed or implied, of the National Science Foundation, ARPA, or the U.S. Government. References Cottrell, G.W. & Munro, P. (1988) Principal Component Analysis ofImages via back-propagation. Proc Soc. of Photo-Opticallnstr. Eng., Cambridge, MA. Jordan, M.I., (1989). Serial Order: A Parallel, Distributed Processing Approach. In Advances in Connectionist Theory: Speech, eds. J.L. Elman and D.E. Rumerlhart. Hillsdale: Erlbaum. Jacobs, R.A., Jordan, M.I., Nowlan, S.J. & Hinton, G.E. (1991). Adaptive Mixtures of Local Experts. Neural Computation, 3:1. LeCun, Y., Boser, B., Denker, J.S., Henderson, D. Howard, R.E., Hurnmand W., and Jackel, L.D. (1989) Backpropagation Applied to Handwritten Zip Code Recognition. Neural Computation 1,541-551. MIT, 1989. Mozer, M.C. (1988) A Connectionist Model of Selective Attention in Visual Perception. Technical Report, University of Toronto, CRG-TR-88-4. Pashler, H. & Badgio, P. (1985). Visual Attention and Stimulus Identification. Journal of Experimental Psychology: Human Perception and Performance, II 105-121. Pomerleau, D.A. (1993) Input Reconstruction Reliability Estimation. In Giles, C.L. Hanson, S.J. and Cowan, J.D. (eds). Advances in Neurallnfol71Ultion Processing Systems 5, CA: Morgan Kaufmann Publishers. Pomerleau, D.A. (1993b) Neural Network Perception for Mobile Robot Guidance, Kluwer Academic Publishing. Olshausen, B., Anderson, C., & Van Essen; D. (1992) A Neural Model of Visual Attention and Invariant Pattern Recognition. California Institute of Technology, CNS Program, memo-18. 

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Analysis of Unstandardized Contributions in Cross Connected Networks Thomas R. Shultz Yuriko Oshima-Takane Yoshio Takane shultz@psych.mcgill.ca yuriko@psych.mcgill.ca takane@psych.mcgill.ca Department of Psychology McGill University Montreal, Quebec, Canada H3A IBI Abstract Understanding knowledge representations in neural nets has been a difficult problem. Principal components analysis (PCA) of contributions (products of sending activations and connection weights) has yielded valuable insights into knowledge representations, but much of this work has focused on the correlation matrix of contributions. The present work shows that analyzing the variance-covariance matrix of contributions yields more valid insights by taking account of weights. 1 INTRODUCTION The knowledge representations learned by neural networks are usually difficult to understand because of the non-linear properties of these nets and the fact that knowledge is often distributed across many units. Standard network analysis techniques, based on a network's connection weights or on its hidden unit activations, have been limited. Weight diagrams are typically complex and weights vary across mUltiple networks trained on the same problem. Analysis of activation patterns on hidden units is limited to nets with a single layer of hidden units without cross connections. Cross connections are direct connections that bypass intervening hidden unit layers. They increase learning speed in static networks by focusing on linear relations (Lang & Witbrock, 1988) and are a standard feature of generative algorithms such as cascadecorrelation (Fahlman & Lebiere, 1990). Because such cross connections do so much of the work, analyses that are restricted to hidden unit activations furnish only a partial picture of the network's knowledge. Contribution analysis has been shown to be a useful technique for multi-layer, cross connected nets. Sanger (1989) defined a contribution as the product of an output weight, the activation of a sending unit, and the sign of the output target for that input. Such contributions are potentially more informative than either weights alone or hidden unit activations alone since they take account of both weight and sending activation. Shultz and Elman (1994) used PCA to reduce the dimensionality of such contributions in several different types of cascade-correlation nets. Shultz and Oshima-Takane (1994) demonstrated that PCA of unscaled contributions produced even better insights into cascade-correlation solutions than did comparable analyses of contributions scaled by the sign of output targets. Sanger (1989) had recommended scaling contributions by the signs of output targets in order to determine whether the contributions helped or hindered the network's solution. But since the signs of output targets are only available to networks during error 602 Thomas R. Shultz, Yuriko Oshima-Takane, Yoshio Takane correction learning, it is more natural to use unscaled contributions in analyzing knowledge representations. There is an issue in PCA about whether to use the correlation matrix or the variancecovariance matrix. The correlation matrix contains Is in the diagonal and Pearson correlation coefficients between contributions off the diagonal. This has the effect of standardizing the variables (contributions) so that each has a mean of 0 and standard deviation of 1. Effectively, this ensures that the PCA of a correlation matrix exploits variation in input activation patterns but ignores variation in connection weights (because variation in connection weights is eliminated as the contributions are standardized). Here, we report on work that investigates whether more useful insights into network knowledge structures can be revealed by PCA of un standardized contributions. To do this, we apply PCA to the variance-covariance matrix of contributions. The variance-covariance matrix has contribution variances along the diagonal and covariances between contributions off the diagonal. Taking explicit account of the variation in connection weights in this way may produce a more valid picture of the network's knowledge. We use some of the same networks and problems employed in our earlier work (Shultz & Elman, 1994; Shultz & Oshima-Takane, 1994) to facilitate comparison of results. The problems include continuous XOR, arithmetic comparisons involving addition and mUltiplication, and distinguishing between two interlocking spirals. All of the nets were generated with the cascade-correlation algorithm (Fahlman & Lebiere, 1990). Cascade-correlation begins as a perceptron and recruits hidden units into the network as it needs them in order to reduce error. The recruited hidden unit is the one whose activations correlate best with the network's current error. Recruited units are installed in a cascade, each on a separate layer and receiving input from the input units and from any previously existing hidden units. We used the default values for all cascade-correlation parameters. The goal of understanding knowledge representations learned by networks ought to be useful in a variety of contexts. One such context is cognitive modeling, where the ability of nets to merely simulate psychological phenomena is not sufficient (McCloskey, 1991). In addition, it is important to determine whether the network representations bear any systematic relation to the representations employed by human subjects . 2 PCA OF CONTRIBUTIONS Sanger's (1989) original contribution analysis began with a three-dimensional array of contributions (output unit x hidden unit x input pattern). In contrast, we start with a twodimensional output weight x input pattern array of contributions. This is more efficient than the slicing technique used by Sanger to focus on particular output or hidden units and still allows identification of the roles of specific contributions (Shultz & Elman, 1994; Shultz & Oshima-Takane, 1994). We subject the variance-covariance matrix of contributions to PCA in order to identify the main dimensions of variation in the contributions (Jolliffe, 1986). A component is a line of best fit to a set of data points in multi-dimensional space. The goal of PCA is to summarize a multivariate data set with a relatively small number of components by capitalizing on covariance among the variables (in this case, contributions). We use the scree test (Cattell, 1966) to determine how many components are useful to include in the analysis. Varimax rotation is applied to improve the interpretability of the solution. Component scores are plotted to identify the function of each component 3 APPLICATION TO CONTINUOUS XOR The classical binary XOR problem does not have enough training patterns to make contribution analysis worthwhile. However, we constructed a continuous version of the XOR problem by dividing the input space into four quadrants. Starting from 0.1, input values were incremented in steps of 0.1, producing 100 x, y input pairs that can be partitioned into four quadrants of the input space. Quadrant a had values of x less than Analysis of Unstandardized Contributions in Cross Connected Networks 603 0.55 combined with values of y above 0.55. Quadrant b had values of x and y greater than 0.55. Quadrant c had values of x and y less than 0.55. Quadrant d had values of x greater than 0.55 combined with values of y below 0.55. Similar to binary XOR, problems from quadrants a and d had a positive output target (0.5) for the net, whereas problems from quadrants band c had a negative output target (-0.5). There was a single output unit with a sigmoid activation. Three cascade-correlation nets were trained on continuous XOR. Each of these nets generated a unique solution, recruiting five or six hidden units and taking from 541 to 765 epochs to learn to correctly classify all of the input patterns. Generalization to test patterns not in the training set was excellent. PCA of unscaled, unstandardized contributions yielded three components. A plot of rotated component scores for the 100 training patterns of net 1 is shown in Figure 1. The component scores are labeled according to their respective quadrant in the input space. Three components are required to account for 96.0% of the variance in the contributions. Figure 1 shows that component 1, with 44.3% of the variance in contributions, has the role of distinguishing those quadrants with a positive output target (a and d) from those with a negative output target (b and c). This is indicated by the fact that the black shapes are at the top of the component space cube in Figure 1 and the white shapes are at the bottom. Components 2 and 3 represent variation along the x and y input dimensions, respectively. Component 2 accounted for 26.1 % of the variance in contributions, and component 3 accounted for 25.6% of the variance in contributions. Input pairs from quadrants b and d (square shapes) are concentrated on the negative end of component 2, whereas input pairs from quadrants a and c (circle shapes) are concentrated on the positive end of component 2. Similarly, input pairs from quadrants a and b cluster on the negative end of component 3, and input pairs from quadrants c and d cluster on the positive end of component 3. Although the network was not explicitly trained to represent the x and y input dimensions, it did so as an incidental feature of its learning the distinction between quadrants a and d vs. quadrants band c. Similar results were obtained from the other two nets learning the continuous XOR problem. In contrast, PCA of the correlation matrix from these nets had yielded a somewhat less clear picture with the third component separating quadrants a and d from quadrants b and c, and the first two components representing variation along the x and y input dimensions (Shultz & Oshima-Takane, 1994). PCA of the correlation matrix of scaled contributions had performed even worse, with plots of component scores indicating interactive separation of the four quadrants, but with no clear roles for the individual components (Shultz & Elman, 1994). Standardized, rotated component loadings for net 1 are plotted in Figure 2. Such plots can be examined to determine the role played by each contribution in the network. For example, hidden units 2, 3, and 4 all playa major role in the job done by component 1, distinguishing positive from negative outputs. 4 APPLICATION TO COMPARATIVE ARITHMETIC Arithmetic comparison requires a net to conclude whether a sum or a product of two integers is greater than, less than, or equal to a comparison integer. Several psychological simulations have used neural nets to make additive and multiplicative comparisons and this has enhanced interest in this type of problem (McClelland, 1989; Shultz, Schmidt, Buckingham, & Mareschal, in press). The first input unit coded the type of arithmetic operation to be performed: 0 for addition and 1 for multiplication. Three additional linear input units encoded the integers. Two of these input units each coded a randomly selected integer in the range of 0 to 9, inclusive; another input unit coded a randomly selected comparison integer. For addition problems, comparison integers ranged from 0 to i9, inclusive; for multiplication, comparison integers ranged from 0 to 82, inclusive. Two sigmoid output units coded the results of the comparison operation. Target outputs of 0.5, -0.5 represented a greater than result, targets of -0.5, 0.5 represented less than, and targets of 0.5,0.5 represented equal to. 604 Thomas R. Shultz, Yuriko Oshima-Takane, Yoshio Takane 2 Component 1 o -1 3 2 Component 2 -2 Component 3 -2 Figure 1. Rotated component scores for a continuous XOR net. Component scores for the x, y input pairs in quadrant a are labeled with black circles, those from quadrant b with white squares, those from quadrant c with white circles, and those from quadrant d with black squares. The network's task is to distinguish pairs from quadrants a and d (the black shapes) from pairs from quadrants b and c (the white shapes). Some of the white shapes appear black because they are so densely packed, but all of the truly black shapes are relatively high in the cube. Hidden6 Hidden5 Hidden4 c: 0 '5 Hidden3 C Hidden2 :g 0 () Hidden1 Component III 3 II 2 Input2 ? Input1 -1.0 -0.5 0.0 0.5 1.0 Loading Figure 2. Standardized, rotated component loadings for a continuous XOR net. Rotated loadings were standardized by dividing them by the standard deviation of the respective contribution scores. Analysis of Unstandardized Contributions in Cross Connected Networks 605 The training patterns had 100 addition and 100 multiplication problems, randomly selected, with the restriction that 45 of each had correct answers of greater than, 45 of each had correct answers of less than, and 10 of each had correct answers of equal to. These constraints were designed to reduce the natural skew of comparative values in the high direction on multiplication problems. We ran three nets for 1000 epochs each, at which point they were very close to mastering the training patterns. Either seven or eight hidden units were recruited along the way. Generalization to previously unseen test problems was very accurate. Four components were sufficient to account for most the variance in un standardized contributions, 88.9% in the case of net 1. Figure 3 displays the rotated component scores for the first two components of net 1. Component I, accounting for 51.1 % of the variance, separated problems with greater than answers from problems with less than answers, and located problems with equal to answers in the middle, at least for addition problems. Component 2, with 20.2% of the variance, clearly separated multiplication from addition. Contributions from the first input unit were strongly associated with component 2. Similar results obtained for the other two nets. Components 3 and 4, with 10.6% and 7.0% of the variance, were sensitive to variation in the second and third inputs, respectively. This is supported by an examination of the mean input values of the 20 most extreme component scores on these two components. Recall that the second and third inputs coded the two integers to be added or multiplied. The negative end of component 3 had a mean second input value of 8.25; the positive end of this component had a mean second input value of 0.55. Component 4 had mean third input value of 2.00 on the negative end and 7.55 on the positive end. In contrast, PCA of the correlation matrix for these nets had yielded a far more clouded picture, with the largest components focusing on input variation and lesser components doing bits and pieces of the separation of answer types and operations in an interactive manner (ShUltz & Oshima-Takane, 1994). Problems with equal to answers were not isolated by any of the components. PCA of scaled contributions had produced three components that interactively separated the three answer types and operations, but failed to represent variation in input integers (ShUltz & Elman, 1994). Essentially similar advantages for using the variance-covariance matrix were found for nets learning either addition alone or multiplication alone. 5 APPLICATION TO THE TWO-SPIRALS PROBLEM The two-spirals problem requires a particularly difficult discrimination and a large number of hidden units. The input space is defined by two interlocking spirals that wrap around their origin three times. There are two sets of 97 real-valued x, y pairs, with each set representing one of the spirals, and a single sigmoid output unit coded for the identity of the spiral. Our three nets took between 1313 and 1723 epochs to master the distinction, and recruited from 12 to 16 hidden units. All three nets generalized well to previously unseen input pairs on the paths of the two spirals. PCA of the variance-covariance matrix for net 1 revealed that six components accounted for a total of 97.9% of the variance in contributions. The second and fourth of these components together distinguished one spiral from the other, with 20.7% and 9.8% of the variance respectively. Rotated component scores for these two components are plotted in Figure 4. A diagonal line drawn on Figure 4 from coordinates -2,2 to 2, -2 indicates that 11 points from each spiral were misclassified by components 2 and 4. This is only 11.3% of the data points in the training patterns. The fact that the net learned all of the training patterns implies that these exceptions were picked up by other components. Components 1 and 6, with 40.7% and 6.4% of the variance, were sensitive to variation in the x and y inputs, respectively. Again, this was confirmed by the mean input values of the 20 most extreme component scores on these two components. On component I, the negative end had a mean x value of 3.55 and the positive end had a mean y value of -3.55. 606 Thomas R. Shultz. Yuriko Oshima-Takane. Yoshio Takane 2 x> It" #~ ? . ?x Component 1 o -1 +< -2 L..-_ _......L_ _ _....L..._ _--iL..-_ _-J -2 -1 0 2 Component 2 Figure 3. Rotated component scores for an arithmetic comparison net. Greater than problems are symbolized by circles, less than problems by squares, addition by white shapes, and multiplication by black shapes. For equal to problems only, addition is represented by + and multiplication by X. Although some densely packed white shapes may appear black, they have no overlap with truly black shapes. All of the black squares are concentrated around coordinates -1, -1. 2 Component 2 o o o -1 o o Spiral 1 -2 1....-_ _- - '_ _ _......._ _ _........_ __ -2 -1 o 2 Component 4 Figure 4. Rotated component scores for a two-spirals net. Squares represent data points from spiral 1, and circles represent data points from spiral 2. Analysis of Unstandardized Contributions in Cross Connected Networks 607 On component 6, the negative end had a mean x value of 2.75 and the positive end had a mean y value of -2.75. The skew-symmetry of these means is indicative of the perfectly symmetrical representations that cascade-correlation nets achieve on this highly symmetrical problem. Every data point on every component has a mirror image negative with the opposite signed component score on that same component. This -x, -y mirror image point is always on the other spiral. Other components concentrated on particular regions of the spirals. The other two nets yielded essentially similar results. These results can be contrasted with our previous analyses of the two-spirals problem, none of which succeeded in showing a clear separation of the two spirals. PCAs based on scaled (Shultz & Elman, 1994) or unscaled (Shultz & Oshima-Takane, 1994) correlation matrices showed extensive symmetries but never a distinction between one spiral and another.1 Thus, although it was clear that the nets had encoded the problem's inherent symmetries, it was still unclear from previous work how the nets used this or other information to distinguish points on one spiral from points on the other spiral. 6 DISCUSSION On each of these problems, there was considerable variation among network solutions, as revealed, for example, by variation in numbers of hidden units recruited and signs and sizes of connection weights. In spite of such variation, the present technique of applying peA to the variance-covariance matrix of contributions yielded results that are sufficiently abstract to characterize different nets learning the same problem. The knowledge representations produced by this analysis clearly identify the essential information that the net is being trained to utilize as well as more incidental features of the training patterns such as the nature of the input space. This research strengthens earlier conclusions that PCA of network contributions is a useful technique for understanding network performance (Sanger, 1989), including relatively intractable multi-level cross connected nets (Shultz & Elman, 1994; Shultz & Oshima-Takane, 1994). However, the current study underscores the point that there are several ways to prepare a contribution matrix for PCA, not all of which yield equally valid or useful results. Rather than starting with a three dimensional matrix of output unit x hidden unit x input pattern and focusing on either one output unit at a time or one hidden unit at a time (Sanger, 1989), it is preferable to collapse contributions into a two dimensional matrix of output weight x input pattern. The latter is not only more efficient, but yields more valid results that characterize the network as a whole, rather than small parts of the network. Also, rather than scaling contributions by the sign of the output target (Sanger, 1989), it is better to use unsealed contributions. Unsealed contributions are not only more realistic, since the network has no knowledge of output targets during its feed-forward phase, but also produce clearer interpretations of the nefs knowledge representations (Shultz & Oshima-Takane, 1994). The latter claim is particularly true in terms of sensitivity to input dimensions and to operational distinctions between adding and multiplying. Plots of component scores based on unscaled contributions are typically not as dense as those based on sealed contributions but are more revealing of the network's knowledge. Finally, rather than applying peA to the correlation matrix of contributions, it makes more sense to apply it to the variance-covariance matrix. As noted in the introduction, using the correlation matrix effectively standardizes the contributions to have identical means and variances, thus obseuring the role of network connection weights. The present results indicate much clearer knowledge representations when the variance-covariance matrix is used since connection weight information is explicitly retained. Matrix differences were especially marked on the more difficult problems, such as two-spirals, where the only peAs to reveal how nets distinguished the spirals were those based on 1Results from un scaled contributions on the two-spirals problem were not actually presented in Shultz & Oshima-Takane (1994) since they were not very clear. 608 Thomas R. Shultz, Yuriko Oshima-Takane, Yoshio Takane variance-covariance matrices. But the relative advantages of using the variance-covariance matrix were evident on the easier problems too. There has been recent rapid progress in the study of the knowledge representations leamed by neural nets. Feed-forward nets can be viewed as function approximators for relating inputs to outputs. Analysis of their knowledge representations should reveal how inputs are encoded and transformed to produce the correct outputs. PCA of network contributions sheds light on how these function approximations are done. Components emerging from PCA are orthonormalized ingredients of the transformations of inputs that produce the correct outputs. Thus, PCA helps to identify the nature of the required transformations. Further progress might be expected from combining PCA with other matrix decomposition techniques. Constrained PCA uses external information to decompose multivariate data matrices before applying PCA (Takane & Shibayama, 1991). Analysis techniques emerging from this research will be useful in understanding and applying neural net research. Component loadings, for example, could be used to predict the results of lesioning experiments with neural nets. Once the role of a hidden unit has been identified by virtue of its association with a particular component, then one could predict that lesioning this unit would impair the function served by the component. Acknowledgments This research was supported by the Natural Sciences and Engineering Research Council of Canada. References Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, 1,245-276. Fahlman, S. E., & Lebiere, C. (1990.) The Cascade-Correlation learning architecture. In D. Touretzky (Ed.), Advances in neural information processing systems 2, (pp. 524532). Mountain View, CA: Morgan Kaufmann. Jolliffe, I. T. (1986). Principal component analysis. Berlin: Springer Verlag. Lang, K. J., & Wi tbrock , M. J. (1988). Learning to tell two spirals apart. In D. Touretzky, G. Hinton, & T. Sejnowski (Eds)., Proceedings of the Connectionist Models Summer School, (pp. 52-59). Mountain View, CA: Morgan Kaufmann. McClelland, J. L. (1989). Parallel distributed processing: Implications for cognition and development. In Morris, R. G. M. (Ed.), Parallel distributed processing: Implications for psychology and neurobiology, pp. 8-45. Oxford University Press. McCloskey, M. (1991). Networks and theories: The place of connectionism in cognitive science. Psychological Science, 2, 387-395. Sanger, D. (1989). Contribution analysis: A technique for assigning responsibilities to hidden units in connectionist networks. Connection Science, 1, 115-138. Shultz, T. R., & Elman, J. L. (1994). Analyzing cross connected networks. In J. D. Cowan, G. Tesauro, & J. Alspector (Eds.), Advances in Neural Information Processing Systems 6. San Francisco, CA: Morgan Kaufmann. ShUltz, T. R., & Oshima-Takane, Y. (1994). Analysis of un scaled contributions in cross connected networks. In Proceedings of the World Congress on Neural Networks (Vol. 3, pp. 690-695). Hillsdale, NJ: Lawrence Erlbaum. Shultz, T. R., Schmidt, W. C., Buckingham, D., & Mareschal, D. (In press). Modeling cognitive development with a generative connectionist algorithm. In G. Halford & T. Simon (Eds.), Developing cognitive competence: New approaches to process modeling. Hillsdale, NJ: Erlbaum. Takane, Y., & Shibayama, T. (1991). Principal component analysis with external information on both subjects and variables. Psychometrika, 56, 97-120. 

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Effects of Noise on Convergence and Generalization in Recurrent Networks Kam Jim Bill G. Horne c. Lee Giles* NEC Research Institute, Inc., 4 Independence Way, Princeton, NJ 08540 {kamjim,horne,giles}~research.nj.nec.com *Also with UMIACS, University of Maryland, College Park, MD 20742 Abstract We introduce and study methods of inserting synaptic noise into dynamically-driven recurrent neural networks and show that applying a controlled amount of noise during training may improve convergence and generalization. In addition, we analyze the effects of each noise parameter (additive vs. multiplicative, cumulative vs. non-cumulative, per time step vs. per string) and predict that best overall performance can be achieved by injecting additive noise at each time step. Extensive simulations on learning the dual parity grammar from temporal strings substantiate these predictions. 1 INTRODUCTION There has been much research in applying noise to neural networks to improve network performance. It has been shown that using noisy hidden nodes during training can result in error-correcting codes which increase the tolerance of feedforward nets to unreliable nodes (Judd and Munro, 1992) . Also, randomly disabling hidden nodes during the training phase increases the tolerance of MLP's to node failures (Sequin and Clay, 1990). Bishop showed that training with noisy data is equivalent to Tikhonov Regularization and suggested directly minimizing the regularized error function as a practical alternative (Bishop , 1994) . Hanson developed a stochastic version of the delta rule which adapt weight means and standard deviations instead 650 Kam Jim, Bill G. Home, C. Lee Giles of clean weight values (Hanson, 1990). (Mpitsos and Burton, 1992) demonstrated faster learning rates by adding noise to the weight updates and adapting the magnitude of such noise to the output error. Most relevant to this paper, synaptic noise has been applied to MLP's during training to improve fault tolerance and training quality. (Murray and Edwards, 1993) In this paper, we extend these results by introducing several methods of inserting synaptic noise into recurrent networks, and demonstrate that these methods can improve both convergence and generalization. Previous work on improving these two performance measures have focused on ways of simplifying the network and methods of searching the coarse regions of state space before the fine regions. Our work shows that synaptic noise can improve convergence by searching for promising regions of state space, and enhance generalization by enforcing saturated states. 2 NOISE INJECTION IN RECURRENT NETWORKS In this paper, we inject noise into a High Order recurrent network (Giles et al., 1992) consisting of N recurrent state neurons Sj, L non-recurrent input neurons lie, and N 2 L weights Wijle. (For justification ofits use see Section 4.) The recurrent network g(Lj,1e WijleSJlD, where g(.) is a operation is defined by the state process S:+1 sigmoid discriminant function. During training, an error function is computed as Ep where Sb -dp, Sb is the output neuron, and dp is the target output value tor pattern p. = = !f;, fp = Synaptic noise has been simulated on Multi-Layered-Perceptrons by inserting noise to the weights of each layer during training (Murray et al., 1993). Applying this method to recurrent networks is not straightforward because effectively the same weights are propagated forward in time. This can be seen by recalling the BPTT representation of unrolling a recurrent network in time into T layers with identical weights, where T is the length of the input string. In Tables 2 and 3, we introduce the noise injection steps for eight recurrent network noise models representing all combinations of the following noise parameters: additive vs. multiplicative, cumulative vs. non-cumulative, per time step vs. per string. As their name imply, additive and multiplicative noise add or multiply the weights by a small noise term. In cumulative noise, the injected noise is accumulated, while in non-cumulative noise the noise from the current step is removed before more noise is injected on the next step. Per time step and per string noise refer to when the noise is inserted: either at each time step or only once for each training string respectively. Table 1 illustrates noise accumulation examples for all additive models (the multiplicative case is analogous). 3 ANALYSIS ON THE EFFECTS OF SYNAPTIC NOISE The effects of each noise model is analyzed by taking the Taylor expansion on the error function around the noise-free weight set. By truncating this expansion to second and lower order terms, we can interpret the effect of noise as a set of regularization terms applied to the error function. From these terms predictions can be made about the effects on generalization and convergence. A similar analysis was Effects of Noise on Convergence and Generalization in Recurrent Networks 651 performed on MLP's to demonstrate the effects of synaptic noise on fault tolerance and training quality (Murray et. al., 1993). Tables 2 and 3 list the noise injection step and the resulting first and second brder Taylor expansion terms for all noise models. These results are derived by assuming the noise to be zero-mean white with variance (F2 and uncorrelated in time. 3.1 Predictions on Generalization One common cause of bad generalization in recurrent networks is the presence of unsaturated state representations. Typically, a network cannot revisit the exact same point in state space, but tends to wander away from its learned state representation. One approach to alleviate this problem is to encourage state nodes to operate in the saturated regions of the sigmoid. The first order error expansion terms of most noise models considered are capable of encouraging the network to achieve saturated states. This can be shown by applying the chain rule to the partial derivative in the first order expansion terms: (1) where e~ is the net input to state node i at time step t. The partial derivatives g~ favor internal representations such that the effects of perturbations to the net are minimized. inputs e: Multiplicative noise implements a form of weight decay because the error expansion terms include the weight products Wt~ijk or Wt ,ijk Wu,ijk' Although weight decay has been shown to improve generalization on feedforward networks (Krogh and Hertz, 1992) we hypothesize this may not be the case for recurrent networks that are learning to solve FSA problems. Large weights are necessary to saturate the state nodes to the upper and lower limits of the sigmoid discriminant function. Therefore, we predict additive noise will allow better generalization because of its absence of weight decay. att Sb . Noise models whose first order error term contain the expression a~rb i , l,k ",'mn. will favor saturated states for those partials whose sign correspond to the sign of a majority of the partials. It will favor unsaturated states, operating in the linear region of the sigmoid, for partials whose sign is the minority. Such sign-dependent enforcement is not optimal. The error terms for cumulative per time step noises sum a product with the expresSb . Sb ,where = min(t + 1, U + 1). The effect of cumulative noise sion lm " increases more rapidly because of v and thus optimal generalization and detrimental noise effects will occur at lower amplitudes than non-cumulative noise. va:.r,.". a:.t v For cumulative per string noise models, the products (t+ l)(u+ 1) and Wt,ijk Wu,lmn in the expansion terms rapidly overwhelm the raw error term. Generalization improvement is not expected for these models. We also reason that all generalization enhancements will be valid only for a range of noise values, above which noise overwhelms the raw error information. 652 3.2 Kam Jim, Bill G. Horne, C. Lee Giles Predictions on Convergence Synaptic noise can improve convergence by favoring promising weights in the beginning stages of training. This can be demonstrated by examining the second order error expansion term for non-cumulative, multiplicative, per time step noise: When fp is negative, solutions with a negative second order state-weight partial derivative will be de-stabilized. In other words, when the output Sb is too small the network will favor updating in a direction such that the first order partial derivative is increasing. A corresponding relationship can be observed for the case when fp is positive. Thus the second order term of the error function will allow a higher raw error fp to be favored if such an update will place the weights in a more promising area, i.e. a region where weight changes are likely to move Sb in a direction to reduce the raw error. The anticipatory effect of this term is more important in the beginning stages of training where fp is large, and will become insignificant in the finishing stages of training as fp approaches zero. Similar to arguments in Section 3.1, the absence of weight decay will make the learning task easier and improve convergence. From this discussion it can be inferred that additive per time step noise models should yield the best generalization and convergence performance because of their sign-independent favoring of saturated states and the absence of weight decay. Furthermore, convergence and generalization performance is more sensitive to cumulative noise, i.e. optimal performance and detrimental effects will occur at lower amplitudes than in non-cumulative noise. 4 SIMULATION RESULTS In order to perform many experiments in a reasonable amount of computation time, we attempt to learn the simple "hidden-state" dual parity automata from sample strings encoded as temporal sequences. (Dual parity is a 4-state automata that recognizes binary strings containing an even number of ones and zeroes.) We choose a second-order recurrent network since such networks have demonstrated good performance on such problems (Giles et. al., 1992). Thus our experiments consist of 500 simulations for each data point and achieve useful (90%) confidence levels. Experiments are performed with both 3 and 4 state networks, both of which are adequate to learn the automata. The learning rate and momentum are set to 0.5, and the weights are initialized to random values between [-1.0, 1.0]. The data consists of 8191 strings of lengths 0 to 12. The networks are trained on a subset of the training set, called the working set, which gradually increases in size until the entire training set is classified correctly. Strings from the working set are presented in alphabetical order. The training set consists of the first 1023 strings of lengths 0 to 9, while the initial working set consists of 31 strings of lengths 0 to 4. During testing no noise is added to the weights of the network. Effects of Noise on Convergence and Generalization in Recurrent Networks 653 300rT-----,--~--rT----,_----. e ? ?? ???? ? ? b 2150 1 i 200 . ? S; ~150 ~ ~OO . 150 . ~ 2 TraininG Nole. Std Oev 3 o 0 0_15 ~ ~.15 Traln'nG Nol_. Std Oev 2 Figure 1: Best Convergence/Generalization for Additive and Multiplicative Noises. (a) multiplicative non-cumulative per time step; (b) additive cumulative per time step. 4.1 Convergence and Generalization Performance Simulated performance closely mirror our predictions. Improvements were observed for all noise models except for cumulative per string noises which failed to converge for all runs. Generalization improvement was more emphasized on networks with 4 states, while convergence enhancement was more noticeable on 3-state networks. The simulations show the following results: ? Additive noise is better tolerated than multiplicative noise, and achieves better convergence and generalization (Figure 1). ? Cumulative noise achieves optimal generalization and convergence at lower amplitudes than non-cumulative noise. Cumulative noise also has a narrower range of beneficial noise, which is defined as the range of noise amplitudes which yields better performance than that of a noiseless network (Figure 2a illustrates this for generalization). ? Per time step noise achieves better convergence/generalization and has a wider range of beneficial values than per string noise (Figure 2b). Overall, the best performance is obtained by applying cumulative and noncumulative additive noise at each time step. These results closely match the predictions of section 3.1. The only exceptions are that all multiplicative noise models seem to yield equivalent performance. This discrepancy between prediction and simulation may be due to the detrimental effects of weight decay in multiplicative noise , which can conflict with the advantages of cumulative and per time step noise. 4.2 The Payoff Picture: Generalization vs. Convergence Generalization vs. Convergence results are plotted in Figure 3. Increasing noise amplitudes proceed from the left end-point of each curve to the right end-point. 654 Kam Jim, Bill G. Horne, C. Lee Giles Table 1: Examples: Additive Noise Accumulation. ~i is the noise at time step ti TIME STEPS NOISE MODEL per time step non-cumulative per time step cumulative per sequence non-cumulative per sequence cumulative t1 t2 t3 W+~l W+~2 W +~1 +~2 W+~l W+2~1 W+~3 W+~l W+~l W+~l W ... ... +~1 +~2 + ~3 W+~l W+3~1 ... ... ... 300rT------r---r-~r---~----_. 2150 j200 .5 , 1150 ~ 100 1 00 1 2 Training Not_. S'td D_v Figure 2: (I) Best Generalization for Cumulative and Non-Cumulative Noises: a) cumulative additive per time step; b) non-cumulative additive per time step. (II) Best Generalization for Per Time Step and Per String Noises: a) non-cumulative per string additive; b) non-cumulative per time step additive. e e 15.15 5.5 5 .) a. .. .. .. 15 4 .5 ~ ~ .... i ~ ~ II 3.15 ~ 'l\J 1>5 Iii ~ 2.5 3 4 3 .5 3 CJ 2 .5 2 2 c 1.15 ~oo 1 .5 200 300 Convergence In epocha 400 ~OO 200 300 Convergence 'n epochs 400 Figure 3: Payoff: Mean Generalization vs. Convergence for 4-state (I) and 3state(lI) recurrent network. (I a) Worst i-state - non-cumulative multiplicative per string; (Ib, Ic) Best 4-state - cumulative and non-cumulative additive per time step, respectively; (lIa) Worst 3-state - non-cumulative multiplicative per string; (lib) Best 3-state - cumulative additive per time step. Effects of Noise on Convergence and Generalization in Recurrent Networks 655 Table 2: Noise injection step and error expansion terms for per time step nOIse models. v = min(t + 1, U + 1). W'" is the noise-free weight set. NON-CUMULATIVE Addlhve MU bpncahve Noise step l.tt order 2nd order ClIM~LATIVE Additive M ulhplicati ve t Noise step Wt,ijk = wtjl< + L t AT,iJI< = wtjl< Wt,ijl< T-O II (1 + AThl<) T-O l.tt order L T-1 ~~ ~2 2 P 2nd order ~ T ~ "Wt ,.JI< .. W ._ So Lu,.JI< 8W .. 8W .. .. t,.JI< u,.JI< t ,u=O .JI< NON-CUMULATIVE AddItIve Multlpllcahve Noise step 1 1st order -~ 2 2 1 2 P 2nd order -~ T-1 2: L T W .. t,u=O .JI< T 1 ~2 L 2: " t,.JI< T 850 u,.JI< 8W . - .. t,.JI< 8S0 8W . . _2 W t,u=O ijl< ;UM ILA IVE Additive W u,.JI< T crSO .. W .. t,.JI< u,.JI< 8W .. W . . t,.Jk u,.JI< Multiplicative Noise step Wt,ijl< = W i jl?l + Aijl<)' t = wijl< + 2: Ct,T(Aijl<)T+1 T-O 2: 2: TIT 1st order 1 2 -~ 2 .. t,u=O .JI< 0 w s85 -- T 85 0 8Wt ,ijl< 8Wu ,ijl< T-1 -2: 1 2 " .. " ~ L- t,.JI< t,u=O ijl<,lmn T-1 2~~ +2~p T T 85 85 u,lmn _ 8W0._ . 8W 0 t,.JI< u,lmn T L- L"t ' i j l8Wt <--.. ,ijl< 8S 0 t=O .JI< 2: L T-1 2nd order 1 -~ 2 P .. t,u=O .JI<,lmn _2 "t,.JI< .. "u ,lmn 8w .' t,.JI< T crS O 8W u,lmn 656 Kam Jim, Bill G. Home, C. Lee Giles These plots illustrate the cases where both convergence and generalization are improved. In figure 311 the curves clearly curl down and to the left for lower noise amplitudes before rising to the right at higher noise amplitudes. These lower regions are important because they represent noise values where generalization and convergence improve simultaneously and do not trade off. 5 CONCLUSIONS We have presented several methods of injecting synaptic noise to recurrent neural networks. We summarized the results of an analysis of these methods and empirically tested them on learning the dual parity automaton from strings encoded as temporal sequences. (For a complete discussion of results, see (Jim, Giles, and Horne, 1994) ). Results show that most of these methods can improve generalization and convergence simultaneously - most other methods previously discussed in literature cast convergence as a cost for improved generalization performance. References [1] Chris M. Bishop. Training with noise is equivalent to Tikhonov Regularization. Neural Computation, 1994. To appear. [2] Robert M. Burton, Jr. and George J. Mpitsos. Event-dependent control of noise enhances learning in neural networks. Neural Networks, 5:627-637, 1992. [3] C.L. Giles, C.B. Miller, D. Chen, H.H. Chen, G.Z . Sun, and Y.C. Lee. Learning and extracting finite state automata with second-order recurrent neural networks. Neural Computation, 4(3):393-405, 1992. [4] Stephen Jose Hanson. A stochastic version ofthe delta rule. Physica D., 42:265272, 1990. [5] Kam Jim, C.L. Giles, and B.G. Horne. Synaptic noise in dynamically-driven recurrent neural networks: Convergence and generalization. Technical Report UMIACS-TR-94-89 and CS-TR-3322, Institute for Advanced Computer Studies, University of Maryland, College Park, MD, 1994. [6] Stephen Judd and Paul W . Munro. Nets with unreliable hidden nodes learn error-correcting codes. In S.J Hanson, J.D. Cowan, and C.L. Giles, editors, Advances in Neural Information Processing Systems 5, pages 89-96, San Mateo, CA, 1993. Morgan Kaufmann Publishers. [7] Anders Krogh and John A. Hertz. A simple weight decay can improve generalization. In J .E. Moody, S.J. Hanson, and R.P. Lippmann, editors, Advances in Neural Information Processing Systems 4, pages 450-957, San Mateo, CA, 1992. Morgan Kaufmann Publishers. [8] Alan F. Murray and Peter J. Edwards. Synaptic weight noise during multilayer perceptron training: Fault tolerance and training improvements. IEEE Trans. on Neural Networks, 4(4):722-725, 1993. [9] Carlo H. Sequin and Reed D. Clay. Fault tolerance in artificial neural networks. In Proc. of IJCNN, volume I, pages 1-703-708, 1990. 

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Pairwise Neural Network Classifiers with Probabilistic Outputs David Price A2iA and ESPCI 3 Rue de l'Arrivee, BP 59 75749 Paris Cedex 15, France a2ia@dialup.francenet.fr Stefan Knerr ESPCI and CNRS (UPR AOOO5) 10, Rue Vauquelin, 75005 Paris, France knerr@neurones.espci.fr Leon Personnaz, Gerard Dreyfus ESPeI, Laboratoire d'Electronique 10, Rue Vauquelin, 75005 Paris, France dreyfus@neurones.espci.fr Abstract Multi-class classification problems can be efficiently solved by partitioning the original problem into sub-problems involving only two classes: for each pair of classes, a (potentially small) neural network is trained using only the data of these two classes. We show how to combine the outputs of the two-class neural networks in order to obtain posterior probabilities for the class decisions. The resulting probabilistic pairwise classifier is part of a handwriting recognition system which is currently applied to check reading. We present results on real world data bases and show that, from a practical point of view, these results compare favorably to other neural network approaches. 1 Introduction Generally, a pattern classifier consists of two main parts: a feature extractor and a classification algorithm. Both parts have the same ultimate goal, namely to transform a given input pattern into a representation that is easily interpretable as a class decision. In the case of feedforward neural networks, the interpretation is particularly easy if each class is represented by one output unit. For many pattern recognition problems, it suffices that the classifier compute the class of the input pattern, in which case it is common practice to associate the pattern to the class corresponding to the maximum output of the classifier. Other problems require graded (soft) decisions, such as probabilities, at the output of the 1110 David Price, Stefan Knerr, Leon Personnaz, Gerard Dreyfus classifier for further use in higher context levels: in speech or character recognition for instance, the probabilistic outputs of the phoneme (character) recognizer are often used by a Hidden-Markov-Model algorithm or by some other dynamic programming algorithm to compute the most probable word hypothesis. In the context of classification, it has been shown that the minimization of the Mean Square Error (MSE) yields estimates of a posteriori class probabilities [Bourlard & Wellekens, 1990; Duda & Hart, 1973]. The minimization can be performed by a feedforward multilayer perceptrons (MLP's) using the backpropagation algorithm, which is one of the reasons why MLP's are widely used for pattern recognition tasks. However, MLPs have well-known limitations when coping with real-world problems, namely long training times and unknown architecture. In the present paper, we show that the estimation of posterior probabilities for a K-class problem can be performed efficiently using estimates of posterior probabilities for K(K-1 )/2 two-class sub-problems. Since the number of sub-problems increases as K2, this procedure was originally intended for applications involving a relatively small number of classes, such as the 10 classes for the recognition of handwritten digits [Knerr et aI., 1992]. In this paper we show that this approach is also viable for applications with K? 10. The probabilistic pairwise classifier presented in this paper is part of a handwriting recognition system, discussed elsewhere [Simon, 1992], which is currently applied to check reading. The purpose of our character recognizer is to classify pre-segmented characters from cursive handwriting. The probabilistic outputs of the recognizer are used to estimate word probabilities. We present results on real world data involving 27 classes, compare these results to other neural network approaches, and show that our probabilistic pairwise classifier is a powerful tool for computing posterior class probabilities in pattern recognition problems. 2 Probabilistic Outputs from Two-class Classifiers Multi-class classification problems can be efficiently solved by "divide and conquer" strategies which partition the original problem into a set of K(K-l)/2 two-class problems. For each pair of classes (OJ and (OJ, a (potentially small) neural network with a single output unit is trained on the data of the two classes [Knerr et aI., 1990, and references therein]. In this section, we show how to obtain probabilistic outputs from each of the two-class classifiers in the pairwise neural network classifier (Figure 1). K(K-I)12 two-class networks inputs Figure 1: Pairwise neural network classifier. Pairwise Neural Network Classifiers with Probabilistic Outputs 1111 It has been shown that the \llinimization of the MSE cost function (or likewise a cost function based on an entropy measure, [Bridle, 1990]) leads to estimates of posterior probabilities. Of course, the quality of the estimates depends on the number and distribution of examples in the training set and on the minimization method used. In the theoretical case of two classes <01 and <02, each Gaussian distributed, with means m 1 and m2, a priori probabilities Pq and Pr2, and equal covariance matrices ~, the posterior probability of class <01 given the pattern x is: Pr(class=<o\ I X=x) = _ _ _ _ _ _ _ _ _ _--'1'----_ _ _ _ _ _ _ _ __ 1 + Pr2 exp( _ !-(2xT~-\(m\-m2) + m!~-\m2 - m T~-lm\)) PrJ 2 (1) Thus a single neuron with a sigmoidal transfer function can compute the posterior probabilities for the two classes. However, in the case of real world data bases, classes are not necessarily Gaussian distributed, and therefore the transformation of the K(K-l )/2 outputs of our pairwise neural network classifier to posterior probabilities proceeds in two steps. In the first step, a class-conditional probability density estimation is performed on the linear output of each two-class neural network: for both classes <OJ and <OJ of a given twoclass neural network, we fit the probability density over Vjj (the weighted sum of the inputs of the output neuron) to a function. We denote by <Ojj the union of classes <OJ and <OJ. The resulting class-conditional densities p(vij I <OJ) and p(Vjj I <OJ) can be transformed to probabilities Pr(<Oj I <OJ' /\ (Vij=Vjj? and Pr(<Oj I <Ojj /\ (Vjj=Vjj? via the Bayes rule (note that Pr(<Ojj /\ (Vij=Vjj) <OJ) = Pr?Vij=Vjj) I <OJ)): 1 p( VjJ" I <OJ) Pr( <OJ) Pr(<Oj I <Ojj/\(Vij=Vij? = - - - - - " - - - - - p(Vjj I <Ok) Pr(<Ok) ke{j,j} L (2) It is a central assumption of our approach that the linear classifier output Vij is as informative as the input vector x. Hence, we approximate Prij = Pr(<Oj I <Ojj /\ (X=x? by Pr(<Oi I <Ojj /\ (V=Vjj?. Note that Pji = I-Pjj. In the second step, the probabilities Prij are combined to obtain posterior probabilities Pr(<Oj I (X=x? for all classes <Oi given a pattern x. Thus, the network can be considered as generating an intermediate data representation in the recognition chain, subject to further processing [Denker & LeCun, 1991]. In other words, the neural network becomes part of the preprocessing and contributes to dimensionality reduction. 3 Combining the Probabilities Prij of the Two-class Classifiers to a posteriori Probabilities The set of two-class neural network classifiers discussed in the previous section results in probabilities Prjj for all pairs (i, j) with i j. Here, the task is to express the posterior probabilities Pr(<Oj I (X=x? as functions of the Prjj- * 1112 David Price, Stefan Knerr, Leon Personnaz, Gerard Dreyfus We assume that each pattern belongs to only one class: K (3) Pr( U Olj I (X=x? = 1 j=1 From the definition of Olij. it follows for any given i: K K Pr(U Olj I (X=x? = Pr( U Olij I (X=x? = 1 j=l,j*i J=I (4) U sing the closed form expression for the probability of the union of N events Ei: N N N L L Pr(U Ei) = Pr(Ej) +... + (_I)k.1 Pr(EhA ... AEh) +...+ (-I)N.l pr(EIA ... AEN) i= 1 i= 1 i}< ... <ik it follows from (4): K L Pr(Olij I (X=x? - (K-2) Pr(Oli I (X=x? = 1 (5) j=l,j*i With Prij = Pr(Oli I OlijA(X=X? = Pr(OliAOli'A(X=X? J Pr(OlijA(X=X? Pr(Oli I (X=x? = --'----~ Pr(Olij I (X=x? (6) one obtains the final expression for the K posterior probabilities given the K(K-l)12 twoclass probabilities Prji : Pr(Oli I (X=x? = _ _--"-1_ __ f (7) _1__ (K-2) j=I,#i Prij In [Refregier et aI., 1991], a method was derived which allows to compute the K posterior probabilities from only (K-l) two-class probabilities using the following relation between posterior probabilities and two-class probabilities: Prij = Pr(Oli I (X=x? Prji Pr(Olj I (X=x? (8) However, this approach has several practical drawbacks. For instance, in practice, the quality of the estimation of the posterior probabilities depends critically on the choice of the set of (K-l) two-class probabilities, and finding the optimal subset of (K-l) Prij is costly, since it has to be performed for each pattern at recognition time. Pairwise Neural Network Classifiers with Probabilistic Outputs 4 1113 Application to Cursive Handwriting Recognition We applied the concepts described in the previous sections to the classification of presegmented characters from cursive words originating from real-world French postal checks. For cursive word recognition it is important to obtain probabilities at the output of the character classifier since it is necessary to establish an ordered list of hypotheses along with a confidence value for further processing at the word recognition level: the probabilities can be passed to an Edit Distance algorithm [Wagner et at, 1974] or to a Hidden-Markov-Model algorithm [Kundu et aI., 1989] in order to compute recognition scores for words. For the recognition of the amounts on French postal checks we used an Edit Distance algorithm and made extensive use of the fact that we are dealing with a limited vocabulary (28 words). The 27 character classes are particularly chosen for this task and include pairs of letters such as "fr", "gttl, and "tr" because these combinations of letters are often difficult to presegment. Other characters, such as tlk" and "y" are not included because they do not appear in the given 28 word vocabulary. 0~)(" C.A~ t:; ~"U::l._ r;.., ~ \\:..~~~ (~tSl\h~~ \J'~ ~~~ ~~~~~~ &.nl' tL'i.upr rm'A Figure 2: Some examples of literal amounts from live French postal checks. A data base of about 3,300 literal amounts from postal checks (approximately 16,000 words) was annotated and, based on this annotation, segmented into words and letters using heuristic methods [Simon et aI., 1994]. Figure 2 shows some examples of literal amounts. The writing styles vary strongly throughout the data base and many checks are difficult to read even for humans. Note that the images of the pre-segmented letters may still contain some of the ligatures or other extraneous parts and do not in general resemble hand-printed letters. The total of about 55,000 characters was divided into three sets: training set (20,000), validation set (20,000), and test set (15,000). All three sets were used without any further data base cleaning. Therefore, many letters are not only of very bad quality, but they are truly ambiguous: it is not possible to recognize them uniquely without word context. Figure 3: Reference lines indicating upper and lower limit of lower case letters. Before segmentation, two reference lines were detected for each check (Figure 3). They indicate an estimated upper and lower limit of the lower case letters and are used for 1114 David Price, Stefan Knerr, Leon Personnaz, Gerard Dreyfus nonnalization of the pre-segmented characters (Figure 4) to 10 by 24 pixel matrices with 16 gray values (Figure 5). This is the representation used as input to the classifiers. Figure 4: Segmentation of words into isolated letters (ligatures are removed later). Figure 5: Size nonnalized letters: 10 by 24 pixel matrices with 16 gray values. The simplest two-class classifier is a single neuron; thus, 351 neurons of the resulting pairwise classifier were trained on the training data using the generalized delta rule (sigmoidal transfer function). In order to avoid overfitting, training was stopped at the minimum of MSE on the validation set. The probability densities P(Vij I IDi) were estimated on the validation set: for both classes IDi and IDj of a given neuron, we fitted the probability densities over the linear output Vij to a Gaussian. The two-class probabilities Prij and Prji were then obtained via Bayes rule. The 351 probabilities Prij were combined using equation (7) in order to obtain a posteriori probabilities Pr(IDi I (X=x?, i E {1, .. ,27}. However, the a priori probabilities for letters as given by the training set are different from the prior probabilities in a given word context [Bourlard & Morgan, 1994]. Therefore, we computed the posterior probabilities either by using, in Bayes rule, the prior probabilities of the letters in the training set, or by assuming that the prior probabilities are equal. In the first case, many infonnative letters, for instance those having ascenders or descenders, have little chance to be recognized at all due to small a priori probabilities. Table 1 gives the recognition perfonnances on the test set for classes assumed to have equal a priori probabilities as well as for the true a priori probabilities of the test set. For each pattern, an ordered list (in descending order) of posterior class probabilities was generated; the recognition perfonnance is given (i) in tenns of percentage of true classes found in first position, and (ii) in tenns of average position of the true class in the ordered list. As mentioned above, the results of the first column are the most relevant ones, since the classifier outputs are subsequently used for word recognition. Note that the recognition rate (first position) of isolated letters without context for a human reader can be estimated to be around 70% to 80%. We compared the results of the pairwise classifier to a number of other neural network classification algorithms. First, we trained MLPs with one and two hidden layers and various numbers of hidden units using stochastic backpropagation. Here again, training was stopped based on the minimum MSE on the validation set. Second, we trained MLPs with a single hidden layer using the Softmax training algorithm [Bridle, 1990]. As a third approach, we trained 27 MLPs with 10 hidden units each, each MLP separating one class from all others. Table 1 gives the recognition perfonnances on the test set. The Softmax 1115 Pairwise Neural Network Classifiers with Probabilistic Outputs training algorithm clearly gives the best results in terms of recognition performance. However, the pairwise classifier has three very attractive features for classifier design: (i) training is faster than for MLP's by more than one order of magnitude; therefore, many different designs (changing pattern representations for instance) can be tested at a small computational cost; (ii) in the same spirit, adding a new class or modifying the training set of an existing one can be done without retraining all two-class classifiers; (iii) at least as importantly, the procedure gives more insight into the classification problem than MLP's do. Classifier Pairwise Classifier MLP (100 hid. units) Softmax (100 hid. units) 27 MLPs AveragePosition First Position equal prior probs equal prior probs 2.9 48.9 % AveragePosition true prior probs 2.6 First Position true prior probs 52.2 % 3.6 48.9 % 2.7 60.0 % 2.6 54.9 % 2.2 61.9 % 3.2 41.6 % 2.4 55.8 % Table I: Recognition performances on the test set in terms of average position and recognition rate (first position) for the various neural networks used. Our pairwise classifier is part of a handwriting recognition system which is currently applied to check reading. The complete system also incorporates other character recognition algorithms as well as a word recognizer which operates without pre-segmentation. The result of the complete check recognition chain on a set of test checks is the following: (i) at the word level, 83.3% of true words are found in first position; (ii) 64.1 % of well recognized literal amounts are found in first position [Simon et al., 1994]. Recognizing also the numeral amount, we obtained 80% well recognized checks for 1% error. 5 Conclusion We have shown how to obtain posterior class probabilities from a set of pairwise classifiers by (i) performing class density estimations on the network outputs and using Bayes rule, and (ii) combining the resulting two-class probabilities. The application of our pairwise classifier to the recognition of real world French postal checks shows that the procedure is a valuable tool for designing a recognizer, experimenting with various data representations at a small computational cost and, generally, getting insight into the classification problem. Acknowledgments The authors wish to thank J.C. Simon, N. Gorsky, O. Baret, and J.C. Deledicq for many informative and stimulating discussions. 1116 David Price, Stefan Knerr, Leon Personnaz, Gerard Dreyfus References H.A. Bourlard, N. Morgan (1994). Connectionist Speech Recognition. Kluwer Academic Publishers. H.A. Bourlard, C. Wellekens (1990). Links between Markov Models and Multilayer Perceptrons. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 12, No. 12, 1167-1178. J.S. Bridle (1990). Probabilistic Interpretation of Feedforward Classification Network Outputs, with Relationships to Statistical Pattern Recognition. In Neurocomputing: Algorithms, Architectures and Applications, Fogelman-Soulie, and Herault (eds.). NATO ASI Series, Springer. J.S. Denker, Y.LeCun (1991). Transforming Neural-Net Output Levels to Probability Distributions. In Advances in Neural Information Processing Systems 3, Lippmann, Moody, Touretzky (eds.). Morgan Kaufman. R.O. Duda, P.E. Hart (1973). Pattern Classification and Scene Analysis. Wiley. S. Knerr, L. Personnaz, G. Dreyfus (1990). Single-Layer Learning Revisited: A Stepwise Procedure for Building and Training a Neural Network. In Neurocomputing: Algorithms, Architectures and Applications, Fogelman-Soulie and Herault (eds.). NATO AS! Series, Springer. S. Knerr, L. Personnaz, G. Dreyfus (1992). Handwritten Digit Recognition by Neural Networks with Single-Layer Training. IEEE Transactions on Neural Networks, Vol. 3, No.6, 962-968. A. Kundu, Y. He, P. Bahl (1989). Recognition of Handwritten Words: First and Second Order Hidden Markov Model Based Approach. Pattern Recognition, Vol. 22, No.3. lC. Simon (1992). Off-Line Cursive Word Recognition. Proceedings of the IEEE, Vol. 80, No.7, 1150-1161. Ph. Refregier, F. Vallet (1991). Probabilistic Approach for Multiclass Classification with Neural Networks. Int. Conference on Artificial Networks, Vol. 2, 1003-1007. J.C. Simon, O. Baret, N. Gorski (1994). Reconnaisance d'ecriture manuscrite. Compte Rendu Academie des Sciences, Paris, t. 318, Serie II, 745-752. R.A. Wagner, M.J. Fisher (1974). The String to String Correction Problem. J.A.C.M. Vol. 21, No.5, 168-173. . 

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Spatial Representations in the Parietal Cortex May Use Basis Functions Alexandre Pouget alex@salk.edu Terrence J. Sejnowski terry@salk.edu Howard Hughes Medical Institute The Salk Institute La Jolla, CA 92037 and Department of Biology University of California, San Diego Abstract The parietal cortex is thought to represent the egocentric positions of objects in particular coordinate systems. We propose an alternative approach to spatial perception of objects in the parietal cortex from the perspective of sensorimotor transformations. The responses of single parietal neurons can be modeled as a gaussian function of retinal position multiplied by a sigmoid function of eye position, which form a set of basis functions. We show here how these basis functions can be used to generate receptive fields in either retinotopic or head-centered coordinates by simple linear transformations. This raises the possibility that the parietal cortex does not attempt to compute the positions of objects in a particular frame of reference but instead computes a general purpose representation of the retinal location and eye position from which any transformation can be synthesized by direct projection. This representation predicts that hemineglect, a neurological syndrome produced by parietal lesions, should not be confined to egocentric coordinates, but should be observed in multiple frames of reference in single patients, a prediction supported by several experiments. Alexandre Pouget, Terrence J. Sejnowski 158 1 Introduction The temporo-parietal junction in the human cortex and its equivalent in monkeys, the inferior parietal lobule, are thought to playa critical role in spatial perception. Lesions in these regions typically result in a neurological syndrome, called hemineglect, characterized by a lack of motor exploration toward the hemispace contralateral to the site of the lesion. As demonstrated by Zipser and Andersen [11), the responses of single cells in the monkey parietal cortex are also consistent with this presumed role in spatial perception. In the general case, recovering the egocentric position of an object from its multiple sensory inputs is difficult because of the multiple reference frames that must be integrated . In this paper, we consider a simpler situation in which there is only visual input and all body parts are fixed but the eyes, a condition which has been extensively used for neurophysiological studies in monkeys. In this situation, the head-centered position of an object, X, can be readily recovered from the retinal location,R, and current eye position, E, by vector addition: (1) If the parietal cortex contains a representation of the egocentric position of objects, then one would expect to find a representation of the vectors, X, associated with these objects. There is an extensive literature on how to encode a vector with a population of neurons, and we first present two schemes that have been or are used as working hypothesis to study the parietal cortex. The first scheme involves what is typically called a computational map, whereas the second uses a vectorial representation [9]. This paper shows that none of these encoding schemes accurately accounts for all the response properties of single cells in the parietal cortex. Instead, we propose an alternative hypothesis which does not aim at representing X per se; instead, the inputs Rand E are represented in a particular basis function representation. We show that this scheme is consistent with the way parietal neurons respond to the retinal position of objects and eye position, and we give computational arguments for why this might be an efficient strategy for the cortex. 2 Maps and Vectorial Representations One way to encode a two-dimensional vector is to use a lookup table for this vector which, in the case of a two-dimensional vector, would take the form of a twodimensional neuronal map. The parietal cortex may represent the egocentric location of object, X, in a similar fashion. This predicts that the visual receptive field of parietal neurons have a fixed position with respect to the head (figure IB). The work of Andersen et al. (1985) have clearly shown that this is not the case. As illustrated in figure 2A, parietal neurons have retinotopic receptive fields . In a vectorial representation, a vector is encoded by N units, each of them coding for the projection of the vector along its preferred direction. This entails that the activity, h, of a neuron is given by: Spatial Representations in the Parietal Cortex May Use Basis Functions B Map Representation A ~ V 159 C Vectorial Representation ?io j[ o 0 0 o vx ?1110 -'JO 0 90 180 a(I)'gr") Figure 1: Two neural representations of a vector . A) A vector if in cartesian and polar coordinates. B) In a map representation, units have a narrow gaussian tuning to the horizontal and vertical components of if . Moreover, the position of the peak response is directly related to the position of the units on the map. C) In a vectorial representation , each unit encodes the projection of if along its preferred direction ( central arrows) . This results in a cosine tuning to the vector angle, () . (2) Wa is usually called the preferred direction of the cells because the activity is maximum whenever () = 0; that is, when A points in the same direction as Wa. Such neurons have a cosine tuning to the direction of the egocentric location of objects, as shown also in figure lC. Cosine tuning curves have been reported in the motor cortex by Georgopoulos et al. (1982) , suggesting that the motor cortex uses a vectorial code for the direction of hand movement in extrapersonal space. The same scheme has been also used by Goodman and Andersen (1990), and Touretzski et al. (1993) to model the encoding of egocentric position of objects in the parietal cortex. Touretzski et al. (1993) called their representation a sinusoidal array instead of a vectorial representation. Using Eq. 1, we can rewrite Eq. 2: (3) This second equation is linear in Ii and if and uses the same vectors , dot products. This leads to three important predictions: Wa , in both 1) The visual receptive fields of parietal neurons should be planar. 2) The eye position receptive fields of parietal neurons should also be planar; that is, for a given retinal positions, the response of parietal neuron should be a linear function of eye position. 160 Alexandre Pouget, Terrence J. Sejnowski B A @ ? o -101'--'---'--~-'--'--'---'---' -40 -20 0 20 40 Retinal Position (Deg) Figure 2: Typical response of a neuron in the parietal cortex of a monkey. A) Visual receptive field has a fixed position on the retina, but the gain of the response is modulated by eye position (ex). (Adpated from Andersen et al., 1985) B) Example of an eye position receptive field, also called gain field, for a parietal cell. The nine circles indicate the amplitude of the response to an identical retinal stimulation for nine different eye positions. Outer circles show the total activity, whereas black circles correspond to the total response minus spontaneous activity prior to visual stimulation. (Adpated from Zipser et al., 1988) 3) The preferred direction for retinal location and eye position should be identical. For example, if the receptive field is on the right side of the visual field , the gain field should also increase with eye positon to the right side. The visual receptive fields and the eye position gain fields of single parietal neurons have been extensively studied by Andersen et al. [2]. In most cases, the visual receptive fields were bell-shaped with one or several peaks and an average radius of 22 degrees of visual angle [1], a result that is clearly not consistent with the first prediction above . We show in figure 2A an idealized visual receptive field of a parietal neuron. The effect of eye position on the visual receptive field is also illustrated. The eye position clearly modulates the gain of the visual response. The prediction regarding the receptive field for eye position has been borne out by statistical analysis. The gain fields of 80% of the cells had a planar component [1 , 11] . One such gain field is shown in figure 2B. There is not enough data available to determine whether or not the third prediction is valid. However, indirect evidence suggests that if such a correlation exists between preferred direction for retinal location and for eye position, it is probably not strong. Cells with opposite preferred directions [2, 3] have been observed. Furthermore, although each hemisphere represents all possible preferred eye position directions, there is a clear tendency to overrepresent the contralateral retinal hemifield [1]. In conclusion, the experimental data are not fully consistent with the predictions of the vectorial code. The visual receptive fields, in particular, are strongly nonlinear. If these nonlinearities are computationally neutral, that is, they are averaged out in subsequent stages of processing in the cortex, then the vectorial code could capture Spatial Representations in the Parietal Cortex May Use Basis Functions 161 the essence of what the parietal cortex computes and, as such, would provide a valid approximation of the neurophysiological data. We argue in the next section that the nonlinearities cannot be disregarded and we present a representational scheme in which they have a central computational function. 3 3.1 Basis Function Representation Sensorimotor Coordination and Nonlinear Function Approximation The function which specified the pattern of muscle activities required to move a limb, or the body, to a specific spatial location is a highly nonlinear function of the sensory inputs. The cortex is not believed to specify patterns of muscle activation, but the intermediate transformations which are handled by the cortex are often themselves nonlinear . Even if the transformations are actually linear, the nature of cortical representations often makes the problem a nonlinear mapping. For example, there exists in the putamen and premotor cortex cells with gaussian head-centered visual receptive fields [7J which means that these cells compute gaussians of A or, equivalently, gaussians of R + E, which is nonlinear in Rand E. There are many other examples of sensory remappings involving similar computations. If the parietal cortex is to have a role in these remappings, the cells should respond to the sensory inputs in a way that can be used to approximate the nonlinear responses observed elsewhere. One possibility would be for parietal neurons to represent input signals such as eye position and retinal location with basis functions. A basis function decomposition is a well-known method for approximating nonlinear functions which is, in addition, biologically plausible [8J. In such a representation, neurons do not encode the head-centered locations of objects, A; instead, they compute functions of the input variables, such as Rand E, which can be used subsequently to approximate any functions of these variables. 3.2 Predictions of the Basis Function Representation Not all functions are basis functions. Linear functions do not qualify, nor do sums of functions which, individually, would be basis functions, such as gaussian functions of retinal location plus a sigmoidal functions of eye position. If the parietal cortex uses a basis function representation, two conditions have to be met: 1) The visual and the eye position receptive fields should be smooth nonlinear function of Rand E. 2) The selectivities to Rand E should interact nonlinearly The visual receptive fields of parietal neurons are typically smooth and nonlinear. Gaussian or sum of gaussians appear to provide good models of their response profiles [2]. The eye position receptive field on the other hand, which is represented by the gain field, appears to be approximately linear. We believe, however, that the published data only demonstrate that the eye position receptive field is monotonic, 162 Alexandre Pouget, Terrence J. Sejnowski Head-Centered Retinotopic o o Figure 3: Approximation of a gaussian head-centered (top-left) and a retinotopic (top-right) receptive field, by a linear combination of basis function neurons. The bottom 3-D plots show the response to all possible horizontal retinal position, r x , and horizontal eye positions, ex, of four typical basis function units. These units are meant to model actual parietal neurons but not necessarily linear. In published experiments, eye position receptive fields (gain fields) were sampled at only nine points, which makes it difficult to distinguish between a plane and other functions such as a sigmoidal function or a piecewise linear function. The hallmark of a nonlinearity would be evidence for saturation of activity within working range of eye position. Several published gain fields show such saturations [3, 11], but a rigorous statistical analysis would be desirable. Andersen et al. (1985) have have shown that the responses of parietal neurons are best modeled by a multiplication between the retinal and eye position selectivities which is consistent with the requirements for basis functions. Therefore, the experimental data are consistent with our hypothesis that the parietal cortex uses a basis function representation. The response of most gain-modulated neurons in the parietal cortex could be modeled by multiplying a gaussian tuning to retinal position by a sigmoid of eye position, a function which qualifies as a basis function. 3.3 Simulations We simulated the response of 121 parietal gain-modulated neurons modeled by multiplying a gaussian of retinal position, r x , with a sigmoid of eye position, ex : Spatial Representations in the Parietal Cortex May Use Basis Functions e- (rz-rra):il ~ ~ .. h,o=----1 +e- t e,,;-e,l'j 163 (4) where the centers of the gaussians for retinalloction rxi and the positions of the sigmoids for eye postions exi were uniformly distributredo The widths of the gaussian (T and the sigmoid t were fixed. Four of these functions are shown at the bottom of figure 3. We used these basis functions as a hidden layer to approximate two kinds of output functions: a gaussian head-centered receptive field and a gaussian retinotopic receptive field . Neurons with these response properties are found downstream of the parietal cortex in the premotor cortex [7] and superior colliculus, two structures believed to be involved in the control of, respectively, arm and eye movements. The weights for a particular output were obtained by using the delta rule. Weights were adjusted until the mean error was below 5% of the maximum output value. Figure 3 shows our best approximations for both the head-centered and retinotopic receptive fields. This demonstrates that the same pool of neurons can be used to approximate several diffferent nonlinear functions. 4 Discussion Neurophysiological data support our hypothesis that the parietal cortex represents its inputs, such as the retinal location of objects and eye position, in a format suitable to non-linear function approximation, an operation central to sensorimotor coordination. Neurons have gaussian visual receptive fields modulated by monotonic function of eye position leading to response function that can be modeled by product of gaussian and sigmoids. Since the product of gaussian and sigmoids forms basis functions, this representation is good for approximating nonlinear functions of the input variables. Previous attempts to characterize spatial representations have emphasized linear encoding schemes in which the location of objects is represented in egocentric coordinates. These codes cannot be used for nonlinear function approximation and, as such, may not be adequate for sensorimotor coordination [6, 10]. On the other hand, such representations are computationally interesting for certain operations, like addition or rotation. Some part of the brain more specialized in navigation like the hippocampus might be using such a scheme [10]. In figure 3, a head-centered or a retinotopic receptive field can be computed from the same pool of neurons. It would be arbitrary to say that these neurons encode the positions of objects in egocentric coordinates. Instead, these units encode a position in several frames of reference simultaneously. If the parietal cortex uses this basis function representation, we predict that hemineglect, the neurological syndrome which results from lesions in the parietal cortex, should not be confined to any particular frame of reference. This is precisely the conclusion that has emerged from recent studies of parietal patients [4]. Whether the behavior of parietal patients can be fully explained by lesions of a basis function representation remains to be investigated. 164 Alexandre Pouget, Terrence J. Sejnowski Acknowledgments We thank Richard Andersen for helpful conversations and with access to unpublished data. References [1] R.A. Andersen, C. Asanuma, G. Essick, and R.M. Siegel. Corticocortical connections of anatomically and physiologically defined subdivisions within the inferior parietal lobule. Journal of Comparative Neurology, 296(1):65-113, 1990. [2] R.A . Andersen, G.K. Essick, and R.M. Siegel. Encoding of spatial location by posterior parietal neurons. Science, 230:456-458 , 1985. [3] R.A. Andersen and D. Zipser. The role of the posterior parietal cortex in coordinate transformations for visual-motor integration. Canadian Journal of Physiology and Pharmacology, 66:488-501, 1988. [4] M. Behrmann and M. Moscovitch. Object-centered neglect in patient with unilateral neglect: effect of left-right coordinates of objects. Journal of Cognitive Neuroscience , 6:1-16, 1994. [5] A.P. Georgopoulos, J.F. Kalaska, R. Caminiti, and J.T. Massey. On the relations between the direction of two-dimensional arm movements and cell discharge in primate motor cortex. Journal of Neuroscience, 2(11):1527-1537, 1982. [6] S.J. Goodman and R.A. Andersen. Algorithm programmed by a neural model for coordinate transformation. In International Joint Conference on Neural Networks, San Diego, 1990. [7] M.S. Graziano, G.s. Yap, and C.G. Gross. Coding of visual space by premotor neurons. Science, 266:1054-1057, 1994. [8] T. Poggio. A theory of how the brain might work. Cold Spring Harbor Symposium on Quantitative Biology, 55:899-910, 1990. [9] J .F. Soechting and M. Flanders. Moving in three-dimensional space: frames of reference, vectors and coordinate systems. Annual Review in Neuroscience, 15:167-91, 1992. [10] D.S. Touretzky, A.D . Redish, and H.S . Wan. Neural representation of space using sinusoidal arrays. Neural Computation, 5:869-884, 1993. [11] D. Zipser and R.A. Andersen. A back-propagation programmed network that stimulates reponse properties of a subset of posterior parietal neurons. Nature, 331:679- 684, 1988. 

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A Comparison of Discrete-Time Operator Models for Nonlinear System Identification Andrew D. Back, Ah Chung Tsoi Department of Electrical and Computer Engineering, University of Queensland St. Lucia, Qld 4072. Australia. e-mail: {back.act}@elec.uq.oz.au Abstract We present a unifying view of discrete-time operator models used in the context of finite word length linear signal processing. Comparisons are made between the recently presented gamma operator model, and the delta and rho operator models for performing nonlinear system identification and prediction using neural networks. A new model based on an adaptive bilinear transformation which generalizes all of the above models is presented. 1 INTRODUCTION The shift operator, defined as qx(t) ~ x(t + 1), is frequently used to provide time-domain signals to neural network models. Using the shift operator, a discrete-time model for system identification or time series prediction problems may be constructed. A common method of developing nonlinear system identification models is to use a neural network architecture as an estimator F(Y(t), X(t); 0) of F(Y(t), X(t?, where 0 represents the parameter vector of the network. Shift operators at the input of the network provide the regression vectors Y(t-l) [yet-I), ... , y(t-N)]', andX(t) [x(t), ... , x(t-M)]' in a manner analogous to linear filters, where [.], represents the vector transpose. = = It is known that linear models based on the shift operator q suffer problems when used to modellightly-damped-Iow-frequency (LDLF) systems, with poles near (1,0) on the unit circle in the complex plane [5]. As the sampling rate increases, coefficient sensitivity and round-off noise become a problem as the difference between successive sampled inputs becomes smaller and smaller. 884 Andrew D. Back, Ah Chung Tsoi A method of overcoming this problem is to use an alternative discrete-time operator. Agarwal and Burrus first proposed the use of the delta operator in digital filters to replace the shift operator in an attempt to overcome the problems described above [1]. The delta operator is defined as q-l {) = (1) ~ where ~ is the discrete-time sampling interval. Williamson showed that the delta operator allows better performance in terms of coefficient sensitivity for digital filters derived from the direct form structure [19], and a number of authors have considered using it in linear filtering, estimation and control [5, 7, 8] More recently, de Vries, Principe at. al. proposed the gamma operator [2, 3] as a means of studying neural network models for processing time-varying patterns. This operator is defined by = 'Y q - (1 - c) (2) C It may be observed that it is a generalization of the delta operator with adjustable parameters c. An extension to the basic gamma operator introducing complex poles using a second order operator, was given in [18]. This raises the question, is the gamma operator capable of providing better neural network modelling capabilities for LDLF systems ? Further, are there any other operators which may be better than these for nonlinear modelling and prediction using neural networks? In the context of robust adaptive control, Palaniswami has introduced the rho operator which has shown useful improvements over the performance ofthe delta operator [9, 10]. The rho operator is defined as p = (3) where Cl, C2 are adjustable parameters. The rho operator generalizes the delta and gamma operators. For the case where Cl~ C2~ 1, the rho operator reduces to the usual shift 0, and C2 1, the rho operator reduces to the delta operator [10]. operator. When c) For Cl ~ = C2~ = c, the rho operator is equivalent to the gamma operator. = = = = One advantage of the rho operator over the delta operator is that it is stably invertible, allowing the derivation of simpler algorithms [9]. The p operator can be considered as a stable low pass filter, and parameter estimation using the p operator is low frequency biased. For adaptive control systems, this gives robustness advantages for systems with unmodelled high frequency characteristics [9] . By defining the bilinear transformation (BLT) as an operator, it is possible to introduce an operator which generalizes all of the above operators. We can therefore define the pi operator as 11" = 2 (Clq - ~ (C3Q C2) + C4) (4) with the restriction that Cl C4 f C2C3 (to ensure 11" is not a constant function [14]). The bilinear mapping produced has a pole at q = -C4/C3. By appropriate setting of the Cl, C2, C3, C4 parameters each operator, the pi operator can be reduced to each of the previous operators. In the work reported here, we consider these alternative discrete-time operators in feedforward neural network models for system identification tasks. We compare the popular A Comparison of Discrete-Time Operator Models for Nonlinear System Identification 885 gamma model [4] with other models based on the shift, delta, rho and pi operators. A framework of models and Gauss-Newton training algorithms is provided, and the models are compared by simulation experiments. 2 OPERATOR MODELS FOR NONLINEAR SIGNAL PROCESSING A model which generalizes the usual discrete-time linear moving average model, ie, a single layer network is given by yet) G(v,O)x(t) (5) M C(v,O) L: bw- I i i=O q-~ 8-' ,-i p- i 7r- i shift operator delta operator gamma operator rho operator pi operator (6) This general class of moving average model can be termed MA(v). We define uo(t) ~ x(t), and Ui(t) ~ V-IUi_l (t) I and hence obtain x(t - i) - 1) + Ui(t - 1) CUi-l(t - 1) + (1 - C)Ui(t - 1) C2~Ui-l(t - 1) + (1- Cl~)Ui(t - 1) 2~1 (C3 Ui-l(t) + C4 Ui-l(t - 1?) - ~Ui(t - 1) ~Ui-l(t shift operator delta operator gamma operator rho operator pi operator (7) A nonlinear model may be defined using a multilayer perceptron (MLP) with the v-operator elements at the input stage. The input vector ZP( t) to the network is Z?(t) = [Xi(t), V-1Xi(t), ... , V-MXi(t)]' (8) where Xi(t) is the ith input to the system. This model is termed the v-operator multilayer perceptron or MLP(v) model. An MLP(v) model having L layers with No, N I , ... , NL nodes per layer, is defined in the same manner as a usual MLP, with (9) Nz L WiiZJ-I(t) (10) i=l where each neuron i in layer 1 has an output of z!(t); a layer consists of N/ neurons (1 = 0 denotes the input layer, and 1 = L denotes the output layer, zJvz 1.0 may be used for a bias); 10 is a sigmoid function typically evaluated as tanh(?), and a synaptic connection between unit i in the previous layer and unit k in the current layer is represented by The notation t may be used to represent a discrete time or pattern instance. While the case = wt. 886 Andrew D. Back, Ah Chung Tsoi we consider employs the v-operator at the input layer only, it would be feasible to use the operators throughout the network as required. On-line algorithms to update the operator parameters in the MA(v) model can be found readily. In the case of the MLP(v) model, we approach the problem by backpropagating the error information to the input layer and using this to update the operator coefficients. de Vries and Principe et. al., proposed stochastic gradient descent type algorithms for adjusting the c operator coefficient using a least-squares error criterion [2, 12]. For brevity we omit the updating procedures for the MLP network weights; a variety of methods may be applied (see for example [13, 15]). We define an instantaneous output error criterion J(t) = !e 2(t), where e(t) = y(t) - f)(t). Defining 0 as the estimated operator parameter vector at time t of the parameter vector 0, we have c gamma operator { [CI , C2]' rho operator (11) [CI, C2, C3, 134 ]' pi operator o= A first order algorithm to update the coefficients is Oi(t + 1) ~Oi (t) Oi(t) + ~Oi(t) -71\1 eJ ((}; t) = (12) (13) where the adjustment in weights is found as ~Oi(t) = -71 oJ(t) o(}j M 71 I: 1/J1'(t)Oj(t) (14) i=1 where OJ (t) is the backpropagated error at the jth node of input layer, and 1/J1' (t) is the first order sensitivity vector of the model operator parameters, defined by &Ui(t) gamma operator ./~ (t) !Pi = I [::~(t) &Cjl &Cj2 [ &Ui(t) &Ui(t) &Cjl rho operator &Ui(t)]' ' ' &Cj2 &Ui(t) ' &Cj3 &Ui(t)] ' &Cj4 I (15) Pi operator Substituting Ui(t) in from (7), the recursive equations for 1/J1 (t) (noting that 1/J1 (t)= tPHt) 'Vj) are tPi(t) = Ui_l(t - 1) - Ui(t - 1) + CitPi-l(t - 1) + (1 - C)tPi(t - 1) gamma operator tPi(t) = - 1) + (1- CI~)tf;i I(t - 1) - ~Ui(t - 1) ] [ C2~tPi-I,I(t ~Ui-I(t - 1) + C2~tPi-I,2(t - 1) + (1 - CI~)tPi,2(t _ 1) rho operator 2t (C3tPi-I,I(t) + C4tPi-l,1(t - 1?) + ~tPi,l(t - 1) + C4Ui_l(t - 1?) - ~Ui(t - 1), ~C3tPi-I,2(t) + C4tPi-I,2(t - 1?) + ~tPi,2(t - 1) + tUi(t - 1), (Ui_l(t) + C3tPi-l,3(t) + C4tPi-I,3(t - 1?) + ~tPi~3(t - 1), (C3tPi-I,4(t) + Ui-I(t - 1) + C4tPi-l,4(t - 1? + T.-tPi,4(t - 1) -2~2 (C3Ui-l(t) c} 2~1 2t 2t C1 pi operator A Comparison of Discrete-Time Operator Models for Nonlinear System Identification 887 for the gamma, rho, and pi operators respectively, and where ""i ,j (t) refers to the jth element of the ith "" vector, with ""i ,o(t) = O. A more powerful updating procedure can be obtained by using the Gauss-Newton method [6]. In this case, we replace (14) with (omitting i subscripts for clarity), OCt + 1) = OCt) + 'Y(t)R- 1(t)",,(t)A -16(t) (16) where 'Y(t) is the gain sequence (see [6] for details), A-I is a weighting matrix which may be replaced by the identity matrix [16], or estimated as [6] A(t) = A(t - 1) + 'Y(t) (6 2(t) - A(t - 1?) (17) R( t) is an approximate Hessian matrix, defined by R(t + 1) = A(t)R(t) + ?(t)""(t),,,,'(t) (18) where A(t) = 1 - ?(t). Efficient computation of R- 1 may be performed using the matrix inversion lemma [17], factorization methods such as Cholesky decomposition or other fast algorithms. Using the well known matrix inversion lemma [6], we substitute pet) = R-l(t), where Pt () - _1_p t _ ?(t) ( P(t)",,(t)""'(t)P(t) ) A(t) () A(t) A(t) + ?(t)""'(t)P(t)",,(t) (19) The initial values of the coefficients are important in determining convergence. Principe et. al. [12] note that setting the coefficients for the gamma operator to unity provided the best approach for certain problems. 3 SIMULATION EXAMPLES We are primarily interested in the differences between the operators themselves for modelling and prediction, and not the associated difficulties of training multilayer perceptrons (recall that our models will only differ at the input layer). For the purposes of a more direct comparison, in this paper we test the models using a single layer network. Hence these linear system examples are used to provide an indication of the operators' performance. 3.1 EXPERIMENT 1 The first problem considered is a system identification task arising in the context of high bit rate echo cancellation [5]. In this case, the system is described by H(z) = 0.0254- 0.0296z- 1+ 0.00425z- 2 1 _ 1.957z-1 + 0.957z- 2 (20) This system has poles on the real axis at 0.9994, and 0.9577, thus it is an LDLF system. The input signal to the system in each case consisted of uniform white noise with unit variance. A Gauss-Newton algorithm was used to determine all unknown weights. We conducted Monte-Carlo tests using 20 runs of differently seeded training samples each of 2000 points to obtain the results reported. We assessed the performance of the models by using the Signal-to-NoiseRatio (SNR) defined as 1010g( E[d(t)2Jj E[e(t)2D, where E[?l is the expectation operator, and d(t) is the desired signal. For each run, we used the last 500 samples to compute a SNR figure. 888 Andrew D. Back, Ah Chung Tsoi 0.04 0.03 ? 0.02 0.01 0.04 0.03 0.02 ? o .~.. -0.01 -0.02 -0.03 -O''?900 ~. .~ ? o.ot o.ot 0 -0.01 -0.02 -0.03 0 -0.01 -0.02 -0.03 -0?11900 1920 (a) 0.04 0.03 0.02 0.01 0 -{l.01 -0.02 -0.03 -0?11900 0.04 0.03 0.02 ..? -0?11900 1920 1920 (b) (c) 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -O''?900 1920 (d) 1920 (e) Figure 1: Comparison of typical model output results for Experiment 1 with models based on the following operators: (a) shift, (b) delta (c) gamma, (d) rho, and (e) pi. Table 1: System Identification Experiment 1 Results For the purposes of this experiment, we conducted several trials and selected 0(0) values which provided stable convergence. The values chosen for this experiment were: 0(0) {0.75, [0.5,0.75]' [0.75,0.7,0.35,-0.25]} for the gamma, rho and pi operator models respectively. In each case we used model order M = 8. = Results for this experiment are shown in Table 1 and Figure 1. We observe that the pi operator gives the best performance overall. Some difficulties with instability occurring were encountered, thereby requiring a stability correction mechanism to be used on the operator updates. The next best performance was observed in the rho and then gamma models, with fewer instability problems occurring. 3.2 EXPERIMENT 2 The second experiment used a model described by H(z) = 1 - 0.8731z- 1 - 0.8731z-2 + z-3 1 - 2.8653z- 1 + 2.7505z- 2 - 0.8843z- 3 (21) This system is a 3rd order lowpass filter tested in [11]. The same experimental procedures as used in Experiment 1 were followed in this case. For the second experiment (see Table 2), it was found that the pi operator gave the best results A Comparison of Discrete-Time Operator Models for Nonlinear System Identification 889 Table 2: System Identification Experiment 2 Results recorded over all the tests. On average however, the improvement for this identification problem is less. It is observed that that the pi model is only slightly better than the gamma and rho models. Interestingly, the gamma and rho models had no problems with stability, while the pi model still suffered from convergence problems due to instability. As before, the delta model gave a wide variation in results and performed poorly. From these and other experiments performed it appears that performance advantages can be obtained through the use of the more complex operators. As observed from the best recorded runs, the extra degrees of freedom in the rho and pi operators appear to provide the means to give better performance than the gamma model. The improvements of the more complex operators come at the expense of potential convergence problems due to instabilities occurring in the operators and a potentially multimodal mean square output error surface in the operator parameter space. Clearly, there is a need for further investigation into the performance of these models on a wider range of tasks. We present these preliminary examples as an indication of how these alternative operators perform on some system identification problems. 4 CONCLUSIONS Models based on the delta operator, rho operator, and pi operator have been presented and new algorithms derived. Comparisons have been made to the previously presented gamma model introduced by de Vries, Principe et. al. [4] for nonlinear signal processing applications. While the simulation examples considered show are only linear, it is important to realize that the derivations are applicable for multilayer perceptrons, and that the input stage of these networks is identical to what we have considered here. We treat only the linear case in the examples in order not to complicate our understanding of the results, knowing that what happens in the input layer is important to higher layers in network structures. The results obtained indicate that the more complex operators provide a potentially more powerful modelling structure, though there is a need for further work into mechanisms of maintaining stability while retaining good convergence properties. The rho model was able to perform better than the gamma model on the problems tested, and gave similar results in terms of susceptibility to convergence and instability problems. The pi model appears capable of giving the best performance overall, but requires more attention to ensure the stability of the coefficients. For future work it would be of value to analyse the convergence of the algorithms, in order to design methods which ensure stability can be maintained, while not disrupting the convergence of the model. 890 Andrew D. Back, Ah Chung Tsoi Acknowledgements The first author acknowledges financial support from the Australian Research Council. The second author acknowledges partial support from the Australian Research Council. References [1] R.C. Agarwal and C.S. Burrus, ''New recursive digital filter structures having very low sensitivity and roundoff noise", IEEE Trans. Circuits and Systems, vol. cas-22, pp. 921-927, Dec. 1975. [2] de Vries, B. Principe, J.C. "A theory for neural networks with time delays", Advances in Neural Information Processing Systems, 3, R.P. Lippmann (Ed.), pp 162 - 168, 1991. [3] de Vries, B., Principe, J. and P.G. de Oliveira "Adaline with adaptive recursive memory", Neural Networks for Signal Processing I. Juang, B.H., Kung, S.Y., Kamm, C.A. (Eds) IEEE Press, pp. 101-110, 1991. [4] de Vries, B. Principe, J. "The Gamma Model - a new neural model for temporal processing". Neural Networks. Vol 5, No 4, pp 565 - 576, 1992. [5] H. Fan and Q. Li, "A () operator recursive gradient algorithm for adaptive signal processing", Proc. IEEE Int. Conf. Acoust. Speech and Signal Proc., vol. nI, pp. 492-495, 1993. [6] L. Ljung, and T. SOderstrtlm, Theory and Practice of Recursive Identification, Cambridge, Massachusetts: The MIT Press, 1983. [7] R.H. Middleton, and G.C. Goodwin, Digital Control and Estimation, Englewood Cliffs: Prentice Hall, 1990. [8] V. Peterka, "Control of Uncertain Processes: Applied Theory and Algorithms", Kybernetika, vol. 22, pp. 1-102, 1986. [9] M. Palaniswami, "A new discrete-time operator for digital estimation and control". The Uni versity of Melbourne, Department of Electrical Engineering, Technical Report No.1, 1989. [10] M. Palaniswami, "Digital Estimation and Control with a New Discrete Time Operator", Proc. 30th IEEE Conf. Decision and Control, pp. 1631-1632, 1991. [11] J.C. Principe, B. de Vries, J-M. Kuo and P. Guedes de Oliveira, "Modeling Applications with the Focused Gamma Net", Advances in Neural Information Processing Systems, vol. 4, pp. 143-150,1991. [12] J.C. Principe, B. de Vries, and P. Guedes de Oliveira, "The Gamma Filter - a new class of adaptive IIR filters with restricted feedback", IEEE Trans. Signal Processing, vol. 41, pp. 649-656, 1993. [13] G. V. Puskorius, and L.A. Feldkamp, "Decoupled Extended Kalman Filter Training of Feedforward Layered Networks", Proc. Int Joint Conf. Neural Networks, Seattle, vol I, pp. 771-777,1991. [14] E.B. Saff and A.D. Snider, Fundamentals of Complex Analysis for Mathematics, Science and Engineering. Englewood Cliffs, NJ: Prentice-Hall, 1976. [15] S. Shah and F. Palmieri, "MEKA - A Fast Local Algorithm for Training Feedfoward Neural Networks", Proc Int Joint Conf. on Neural Networks, vol m, pp. 41-46, 1990. [16] J.J. Shynk, "Adaptive IIR filtering using parallel-form realizations", IEEE Trans. Acoust. Speech Signal Proc., vol. 37, pp. 519-533,1989. [17] Soderstrom and Stoica, "System Identification", London: Prentice Hall, 1989. [18] T.O. de Silva, P.G. de Oliveira, J.e. Principe, and B. de Vries, " Generalized feedforward filters with complex poles", Neural Networks for Signal Processing n, S.Y. Kung et. al. (Eds) Piscataway,NJ: IEEE Press, 1992. [19] D. Williamson, "Delay replacement in direct form structures", IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 453-460, Aprl. 1988. PARTvm VISUAL PROCESSING 

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Implementation of Neural Hardware with the Neural VLSI of URAN in Applications with Reduced Representations ll-Song Han Korea Telecom Research Laboratories 17, Woomyun-dong, Suhcho-ku Seoul 137-140, KOREA Ki-Chul Kim Dept. of Info and Comm KAIST Seoul, 130-012, Korea Hwang-Soo Lee Dept. of Info and Comm KAIST Seoul, 130-012, Korea Abstract This paper describes a way of neural hardware implementation with the analog-digital mixed mode neural chip. The full custom neural VLSI of Universally Reconstructible Artificial Neural network (URAN) is used to implement Korean speech recognition system. A multi-layer perceptron with linear neurons is trained successfully under the limited accuracy in computations. The network with a large frame input layer is tested to recognize spoken korean words at a forward retrieval. Multichip hardware module is suggested with eight chips or more for the extended performance and capacity. 812 ll-Song Han, Hwang-Soo Lee, Ki-Chul Kim 1 INTRODUCTION In general, the neural network hardware or VLSI has been preferred in respects of its relatively fast speed, huge network size and effective cost comparing to software simulation. Universally Reconstructible Artificial Neural-network(URAN), the new analog-digital mixed VLSI neural network, can be used for the implementation of the real world neural network applications with digital interface. The basic electronic synapse circuit is based on the electrically controlled MOSFET resistance and is operated with discrete pulses. The URAN's adaptability is tested for the multi-layer perceptron with the reduced precision of connections and states. The linear neuron function is also designed for the real world applications. The multi-layer network with back propagation learning is designed for the speaker independent digit/word recognition. The other case of application is for the servo control, where the neural input and output are extended to 360 levels for the suitable angle control. With the servo control simulation, the flexibility of URAN is proved to extend the accuracy of input and output from external. 2. Analog-Digital Mixed Chip - URAN In the past, there have been improvements in analog or analog-digital mixed VLSI chips. Analog neural chips or analog-digital mixed neural chips are still suffered from the lack of accuracy, speed or flexibility. With the proposed analog-digital mixed neural network circuit of URAN, the accuracy is improved by using the voltage-controlled linear MOSFET resistance for the synapse weight emulation. The speed in neural computation is also improved by using the simple switch controlled by the neural input as described in previous works. The general flexibility is attained by the independent characteristic of each synapse cell and the modular structure of URAN chip. As in Table 1 of URAN chip feature, the chip is operated under the flexible control, that is, the various mode of synaptic connection per neuron or the extendable weight accuracy can be implemented. It is not limited for the asynchronous/direct interchip expansion in size or speed. In fact, 16 fully connected module of URAN is selected from external and independently - it is possible to select either one by one or all at once. Table 1. URAN Chip Features Total Synapses Computation Speed Weight Accuracy Module No. Module Size 135,424 connections 200 Giga Connections Per S 8 Bit 16 92 X 92 Implementation of Neural Hardware with the Neural VLSI of URAN 813 As all circuits over the chip except digital decoder unit are operated in analog transistor level, the computation speed is relatively high and even can be improved substantially. The cell size including interconnection area in conventional short-channel technology is reduced less than 900 1.1 m2 . From its expected and measured linear characteristic, URAN has the accuracy more than 256 linear levels. The accuracy extendability and flexible modularity are inherent in electrical wired- OR characteristics as each synapse is an independent bipolar current source with switch. No additional clocking or any limited synchronous operation is required in this case, while it is indispensible in most of conventional digital neural hardware or analog-digital neural chip. Therefore, any size of neural network can be integrated in VLSI or module hardware merely by placing the cell in 2 dimensional array without any timing limitation or loading effect. 3. Neural Hardware with URAN - Module Expansion URAN is the full custom VLSI of analog-digital mixed operation. The prototype of URAN chip is fabricated in 1.0tI digital CMOS technology. The chip contains 135,424 synapses with 8 bit weight accuracy on a 13 X 13 mm2 die size using single poly double metal technology. As summarized in Table 1 of chip features, the chip allows the variety of configuration. In the prototype chip, 16 fully connected module of 92 X 92 can be selected from external and independently - selecting independent module either one by one, several or all at a time IS possible. With URAN's synapse circuit of linear Voltage-controlled bipolar current source, the synaptic multiplication with weight value is done with the switching transistor, in a similar way of analog-sampled data type. The accuracy enhancement and flexible modularity of URAN are inherent in its electrical wired-OR interface from each independent bipolar current source. And the neural network hardware module can be realized in any size with the multi URAN chips. 4. Considerations on the Reduced Precision URAN chip is applied for the case of Korean speaker independent speech recognition. By changing numbers of hidden units and input accuracy, the result of simulations has not shown any problems in It means that the overall performance is not recognition accuracy. severely affected from the accuracy of weight, input, and output with URAN. Also, it was possible to train with 2 or 1 decimal accuracy for input and output, which is equivalent to 8 bit or 4 bit precisions. With 20 hidden units for the Korean spoken 10 digit recognition, 2 decimal input accuracy yields 99.2% and l ' decimal input accuracy yields 98.6%, 814 II-Song Han, Hwang-Soo Lee, Ki-Chul Kim while binary I-bit input results 96.6%. The following is the condition for the experimentation. The general result is summarized in Table 2. Conditions for Training and Test ? 2,000 samples from 10 women and 10 men ( 10 times X 10 digits X 20 persons) [] Training with 500 spoken samples of 10 digits in Korean from 10 persons (5 times X 10 digits X [ 5 women and 5 men ] ) from 2,000 samples [] Recognition Test with 1,000 spoken samples from the other 10 persons of women and men. Preprocessing of samples [] sampled at 10KHz with 12bit accuracy [] preemphasis with 0.95 [] Hamming window of 20ms [] 17 channel critical-band filter bank [] noise added for the SNR of 3OdB, 2OdB, 10dB, OdB Table 2. Low Accuracy Connection with Linear Neuron Input / Output Accuracy SNR Ratio clean 30 dB 20 dB 10 dB o dB 2 decimal 1 decimal 1 bit 97.5% 96.2% 90.1% 59.8% 30.8% 97.2% 96.6% 91.3% 59.9% 29.5% 90.7% 90.5% 86.6% 68.0% 38.5% In case of servo control, the digital VCR for industrial purpose is Six inputs are used to minimize the modelled for the application. number of hidden units and 20 hidden units are configured for one output. For the adaptation to URAN, the linear neuron function is used during the simulation. The weight accuracy during the learning phase using conventional computer is 4 byte and that in the recall phase using URAN chip is 1 byte. With this limitation, the overall performance is not severely degraded, that is, the reduction of error is attained up to 70% improvement comparing to the conventional method. The nonideal factor of 30% results from the limitation in learning data as well as the limited hardware. Current results are suitable for the digital VCR or compact cam coder in noisy environment Implementation of Neural Hardware with the Neural VLSI of URAN 815 5. Conclusion In this paper, it is proved to be suitable for the application to the multi-layer perceptron with the use of URAN chip, which is fabricated in conventional digital CMOS technology - 1.0# single poly double metal. The reduced weight accuracy of 1 byte is proved to be enough to obtain high perfonnance using the linear neuron and URAN. With 8 test chips of 135,424 connections, it is now under development of the practical module of neural hardware with million connections and tera connections per second - comparable to the power of biological neuro-system of some insects. The size of the hardware is smaller than A4 size and is designed for more general recognition system. The flexible modularity of URAN makes it possible to realize a 1,000,000 connections neural chip in 0.5# CMOS technology and a general purpose neural hardware of hundreds of tera connections or more. References II Song Han and Ki-Hwan Ahn, "Neural Network VLSI Chip Implementation of Analog-Digital Mixed Operation for more than 100,000 Connections" MicroNeuro'93, pp. 159-162, 1993 M. Brownlow, L. Tara s senko , A. F. Murray, A. Hamilton, I SHan, H. M. Reekie, "Pulse Firing Neural Chips Implementing Hundreds of Neurons," NIPS2, pp. 785-792, 1990 

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Finding Structure in Reinforcement Learning Sebastian Thrun University of Bonn Department of Computer Science nr R6merstr. 164, D-53117 Bonn, Germany E-mail: thrun@carbon.informatik.uni-bonn.de Anton Schwartz Dept. of Computer Science Stanford University Stanford, CA 94305 Email: schwartz@cs.stanford.edu Abstract Reinforcement learning addresses the problem of learning to select actions in order to maximize one's performance in unknown environments. To scale reinforcement learning to complex real-world tasks, such as typically studied in AI, one must ultimately be able to discover the structure in the world, in order to abstract away the myriad of details and to operate in more tractable problem spaces. This paper presents the SKILLS algorithm. SKILLS discovers skills, which are partially defined action policies that arise in the context of multiple, related tasks. Skills collapse whole action sequences into single operators. They are learned by minimizing the compactness of action policies, using a description length argument on their representation. Empirical results in simple grid navigation tasks illustrate the successful discovery of structure in reinforcement learning. 1 Introduction Reinforcement learning comprises a family of incremental planning algorithms that construct reactive controllers through real-world experimentation. A key scaling problem of reinforcement learning, as is generally the case with unstructured planning algorithms, is that in large real-world domains there might be an enormous number of decisions to be made, and pay-off may be sparse and delayed. Hence, instead of learning all single fine-grain actions all at the same time, one could conceivably learn much faster if one abstracted away the myriad of micro-decisions, and focused instead on a small set of important decisions. But this immediately raises the problem of how to recognize the important, and how to distinguish it from the unimportant. This paper presents the SKILLS algorithm. SKILLS finds partially defined action policies, called skills, that occur in more than one task. Skills, once found, constitute parts of solutions to multiple reinforcement learning problems. In order to find maximally useful skills, a description length argument is employed. Skills reduce the number of bytes required to describe action policies. This is because instead of having to describe a complete action policy for each task separately, as is the case in plain reinforcement learning, skills constrain multiple task to pick the same actions, and thus reduce the total number of actions required 386 Sebastian Thrun, Anton Schwartz for representing action policies. However, using skills comes at a price. In general, one cannot constrain actions to be the same in multiple tasks without ultimately suffering a loss in performance. Hence, in order to find maximally useful skills that infer a minimum loss in performance, the SKILLS algorithm minimizes a function of the form E = PERFORMANCE LOSS + 1J' DESCRIPTION LENGTH. (1) This equation summarizes the rationale of the SKILLS approach. The reminder of this paper gives more precise definitions and learning rules for the terms" PERFORMANCE LOSS" and "DESCRIPTION LENGTH," using the vocabulary of reinforcement learning. In addition, experimental results empirically illustrate the successful discovery of skills in simple grid navigation domains. 2 Reinforcement Learning Reinforcement learning addresses the problem of learning, through experimentation, to act so as to maximize one's pay-off in an unknown environment. Throughout this paper we will assume that the environment of the learner is a partially controllable Markov chain [1]. At any instant in time the learner can observe the state of the environment, denoted by s E S, and apply an action, a E A. Actions change the state of the environment, and also produce a scalar pay-off value, denoted by rs,a E ~. Reinforcement learning seeks to identify an action policy, 7f : S --+ A, i.e., a mapping from states s E S to actions a E A that, if actions are selected accordingly, maximizes the expected discounted sum offuture pay-off R = E [f ,t-til (2) r t ]. t=tu Here I (with 0 S,S 1) is a discount factor that favors pay-offs reaped sooner in time, and rt refers to the expected pay-off at time t. In general, pay-off might be delayed. Therefore, in order to learn an optimal 7f, one has to solve a temporal credit assignment problem [11]. To date, the single most widely used algorithm for learning from delayed pay-off is QLearning [14]. Q-Learning solves the problem of learning 7f by learning a value function, denoted by Q : S x A --+~. Q maps states s E S and actions a E A to scalar values. After learning, Q(s, a) ranks actions according to their goodness: The larger the expected cumulative pay-offfor picking action a at state s, the larger the value Q(s, a). Hence Q, once learned, allows to maximize R by picking actions greedily with respect to Q: 7f(s) = argmax Q(s, a) aEA The value function Q is learned on-line through experimentation. Initially, all values Q( s, a) are set to zero. Suppose during learning the learner executes action a at state s, which leads to a new state s' and the immediate pay-off rs ,a' Q-Learning uses this state transition to update Q(s, a): Q(s, a) with V(s') ;- (1 - a) . Q(s, a) m~x Q(s', a + a? (rs,a + I' V(s')) (3) a) The scalar a (O<aSl) is the learning rate, which is typically set to a small value that is decayed over time. Notice that if Q(s, a) is represented by a lookup-table, as will be the case throughout this paper, the Q-Learning rule (3) has been shown] to converge to a value function Qopt( s, a) which measures the future discounted pay-off one can expect to receive upon applying action a in state s, and acting optimally thereafter [5, 14]. The greedy policy 7f(s) = argmax a Qopt(s, a) maximizes R. I under certain conditions concerning the exploration scheme, the environment and the learning rate Finding Structure in Reinforcement Learning 3 387 Skills Suppose the learner faces a whole collection of related tasks, denoted by B, with identical states 5 and actions A. Suppose each task b E B is characterized by its individual payoff function, denoted by rb(s, a). Different tasks may also face different state transition probabilities. Consequently, each task requires a task-specific value function, denoted by Qb(S, a), which induces a task-specific action policy, denoted by '!rb. Obviously, plain QLearning, as described in the previous section, can be employed to learn these individual action policies. Such an approach, however, cannot discover the structure which might inherently exist in the tasks. In order to identify commonalities between different tasks, the SKILLS algorithm allows a learner to acquire skills. A skill, denoted by k, represents an action policy, very much like '!rb. There are two crucial differences, however. Firstly, skills are only locally defined, on a subset Sk of all states S. Sk is called the domain of skill k. Secondly, skills are not specific to individual tasks. Instead, they apply to entire sets of tasks, in which they replace the task-specific, local action policies. Let f{ denote the set of all skills. In general, some skills may be appropriate for some tasks, but not for others. Hence, we define a vector of usage values Uk,b (with 0 ~ Uk,b ~ 1 for all kEf{ and all b E B). Policies in the SKILLS algorithm are stochastic, and usages Uk,b determine how frequently skill k is used in task b. At first glance, Uk,b might be interpreted as a probability for using skill k when performing task b, and one might always want to use skill k in task b if Uk ,b = 1, and never use skill k if Uk ,b = 0. 2 However, skills might overlap, i.e., there might be states S which occurs in several skill domains, and the usages might add to a value greater than 1. Therefore, usages are normalized, and actions are drawn probabilistical1y according to the normalized distribution: U~,k . mk(s) (4) (with 8= 0) k'EK Here Pb( kls) denotes the probability for using skill k at state s, if the learner faces task b. The indicator function mk (s) is the membership function for skill domains, which is 1 if s E Sk and 0 otherwise. The probabilistic action selection rule (4) makes it necessary to redefine the value Vb (s) of a state s. If no skill dictates the action to be taken, actions will be drawn according to the Qb-optimal policy '!r;(s) argmax Qb(S, a) , .lEA as is the case in plain Q-Learning. The probability for this to happen is Pb*(s) L Pb(kls) . kEK Hence, the value of a state is the weighted sum 1- Vb(S) P;(s) . vt(s) +L Pb(kls) . Qb(S, '!rk(S)) (5) kEK with vt(s) Qb(S, '!rb(s)) = TEa; Qb(S, a) Why should a learner use skills, and what are the consequences? Skills reduce the freedom to select actions, since mUltiple policies have to commit to identical actions. Obviously, such 2This is exactly the action selection mechanism in the SKILLS algorithm if only one skill is applicable at any given state s. 388 Sebastian Thrun, Anton Schwartz a constraint will generally result in a loss in peiformance. This loss is obtained by comparing the actual value of each state s, Vb(S), and the value ifno skill is used, VtCs): LOSS = ~ ~ Vb*(s) - Vb(S) sES bEB '''---v,----' = LO S S( s ) (6) If actions prescribed by the skills are close to optimal, i.e., if Vb*(s) ~ Vb(S)('v'S E S), the loss will be small. If skill actions are poor, however, the loss can be large. Counter-balancing this loss is the fact that skills give a more compact representation of the learner's policies. More specifically, assume (without loss of generality) actions can be represented by a single byte, and consider the total number of bytes it takes to represent the policies of all tasks b E B. In the absence of skills, representing all individual policies requires IBI . lSI bytes, one byte for each state in S and each task in B. If skills are used across multiple tasks, the description length is reduced by the amount of overlap between different tasks. More specifically, the total description length required for the specification of all policies is expressed by the following term : ~ ~ P;(s) + ~ ISkl DL sE S bEB (7) kEK ~~-----v~---------j = DL( s ) If all probabilities are binary, i.e. , Pb(kls) and P;(s) E {O, I}, DL measures precisely the number of bytes needed to represent all skill actions, plus the number of bytes needed to represent task-specific policy actions where no skill is used. Eq. (7) generalizes this measure smoothly to stochastic policies. Notice that the number of skills If{ I is assumed to be constant and thus plays no part in the description length DL. Obviously, minimizing LOSS maximizes the pay-off, and minimizing DL maximizes the compactness of the representation of the learner's policies. In the SKILLS approach, one seeks to minimize both (cf Eq. (1? E = LOSS + TJDL = ~ LOSS(s) + TJDL(s) . (8) sES 11 > 0 is a gain parameter that trades off both target functions. E-optimal policies make heavily use of large skills, yet result in a minimum loss in performance. Notice that the state space may be partitioned completely by skills, and solutions to the individual tasks can be uniquely described by the skills and its usages. If such a complete partitioning does not exist, however, tasks may instead rely to some extent on task-specific, local pOlicies. 4 Derivation of the Learning Algorithm Each skill k is characterized by three types of adjustable variables: skill actions 7rk (s), the skill domain Sk, and skill usages Ub,k. one for each task b E B. In this section we will give update rules that perform hill-cli:nbing in E for each of these variables. As in Q-Learning these rules apply only at the currently visited state (henceforth denoted by s) . Both learning action policies (cf Eq. (3? and learning skills is fully interleaved. Actions. Determining skill actions is straightforward, since what action is prescribed by a skill exclusively affects the performance loss, but does not play any part in the description length. Hence, the action policy 7rk(S) minimizes LOSS(s) (cf Eqs. (5) and (6?: 7rk(S) argmax ~ Pb(kls) . Qb(S. a) .lEA bEB (9) 389 Finding Structure in Reinforcement Learning Domains. Initially, each skill domain Sk contains only a single state that is chosen at random. Sk is changed incrementally by minimizing E(s) for states s which are visited during learning. More specifically, for each skill k, it is evaluated whether or not to include sin Sk by considering E(s) = LOSS(s) + TJDL(s). s E Sk, ifandonlyif E(s)lsESk < E(s)lsf'Sk (otherwises ~ Sk) (10) If the domain of a skill k vanishes completely, i.e., if Sk = 0, it is re-initialized by a randomly selected state. In addition all usage values {Ub ,klb E B} are initialized randomly. This mechanism ensures that skills, once overturned by other skills, will not get lost forever. Usages. Unlike skill domains, which are discrete quantities, usages are real-valued numbers. Initially, they are chosen at random in [0, 1]. Usages are optimized by stochastic gradient descent in E. According to Eq. (8), the derivative of E(s) is the sum of aL~SS(s ) and Ub,k a~uL(s ) . The first term is governed by b,k 8LOSS(s) 8 Ub ,k _ 8Vb(S) 8 'Ub ,k _ - _ 8P;(s) . Q ( *() ) _ " 8Pb(j!S) . Q ( .(? 8 b 7rb s , s ~ 8 b S,7rJ S Ub,k jEK Ub ,k with 8Pb (j!s) (11 ) 8Ub ,k and 8Pb*(s) (12) Here Dkj denotes the Kronecker delta function, which is 1 if k second term is given by 8DL(s) 8Ubk , 8Pb*(s) 8Ukb, ' =j and 0 otherwise. The (13) which can be further transformed using Eqs. (12) and (11). In order to minimize E, usages are incrementally refined in the opposite direction of the gradients: Uk ,b ;....-.- Uk,b - {J. (8V(s) 8Ukb, + TJ 8DL(S?) 8Ukb, (14) Here {J > 0 is a small learning rate. This completes the derivation of the SKILLS algorithm. After each action execution, Q-Learning is employed to update the Q-function. SKILLS also re-calculates, for any applicable skill, the skill policy according to Eq . (9), and adjusts skill domains and usage values based upon Eqs. (10) and (14). 5 Experimental Results The SKILLS algorithm was applied to discover skills in a simple, discrete grid-navigation domain, depicted in Fig. 1. At each state, the agent can move to one of at most eight adjacent grid cells. With a 10% chance the agent is carried to a random neighboring state, regardless of the commanded action. Each corner defines a starting state for one out of four task, with the corresponding goal state being in the opposite corner. The pay-off (costs) for executing actions is -1, except for the goal state, which is an absorbing state with zero pay-off. In a first experiment, we supplied the agent with two skills f{ = {kJ, k2 }. All four tasks were trained in a time-shared manner, with time slices being 2,000 steps long. We used the following parameter settings: TJ = 1.2, 'Y = 1, a = 0.1, and {J = 0.001. After 30000 training steps for each task, the SKILLS algorithm has successfully discovered the two ski11s shown in Figure 1. One of these skills leads the agent to the right door, and 390 Sebastian Thrun. Anton Schwartz 1 I 1/ I 1'/ I //~ I // - /// ",. - - / / r, / - - .- - - - Figure 1: Simple 3-room environment. Start and goal states are marked by circles. The diagrams also shows two skills (black states), which lead to the doors connecting the rooms. the second to the left. Each skill is employed by two tasks. By forcing two tasks to adopt a single policy in the region of the skill, they both have to sacrifice performance, but the loss in performance is considerably small. Beyond the door, however, optimal actions point into opposite directions. There, forcing both tasks to select actions according to the same policy would result in a significant performance loss, which would clearly outweigh the savings in description length. The solution shown in Fig. 1 is (approximately) the global minimum of E, given that only two skills are available. It is easy to be seen that these skills establish helpful building blocks for many navigation tasks. When using more than two skills, E can be minimized further. We repeated the experiment using six skills, which can partition the state space in a more efficient way. Two of the resulting skills were similar to the skills shown in Fig. 1, but they were defined only between the doors. The other four skills were policies for moving out of a corner, one for each corner. Each of the latter four skills can be used in three tasks (unlike two tasks for passing through the middle room), resulting in an improVed description length when compared to the two-skill solution shown in Fig. 1. We also applied skill learning to a more complex grid world, using 25 skills for a total of 20 tasks. The environment, along with one of the skills, is depicted in Fig. 2. Different tasks were defined by different starting positions, goal positions and door configurations which could be open or closed. The training time was typically an order of magnitude slower than in the previous task, and skills were less stable over time. However, Fig. 2 illustrates that modular skills could be discovered even in such complex a domain. 6 Discussion This paper presents the SKILLS algorithm. SKILLS learns skills, which are partial policies that are defined on a subset of all states. Skills are used in as many tasks as possible, while affecting the performance in these tasks as little as possible. They are discovered by minimizing a combined measure, which takes a task performance and a description length argument into account. While our empirical findings in simple grid world domains are encouraging, there are several open questions that warrant future research. Learning speed. In our experiments we found that the time required for finding useful skills is up to an order of magnitude larger than the time it takes to find close-to-optimal policies. Finding Structure in Reinforcement Learning 391 Figure 2: SkiIl found in a more complex grid navigation task. Similar findings are reported in [9]. This is because discovering skills is much harder than learning control. Initially, nothing is know about the structure of the state space, and unless reasonably accurate Q-tables are available, SKILLS cannot discover meaningful skills. Faster methods for learning skills, which might precede the development of optimal value functions, are clearly desirable. Transfer. We conjecture that skills can be helpful when one wants to learn new, related tasks. This is because if tasks are related, as is the case in many natural learning environments, skills allow to transfer knowledge from previously learned tasks to new tasks. In particular, if the learner faces tasks with increasing complexity, as proposed by Singh [10], learning skills could conceivable reduce the learning time in complex tasks, and hence scale reinforcement learning techniques to more complex tasks. Using function approximators. In this paper, performance loss and description length has been defined based on table look-up representations of Q. Recently, various researchers have applied reinforcement learning in combination with generalizing function approximators, such as nearest neighbor methods or artificial neural networks (e.g., [2, 4, 12, 13]). In order to apply the SKILLS algorithm together with generalizing function approximators, the notions of skill domains and description length have to be modified. For example, the membership function mk, which defines the domain of a skill, could be represented by a function approximator which allows to derive gradients in the description length. Generalization in state space. In the current form, SKILLS exclusively discovers skills that are used across mUltiple tasks. However, skills might be useful under multiple circumstances even in single tasks. For example, the (generalized) skill of climbing a staircase may be useful several times in one and the same task. SKILLS, in its current form, cannot represent such skills. The key to learning such generalized skills is generalization. Currently, skills generalize exclusively over tasks, since they can be applied to entire sets of tasks. However, they cannot generalize over states. One could imagine an extension to the SKILLS algorithm, in which skills are free to pick what to generalize over. For example, they could chose to ignore certain state information (like the color of the staircase). It remains to be seen if effective learning mechanisms can be designed for learning such generalized skills. Abstractions and action hierarchies. In recent years, several researchers have recognized the importance of structuring reinforcement learning in order to build abstractions and action 392 Sebastian Thrun, Anton Schwartz hierarchies. Different approaches differ in the origin of the abstraction, and the way it is incorporated into learning. For example, abstractions have been built upon previously learned, simpler tasks [9, 10], previously learned low-level behaviors [7], subgoals, which are either known in advance [15] or determined at random [6], or based on a pyramid of different levels of perceptual resolution, which produces a whole spectrum of problem solving capabilities [3]. For all these approaches, drastically improved problem solving capabilities have been reported, which are far beyond that of plain, unstructured reinforcement learning. This paper exclusively focuses on how to discover the structure inherent in a family of related tasks. Using skills to form abstractions and learning in the resulting abstract problem spaces is beyond the scope of this paper. The experimental findings indicate, however, that skills are powerful candidates for operators on a more abstract level, because they collapse whole action sequences into single entities. References [I] A. G. Barto, S. J. Bradtke, and S. P. Singh. Learning to act using real-time dynamic programming. Artijiciallntelligence, to appear. [2] J. A. Boyan. Generalization in reinforcement learning: Safely approximating the value function. Same volume. [3] P. Dayan and G. E. Hinton. Feudal reinforcement learning. In J. E. Moody, S. J. Hanson, and R. P. Lippmann, editors, Advances in Neural Information Processing Systems 5,1993. Morgan Kaufmann. [4) V. Gullapalli, 1. A. Franklin, and Hamid B. Acquiring robot skills via reinforcement learning. IEEE Control Systems, 272( 1708), 1994. [5) T. Jaakkola, M. I. Jordan, and S. P. Singh. On the convergence of stochastic iterative dynamic programming algorithms. Technical Report 9307, Department of Brain and Cognitive Sciences, MIT, July 1993. [6] L. P. Kaelbling. Hierarchical learning in stochastic domains: Preliminary results. In Paul E. Utgoff, editor, Proceedings of the Tenth International Conference on Machine Learning, 1993. Morgan Kaufmann. [7] L.-J. Lin. Self-supervised Learning by Reinforcementand Artijicial Neural Networks. PhD thesis, Carnegie Mellon University, School of Computer Science, 1992. [8) M. Ring. Two methods for hierarchy learning in reinforcement environments. In From Animals to Animates 2: Proceedings of the Second International Conference on Simulation of Adaptive Behavior. MIT Press, 1993. [9] S. P. Singh. Reinforcement learning with a hierarchy of abstract models. In Proceeding of the Tenth National Conference on Artijiciallntelligence AAAI-92, 1992. AAAI Pressffhe MIT Press. [10] S. P. Singh. Transfer of learning by composing solutions for elemental sequential tasks. Machine Learning, 8,1992. [II] R. S. Sutton. Temporal Credit Assignment in Reinforcement Learning. PhD thesis, Department of Computer and Information Science, University of Massachusetts, 1984. [12) G. J. Tesauro. Practical issues in temporal difference learning. Machine Learning, 8, 1992. [13] S. Thrun and A. Schwartz. Issues in using function approximation for reinforcement learning. In M. Mozer, Pa. Smolensky, D. Touretzky, J. Elman, and A. Weigend, editors, Proceedings of the J993 Connectionist Models Summer School, 1993. Erlbaum Associates. [14] C. J. C. H. Watkins. Learningfrom Delayed Rewards. PhD thesis, King's College, Cambridge, England, 1989. [15] S. Whitehead, J. Karlsson, and J. Tenenberg. Learning multiple goal behavior via task decomposition and dynamic policy merging. In J. H. Connell and S. Mahadevan, editors, Robot Learning. Kluwer Academic Publisher, 1993. 

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A Novel Reinforcement Model of Birdsong Vocalization Learning Kenji Doya ATR Human Infonnation Processing Research Laboratories 2-2 Hikaridai, Seika, Kyoto 619-02, Japan Terrence J. Sejnowski Howard Hughes Medical Institute UCSD and Salk Institute, San Diego, CA 92186-5800, USA Abstract Songbirds learn to imitate a tutor song through auditory and motor learning. We have developed a theoretical framework for song learning that accounts for response properties of neurons that have been observed in many of the nuclei that are involved in song learning. Specifically, we suggest that the anteriorforebrain pathway, which is not needed for song production in the adult but is essential for song acquisition, provides synaptic perturbations and adaptive evaluations for syllable vocalization learning. A computer model based on reinforcement learning was constructed that could replicate a real zebra finch song with 90% accuracy based on a spectrographic measure. The second generation of the birdsong model replicated the tutor song with 96% accuracy. 1 INTRODUCTION Studies of motor pattern generation have generally focussed on innate motor behaviors that are genetically preprogrammed and fine-tuned by adaptive mechanisms (Harris-Warrick et al., 1992). Birdsong learning provides a favorable opportunity for investigating the neuronal mechanisms for the acquisition of complex motor patterns. Much is known about the neuroethology of birdsong and its neuroanatomical substrate (see Nottebohm, 1991 and Doupe, 1993 for reviews), but relatively little is known about the overall system from a computational viewpoint. We propose a set of hypotheses for the functions of the brain nuclei in the song system and explore their computational strength in a model based on biological constraints. The model could reproduce real and artificial birdsongs in a few hundred learning trials. 102 Kenji Doya, Terrence 1. Sejnowski ----+ Direct Motor Pathway ----+ Anterior Forebrain Pathway Figure 1: Major songbird brain nuclei involved in song control. The dark arrows show the direct motor control pathway and the gray arrows show the anterior forebrain pathway. Abbreviations: Uva, nucleus uvaeforrnis of the thalamus; NIf, nucleus interface of the neostriatum; L, field L (primary auditory are of the forebrain); HVc, higher vocal center; RA, robust nucleus of the archistriatum; DM, dorso-medial part of the nucleus intercollicularis; nXllts, tracheosyringeal part of the hypoglossal nucleus; AVT, ventral area of Tsai of the midbrain; X, area X of lobus parolfactorius; DLM, medial part of the dorsolateral nucleus of the thalamus; LMAN, lateral magnocellular nucleus of the anterior neostriatum. 2 NEUROETHOLOGY OF BIRDSONG Although songs from individual birds of the same species may sound quite similar, a young male songbird learns to sing by imitating the song of a tutor, which is usually the father or another adult male in the colony. If a young bird does not hear a tutor song during a critical period, it will sing short, poorly structured songs, and if a bird is deafened in the period when it practices vocalization, it develops highly abnormal songs. These observations indicate that there are two phases in song learning: the sensory learning phase when a young bird memorizes song templates and the motor learning phase in which the bird establishes the motor programs using auditory feedback (Konishi, 1965). These two phases can be separated by several months in some species, implying that birds have remarkable capability for memorizing complex temporal sequences. Once a song is crystallized, its pattern is very stable. Even deafening the bird has little immediate effect. The brain nuclei involved in song learning are shown in Figure 1. The primary motor control pathway is composed ofUva, NIf, HVc, RA, DM, and nXllts. If any of these nuclei is lesioned, a bird cannot sing normally. Experimental studies suggest that HVc is involved in generating syllable sequences and that RA produces motor commands for each syllable (Vu et aI., 1994). Interestingly, neurons in HVc, RA and nXllts show vigorous auditory responses, suggesting that the motor control system is closely coupled with the auditory system (Nottebohm, 1991). There is also a "bypass" from HVc to RA which consists of area X, DLM, and LMAN called the anterior forebrain pathway (Doupe, 1993). This pathway is not directly involved A Novel Reinforcement of Birdsong Vocalization Learning auditory syllable encoding preprocessing sequence generation reinforcement attention 103 motor pattern generation memory of tutor song normalized evaluation synaptic perturbation gradient estimate Figure 2: Schematic of primary song control nuclei and their proposed functions in the present model of bird song learning. in vocalization because lesions in these nuclei in adult birds do not impair their crystallized songs. However, lesions in area X and LMAN during the motor learning phase result in contrasting deficits. The songs of LMAN-Iesioned birds crystallize prematurely, whereas the songs of area X-Iesioned birds remain variable (Scharff and Nottebohm, 1991). It has been suggested that this pathway is responsible for the storage of song templates (Doupe and Konishi, 1991) or guidance of the synaptic connection from HVc to RA (Mooney, 1992). 3 FUNCTIONAL NEUROANATOMY OF BIRDSONG The song learning process can be decomposed into three stages. In the first stage, suitable internal acoustic representations of syllables and syllable combinations are constructed. This "auditory template" can be assembled by unsupervised learning schemes like clustering and principal components analysis. The second stage involves the encoding of phonetic sequences using the internal representation. If the representation is sparse or nearly orthogonal, sequential transition can be easily encoded by Hebbian learning. The third stage is an inverse mapping from the internal auditory representation into spatio-temporal patterns of motor commands. This can be accomplished by exploration in the space of motor commands using reinforcement learning. The responses of the units that encode the acoustic primitives can be used to the evaluate the resulting auditory signal and direct the exploration. How are these three computational stages organized within the brain areas and pathways of the songbird? Figure 2 gives an overview of our current working hypothesis. Auditory inputs are pre-processed in field L. Some higher-order representations, such as syllables and syllable combinations, are established in HVc depending on the bird's auditory experience. Moreover, transitions between syllables are encoded in the HVc network. The sequential activation of syllable coding units in HVc are transformed into the time course of motor commands in RA. DM and nXIIts control breathing and the muscles in syrinx, bird's vocal organ. 104 Kenji Doya, Terrence 1. Sejnowski a bronchus Figure 3: (a) The syrinx of songbirds. (b) The model syrinx. The consequences of selective lesions of areas in the anterior forebrain pathway (Scharff and Nottebohm, 1991) are consistent with the failures expected for a reinforcement learning system. In particular, we suggest that this pathway serves the function of an adaptive critic with stochastic search elements (Barto et al., 1983). We propose that LMAN perturbs the synaptic connections from HVc to RA and area X regulates LMAN by the song evaluation. Modulation of HVc to RA connection by LMAN is biologically plausible since LMAN input to RA is mediated mainly by NMDA type synapses, which can modulate the amplitude of mainly non-NMDA type synaptic input from HVc (Mooney, 1992). The assumption that area X provides evaluation is supported by the fact that it receives catecholaminergic projection (dopamine of norepinephrine) from a midbrain nucleus AVT (Lewis et al., 1981). These neurotransmitters are used in many species for reinforcement or attention signals. It is known that auditory learning is enhanced when associated with visual or social interaction with the tutor. Area X is a candidate region where auditory inputs from HVc are associated with reinforcing input from AVT during auditory learning. 4 CONSTRUCTION OF SONG LEARNING MODEL In order to test the above hypothesis, we constructed a computer model of the bird song learning system. The specific aim was to simulate the process of explorative motor learning, in which the time course of motor command for each syllable is determined by auditory template matching. We assumed that orthogonal representations for syllables and their sequential activation were already established in HVc and that an auditory template matching mechanism exists in area X. 4.1 The syrinx The bird's syrinx is located near the junction of the trachea and the bronchi (Vicario, 1991). Its sound source is the tympani form membrane which faces to the bronchus on one side and the air sac on the other (Figure 3a). When some of the syringeal muscles contract, the lumen of the bronchus is throttled and produces vibration in the membrane. When stretched along one dimension, the membrane produces harmonic sounds, but when stretched along two dimensions, the sound contains non-harmonic components (Casey and Gaunt, 1985). Accordingly, we provided two sound sources for the model syrinx (Figure 3b). The fundamental frequency of the harmonic component was controlled by the membrane tension in one direction (Tl). The amplitu6e of the noisy component was proportional to the A Novel Reinforcement of Birdsong Vocalization Learning 105 Ex ! ! ! ~ ~Th ? ~T' ? T2 HVc RA OM nXlIts Figure 4: RA units with different spatio-temporal output profiles are driven by locallycoded RVc units. LMAN perturbs the weights W between RVc and RA. The output units in DM and nXIIts drive the model syrinx. membrane tension in an orthogonal direction (T2). Mixture of these sounds went through a bandpass filter whose resonance frequency was controlled by the throttling of the bronchus (Th). The overall sound amplitude was determined by the strength of expiration (Ex). By controlling the time course of these four variables (Ex,Th,Tl,T2), the model could produce wide variety of "bird-like" chirps and warbles. 4.2 Motor pattern generation in RA RA is topographically organized, each part projecting to different motoneuron pools in nXIIts (Vicario, 1991). Also, RA neurons have complex temporal responses to the inputs from RVc (Mooney, 1992). Therefore, we assumed that RA consists of groups of neurons with specific spatial and temporal output properties, as shown in Figure 4. For each of the four motor command variables, we provided several units with different temporal response kernels. The sequential activation of syllable coding units in RVc drove the RA units through the weights W. Their responses were linearly combined and squashed between 0 and 1 to make the final motor commands. 4.3 Weight space search by LMAN and area X With the above model of the motor output, the task is to find a connection matrix W that maximizes a template matching measure. One way for doing this is to perturb the output of the units and correlate it with the input and the evaluation (Barto et aI., 1983). An alternative way, adopted here, is to perturb the weights and correlate them with the evaluation. We used the following stochastic gradient ascent algorithm. The weight matrix W is modulated by ~ W, given by the sum of the evaluation gradient estimate G and a random component. The modulated weight persists if the resulting vocalization is better than the recent average evaluation E[r]. The evaluation gradient estimate G is updated by the sum 106 Kenji Doya, Terrence J. Sejnowski of the perturbations ~ W weighted with the normalized evaluation. ~ W := r := G + random perturbation evaluation of the song generated with W W:= G := W+~W if r + ~W > E[r] E[r] JVR" ~W + (1- o:)G, V[r] r 0: where 0 < 0: < 1 provides a form of "momentum" in weight space similar to that used in supervised learning. The average and the variance of evaluation are also estimated on-line as follows. 4.4 + (1 - 0: )E[r] E[r] := o:r V[r] := o:(r - E[r]? + (1 - o:)V[r] . Spectrographic template matching We assumed that evaluation for vocalization is given separately for each of the syllables in a song. The sound signal was analyzed by an 80 channel spectrogram. Each output channel was sent to an analog delay line similar to the gamma filter (de Vries and Principe, 1992). The snapshot image of this (80 channels) x (12 steps) delay line at the end of each syllable was stored as the template. The same delay line image of the syllable generated by the model was compared with the template. This allowed some compensation for variable syllable duration. The direction cosine between the two delay line images was used for the reinforcement signal, r. 5 SIMULATION RESULTS One phrase of a recorded zebra finch song (Figure 5a) was the target. Templates were stored for the five syllables in the phrase. Five HVc units coded the five different syllables and 16 RA units represented the four output variables and four different temporal kernels. Learning was started with small random weights. After 200 to 300 trials, the syllable evaluation by direction cosine reached 0.9 (Figure 5d, solid line). The syllables of the learned song resembled the overall frequency profiles of the original syllables. The complex spectrographic structure of the original syllables were, however, not accurate (Figure 5b). One reason for this imperfect replication could be the difference between the vocal organs of the real zebra finch syrinx and our model syrinx. In order to check the significance of this difference, we took syllable templates from the model song (Figure 5b) and trained another model with random initial weights. In this case, the direction cosine went up to 0.96 (Figure 5d, dotted line) and they sounded quite similar to human ears (Figure 5c). We also checked the importance of the gradient estimate G in our algorithm. The dashed line in Figure 5d shows the performance of the model with G 0: a simple random walk learning. The learning was hopelessly slow and resembles the deficit seen after lesion of area X (Figure 2). = A Novel Reinforcement of Birdsong Vocalization Learning 1.0 107 ~-~--~--~-~---, d - zebra finch JOng model song - - - - random walk trilla Figure 5: Spectrograms of (a) the original zebra finch song, (b) the learned song based on the tutor in (a), and (c) the second generation learned song based on the tutor in (b). (d) Learning curves for two tutors: a zebra finch song (solid line) and a model song (dotted line) compared with an undirected search in weight space (dashed line). Weight perturbation was given by a Gaussian distribution with (J 0.1. The averaging parameter was a 0.2. Simulating 500 trials took 30 minutes on Sparc Station 10. = 6 = Discussion We have assumed that each vocalized syllable was separately evaluated. If the evaluation is given only at the end of one song or a phrase, learning can be much more difficult because of the temporal credit assignment problem. If we assume that birds take the easiest strategy available, there should be syllable specific evaluation and separate perturbation mechanisms. In some songbirds, individual syllables are practiced out of order at an early stage, and only later is the sequence matched to the auditory template. Selectivity of auditory responses in both HVc and area X develop during motor learning (Volman, 1993; Doupe, 1993). We can expect such change in response tuning in area X if the evaluations of syllables or syllable sequences are normalized with respect to recent average performance, as we assumed in our model. Many simplifying assumptions were made in the present model: syllables were unary coded in HVc; simple spectrographic template matching was used; the number of motor output variables and temporal kernels were fairly small; and the sound synthesizer was much simpler than a real syrinx. However, it is not difficult to replace these idealizations with more biologically accurate models. Since the number of learning trials needed in the present model was much less than in the real birdsong learning (tens of thousands of trials), there is margin for further elaboration. 108 Kenji Doya, Terrence 1. Sejnowski Acknowledgments We thank M. Lewicki for the zebra finch song data and M. Konishi, A. Doupe, M. Lewicki, E. Vu, D. Perkel and G. Striedter for their helpful discussions. References Barto, A. G., Sutton, R. S., and Anderson, C. W. (1983). Neuronlike adaptive elements that can solve difficult learning control problems. IEEE Transactions on System, Man, and Cybernetics, SMC-13:834-846. Casey, R. M. and Gaunt, A. S. (1985). Theoretical models of the avian syrinx. Journal of Theoretical Biology, 116:45-64. de Vries, B. and Principe, J. C. (1992). The gamma model-A new neural model for temporal processing. Neural Networks, 5:565-576. Doupe, A. J. (1993). A neural circuit specialized for vocal learning. Current Opinion in Neurobiology, 3:104-111. Doupe, A. J. and Konishi, M. (1991). Song-selective auditory circuits in the vocal control system of the zebra finch. Proceedings of the National Academy of Sciences, USA, 88: 11339-11343. Harris-Warrick, R. M., Marder, E., Selverston, A. I., and Moulins, M. (1992). Dynamic Biological Networks-The Stomatogastric Nervous System. MIT Press, Cambridge, MA. Konishi, M. (1965). fhe role of auditory feedback in the control of vocalization in the white-crowned sparrow. Zeitschrift fur Tierpsychologie, 22:770-783. Lewis, J. W., Ryan, S. M., Arnold, A. P., and Butcher, L. L. (1981). Evidence for a catecholarninergic projection to area x in the zebra finch. Journal of Comparative Neurology, 196:347-354. Mooney, R. (1992). Synaptic basis of developmental plasticity in a birdsong nucleus. Journal of Neuroscience, 12:2464-2477. Nottebohm, F. (1991). Reassessing the mechanisms and origins of vocal learning in birds. Trends in Neurosciences, 14:206-211. Scharff, C. and Nottebohm, F. (1991). A comparative study of the behavioral deficits following lesions of various parts of the zebra finch song systems: Implications for vocal learning. Journal of Neuroscience, 11 :2896-2913. Vicario, D. S. (1991). Neural mechanisms of vocal production in songbirds. Current Opinion in Neurobiology, 1:595-600. Volman, S. F. (1993). Development of neural selectivity for birdsong during vocal learning. Journal of Neuroscience, 13:4737--4747. Vu, E. T., Mazurek, M. E., and Kuo, Y.-C. (1994). Identification of a forebrain motor programming network for the learned song of zebra finches. Journal of Neuroscience, 14:6924-6934. 

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Reinforcement Learning Methods for Continuous-Time Markov Decision Problems Steven J. Bradtke Computer Science Department University of Massachusetts Amherst, MA 01003 bradtkeGcs.umass.edu Michael O. Duff Computer Science Department University of Massachusetts Amherst, MA 01003 duffGcs.umass.edu Abstract Semi-Markov Decision Problems are continuous time generalizations of discrete time Markov Decision Problems. A number of reinforcement learning algorithms have been developed recently for the solution of Markov Decision Problems, based on the ideas of asynchronous dynamic programming and stochastic approximation. Among these are TD(,x), Q-Iearning, and Real-time Dynamic Programming. After reviewing semi-Markov Decision Problems and Bellman's optimality equation in that context, we propose algorithms similar to those named above, adapted to the solution of semi-Markov Decision Problems. We demonstrate these algorithms by applying them to the problem of determining the optimal control for a simple queueing system. We conclude with a discussion of circumstances under which these algorithms may be usefully applied. 1 Introduction A number of reinforcement learning algorithms based on the ideas of asynchronous dynamic programming and stochastic approximation have been developed recently for the solution of Markov Decision Problems. Among these are Sutton's TD(,x) [10], Watkins' Q-Iearning [12], and Real-time Dynamic Programming (RTDP) [1, 394 Steven Bradtke, Michael O. Duff 3]. These learning alogorithms are widely used, but their domain of application has been limited to processes modeled by discrete-time Markov Decision Problems (MDP's). This paper derives analogous algorithms for semi-Markov Decision Problems (SMDP's) - extending the domain of applicability to continuous time. This effort was originally motivated by the desire to apply reinforcement learning methods to problems of adaptive control of queueing systems, and to the problem of adaptive routing in computer networks in particular. We apply the new algorithms to the well-known problem of routing to two heterogeneous servers [7]. We conclude with a discussion of circumstances under which these algorithms may be usefully applied. 2 Semi-Markov Decision Problems A semi-Markov process is a continuous time dynamic system consisting of a countable state set, X, and a finite action set, A. Suppose that the system is originally observed to be in state z EX, and that action a E A is applied. A semi-Markov process [9] then evolves as follows: ? The next state, y, is chosen according to the transition probabilities Pz,(a) ? A reward rate p(z, a) is defined until the next transition occurs ? Conditional on the event that the next state is y, the time until the transition from z to y occurs has probability distribution Fz,(?Ja) One form of the SMDP is to find a policy the minimizes the expected infinite horizon discounted cost, the "value" for each state: e {IoOO e-.B t p(z(t), a(t?dt}, where z(t) and aCt) denote, respectively, the state and action at time t. For a fixed policy 71', the value of a given state z must satisfy v,..(z) L P ,(7I'(z? 10(00 10re-.B? p(z, 71'(z?dsdFz,(tJ7I'(z? + z ,E X L Pz,(7I'(Z? fooo e-.B t V,..(y)dFz,(tJ7I'(z?. (1) ,EX Defining R(z, y, a) = foOO fot e-.B? p(z, 71'(z?dsdFz, (tJ7I'(z?, the expected reward that will be received on transition from state z to state y on action a, and Reinforcement Learning Methods for Continuous-Time Markov Decision Problems 395 the expected discount factor to be applied to the value of state y on transition from state z on action a, it is clear that equation (1) is nearly identical to the value-function equation for discrete time Markov reward processes, Vw(z) = R(z, 1I"(z? + "Y I: Pzr (1I"(z?Vw (Y), (2) rEX where R(z, a) = :ErEx Pzr(a)R(z, y, a). If transition times are identically one for an SMDP, then a standard discrete-time MDP results. Similarly, while the value function associated with an optimal policy for an MDP satisfies the Bellman optimality equation I: Ve(z) = max {R(Z' a) + "Y pzr(a)v*(y)} , ilEA X rE (3) the optimal value function for an SMDP satisfies the following version of the Bellman optimality equation: V*(z) = max { ilEA I:X Pzr(a) 1 re-fJa p(z, a)dsdFzr(tJa) 0 10 00 + rE I: Pzr(a) loo e-fJtv*(y)dFzr(tJa)} . (4) rEX 3 Temporal Difference learning for SMDP's Sutton's TD(O) [10] is a stochastic approximation method for finding solutions to the system of equations (2). Having observed a transition from state z to state y with sample reward r(z, y, 1I"(z?, TD(O) updates the value function estimate V(A:)(z) in the direction of the sample value r(z, y, 1I"(z?+"YV(A:)(y). The TD(O) update rule for MDP's is V(A:+l)(Z) = V(A:)(z) + QA:[r(z, y, 1I"(z? + "YV(A:)(y) - V(A:)(z)], (5) where QA: is the learning rate. The sequence of value-function estimates generated by the TD(O) proceedure will converge to the true solution, Vw , with probability one [5,8, 11] under the appropriate conditions on the QA: and on the definition of the MDP. The TD(O) learning rule for SMDP's, intended to solve the system of equations (1) given a sequence of sampled state transitions, is: 1 -fJT ] V(A:+1)(z) = V(A:)(z) + QA: [ - ; r(z, y, 1I"(z? + e-fJTV(A:)(y) - V(A:)(z) , (6) where the sampled transition time from state z to state y was T time units, r(z, y, 1I"(z? is the sample reward received in T time units, and e- fJT is the sample discount on the value of the next state given a transition time of T time units. The TD(>.) learning rule for SMDP's is straightforward to define from here. I_p-tl. Steven Bradtke. Michael 0. Duff 396 4 Q-Iearning for SMDP's Denardo [6] and Watkins [12] define Q.f) the Q-function corresponding to the policy as (7) Q'II"(z, a) = R(z, a) + 'Y PzJ(a)V'II"(Y) 71", 2: YEX Notice that a can be any action. It is not necesarily the action 7I"(z) that would be chosen by policy 71". The function Q. corresponds to the optimal policy. Q'II"(z, a) represents the total discounted return that can be expected if any action is taken from state z, and policy 71" is followed thereafter. Equation (7) can be rewritten as Q'II"(z, a) = R(z, a) + 'Y 2: PZJ(a)Q'II"(Y' 7I"(Y?, (8) yEX and Q. satisfies the Bellman-style optimality equation Q?(z, a) = R(z, a) + 'Y 2: Pzy(a) max Q.(y, a'), JEX (9) A'EA Q-Iearning, first described by Watkins [12], uses stochastic approximation to iteratively refine an estimate for the function Q ?. The Q-Iearning rule is very similar to TD(O). Upon a sampled transition from state z to state y upon selection of a, with sampled reward r(z, y, a), the Q-function estimate is updated according to Q(A:+l)(Z, a) = Q(J:)(z, a) + etJ: [r(z, y, a) + 'Y ~~ Q(J:)(y, a') - Q(J:)(z, a)]. (10) Q-functions may also be defined for SMDP's. The optimal Q-function for an SMDP satisfies the equation Q?(z, a) 2: PZJ(a) roo t 10 10 'V JE"- e- tJ ? p(z, a)dsdFzJ(tla) + 2: Pz1I (a) roo e- tJt max Q.(y, a')dFzJ(tla). 10 'V JE"- (11) A'EA This leads to the following Q-Iearning rule for SMDP's: Q(A:+l)(Z, a) = Q(J:)(z, a)+etJ: [1 - ;-tJ'r' r(z, y, a) + e-tJ'r' ~~ Q(J:)(y, a') _ Q(J:)(z, a)] (12) 5 RTDP and Adaptive RTDP for SMDP's The TD(O) and Q-Iearning algorithms are model-free, and rely upon stochastic approximation for asymptotic convergence to the desired function (V'll" and Q., respectively). Convergence is typically rather slow. Real-Time Dynamic Programming (RTDP) and Adaptive RTDP [1,3] use a system model to speed convergence. Reinforcement Learning Methods for Continltolts-Time Markov Decision Problems 397 RTDP assumes that a system model is known a priori; Adaptive RTDP builds a model as it interacts with the system. As discussed by Barto et al. [1], these asynchronous DP algorithms can have computational advantages over traditional DP algorithms even when a system model is given. Inspecting equation (4), we see that the model needed by RTDP in the SMDP domain consists of three parts: 1. the state transition probabilities Pzy(a), 2. the expected reward on transition from state z to state y using action a, R(z, y, a), and 3. the expected discount factor to be applied to the value of the next state on transition from state z to state y using action a, 'Y(z, y, a). If the process dynamics are governed by a continuous time Markov chain, then the model needed by RTDP can be analytically derived through uniJormization [2]. In general, however, the model can be very difficult to analytically derive. In these cases Adaptive RTD P can be used to incrementally build a system model through direct interaction with the system. One version of the Adaptive RTDP algorithm for SMDP's is described in Figure 1. 1 2 3 4 Set k = 0, and set Zo to some start state. Initialize P, R, and ~. repeat forever { For all actions a, compute Q(Ie)(ZIe,a) = L P.."v(a) [ R(zIe,y,a) +~(zIe,y,a)V(Ie)(y) ] veX Perform the update V(le+l)(ZIe) = minoeA Q(Ie)(zIe,a) Select an action, ale. Perform ale and observe the transition to ZIe+l after T time units. Update P. Use the sample reward 1__;;11'" r(ZIe,Zle+l,ale) and the sample discount 5 6 7 factor e- f3T to update k=k+l 8 9 R and ~. } Figure 1: Adaptive RTDP for SMDP's. by Adaptive RTDP of P, R, and 'Y. P, il, and .y are the estimates maintained Notice that the action selection procedure (line 6) is left unspecified. Unlike RTDP, Adaptive RTDP can not always choose the greedy action. This is because it only has an e8timate of the system model on which to base its decisions, and the estimate could initially be quite inaccurate. Adaptive RTDP needs to explore, to choose actions that do not currently appear to be optimal, in order to ensure that the estimated model converges to the true model over time. 398 6 Steven Bradtke, Michael O. Duff Experiment: Routing to two heterogeneous servers Consider the queueing system shown in Figure 2. Arrivals are assumed to be Poisson with rate ).. Upon arrival, a customer must be routed to one of the two queues, whose servers have service times that are exponentially distributed with parameters J.l.1 and J.l.2 respectively. The goal is compute a policy that minimizes the objective function: e {foOO e-tJ t [c1n1(t) + C2n2(t)]dt}, where C1 and C2 are scalar cost factors, and n1(t) and n2(t) denote the number of customers in the respective queues at time t. The pair (n1(t), n2(t)) is the state of the system at time t; the state space for this problem is countably infinite. There are two actions available at every state: if an arrival occurs, route it to queue 1 or route it to queue 2. -.<-~ ___ -.J~ Figure 2: Routing to two queueing systems. It is known for this problem (and many like it [7]), that the optimal policy is a threshold policy; i.e., the set of states Sl for which it is optimal to route to the first queue is characterized by a monotonically nondecreasing threshold function F via Sl {(nl,n2)ln1 $ F(n2)}' For the case where C1 C2 1 and J.l.1 J.l.2, the policy is simply to join the shortest queue, and the theshold function is a line slicing diagnonally through the state space. = = = = We applied the SMDP version of Q-Iearning to this problem in an attempt to find the optimal policy for some subset of the state space. The system parameters were set to ). J.l.1 J.l.2 1, /3 0.1, and C1 C2 1. We used a feedforward neural network trained using backpropagation as a function approximator. = = = = = = Q-Iearning must take exploratory actions in order to adequately sample all of the available state transitions. At each decision time k, we selected the action aA: to be applied to state ZA: via the Boltzmann distribution where TA: is the "computational temperature." The temperature is initialized to a relatively high value, resulting in a uniform distribution for prospective actions. TA: is gradually lowered as computation proceeds, raising the probability of selecting actions with lower (and for this application, better) Q-values. In the limit, the action that is greedy with respect to the Q-function estimate is selected. The temperature and the learning rate erA: are decreased over time using a "search then converge" method [4]. Reinforcement Learning Methods for Continuous-Time Markov Decision Problems 399 Figure 3 shows the results obtained by Q-Iearning for this problem. Each square denotes a state visited, with nl(t) running along the z-axis, and n2(t) along the yaxis. The color of each square represents the probability of choosing action 1 (route arrivals to queue 1). Black represents probability 1, white represents probability o. An optimal policy would be black above the diagonal, white below the diagonal, and could have arbitrary colors along the diagonal. == == == == == = = ;;= == == == = = == ;;= == == == = = ? . ?? II II ?? ... !m il lUll it ? w? ? @@ ~d r2 E M m@ mi?1@ moo mllw ? ? ? ? ? ? ,.m @ll A ;;;;;; . .. ? ~2 III l1li 0 @oo @ %@Ii Ell =11111 m?? ? ?? IIIIJI III ?? ???????? @w liliiii w???????? ' @ m ? ? ? ? ? ? ? ? ? '. B II ?? mm oow mw m]lw mill lUll lM]lm ?? ? ? ? ? ? ? ? ? lIlIg!WI'm = wji n i?????????? II ...'.'........'. c Figure 3: Results of the Q-Iearning experiment. Panel A represents the policy after 50,000 total updates, Panel B represents the policy after 100,000 total updates, and Panel C represents the policy after 150,000 total updates. One unsatisfactory feature of the algorithm's performance is that convergence is rather slow, though the schedules governing the decrease of Boltzmann temperature TA: and learning rate 0A: involve design parameters whose tweakings may result in faster convergence. If it is known that the optimal policies are of theshold type, or that some other structural property holds, then it may be of extreme practical utility to make use of this fact by constraining the value-functions in some way or perhaps by representing them as a combination of appropriate basis vectors that implicity realize or enforce the given structural property. 7 Discussion In this paper we have proposed extending the applicability of well-known reinforcement learning methods developed for discrete-time MDP's to the continuous time domain. We derived semi-Markov versions of TD(O), Q-Iearning, RTDP, and Adaptive RTDP in a straightforward way from their discrete-time analogues. While we have not given any convergence proofs for these new algorithms, such proofs should not be difficult to obtain if we limit ourselves to problems with finite state spaces. (Proof of convergence for these new algorithms is complicated by the fact that, in general, the state spaces involved are infinite; convergence proofs for traditional reinforcement learning methods assume the state space is finite.) Ongoing work is directed toward applying these techniques to more complicated systems, examining distributed control issues, and investigating methods for incorporating prior 400 Steven Bradtke, Michael 0. Duff knowledge (such as structured function approximators). Acknowledgements Thanks to Professor Andrew Barto, Bob Crites, and to the members of the Adaptive Networks Laboratory. This work was supported by the National Science Foundation under Grant ECS-9214866 to Professor Barto. References [1] A. G. Barto, S. J. Bradtke, and S. P. Singh. Learning to act using real-time dynamic programming. Artificial Intelligence. Accepted. [2] D. P. Bertsekas. Dynamic Programming: Deterministic and Stochastic Models. Prentice Hall, Englewood Cliffs, NJ, 1987. [3] S. J. Bradtke. Incremental Dynamic Programming for On-line Adaptive Optimal Control. PhD thesis, University of Massachusetts, 1994. [4] C. Darken, J. Chang, and J. Moody. Learning rate schedules for faster stochastic gradient search. In Neural Networks for Signal Processing ~ - Proceedings of the 199~ IEEE Workshop. IEEE Press, 1992. [5] P. Dayan and T. J. Sejnowski. Td(A): Convergence with probability 1. Machine Learning, 1994. [6] E. V. Denardo. Contraction mappings in the theory underlying dynamic programming. SIAM Review, 9(2):165-177, April 1967. [7] B. Hajek. Optimal control of two interacting service stations. 29:491-499, 1984. IEEE-TAC, [8] T. Jaakkola, M. I. Jordan, and S. P. Singh. On the convergence of stochastic iterative dynamic programming algorithms. Neural Computation, 1994. [9] S. M. Ross. Applied Probability Models with Optimization Applications. HoldenDay, San Francisco, 1970. [10] R. S. Sutton. Learning to predict by the method of temporal differences. Machine Learning, 3:9-44, 1988. [11] J. N. Tsitsiklis. Asynchronous stochastic approximation and Q-Iearning. Technical Report LIDS-P-2172, Laboratory for Information and Decision Systems, MIT, Cambridge, MA, 1993. [12] C. J. C. H. Watkins. Learning from Delayed Rewards. PhD thesis, Cambridge University, Cambridge, England, 1989. 

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127 Neural Network Implementation Approaches for the Connection Machine Nathan H. Brown, Jr. MRJlPerkin Elmer, 10467 White Granite Dr. (Suite 304), Oakton, Va. 22124 ABSlRACf The SIMD parallelism of the Connection Machine (eM) allows the construction of neural network simulations by the use of simple data and control structures. Two approaches are described which allow parallel computation of a model's nonlinear functions, parallel modification of a model's weights, and parallel propagation of a model's activation and error. Each approach also allows a model's interconnect structure to be physically dynamic. A Hopfield model is implemented with each approach at six sizes over the same number of CM processors to provide a performance comparison. INTRODUCflON Simulations of neural network models on digital computers perform various computations by applying linear or nonlinear functions, defined in a program, to weighted sums of integer or real numbers retrieved and stored by array reference. The numerical values are model dependent parameters like time averaged spiking frequency (activation), synaptic efficacy (weight), the error in error back propagation models, and computational temperature in thermodynamic models. The interconnect structure of a particular model is implied by indexing relationships between arrays defined in a program. On the Connection Machine (CM), these relationships are expressed in hardware processors interconnected by a 16-dimensional hypercube communication network. Mappings are constructed to defme higher dimensional interconnectivity between processors on top of the fundamental geometry of the communication network. Parallel transfers are defined over these mappings. These mappings may be dynamic. CM parallel operations transform array indexing from a temporal succession of references to memory to a single temporal reference to spatially distributed processors. Two alternative approaches to implementing neural network simulations on the CM are described. Both approaches use "data parallelism" 1 provided by the *Lisp virtual machine. Data and control structures associated with each approach and performance data for a Hopfield model implemented with each approach are presented. DATA STRUCTURES The functional components of a neural network model implemented in *Lisp are stored in a uniform parallel variable (pvar) data structure on the CM. The data structure may be viewed as columns of pvars. Columns are given to all CM virtual processors. Each CM physical processor may support 16 virtual processors. In the fust approach described, CM processors are used to represent the edge set of a models graph structure. In the second approach described, each processor can represent a unit, an outgoing link, or an incoming link in a model's structure. Movement of activation (or error) through a model's interconnect structure is simulated by moving numeric values ? American Institute of Physics 1988 128 over the eM's hypercube. Many such movements can result from the execution of a single CM macroinstruction. The CM transparently handles message buffering and collision resolution. However, some care is required on the part of the user to insure that message traffic is distributed over enough processors so that messages don't stack up at certain processors, forcing the CM to sequentially handle large numbers of buffered messages. Each approach requires serial transfers of model parameters and states over the communication channel between the host and the CM at certain times in a simulation. The first approach, "the edge list approach," distributes the edge list of a network graph to the eM, one edge per CM processor. Interconnect weights for each edge are stored in the memory of the processors. An array on the host machine stores the current activation for all units. This approach may be considered to represent abstract synapses on the eM. The interconnect structure of a model is described by product sets on an ordered pair of identification (id) numbers, rid and sid. The rid is the id of units receiving activation and sid the id of units sending activation. Each id is a unique integer. In a hierarchical network, the ids of input units are never in the set of rids and the ids of output units are never in the set of sids. Various set relations (e.g. inverse, reflexive, symmetric, etc.) defined over id ranges can be used as a high level representation of a network's interconnect structure. These relations can be translated into pvar columns. The limits to the interconnect complexity of a simulated model are the virtual processor memory limits of the CM configuration used and the stack space ~uired by functions used to compute the weighted sums of activation. Fig. 1 shows a R -> R2 -> R4 interconnect structure and its edge list representation on the CM. 6 7 8 :z eM PROCESSOR 0 1 2 3 4 9 3 5 6 7 8 9 1 0111213 f if :~ (~?,';)H f HHH ff SAcr ( 8j ) 1 2 3 1 2 3 4 5 4 5 4 5 4 5 Fig. 1. Edge List Representation of a R3_> R2 -> R4 Interconnect Structure This representation can use as few as six pvars for a model with Hebbian adaptation: rid (i), sid (j), interconnect weight (wij), ract (ai), sact (aj), and learn rate (11)? Error back propagation requires the addition of: error (ei), old interconnect weight (wij(t-l?, and the momentum term (ex). The receiver and sender unit identification pvars are described above. The interconnect weight pvar stores the weight for the interconnect. The activation pvar, sact, stores the current activation, aj' transfered to the unit specified by rid from the unit specified by sid. The activation pvar, ract, stores the current weighted activation ajwij- The error pvar stores the error for the unit specified by the sid. A variety of proclaims (e.g. integer, floating point, boolean, and field) exist in *Lisp to define the type and size ofpvars. Proclaims conserve memory and speed up execution. Using a small number of pvars limits the 129 amount of memory used in each CM processor so that maximum virtualization of the hardware processors can be realized. Any neural model can be specified in this fashion. Sigma-pi models require multiple input activation pvars be specified. Some edges may have a different number of input activation pvars than others. To maintain the uniform data structure of this approach a tag pvar has to be used to determine which input activation pvars are in use on a particular edge. The edge list approach allows the structure of a simulated model to "physically" change because edges may be added (up to the virtual processor limit), or deleted at any time without affecting the operation of the control structure. Edges may also be placed in any processor because the subselection (on rid or sid) operation performed before a particular update operation insures that all processors (edges) with the desired units are selected for the update. The second simulation approach, "the composite approach," uses a more complicated data structure where units, incoming links, and outgoing links are represented. Update routines for this approach use parallel segmented scans to form the weighted sum of input activation. Parallel segmented scans allow a MIMD like computation of the weighted sums for many units at once. Pvar columns have unique values for unit, incoming link, and outgoing link representations. The data structures for input units, hidden units, and output units are composed of sets of the three pvar column types. Fig. 2 shows the representation for the same model as in Fig. 1 implemented with the composite approach. 2 o1 3 5 4 6 7 8 9 2 3 4 5 6 7 8 9 101112 1314151617181920212223242526272829303132333435 rr, f~ ~\'~~Ii~ c - -?. c o c - -?. +----+ lol IO~ O.~~ ~ ~~+-t+*~ t -~. - ~ II I( II I( I Fig. 2. Composite Representation of a R3 -> R2 -> R4 Interconnect Structure In Fig. 2, CM processors acting as units, outgoing links, and incoming links are represented respectively by circles, triangles, and squares. CM cube address pointers used to direct the parallel transfer of activation are shown by arrows below the structure. These pointers defme the model interconnect mapping. Multiple sets of these pointers may be stored in seperate pvars. Segmented scans are represented by operation-arrow icons above the structure. A basic composite approach pvar set for a model with Hebbian adaptation is: forward B, forward A, forward transfer address, interconnect weight (Wij), act-l (ai), act-2 (aj), threshold, learn rate (Tl), current unit id (i), attached unit id U), level, and column type. Back progagation of error requires the addition of: backward B, backward A, backward transfer address, error (ei), previous interconnect weight (Wij(t-l?, and the momentum tenn (ex). The forward and backward boolean pvars control the segmented scanning operations over unit constructs. Pvar A of each type controls the plus scanning and pvar B of each type controls the copy scanning. The forward transfer pvar stores cube addresses for 130 forward (ascending cube address) parallel transfer of activation. The backward transfer pvar stores cube addresses for backward (descending cube address) parallel transfer of error. The interconnect weight, activation, and error pvars have the same functions as in the edge list approach. The current unit id stores the current unit's id number. The attached unit id stores the id number of an attached unit. This is the edge list of the network's graph. The contents of these pvars only have meaning in link pvar columns. The level pvar stores the level of a unit in a hierarchical network. The type pvar stores a unique arbitrary tag for the pvar column type. These last three pvars are used to subselect processor ranges to reduce the number of processors involved in an operation. Again, edges and units can be added or deleted. Processor memories for deleted units are zeroed out. A new structure can be placed in any unused processors. The level, column type, current unit id, and attached unit id values must be consistent with the desired model interconnectivity. The number of CM virtual processors required to represent a given model on the CM differs for each approach. Given N units and N(N-1) non-zero interconnects (e.g. a symmetric model), the edge list approach simply distributes N(N-1) edges to N(N-1) CM virtual processors. The composite approach requires two virtual processors for each interconnect and one virtual processor for each unit or N +2 N (N-1) CM virtual processors total. The difference between the number of processors required by the two approaches is N2. Table I shows the processor and CM virtualization requirements for each approach over a range of model sizes. TABLE I Model Sizes and CM Processors Required Run No. Grid Size Number of Units Edge List Quart CM Virt. Procs. Virt. LeveL N(N-1) 1 2 3 4 5 6 82 92 112 13 2 162 192 64 81 121 169 256 361 4032 6480 14520 28392 65280 129960 8192 8192 16384 32768 65536 131072 0 0 0 2 4 8 Run No. Grid Size Number of Units Composite Quart CM Virt. Procs. Virt. LeveL N+2N(N-1) 7 8 9 10 11 12 82 92 112 132 162 192 64 81 121 169 256 361 8128 13041 29161 56953 130816 260281 8192 16384 32768 65536 131072 262144 0 0 2 4 8 16 131 CONTROL STRUCTURES The control code for neural network simulations (in *Lisp or C*) is stored and executed sequentially on a host computer (e.g. Symbolics 36xx and V AX 86xx) connected to the CM by a high speed communication line. Neural network simulations executed in *Lisp use a small subset of the total instruction set: processor selection reset (*all), processor selection (*when), parallel content assignment (*set), global summation (*sum), parallel multiplication (*!! ), parallel summation (+! I), parallel exponentiation (exp! I), the parallel global memory references (*pset) and (pref! I), and the parallel segmented scans (copy!! and +!!). Selecting CM processors puts them in a "list of active processors" (loap) where their contents may be arithmetically manipulated in parallel. Copies of the list of active processors may be made and used at any time. A subset of the processors in the loap may be "subselected" at any time, reducing the loap contents. The processor selection reset clears the current selected set by setting all processors as selected. Parallel content assignment allows pvars in the currently selected processor set to be assinged allowed values in one step. Global summation executes a tree reduction sum across the CM processors by grid or cube address for particular pvars. Parallel multiplications and additions multiply and add pvars for all selected CM processors in one step. The parallel exponential applies the function, eX, to the contents of a specified pvar, x, over all selected processors. Parallel segmented scans apply two functions, copy!! and +!!, to subsets ofCM processors by scanning across grid or cube addresses. Scanning may be forward or backward (Le. by ascending or descending cube address order, respectively). Figs. 3 and 4 show the edge list approach kernels required for Hebbian learning for a R2 -> R2 model. The loop construct in Fig. 3 drives the activation update (1) operation. The usual loop to compute each weighted sum for a particular unit has been replaced by four parallel operations: a selection reset (*all), a subselection of all the processors for which the particular unit is a receiver of activation (*when (=!! rid (!! (1+ u??, a parallel multiplication (*!! weight sact), and a tree reduction sum (*sum ... ). Activation is spread for a particular unit, to all others it is connected to, by: storing the newly computed activation in an array on the host, then subselecting the processors where the particular unit is a sender of activation (*when (=!! sid (!! (1 + u??, and broadcasting the array value on the host to those processors. (dotimes (u 4) (*all (*when (=!! rid (!! (1+ u?) (setf (aref activation u) (some-nonlinearity (*sum (*!! weight sact?? (*set ract (!! (aref activation u?) (*all (*when (=!! sid (!! (1+ u?) (*set sact (!! (aref activation u??? Fig. 3. Activation Update Kernel for the Edge Lst Approach. Fig. 4 shows the Hebbian weight update kernel 132 (2) (*all (*set weight (*!! learn-rate ract sact?? Fig. 4. Hebbian Weight Modification Kernel for the Edge List Approach The edge list activation update kernel is essentially serial because the steps involved can only be applied to one unit at a time. The weight modification is parallel. For error back propagation a seperate loop for computing the errors for the units on each layer of a model is required. Activation update and error back propagation also require transfers to and from arrays on the host on every iteration step incurring a concomitant overhead. Other common computations used for neural networks can be computed in parallel using the edge list approach. Fig. 5 shows the code kernel for parallel computation of Lyapunov engergy equations (3) where i= 1 to number of units (N). (+ (* -.5 (*sum (*!! weight ract sact?) (*sum (*!! input sact?) Fig. 5. Kernel for Computation of the Lyapunov Energy Equation Although an input pvar, input, is defined for all edges, it is only non-zero for those edges associated with input units. Fig. 6 shows the pvar structure for parallel computation of a Hopfield weight prescription, with segmented scanning to produce the weights in one step, IJ -l:S r= I(2ar1?-I)(2arJ?-I) W? ? (4) where wii=O, Wij=Wjh and r=I to the number of patterns, S, to be stored. seg t n ract vII V2 1 ... V I 2 v22' .. sact weight n t n n VSI vII V2 I ... VSI .. . VS2 v13 v23 ... VS3 .. . wI2 w13 Fig. 6. Pvar Structure for Parallel Computation QfHopfield Weight Prescription Fig. 7 shows the *Lisp kernel used on the pvar structure in Fig. 6. (set weight (scan '+!! (*!! (-!! (*!! ract (!! 2? (!! 1? (-!! (*!! sact (!! 2? (!! 1?? :segment-pvar seg :inc1ude-self t) Fig. 7. Parallel Computation of Hopfield Weight Prescription 133 The inefficiencies of the edge list activation update are solved by the updating method used in the composite approach. Fig. 8 shows the *Lisp kernel for activation update using the composite approach. Fig. 9 shows the *Lisp kernel for the Hebbian learning operation in the composite approach. (*a1l (*when (=!! level (!! 1? (*set act (scan!! act-I 'copy!! :segment-pvar forwardb :include-self t? (*set act (*!! act-l weight? (*when (=!! type (!! 2? (*pset :overwrite act-l act-I ftransfer?) (*when (=!! level (!! 2? (*set act (scan!! act-l '+!! :segment-pvar forwarda :include-self t? (*when (=!! type (!! 1? (some-nonlinearity!! act-I?? Fig. 8. Activation Update Kernel for the Composite Approach (*all (*set act-l (scan!! act-I 'copy!! :segment-pvar forwardb :include-self t? (*when (=!! type (!! 2? (*set act-2 (pref!! act-I btransfer?) (*set weight (+!! weight (*!! learn-rate act-l act-2??) Fig. 9. Hebbian Weight Update Kernel for the Composite Approach It is immediately obvious that no looping is invloved. Any number of interconnects may be updated by the proper subselection. However, the more subselection is used the less efficient the computation becomes because less processors are invloved. COMPLEXITY ANALYSIS The performance results presented in the next section can be largely anticipated from an analysis of the space and time requirements of the CM implementation approaches. For simplicity I use a Rn -> Rn model with Hebbian adaptation. The oder of magnitude requirements for activation and weight updating are compared for both CM implementation approaches and a basic serial matrix arithmetic approach. F~r the given model the space requirements on a conventional serial machine are 2n+n locations or O(n 2). The growth of the space requirement is dominated by the nxn weight matrix. defining the system interconnect structure. The edge list appro~ch uses six pvars for each processor and uses nxn processors for the mapping, or 6n locations or O(n2). The composite approach uses 11 pvars. There are 2n processors for units and 2n2 proces~ors for interconnects in the given model. The composite approach uses 11(2n+2n ) locations or O(n2 ). The CM implementations take up roughly the same space as the serial implementation, but the space for the serial implementation is composed of passive memory whereas the space for the CM implementations is composed of interconnected processors with memory . The time analysis for the approaches compares the time order of magnitudes to compute the activation update (1) and the Hebbian weight update (2). On a serial 134 machine, the n weighted sums computed for the ac~vation update require n2 multiplicationsffd n(n-l) additions. There are 2n -n operations or time order of magnitude O(n ~ The time order of magnitude for the weight matrix update is O(n2) since there are n weight matrix elements. The edge list approach forms n weighted sums by performing a parallel product of all of the weights and activations in the model, (*!! weight sact), and then a tree reduction sum, (*sum ... ), of the products for the n uni~ (see Fig. 4). There are 1+n(nlog2n) operations or time order of magnitude O(n ). This is the same order of magnitude as obtained on a serial machine. Further, the performance of the activation update is a function of the number of interconnects to be processed. The composite approach forms n weighted sums in nine steps (see Fig. 8): five .selection operations; the segmented copy scan before the parallel multiplication; the parallel multiplication; the parallel transfer of the products; and the segmented plus scan, which forms the n sums in one step. This gives the composite activation update a time order of magnitude O( 1). Performance is independent of the number of interconnects processed. The next section shows that this is not quite true. The n2 weights in the model can be updated in three parallel steps using the edge list approach (see Fig. 4). The n2 weights in the model can be updated in eight parallel steps using the composite approach (see Fig. 9). In either case, the weight update operation has a time order of magnitude 0(1). The time complexity results obtained for the composite approach apply to computation of the Lyaponov energy equation (3) and the Hopfield weighting prescription (4), given that pvar structures which can be scanned (see Figs. 1 and 6) are used. The same operations performed serially are time order of magnitude 0(n2). The above operations all incur a one time overhead cost for generating the addresses in the pointer pvars, used for parallel transfers, and arranging the values in segments for scanning. What the above analysis shows is that time complexity is traded for space complexity. The goal of CM programming is to use as many processors as possible at every step. PERFORMANCE COMPARISON Simulations of a Hopfield spin-glass model2 were run for six different model sizes over the same number (16,384) of physical CM processors to provide a performance comparison between implementation approaches. The Hopfield network was chosen for the performance comparison because of its simple and well known convergence dynamics and because it uses a small set of pvars which allows a wide range of network sizes (degrees of virtualization) to be run. Twelve treaments are run. Six with the edge list approach and six with the composite approach. Table 3-1 shows the model sizes run for each treatment. Each treatment was run at the virtualization level just necessary to accomodate the number of processors required for each simulation. Two exemplar patterns are stored. Five test patterns are matched against the two exemplars. Two test patterns have their centers removed, two have a row and column removed, and one is a random pattern. Each exemplar was hand picked and tested to insure that it did not produce cross-talk. The number of rows and columns in the exemplars and patterns increase as the size of the networks for the treatments increases. 135 Since the performance of the CM is at issue, rather than the performance of the network model used, a simple model and a simple pattern set were chosen to minimize consideration of the influence of model dynamics on performance. Performance is presented by plotting execution speed versus model size. Size is measured by the number of interconnects in a model. The execution speed metric is interconnects updated per second, N*(N-l )/t, where N is the number of units in a model and t is the time used to update the activations for all of the units in a model. All of the units were updated three times for each pattern. Convergence was determined by the output activation remaining stable over the fmal two updates. The value of t for a treatment is the average of 15 samples of t. Fig. 10 shows the activation update cycle time for both approaches. Fig. 11 shows the interconnect update speed plots for both approaches. The edge list approach is plotted in black. The composite approach is plotted in white. The performance shown excludes overhead for interpretation of the *Lisp instructions. The model size categories for each plot correspond to the model sizes and levels of eM virtualization shown in Table I. Activation Update Cycle Time vs Model Size 1 .6 1 .4 1.2 ? sees O.B 0.6 0.4 ? ? o 0.2 ? OO___~~~__~O~__~O~__.O__~ 1 2 3 4 5 6 Model Size Fig. 10. Activation Update Cycle Times Interconnect Update Speed Comparison Edge Ust Approach vs. Composite Approach 2000000} 0 1500000 0 0 i.p.s. 1000000 0 0 0 500000t o?1 ? 2 ? ? 3 4 Model Size ? 5 ? 6 Fig. 11. Edge List Interconnect Update Speeds Fig. 11 shows an order of magnitude performance difference between the approaches and a roll off in performance for each approach as a function of the number of virtual processors supported by each physical processor. The performance tum around is at 4x virtualization for the edge list approach and 2x virtualization for the composite approach. 136 CONCLUSIONS Representing the interconnect structure of neural network models with mappings defined over the set of fine grain processors provided by the CM architecture provides good performance for a modest programming effort utilizing only a small subset of the instructions provided by *Lisp. Further, the perfonnance will continue to scale up linearly as long as not more than 2x virtualization is required. While the complexity analysis of the composite activation update suggests that its performance should be independent of the number of interconnects to be processed, the perfonnance results show that the performance is indirectly dependent on the number of interconnects to be processed because the level of virtualization required (after the physical processors are exhausted) is dependent on the number of interconnects to be processed and virtualization decreases performance linearly. The complexity analysis of the edge list activation update shows that its perfonnance should be roughly the same as serial implementations on comparable machines. The results suggest that the composite approach is to be prefered over the edge list approach but not be used at a virtualization level higher than 2x. The mechanism of the composite activation update suggest that hierarchical networks simulated in this fashion will compare in perfonnance to single layer networks because the parallel transfers provide a type of pipeline for activation for synchronously updated hierarchical networks while providing simultaneous activation transfers for asynchronously updated single layer networks. Researchers at Thinking Machines Corporation and the M.I.T. AI Laboratory in Cambridge Mass. use a similar approach for an implementation of NETtalk. Their approach overlaps the weights of connected units and simultaneously pipelines activation forward and error backward. 3 Perfonnance better than that presented can be gained by translation of the control code from interpreted *Lisp to PARIS and use of the CM2. In addition to not being interpreted, PARIS allows explicit control over important registers that aren't accessable through *Lisp. The CM2 will offer a number of new features which will enhance perfonnance of neural network simulations: a *Lisp compiler, larger processor memory (64K), and floating point processors. The complier and floating point processors will increase execution speeds while the larger processor memories will provide a larger number of virtual processors at the performance tum around points allowing higher perfonnance through higher CM utilization. REFERENCES 1. "Introduction to Data Level Parallelism," Thinking Machines Technical Report 86.14, (April 1986). 2. Hopfield, J. J., "Neural networks and physical systems with emergent collective computational abilities," Proc. Natl. Acad. Sci., Vol. 79, (April 1982), pp. 2554-2558. 3. Blelloch, G. and Rosenberg, C. Network Learning on the Connection Machine, M.I.T. Technical Report, 1987. 

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A Model of the Neural Basis of the Rat's Sense of Direction William E. Skaggs James J. Knierim Hemant S. Kudrimoti bill@nsma.arizona. edu jim@nsma.arizona. edu hemant@nsma. arizona. edu Bruce L. McNaughton bruce@nsma. arizona. edu ARL Division of Neural Systems, Memory, And Aging 344 Life Sciences North, University of Arizona, 'IUcson AZ 85724 Abstract In the last decade the outlines of the neural structures subserving the sense of direction have begun to emerge. Several investigations have shed light on the effects of vestibular input and visual input on the head direction representation. In this paper, a model is formulated of the neural mechanisms underlying the head direction system. The model is built out of simple ingredients, depending on nothing more complicated than connectional specificity, attractor dynamics, Hebbian learning, and sigmoidal nonlinearities, but it behaves in a sophisticated way and is consistent with most of the observed properties ofreal head direction cells. In addition it makes a number of predictions that ought to be testable by reasonably straightforward experiments. 1 Head Direction Cells in the Rat There is quite a bit of behavioral evidence for an intrinsic sense of direction in many species of mammals, including rats and humans (e.g., Gallistel, 1990). The first specific information regarding the neural basis of this "sense" came with the discovery by Ranck (1984) of a population of "head direction" cells in the dorsal presubiculum (also known as the "postsubiculum") of the rat. A head direction cell 174 William Skaggs, James J. Knierim, Hemant S. Kudrimoti, Bruce L. McNaughton fires at a high rate if and only if the rat's head is oriented in a specific direction. Many things could potentially cause a cell to fire in a head-direction dependent manner: what made the postsubicular cells particularly interesting was that when their directionality was tested with the rat at different locations, the head directions corresponding to maximal firing were consistently parallel, within the experimental resolution. This is difficult to explain with a simple sensory-based mechanism; it implies something more sophisticated. 1 The postsubicular head direction cells were studied in depth by Taube et al. (1990a,b), and, more recently, head direction cells have also been found in other parts of the rat brain, in particular the anterior nuclei of the thalamus (Mizumori and Williams, 1993) and the retrosplenial (posterior cingulate) cortex (Chen et al., 1994a,b). Interestingly, all of these areas are intimately associated with the hippocampal formation, which in the rat contains large numbers of "place" cells. Thus, the brain contains separate but neighboring populations of cells coding for location and cells coding for direction, which taken together represent much of the information needed for navigation. Figure 1 shows directional tuning curves for three typical head direction cells from the anterior thalamus. In each of them the breadth of tuning is on the order of 90 degrees. This value is also typical for head direction cells in the postsubiculum and retrosplenial cortex, though in each of the three areas individual cells may show considerable variability. Figure 1: Polar plots of directional tuning (mean firing rate as a function of head direction) for three typical head direction cells from the anterior thalamus of a rat. Every study to date has indicated that the head direction cells constitute a unitary system, together with the place cells of the hippocampus . Whenever two head direction cells have been recorded simultaneously, any manipulation that caused one of them to shift its directional alignment caused the other to shift by the same amount; and when head direction cells have been recorded simultaneously with place cells , any manipulation that caused the head direction cells to realign either caused the hippocampal place fields to rotate correspondingly or to "remap" into a different pattern (Knierim et al., 1995). Head direction cells maintain their directional tuning for some time when the lights in the recording room are turned off, leaving an animal in complete darkness; the directionality tends to gradually drift, though, especially if the animal moves around (Mizumori and Williams, 1993). Directional tuning is preserved to some degree even 1 Sensitivity to the Earth's geomagnetic field has been ruled out as an explanation of head-directional firing . A Model of the Neural Basis of the Rat's Sense of Direction 175 if an animal is passively rotated in the dark, which indicates strongly that the head direction system receives information (possibly indirect) from the vestibular system. Visual input influences but does not dictate the behavior of head direction cells. The nature of this influence is quite interesting. In a recent series of experiments (Knierim et al., 1995), rats were trained to forage for food pellets in a gray cylinder with a single salient directional cue, a white card covering 90 degrees of the wall. During training, half of the rats were disoriented before being placed in the cylinder, in order to disrupt the relation between their internal sense of direction and the location of the cue card; the other half of the rats were not disoriented. Presumably, the rats that were not disoriented during training experienced the same initial relationship between their internal direction sense and the CUe card each time they were placed in the cylinder; this would not have been true of the disoriented rats. Head direction cells in the thalamus were subsequently recorded from both groups of rats as they moved in the cylinder. All rats were disoriented before each recording session. Under these conditions, the cue card had much weaker control over the head direction cells in the rats that had been disoriented during training than in the rats that had not been disoriented . For all rats the influence of the cue card upon the head direction system weakened gradually over the course of multiple recording sessions, and eventually they broke free, but this happened much sooner in the rats that had been disoriented during training. The authors concluded that a visual cue could only develop a strong influence upon the head direction system if the rat experienced it as stable. Figure 2 illustrates the shifts in alignment during a typical recording session. When the rat is initially placed in the cylinder, the cell's tuning curve is aligned to the west. Over the first few minutes of recording it gradually rotates to SSW, and there it stays. Note the "tail" of the curve. This comes from spikes belonging to another, neighboring head direction cell, which could not be perfectly isolated from the first. Note that, even though they come from different cells, both portions shift alignment synchronously. Figure 2: Shifts in alignment of a head direction cell over the course of a single recording session (one minute intervals). 2 The Model As reviewed above, the most important facts to be accounted for by any model of the head direction system are (1) the shape of the tuning curves for head direction cells, (2) the control of head direction cells by vestibular input, and (3) the stabilitydependent influence of visual cues on head direction cells. We introduce here a 176 William Skaggs, James J. Knierim, Hemant S. Kudrimoti, Bruce L McNaughton model that accounts for these facts. It is a refinement of a model proposed earlier by McNaughton et al. (1991), the main addition being a more specific account of neural connections and dynamics. The aim of this effort is to develop the simplest possible architecture consistent with the available data. The reality is sure to be more complicated than this model. Figure 3 schematically illustrates the architecture of the model. There are four groups of cells in the model: head direction cells, rotation cells (left and right), vestibular cells (left and right), and visual feature detectors. For expository purposes it is helpful to think of the network as a set of circular layers; this does not reflect the anatomical organization of the corresponding cells in the brain. V.oIIbul .. col (rtght) 00 ??00 0 ??0 0 ?0000 @ @ @ @ 00 @ ? @ ~ROt.lOftC~I~etQ 66 ?00---- Rotadon cell (rtght) ?0 0?0 0?0 ? 0 00 ? ? ? ? ? 00 0 ? ? 00000 ? @ @ @ ? ~H~ @ @ @ @ eM.ocUon coM Figure 3: Architecture of the head direction cell model. The head direction cell group has intrinsic connections that are stronger than any other connections in the model, and dominate their dynamics, so that other inputs only provide relatively small perturbations. The connections between them are set up so that the only possible stable state of the system is a single localized cluster of active cells, with all other cells virtually silent. This will occur if there are strong excitatory connections between neighboring cells, and strong inhibitory connections between distant cells. It is assumed that the network of interconnections has rotation and reflection symmetry. Small deviations from symmetry will not impair the model too much; large deviations may cause it to have strong attractors at a few points on the circle, which would cause problems. The crucial property of this network is the following . Suppose it is in a stable state, with a single cluster of activated cells at one point on the circle, and suppose an external input is applied that excites the cells selectively on one side (left or right) A Model of the Neural Basis of the Rat' s Sense of Direction 177 of the peak. Then the peak will rotate toward the side at which the input is applied, and the rate of rotation will increase with the strength of the input. This feature is exploited by the mechanisms for vestibular and visual control of the system. The vestibular mechanism operates via a layer of "rotation" cells, corresponding to the circle of head direction cells (Units with a similar role were referred to as "H x H'" cells in the McNaughton et al. (1991) model). There are two groups of rotation cells, for left and right rotations. Each rotation cell receives excitatory input from the head direction cell at the same point on the circle, and from the vestibular system. The activation function of the rotation cell is sigmoidal or threshold linear, so that the cell does not become active unless it receives input simultaneously from both sources. Each right rotation cell sends excitatory projections to head direction cells neighboring it on the right, but not to those that neighbor it on the left, and contrariwise for left rotation cells. It is easy to see how the mechanism works. When the animal turns to the right, the right vestibular cells are activated, and then the right rotation cells at the current peak of the head direction system are activated. These add to the excitation of the head direction cells to the right of the peak, thereby causing the peak to shift rightward . This in turn causes a new set of rotation cells to become active (and the old ones inactive), and thence a further shift of the peak, and so on. The peak will continue to move around the circle as long as the vestibular input is active, and the stronger the vestibular input, the more rapidly the peak will move. If the gain of this mechanism is correct (but weak compared to the gain of the intrinsic connections of the head direction cells), then the peak will move around the circle at the same rate that the animal turns, and the location of the peak will function as an allocentric compass. This can only be expected to work over a limited range of turning rates, but the firing rates of cells in the vestibular nuclei are linearly proportional to angular velocity over a surprisingly broad range, so there is no reason why the mechanism cannot perform adequately. Of course the mechanism is intrinsically error-prone, and without some sort of external correction, deviations are sure to build up over time. But this is an inevitable feature of any plausible model, and in any case does not conflict with the available data, which, while sketchy, suggests that passive rotation of animals in the dark can cause quite erratic behavior in head direction cells (E. J. Markus, J. J . Knierim, unpublished observations). The final ingredient of the model is a set of visual feature detectors, each of which responds if and only if a particular visual feature is located at a particular angle with respect to the axis of the rat's head. Thus, these cells are feature specific and direction specific, but direction specific in the head-centered frame, not in the world frame . It is assumed that each visual feature detector projects weakly to all of the head direction cells, and that these connections are modifiable according to a Hebbian rule, specifically, ~W = a(Wmaxt(Aposd - W)Apre, where W is the connection weight, W max is its maximum possible value, Apost is the firing rate of the postsynaptic cell, Apre is the firing rate of the presynaptic cell, and the function has the shape shown in figure 4. (Actually, the rule is modified slightly to prevent any of the weights from becoming negative.) The net effect of to 178 William Skaggs, James J. Knierim, Hemant S. Kudrimoti, Bruce L. McNaughton this rule is that the weight will only change when the presynaptic cell (the visual feature detector) is active, and the weight will increase if the postsynaptic cell is strongly active, but decrease ifit is weakly active or silent. Modification rules of this form have previously been proposed in theories of the development of topography in the neocortex (e.g., Bienenstock et al., 1982), and there is considerable evidence for such an effect in the control of LTP /LTD (Singer and Artola, 1994) . 1(rate) rate Figure 4: Dependence of synaptic weight change on postsynaptic firing rate for connections from visual feature detectors to head direction cells in the model. To understand how this works, suppose we have a feature detecting cell that responds to a cue card whenever the cue card is directly in front of the rat. Suppose the rat's motion is restricted to a small area, and the cue card is far away, so that it is always at approximately the same absolute bearing (say, 30 degrees), and suppose the rat's head direction system is working correctly, i.e., functioning as an absolute compass. Then the cell will only be active at moments when the head direction cells corresponding to 30 degrees are active, and the Hebbian learning process will cause the feature detecting cell to be linked by strong weights to these cells, but by vanishing weights to other head direction cells. If the absolute bearing of the cue card were more variable, then the connection strengths from the feature detecting cell would be weaker and more broadly dispersed . In the limit where the bearing of the cue card was completely random, all connections would be weak and equal. Thus the influence of a visual cue is determined by the amount of training and by the variability in its bearing (with respect to the head direction system). It can be seen that the model implements a competition between visual inputs and vestibular inputs for control of the head direction cells. If the visual cues are rotated while the rat is left stationary, then the head direction cells may either rotate to follow the visual cues, or stick with the inertial frame, depending on parameter values and, importantly, on the training regimen imposed on the network. Both of these outcomes have been observed in anterior thalamic head direction cells (McNaughton et al., 1993). 3 Discussion Do the necessary types of cells exist in the brain? Cells in the brainstem vestibular nuclei are known to have the properties required by the model (Precht, 1978). The "rotation" cells would be recognizable from the fact that they would fire only when A Model of the Neural Basis of the Rat's Sense of Direction 179 the rat is facing in a particular direction and turning in a particular direction, with rate at least roughly proportional to the speed ofturning. Cells with these properties have been recorded in the postsubiculum (Markus et al., 1990) and retrosplenial cortex (Chen et ai., 1994a). The visual cells would be recognizeable from the fact that they would respond to visual stimuli only when they come from a particular direction with respect to the animal's head axis. Cells with these properties have been recorded in the inferior parietal cortex, the internal medullary lamina of the thalamus, and the superior colliculus (e.g., Sparks, 1986). The superior colliculus also contains cells that respond in a direction-dependent manner to auditory inputs, thus allowing a possiblility of control of the head direction system by sound sources. There do not seem to be any strong direct projections from the superior colli cui us to the components of the head direction system, but there are numerous indirect pathways. The most general prediction of the model is that the influence of vestibular input upon head direction cells is not susceptible to experience-dependent modification, whereas the influence of visual input is plastic, and is enhanced by the duration of experience, the richness of the visual cue array, and the distance of visual cues from the rat's region of travel. The "rotation" cells should be responsive to stimulation of the vestibular system. It is possible to activate the vestibular system by applying hot or cold water to the ears: if this is done in the dark, and head direction cells are simultaneously recorded, the model predicts that they will show periodic bursts of activity, with a frequency related to the intensity of the stimulus. For another prediction, suppose we train two groups of rats to forage in a cylinder containing a single landmark. For one group, the landmark is placed at the edge of the cylinder; for the other group, the same landmark is placed halfway between the center and the edge. The model predicts that in both cases the landmark will influence the head direction sytem, but the influence will be stronger and more tightly focused when the landmark is at the edge. In some respects the model is flexible, and may be extended without compromising its essence. For example, there is no intrinsic necessity that the vestibular system be the sole input to the rotation cells (other than the head direction cells). The performance of the system might be improved in some ways by sending the rotation cells input about optokinetic flow, or certain types of motor efference copy. But there is as yet no clear evidence for these things. On a more abstract level, the mechanism used by the model for vestibular control may be thought of as a special case of a general-purpose method for integration with neurons. As such, it has significant advantages over some previously proposed neural integrators, in particular, better stability properties. It might be worth considering whether the method is applicable in other situations where integrators are known to exist, for example the control of eye position. 180 William Skaggs, James J. Knierim, Hemant S. Kudrimoti, Bruce L. McNaughton Supported by MH46823 and O.N .R. References Bienenstock, E. L., Cooper, L. N., and Munro, P. W. (1982). Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex. J. Neurosci., 2:32-48. Chen, L. L., Lin, L., Green, E. J., Barnes, C. A., and McNaughton, B. L. (1994b). Head-direction cells in the rat posterior cortex. 1. Anatomical distribution and behavioral modulation. Exp. Brain Res., 101:8-23. Chen, L. L., Lin, L., Barnes, C. A., and McNaughton, B. L. (1994a). Head-direction cells in the rat posterior cortex. II. Contributions of visual and ideothetic information to the directional firing. Exp. Brain Res., 101:24-34. Gallistel, C. R. (1990). The Organization oj Learning. MIT Press, Cambridge, Massachusetts. Knierim, J. J ., Kudrimoti, H. S., and McNaughton, B. L. (1995). Place cells, head direction cells, and the learning of landmark stability. J. Neurosci. (in press). Markus, E. J., McNaughton, B. L., Barnes, C. A., Green, J. C., and Meltzer, J. (1990). Head direction cells in the dorsal presubiculum integrate both visual and angular velocity information. Soc. Neurosci. Abs., 16:44l. McNaughton, B. L., Chen, L. L., and Markus, E. J. (1991). ((Dead reckoning," landmark learning, and the sense of direction: A neurophysiological and computational hypothesis. J. Cognit. Neurosci ., 3: 190-202. McNaughton, B. L., Markus, E. J., Wilson, M. A., and Knierim, J . J. (1993). Familiar landmarks can correct for cumulative error in the inertially based dead-reckoning system. Soc. Neurosci. Abs., 19:795. Mizumori, S. J. and Williams, J. D. (1993). Directionally selective mnemonic properties of neurons in the lateral dorsal nucleus of the thalamus of rats. J. Neurosci., 13:4015-4028. Precht, W. (1978). Neuronal operations in the vestibular system. Springer, New York. Ranck, Jr ., J. B. (1984). Head direction cells in the deep cell layer of dorsal presubiculum in freely moving rats . Soc. Neurosci. Abs., 10:599. Singer, W . and Artola, A. (1994). Plasticity of the mature neocortex. In Selverston, A. I. and Ascher, P., editors, Cellular and molecular mechanisms underlying higher neural junctions, pages 49-69. Wiley. Sparks, D. L. (1986). Translation of sensory signals into commands for control of saccadic eye movements: role of primate superior colliculus. Physiol. Rev., 66:118-17l. Taube, J. S., Muller, R. V., and Ranck, Jr ., J. B. (1990a) . Head direction cells recorded from the postsubiculum in freely moving rats. I. Description and quantitative analysis . J. Neurosci., 10:420-435. Taube, J. S., Muller, R. V., and Ranck, Jr., J. B. (1990b) . Head direction cells recorded from the postsubiculum in freely moving rats. II. Effects of environmental manipulations. J. Neurosci., 10:436-447. PARTm LEARNiNGTHEORYANDDYNANUCS 

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Morphogenesis of the Lateral Geniculate Nucleus: How Singularities Affect Global Structure Svilen Tzonev Beckman Institute University of Illinois Urbana, IL 61801 svilen@ks.uiuc.edu Klaus Schulten Beckman Institute University of Illinois Urbana, IL 61801 kschulte@ks.uiuc.edu Joseph G. Malpeli Psychology Department University of Illinois Champaign, IL 61820 jmalpeli@uiuc.edu Abstract The macaque lateral geniculate nucleus (LGN) exhibits an intricate lamination pattern, which changes midway through the nucleus at a point coincident with small gaps due to the blind spot in the retina. We present a three-dimensional model of morphogenesis in which local cell interactions cause a wave of development of neuronal receptive fields to propagate through the nucleus and establish two distinct lamination patterns. We examine the interactions between the wave and the localized singularities due to the gaps, and find that the gaps induce the change in lamination pattern. We explore critical factors which determine general LGN organization. 1 INTRODUCTION Each side of the mammalian brain contains a structure called the lateral geniculate nucleus (LGN), which receives visual input from both eyes and sends projections to 134 Svilen Tzonev, Klaus Schulten, Joseph G. Malpeli the primary visual cortex. In primates the LG N consists of several distinct layers of neurons separated by intervening layers ofaxons and dendrites. Each layer of neurons maps the opposite visual hemifield in a topographic fashion. The cells comprising these layers differ in terms of their type (magnocellular and parvocellular) , their input (from ipsilateral (same side) and contralateral (opposite side) eyes), and their receptive field organization (ON and OFF center polarity). Cells in one layer receive input from one eye only (Kaas et al., 1972), and in most parts of the nucleus have the same functional properties (Schiller & Malpeli, 1978). The maps are in register, i.e., representations of a point in the visual field are found in all layers, and lie in a narrow column roughly perpendicular to the layers (Figure 1). A prominent a projection column Figure 1: A slice along the plane of symmetry of the macaque LGN. Layers are numbered ventral to dorsal. Posterior is to the left, where foveal (central) parts of the retinas are mapped; peripheral visual fields are mapped anteriorly (right). Cells in different layers have different morphology and functional properties: 6-P /C/ON; 5-P /I/ON; 4-P /C/OFF; 3-P /I/OFF; 2-M/I/ON&OFF; 1-M/C/ON&OFF, where P is parvocellular, M is magnocellular, C is contralateral, I is ipsilateral, ON and OFF refer to polarities of the receptive-field centers. The gaps in layers 6, 4, and 1 are images of the blind spot in the contralateral eye. Cells in columns perpendicular to the layers receive input from the same point in the visual field. feature in this laminar organization is the presence of cell-free gaps in some layers. These gaps are representations of the blind spot (the hole in the retina where the optic nerve exits) of the opposite retina. In the LGN of the rhesus macaque monkey (Macaca mulatta) the pattern of laminar organization drastically changes at the position of the gaps - foveal to the gaps there are six distinct layers, peripheral to the gaps there are four layers. The layers are extended two-dimensional structures whereas the gaps are essentially localized. However, the laminar transition occurs in a surface that extends far beyond the gaps, cutting completely across the main axis of the LGN (Malpeli & Baker., 1975). We propose a developmental model of LGN laminar morphogenesis. In particular, we investigate the role of the blind-spot gaps in the laminar pattern transition, and their extended influence over the global organization of the nucleus. In this model a wave of development caused by local cell interactions sweeps through the system (Figure 2). Strict enforcement of retinotopy maintains and propagates an initially localized foveal pattern. At the position of the gaps, the system is in a metastable Morphogenesis of the Lateral Geniculate Nucleus 135 wavefront maturing cells . immature cells ~;r , F ...... ...... . I? ? ? ? ~ 1 ? I I I .. -->- : I .,Y ~/!.. .".,. ....~ . ... .- X blind spot gap (no cells) Figure 2: Top view of a single layer. As a wave of development sweeps through the LGN the foveal part matures first and the more peripheral parts develop later. The shape of the developmental wave front is shown schematically by lines of "equal development" . state, and the perturbation in retinotopy caused by the gaps is sufficient to change the state of the system to its preferred four-layered pattern. We study the critical factors in this model, and make some predictions about LGN morphogenesis. 2 MODEL OF LGN MORPHOGENESIS We will consider only the upper four (parvocellular) layers since the laminar transition does not involve the other two layers. This transition results simply from a reordering of the four parvocellular strata (Figure 1). Foveal to the gaps, the strata form four morphologically distinct layers (6, 5, 4 and 3) because adjacent strata receive inputs from opposite eyes, which "repel" one another. Peripheral to the gaps, the reordering of strata reduces the number of parvocellular eye alternations to one, resulting in two parvocellular layers (6+4 and 5+3). 2.1 GEOMETRY AND VARIABLES LGN cells Ci are labeled by indices i = 1,2, ... , N. The cells have fixed, quazi-random and uniformly distributed locations ri E V e R3, where V = {(x,y,z) 10 < x < Sz,O < y < Sy,O < z < Sz}, and belong to one projection column Cab, a = 1,2, ... ,A and b = 1,2, ... ,B, (Figure 3). Functional properties of the neurons change in time (denoted by T), and are described by eye specificity and receptive-field polarity, ei(T), and Pi(T), respectively: ei (T), pdT) E [-1,1] C R, i = 1,2, ... ,N, T = 0, 1, ... , T maz ? The values of eye specificity and polarity represent the proportions of synapses from competing types of retinal ganglion cells (there are four type of ganglion cells from different eyes and with ON or OFF polarity). ei = -1 (ei = 1) denotes that the i-th cell is receiving input solely from the opposite (same side) retina. Similarly, Pi = -1 (Pi = 1) denotes that the cell input is pure ON (OFF) center. Intermediate values of ei and Pi imply that the cell does not have pure properties (it receives 136 Svilen Tzonev, Klaus Schulten, Joseph G. Malpeli N) Figure 3: Geometry of the model. LGN cells Ci (i = 1,2, ... , have fixed random, and uniformly-distributed locations Ti within a volume VcR , and belong to one projection column Cab. input from retinal ganglion cells of both eyes and with different polarities). Initially, at r = 0, all LGN cells are characterized by ei, Pi = O. This corresponds to two possibilities: no retinal ganglion cells synapse on any LGN cell, or proportions of synapses from different ganglion cells on all LGN neurons are equal, i.e., neurons possess completely undetermined functionality because of competing inputs of equal strength. As the neurons mature and acquire functional properties, their eye specificity and polarity reach their asymptotic values, ?l. Even when cells are not completely mature, we will refer to them as being of four different types, depending on the signs of their functional properties. Following accepted anatomical notation, we will label them as 6, 5, 4, and 3. We denote eye specificity of cell types 6 and 4 as negative, and cell types 5 and 3 as positive. Polarity of cell types 6 and 5 is negative, while polarity of types 4 and 3 is positive. Cell functional properties are subject to the dynamics described in the following section. The process of LGN development starts from its foveal part, since in the retina it is the fovea that matures first. As more peripheral parts of the retina mature, their ganglion cells start to compete to establish permanent synapses on LGN cells. In this sorting process, each LGN cell gradually emerges with permanent synapses that connect only to several neighboring ganglions of the same type. A wave of gradual development of functionality sweeps through the nucleus. The driving force for this maturation process is described by localized cell interactions modulated by external influences. The particular pattern of the foveal lamination is shaped by external forces, and later serves as a starting point for a "propagation of sameness" of cell properties. Such a sameness propagation produces clustering of similar cells and formation of layers. It should be stressed that cells do not move, only their characteristics change. 2.2 DYNAMICS The variables describing cell functional properties are subject to the following dynamics edr + 1) Pi (r + 1) (r) + ~ei (r) + 1Je Pi (r) + ~Pi (r) + 1Jp, ei - 1,2, ... ,N. (1) Morphogenesis of the Lateral Geniculate Nucleus 137 In Eq. (1), there are two contributions to the change of the intermediate variables and Pi (T). The first is deterministic, given by ei (T) 6.e;( r) = ,,(r,) [ 6.p,(r) - ,,(r,) (t. e; (r)! (Ie; - r;ll) + E", (r,)] (1- c; (r)) f3;. [(t,p;(r)!(lr,_r;ll) +P", (r,)] (I-p;(r)) f3;?. (2) The second is a stochastic contribution corresponding to fluctuations in the growth of the synapses between retinal ganglion cells and LGN neurons. This noise in synaptic growth plays both a driving and a stabilizing role to be explained below. We explain the meaning of the variables in Eq. (2) only for the eye specificity variable ei. The corresponding parameters for polarity Pi have similar interpretations. The parameter a (ri) is the rate of cell development. This rate is the same for eye specificity and polarity. It depends on the position ri of the cell in order to allow for spatially non-uniform development. The functional form of a (ri) is given in the Appendix. The term Eint (ri) = 2:7=1 ejl (h - rjl) is effectively a cell force field. This field influences the development of nearby cells and promotes clustering of same type of cells. It depends on the maturity of the generating cells and on the distance between cells through the interaction function 1(8). We chose for 1(8) a Gaussian form, i.e., 1 (8) = exp ( _6 2 / (T2), with characteristic interaction distance (T. The external influences on cell development are incorporated in the term for the external field Eext(ri). This external field plays two roles: it launches a particular laminar configuration of the system (in the foveal part of the LGN), and determines its peripheral pattern. It has, thus, two contributions Eext (ri) = E!xt (ri) + E~xt (ri). The exact forms of E!xt(ri) and E~xt(ri) are provided in the Appendix. The nonlinear term (1 - e~) in Eq. (2) ensures that ?1 are the only stable fixed points of the dynamics. The neuronal properties gradually converge to either of these fixed points capturing the maturation process. This term also stabilizes the dynamic variables and prevents them from diverging. The last term f3~b (T) reflects the strict columnar organization of the maps. At each step of the development the proportion of all four types of LGN cells is calculated within a single column Cab, and f3~b(T) for different types t is adjusted such that all types are equally represented. Without this term, the cell organization degenerates to a non-laminar pattern (the system tries to minimize the surfaces between cell clusters of different type). The exact form of f3!b(T) is given in the Appendix. At each stage of LGN development, cells receive input from retinal ganglion cells of particular types. This means that eye specificity and polarity of LGN cells are not independent variables. In fact, they are tightly coupled in the sense that lei (T) I = IPi ( T) I should hold for all cells at all times. This gives rise to coupled dynamics described by 138 Svilen Tzonev, Klaus Schulten, Joseph G. Malpeli mi(7+1) ei (7 + 1) Pi (7 + 1) = - min ( Iei (7 + 1) I, IPi (7 + 1) I) mi (7 + 1) sgn ( ei (7 + 1) ) md7+ 1) sgn(Pi(7+ 1)), i = 1,2, ... ,N. (3) The blind spot gaps are modeled by not allowing cells in certain columns to acquire types of functionality for which retinal projections do not exist, e.g., from the blind spot of the opposite eye. Accordingly, ei is not allowed to become negative. Thus, some cells never reach a pure state ei, Pi = ? 1. It is assumed that in reality such cells die out. Of all quantities and parameters, only variables describing the neuronal receptive fields (ei and Pi) are time-dependent. 3 RESULTS We simulated the dynamics described by Eqs. (1, 2, 3), typically for 100,000 time steps. Depending on the rate of cell development, mature states were reached in about 10,000 steps. The maximum value of Q: was 0.0001. We used an interaction function with u = 1. First, we considered a two-dimensional LGN, V = {(x,z)IO < x < Sx,O < z < Sz} with Sx = 10 and Sz = 6. There were ten projection columns (with equal size) along the x axis. An initial pattern was started in the foveal part by the external field. The size of the gaps 9 measured in terms of the interaction distance u was crucial for pattern development. When the developmental wave reached the gaps, layer 6 could" jump" its gap and continued to spread peripherally if the gap was sufficiently narrow (g/u < 1.5). If its gap was not too narrow (g/u > 0.5), layer 4 completely stopped (since cells in the gaps were not allowed to acquire negative eye specificity) and so layers 5 and 3 were able to merge. Cells of type 4 reappeared after the gaps (Figure 4, right side, shows behavior similar to the two-dimensional model) because of the required equal representation of all cell types in the projection columns, and because of noise in cell development. Energetically, the most favorable position of cell type 4 would be on top of type 6, which is inconsistent with experimental observations. Therefore, one must assume the existence of an external field in the peripheral part that will drive the system away from its otherwise preferred state. If the gaps were too large (g/u > 1.5), cells of type 6 and 4 reappeared after the gaps in a more or less random vertical position and caused transitions of irregular nature. On the other hand, if the gaps were too narrow (g/u < 0.5), both layers 6 and 4 could continue to grow past their gaps, and no transition between laminar patterns occurred at all. When g/u was close to the above limits, the pattern after the gaps differed from trial to trial. For the two-dimensional system, a realistic peripheral pattern always occurred for 0.7 < g/u < 1.2. = = = We simulated a three-dimensional system with size Sx 10, Sy 10, and Sz 6, and projection columns ordered in a 10 by 10 grid. The topology of the system is different in two and three dimensions: in two dimensions the gaps interrupt the layers completely and, thus, induce perturbations which cannot be by-passed. In three dimensions the gaps are just holes in a plane and generate localized perturbations: the layers can, in principle, grow around the gaps maintaining the initial laminar pattern. Nevertheless, in the three-dimensional case, an extended transi- Morphogenesis of the Lateral Geniculate Nue/ells 139 6 5 4~~UIIii 3 Figure 4: Left: Mature state of the macaque LG~ - result of the three-dimensional model with 4,800 cells. Spheres with different shades represent cells with different properties. Gaps in strata 6 and 4 (this gap is not visible) are coded by the darkest color, and coincide with the transition surface between 4- and 2-layered patterns. Right: A cut of the three-dimensional structure along its plane of symmetry. A two-dimensional system exhibits similar organization. Compare with upper layers in Figure 1. Spatial segregation between layers is not modeled explicitly. tion was triggered by the gaps. The transition surface, which passed through the gaps and was oriented roughly perpendicularly to the x axis, cut completely across the nucleus (Figure 4). Several factors were critical for the general behavior of the system. As in two dimensions, the size of the gap~ must be within certain limits: typically 0.5 < g / (J" < 1.0. These limits depend on the curvature of the wavefront. The gaps must lie in a certain "inducing" interval along the x axis. If they were too close to the origin, the foveal pattern was still more stable, so no transition could be induced there. However, a spontaneous transition might occur downstream. If the gaps were too far from the origin, a ~pontaneou~ transition might occur before them. The occurrence and location of a spontaneous transition, (therefore, the limits of the "inducing" interval) depended on the external-field parameters. A realistic transition was observed only when the front of the developmental wave had sufficient curvature when it reached the gaps. Underlying anatomical reasons for a sufficiently curved front along the main axis could be the curvature of the nucleus, differences in layer thickness, or differences in ganglion-cell densities in the retinas. Propagation of the developmental wave away from the gaps was quite stable. Before and after the gaps, the wave simply propagated the already established patterns. In a system without gaps, transitions of variable shape and location occurred when the peripheral contribution to the external fields was sufficiently large; a weaker contribution allowed the foveal pattern to propagate through the entire nucleus. 4 SUMMARY \Ve present a model that successfully captures the most important features of macaque LG~ morphogenesis. It produces realistic laminar patterns and supports 140 Svilen Tzonev, Klaus Schulten, Joseph G. Malpeli the hypothesis (Lee & Malpeli, 1994) that the blind spot gaps trigger the transition between patterns. It predicts that critical factors in LGN development are the size and location of the gaps, cell interaction distances, and shape of the front of the developmental wave. The model may be general enough to incorporate the LGN organizations of other primates. Small singularities, similar to the blind spot gaps, may have an extended influence on global organization of other biological systems. Acknowledgements This work has been supported by a Beckman Institute Research Assistantship, and by grants PHS 2P41 RR05969 and NIH EY02695. References J.H. Kaas, R.W. Guillery & J.M. Allman. (1972) Some principles of organization in the dorsal lateral geniculate nucleus, Brain Behav. Evol., 6: 253-299. D.Lee & J.G.Malpeli. (1994) Global Form and Singularity: Modeling the Blind Spot's Role in Lateral Geniculate Morphogenesis, Science, 263: 1292-1294. J.G. Malpeli & F.H. Baker. (1975) The representation of the visual field in the lateral geniculate nucleus of Macaca mulatta, 1. Comp. Neural., 161: 569-594. P.H. Schiller & J.G. Malpeli. (1978) Functional specificity ofLGN ofrhesus monkey, 1. Neurophysiol., 41: 788-797. APPENDIX The form of 0 (x, y, z) (with o(x,y,z) 00 = 0.0001) was chosen as = 00(0.1+exp(-(y-Sy/2)2)). (4) Foveal external fields of the following form were used: E!:z:t(x,y,z) = lO[O(z-d)-20(z-2d)+20(z-3d)-O(d-z)] exp(-x) p!:z:t(x,y,z) = 10[20(z-2d)-1]exp(-x), (5) where d = Sz/4 is the layers' thickness and the "theta" function is defined as O( x) = 1, x > 0 and O( x) = 0, x < O. Peripheral external fields (in fact they are present everywhere but determine the pattern in the peripheral part only) were chosen as E::z:t(x,y,z) = 5[20(z-2d)-1] P::z:t(x,y,z) = 5 [O(z-d)-20(z-2d)+20(z-3d)-O(d-z)]. (6) f3!b (1') was calculated in the following way: at any given time 1', within the column Cab, we counted the number N~b( 1') of cells, that could be classified as one of the four types t = 3,4,5,6. Cells with ei( 1') or Pi( 1') exactly zero were not counted. The total number of classified cells is then Nab( 1') = 2:~=3 N~b( 1'). If there were no classified cells (Nab( 1') = 0), then f3~b( 1') was set to one for all t. Otherwise the ratio of different types was calculated: n~b = N~b(1')/Nab(1'). In this way we calculated f3~b (1') = 4 - 12 nab, t = 3,4,5,6. (7) If f3~b (1') was negative it was replaced by zero. 

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Glove-TalkII: Mapping Hand Gestures to Speech Using Neural Networks S. Sidney Fels Department of Computer Science University of Toronto Toronto, ON, M5S lA4 ssfels@ai.toronto.edu Geoffrey Hinton Department of Computer Science University of Toronto Toronto, ON, M5S lA4 hinton@ai.toronto.edu Abstract Glove-TaikII is a system which translates hand gestures to speech through an adaptive interface. Hand gestures are mapped continuously to 10 control parameters of a parallel formant speech synthesizer. The mapping allows the hand to act as an artificial vocal tract that produces speech in real time. This gives an unlimited vocabulary in addition to direct control of fundamental frequency and volume. Currently, the best version of Glove-TalkII uses several input devices (including a CyberGlove, a ContactGlove, a 3space tracker, and a foot-pedal), a parallel formant speech synthesizer and 3 neural networks. The gesture-to-speech task is divided into vowel and consonant production by using a gating network to weight the outputs of a vowel and a consonant neural network. The gating network and the consonant network are trained with examples from the user. The vowel network implements a fixed, user-defined relationship between hand-position and vowel sound and does not require any training examples from the user. Volume, fundamental frequency and stop consonants are produced with a fixed mapping from the input devices. One subject has trained to speak intelligibly with Glove-TalkII. He speaks slowly with speech quality similar to a text-to-speech synthesizer but with far more natural-sounding pitch variations. 844 1 S. Sidney Fe Is, Geoffrey Hinton Introduction There are many different possible schemes for converting hand gestures to speech. The choice of scheme depends on the granularity of the speech that you want to produce. Figure 1 identifies a spectrum defined by possible divisions of speech based on the duration of the sound for each granularity. What is interesting is that in general, the coarser the division of speech, the smaller the bandwidth necessary for the user. In contrast, where the granularity of speech is on the order of articulatory muscle movements (i.e. the artificial vocal tract [AVT]) high bandwidth control is necessary for good speech. Devices which implement this model of speech production are like musical instruments which produce speech sounds. The user must control the timing of sounds to produce speech much as a musician plays notes to produce music. The AVT allows unlimited vocabulary, control of pitch and non-verbal sounds. Glove-TalkII is an adaptive interface that implements an AVT. Translating gestures to speech using an AVT model has a long history beginning in the late 1700's. Systems developed include a bellows-driven hand-varied resonator tube with auxiliary controls (1790's [9]), a rubber-moulded skull with actuators for manipulating tongue and jaw position (1880's [1]) and a keyboard-footpedal interface controlling a set of linearly spaced bandpass frequency generators called the Yoder (1940 [3]). The Yoder was demonstrated at the World's Fair in 1939 by operators who had trained continuously for one year to learn to speak with the system. This suggests that the task of speaking with a gestural interface is very difficult and the training times could be significantly decreased with a better interface. GloveTalkII is implemented with neural networks which allows the system to learn the user's interpretation of an articulatory model of speaking. This paper begins with an overview of the whole Glove-TalkII system. Then, each neural network is described along with its training and test results. Finally, a qualitative analysis is provided of the speech produced by a single subject after 100 hours of speaking with Glove-TalkI!. Artificial Vocal Tract (AVT) Phoneme Generator Finger Spelling Syllable Generator Word Generator I 10-30 > 100 130 200 600 Approximate time per gesture (msec) Figure 1: Spectrum of gesture-to-speech mappings based on the granularity of speech. 2 Overview of Glove-TalkII The Glove-TalkIl system converts hand gestures to speech , based on a gesture-toformant model. The gesture vocabulary is based on a vocal-articulator model of the hand. By dividing the mapping tasks into independent subtasks , a substantial reduction in network size and training time is possible (see [4]). Figure 2 illustrates the whole Glove-TalkIl system. Important features include the Glove- Talkll 845 Rlabt Hand data ll.y.:r: roll.l!itch. yaw every 1/60 IIOCUld ~~ ~, k-.--'- 10 flell analea 4 abduction anclea thumb and pinkie rotation wrist pitch and yaw every 1/100 aecond )))S~ Fllled Pitch Mappina VIC Doc:ision Network Vowel Network I---~ Combinina Function Ccnsonant Network Fllled SlOp Mappina Figure 2: Block diagram of Glove-TalkII: input from the user is measured by the Cyberglove, polhemus, keyboard and foot pedal, then mapped using neural networks and fixed functions to formant parameters which drive the parallel formant synthesizer [8]. three neural networks labeled vowel/consonant decision (V /C), vowel, and consonant. The V /C network is trained on data collected from the user to decide whether he wants to produce a vowel or a consonant sound . Likewise, the consonant network is trained to produce consonant sounds based on user-generated examples based on an initial gesture vocabulary. In contrast, the vowel network implements a fixed mapping between hand-positions and vowel phonemes defined by the user. Nine contact points measured on the user's left hand by a ContactGlove designate the nine stop consonants (B, D, G, J, P, T, K, CH, NG), because the dynamics of such sounds proved too fast to be controlled by the user. The foot pedal provides a volume control by adjusting the speech amplitude and this mapping is fixed. The fundamental frequency, which is related to the pitch of the speech, is determined by a fixed mapping from the user's hand height. The output of the system drives 10 control parameters of a parallel formant speech synthesizer every 10 msec. The 10 control parameters are: nasal amplitude (ALF), first, second and third formant frequency and amplitude (F1, A1, F2, A2, F3, A3), high frequency amplitude (AHF), degree of voicing (V) and fundamental frequency (FO). Each of the control parameters is quantized to 6 bits. Once trained, Glove-Talk II can be used as follows: to initiate speech, the user forms the hand shape of the first sound she intends to produce. She depresses the foot pedal and the sound comes out of the synthesizer. Vowels and consonants of various qualities are produced in a continuous fashion through the appropriate co-ordination of hand and foot motions. Words are formed by making the correct motions; for example, to say "hello" the user forms the "h" sound, depresses the foot pedal and quickly moves her hand to produce the "e" sound, then the "I" sound and finally the "0" sound. The user has complete control of the timing and quality of the individual sounds. The articulatory mapping between gestures and speech 846 S. Sidney Fe/s, Geoffrey Hinton ... .-),~O""':--+--"'--7-... -~-~c- - - - : - - Figure 3: Hand-position to Vowel Sound Mapping. The coordinates are specified relative to the origin at the sound A. The X and Y coordinates form a horizontal plane parallel to the floor when the user is sitting. The 11 cardinal phoneme targets are determined with the text-to-speech synthesizer. u - ... --!----"~ Y(cm) .. I is decided a priori. The mapping is based on a simplistic articulatory phonetic description of speech (5]. The X,Y coordinates (measured by the polhemus) are mapped to something like tongue position and height l producing vowels when the user's hand is in an open configuration (see figure 2 for the correspondence and table 1 for a typical vowel configuration). Manner and place of articulation for non-stop consonants are determined by opposition of the thumb with the index and middle fingers as described in table 1. The ring finger controls voicing. Only static articulatory configurations are used as training points for the neural networks, and the interpolation between them is a result of the learning but is not explicitly trained. Ideally, the transitions should also be learned, but in the text-to-speech formant data we use for training [6] these transitions are poor, and it is very hard to extract formant trajectories from real speech accurately. 2.1 The Vowel/Consonant (VIC) Network The VIC network decides, on the basis of the current configuration of the user's hand, to emit a vowel or a consonant sound. For the quantitative results reported here, we used a 10-5-1 feed-forward network with sigmoid activations [7]. The 10 inputs are ten scaled hand parameters measured with a Cyberglove: 8 flex angles (knuckle and middle joints of the thumb, index, middle and ring fingers), thumb abduction angle and thumb rotation angle. The output is a single number representing the probability that the hand configuration indicates a vowel. The output of the VIC network is used to gate the outputs of the vowel and consonant networks, which then produce a mixture of vowel and consonant formant parameters. The training data available includes only user-produced vowel or consonant sounds. The network interpolates between hand configurations to create a smooth but fairly rapid transition between vowels and consonants. For quantitative analysis, typical training data consists of 2600 examples of consonant configurations (350 approximants, 1510 fricatives [and aspirant], and 740 nasals) and 700 examples of vowel configurations. The consonant examples were obtained from training data collected for the consonant network by an expert user. The vowel examples were collected from the user by requiring him to move his hand in vowel configurations for a specified amount of time. This procedure was performed in several sessions. The test set consists of 1614 examples (1380 consonants and 234 vowels). After training,2 the mean squared error on the training and test lIn reality, the XY coordinates map more closely to changes in the first two formants, FI and F2 of vowels. From the user's perspective though, the link to tongue movement is useful. 2The VIe network, the vowel network and the consonant network are trained using Glove-Talkll 847 ~.. '.~ ;J(il. ~ ? ??:$??? , ~ ~ '.:~:~:::. DH F H 7?:. .I.!:.:? ~ ~. N R S .:::~::. .. ~ ....... ~ ' '.>' V :<: ? ~:;?::. " ~':" ':: , - - ~ . ..;- ., .... ... L M ''':;;1.. ' ...to:.?.?. ~: ). ":"~"'" SH TH . ~ ~ ~ Z ZH vowel .~ ":'s (, ~~. W , ' .? ~.,,> ? 0:' :~: ????'<1.?? Table 1: Static Gesture-to-Consonant Mapping for all phonemes. Note, each gesture corresponds to a static non-stop consonant phoneme generated by the text-to-speech synthesizer. set was less than 10- 4 . During normal speaking neither network made perceptual errors. The decision boundary feels quite sharp, and provides very predictable, quick transitions from vowels to consonants and back. Also, vowel sounds are produced when the user hyperextends his hand. Any unusual configurations that would intuitively be expected to produce consonant sounds do indeed produce consonant sounds. 2.2 The Vowel Network The vowel network is a 2-11-8 feed forward network . The 11 hidden units are normalized radial basis functions (RBFs) [2] which are centered to respond to one of 11 cardinal vowels. The outputs are sigmoid units representing 8 synthesizer control parameters (ALF, F1, AI, F2, A2, F3, A3, AHF). The radial basis function used is: L(Wji-O.)~ (1) where OJ is the (un-normalized) output of the RBF unit, Wji is the weight from unit i to unit j, 0i is the output of input unit i, and (1/ is the variance of the RBF. The normalization used is: O? nj = L J (2) oj=e- <l'j2 mEpom where nj is the normalized output of unit j and the summation is over all the units in the group of normalized RBF units. The centres of the RBF units are fixed conjugate gradient descent and a line search. 848 S. Sidney Fels, Geoffrey Hinton according to the X and Y values of each of the 11 vowels in the predefined mapping (see figure 2). The variances of the 11 RBF's are set to 0.025. The weights from the RBF units to the output units are trained. For the training data, 100 identical examples of each vowel are generated from their corresponding X and Y positions in the user-defined mapping, providing 1100 examples. Noise is then added to the scaled X and Y coordinates for each example. The added noise is uniformly distributed in the range -0.025 to 0.025. In terms of unscaled ranges, these correspond to an X range of approximately ? 0.5 cm and a Y range of ? 0.26 cm. Three different test sets were created. Each test set had 50 examples of each vowel for a total of 550 examples. The first test set used additive uniform noise in the interval ? 0.025. The second and third test sets used additive uniform noise in the interval ? 0.05 and ? 0.1 respectively. The mean squared error on the training set was 0.0016. The MSE on the additive noise test sets (noise = ? 0.025, 0.05 and 0.01) was 0.0018, 0.0038, 0.0120 which corresponds to expected errors of 1.1 %, 3.1% and 5.5% in the formant parameters, respectively. This network performs well perceptually. The key feature is the normalization of the RBF units. Often, when speaking, the user will overshoot cardinal vowel positions (especially when she is producing dipthongs) and all the RBF units will be quite suppressed. However, the normalization magnifies any slight difference between the activities of the units and the sound produced will be dominated by the cardinal vowel corresponding to the one whose centre is closest in hand space. 2.3 The Consonant Network The consonant network is a 10-14-9 feed-forward network . The 14 hidden units are normalized RBF units. Each RBF is centred at a hand configuration determined from training data collected from the user corresponding to one of 14 static consonant phonemes. The target consonants are created with a text-to-speech synthesizer. Figure 1 defines the initial mapping for each of the 14 consonants. The 9 sigmoid output units represent 9 control parameters of the formant synthesizer (ALF, F1, AI, F2, A2, F3, A3, AHF, V). The voicing parameter is required since consonant sounds have different degrees of voicing. The inputs are the same as for the manager V Ie network. Training and test data for the consonant network is obtained from the user . Target data is created for each of the 14 consonant sounds using the text-to-speech synthesizer . The scheme to collect data for a single consonant is: 1. The target consonant is played for 100 msec through the speech synthesizer; 2. the user forms a hand configuration corresponding to the consonant; 3. the user depresses the foot pedal to begin recording; the start of recording is indicated by the appearance of a green square; 4. 10-15 time steps of hand data are collected and stored with the corresponding formant parameter targets and phoneme identifier; the end of data collection is indicated by turning the green square red; 5. the user chooses whether to save the data to a file, and whether to redo the current target or move to the next one. Glove- Talkll 849 Using this procedure 350 approximants , 1510 fricatives and 700 nasals were collected and scaled for the training data. The hand data were averaged for each consonant sound to form the RBF centres. For the test data, 255 approximants, 960 fricatives and 165 nasals were collected and scaled . The RBF variances were set to 0.05. The mean square error on the training set was 0.005 and on the testing set was 0.01 corresponding to expected errors of 3.3% and 4.7% in the formant parameters, respectively. Listening to the output of the network reveals that each sound is produced reasonably well when the user's hand is held in a fixed position. The only difficulty is that the Rand L sounds are very sensitive to motion of the index finger. 3 Qualitative Performance of Glove-TalkII One subject, who is an accomplished pianist, has been trained extensively to speak with Glove-TalkII. We expected that his pianistic skill in forming finger patterns and his musical training would help him learn to speak with Glove-TalkII. After 100 hours of training, his speech with Glove-TalklI is intelligible and somewhat natural-sounding. He still finds it difficult to speak quickly, pronounce polysyllabic words, and speak spontaneously. During his training, Glove-TalkII also adapted to suit changes required by the subject . Initially, good performance of the VIC network is critical for the user to learn to speak. If the VIe network performs poorly the user hears a mixture of vowel and consonant sounds making it difficult to adjust his hand configurations to say different utterances . For this reason, it is important to have the user comfortable with the initial mapping so that the training data collected leads to the VIC network performing well. In the 100 hours of practice, Glove-Talk II was retrained about 10 times . Four significant changes were made from the original system analysed here for the new subject. First, the NG sound was added to the non-stop consonant list by adding an additional hand shape, namely the user touches his pinkie to his thumb on his right hand. To accomodate this change, the consonant and VIC network had two inputs added to represent the two flex angles of the pinkie. Also, the consonant network has an extra hidden unit for the NG sound. Second, the consonant network was trained to allow the RBF centres to change. After the hidden-to-output weights were trained until little improvement was seen, the input-to-hidden weights (i.e. the RBF centres) were also allowed to adapt . This noticeably improved performance for the user . Third, the vowel mapping was altered so that the I was moved closer to the EE sound and the entire mapping was reduced to 75% of its size. Fourth, for this subject, the VIC network needed was a 10-10-1 feed-forward sigmoid unit network. Understanding the interaction between the user's adaptation and Glove-TalkII's adaptation remains an interesting research pursuit. 4 Summary The initial mapping is loosely based on an articulatory model of speech. An open configuration of the hand corresponds to an unobstructed vocal tract, which in turn generates vowel sounds. Different vowel sounds are produced by movements of the hand in a horizontal X-Y plane that corresponds to movements of the first two formants which are roughly related to tongue position . Consonants other than stops are produced by closing the index, middle, or ring fingers or flexing the thumb, representing constrictions in the vocal tract . Stop consonants are produced by 850 S. Sidney Fels, Geoffrey Hinton contact switches worn on the user's left hand. FO is controlled by hand height and speaking intensity by foot pedal depression. Glove-TaikII learns the user's interpretation of this initial mapping. The VIC network and the consonant network learn the mapping from examples generated by the user during phases of training. The vowel network is trained on examples computed from the user-defined mapping between hand-position and vowels. The FO and volume mappings are non-adaptive. One subject was trained to use Glove-TalkII. After 100 hours of practice he is able to speak intelligibly. His speech is fairly slow (1.5 to 3 times slower than normal speech) and somewhat robotic. It sounds similar to speech produced with a text-tospeech synthesizer but has a more natural intonation contour which greatly improves the intelligibility and naturalness of the speech. Reading novel passages intelligibly usually requires several attempts, especially with polysyllabic words. Intelligible spontaneous speech is possible but difficult. Acknowledgements We thank Peter Dayan, Sageev Oore and Mike Revow for their contributions. This research was funded by the Institute for Robotics and Intelligent Systems and NSERC. Geoffrey Hinton is the Noranda fellow of the Canadian Institute for Advanced Research. References [1] A. G. Bell. Making a talking-machine. In Beinn Bhreagh Recorder, pages 61-72, November 1909~ [2] D. Broomhead and D. Lowe. Multivariable functional interpolation and adaptive networks. Complex Systems, 2:321-355, 1988. [3] Homer Dudley, R. R. Riesz, and S. S. A. Watkins. A synthetic speaker. Journal of the Franklin Institute, 227(6):739-764, June 1939. [4] S. S. Fels. Building adaptive interfaces using neural networks: The Glove-Talk pilot study. Technical Report CRG-TR-90-1, University of Toronto, 1990. [5] P. Ladefoged. A course in Phonetics (2 ed.). Harcourt Brace Javanovich, New York, 1982. [6] E. Lewis. A 'C' implementation of the JSRU text-to-speech system. Technical report, Computer Science Dept., University of Bristol, 1989. [7] D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representations by back-propagating errors. Nature, 323:533-536, 1986. [8] J. M. Rye and J. N. Holmes. A versatile software parallel-formant speech synthesizer. Technical Report JSRU-RR-1016, Joint Speech Research Unit, Malvern, UK, 1982. [9] Wolfgang Ritter von Kempelen. Mechanismus der menschlichen Sprache nebst Beschreibungeiner sprechenden Maschine. Mit einer Einleitung vonHerbert E. Brekle und Wolfgang Wild. Stuttgart-Bad Cannstatt F. Frommann, Stuttgart, 1970. 

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A Growing Neural Gas Network Learns Topologies Bernd Fritzke Institut fur Neuroinformatik Ruhr-Universitat Bochum D-44 780 Bochum Germany Abstract An incremental network model is introduced which is able to learn the important topological relations in a given set of input vectors by means of a simple Hebb-like learning rule. In contrast to previous approaches like the "neural gas" method of Martinetz and Schulten (1991, 1994), this model has no parameters which change over time and is able to continue learning, adding units and connections, until a performance criterion has been met. Applications of the model include vector quantization, clustering, and interpolation. 1 INTRODUCTION In unsupervised learning settings only input data is available but no information on the desired output. What can the goal of learning be in this situation? One possible objective is dimensionality reduction: finding a low-dimensional subspace of the input vector space containing most or all of the input data. Linear subspaces with this property can be computed directly by principal component analysis or iteratively with a number of network models (Sanger, 1989; Oja, 1982). The Kohonen feature map (Kohonen, 1982) and the "growing cell structures" (Fritzke, 1994b) allow projection onto non-linear, discretely sampled subspaces of a dimensionality which has to be chosen a priori. Depending on the relation between inherent data dimensionality and dimensionality of the target space, some information on the topological arrangement of the input data may be lost in the process. 626 Bernd Fritzke This is not astonishing since a reversible mapping from high-dimensional data to lower-dimensional spaces (or structures) does not exist in general. Asking how structures must look like to allow reversible mappings directly leads to another possible objective of unsupervised learning which can be described as topology learning: Given some high-dimensional data distributionp(e), find a topological structure which closely reflects the topology of the data distribution. An elegant method to construct such structures is "competitive Hebbian learning" (CHL) (Martinetz, 1993). CHL requires the use of some vector quantization method. Martinetz and Schulten propose the "neural gas" (NG) method for this purpose (Martinetz and Schulten, 1991). We will briefly introduce and discuss the approach of Martinetz and Schulten. Then we propose a new network model which also makes use of CHL. In contrast to the above-mentioned CHL/NG combination, this model is incremental and has only constant parameters. This leads to a number of advantages over the previous approach. 2 COMPETITIVE HEBBIAN LEARNING AND NEURAL GAS CHL (Martinetz, 1993) assumes a number of centers in R n and successively inserts topological connections among them by evaluating input signals drawn from a data distribution p(e). The principle of this method is: For each input signal x connect the two closest centers (measured by Euclidean distance) by an edge. The resulting graph is a subgraph of the Delaunay triangulation (fig. 1a) corresponding to the set of centers. This subgraph (fig. 1b), which is called the "induced Delaunay triangulation", is limited to those areas of the input space R n where p(e? O. The "induced Delaunay triangulation" has been shown to optimally preserve topology in a very general sense (Martinetz, 1993). Only centers lying on the input data submanifold or in its vicinity actually develop any edges. The others are useless for the purpose of topology learning and are often called dead units. To make use of all centers they have to be placed in those regions of R n where P (e) differs from zero. This could be done by any vector quantization (VQ) procedure. Martinetz and Schulten have proposed a particular kind of VQ method, the mentioned NG method (Martinetz and Schulten, 1991). The main principle of NG is the following: For each input signal x adapt the k nearest centers whereby k is decreasing from a large initial to a small final value. A large initial value of k causes adaptation (movement towards the input signal) of a large fraction of the centers. Then k (the adaptation range) is decreased until finally only the nearest center for each input signal is adapted. The adaptation strength underlies a similar decay schedule. To realize the parameter decay one has to define the total number of adaptation steps for the NG method in advance. A Growing Neural Gas Network Learns Topologies a) Delaunay triangulation 627 b) induced Delaunay triangulation Figure 1: Two ways of defining closeness among a set of points. a) The Delaunay triangulation (thick lines) connects points having neighboring Voronoi polygons (thin lines). Basically this reduces to points having small Euclidean distance w.r.t. the given set of points. b) The induced Delaunay triangulation (thick lines) is obtained by masking the original Delaunay triangulation with a data distribution P(~) (shaded) . Two centers are only connected if the common border of their Voronoi polygons lies at least partially in a region where P(~? 0 (closely adapted from Martinetz and Schulten, 1994) For a given data distribution one could now first run the NG algorithm to distribute a certain number of centers and then use CHL to generate the topology. It is, however, also possible to apply both techniques concurrently (Martinetz and Schulten, 1991). In this case a method for removing obsolete edges is required since the motion of the centers may make edges invalid which have been generated earlier. Martinetz and Schulten use an edge aging scheme for this purpose. One should note that the CHL algorithm does not influence the outcome of the NG method in any way since the adaptations in NG are based only on distance in input space and not on the network topology. On the other hand NG does influence the topology generated by CHL since it moves the centers around. The combination of NG and CHL described above is an effective method for topology learning. A problem in practical applications, however, may be to determine a priori a suitable number of centers. Depending on the complexity of the data distribution which one wants to model, very different numbers of centers may be appropriate. The nature of the NG algorithm requires a decision in advance and, if the result is not satisfying, one or several new simulations have to be performed from scratch. In the following we propose a method which overcomes this problem and offers a number of other advantages through a flexible scheme for center insertion. Bernd Fritzke 628 3 THE GROWING NEURAL GAS ALGORITHM In the following we consider networks consisting of ? a set A of units (or nodes). Each unit c E A has an associated reference vector We E Rn. The reference vectors can be regarded as positions in input space of the corresponding units. ? a set N of connections (or edges) among pairs of units. These connections are not weighted. Their sole purpose is the definition of topological structure. Moreover, there is a (possibly infinite) number of n-dimensional input signals obeying some unknown probability density function P(~). The main idea of the method is to successively add new units to an initially small network by evaluating local statistical measures gathered during previous adaptation steps. This is the same approach as used in the "growing cell structures" model (Fritzke, 1994b) which, however, has a topology with a fixed dimensionality (e.g., two or three). In the approach described here, the network topology is generated incrementally by CHL and has a dimensionality which depends on the input data and may vary locally. The complete algorithm for our model which we call "growing neural gas" is given by the following: o. Start with two units a and b at random positions Wa and Wb in Rn. 1. Generate an input signal ~ according to P(~). 2. Find the nearest unit 81 and the second-nearest unit 82. 3. Increment the age of all edges emanating from 81. 4. Add the squared distance between the input signal and the nearest unit in input space to a local counter variable: Aerror(8t} = IIWSl - ell 2 5. Move 81 and its direct topological neighbors1 towards ~ by fractions Eb and En, respectively, of the total distance: AW s1 = Eb(e - AWn = En(~ - w n ) W S1 ) for all direct neighbors n of 81 6. If 81 and 82 are connected by an edge, set the age of this edge to zero. If such an edge does not exist, create it. 2 7. Remove edges with an age larger than a maz ? If this results in points having no emanating edges, remove them as well. IThroughout this paper the term neighbors denotes units which are topological neighbors in the graph (as opposed to units within a small Euclidean distance of each other in input space). 2This step is Hebbian in its spirit since correlated activity is used to decide upon insertions. A Growing Neural Gas Network Learns Topologies 629 8. If the number of input signals generated so far is an integer multiple of a parameter A, insert a new unit as follows: ? Determine the unit q with the maximum accumulated error. ? Insert a new unit r halfway between q and its neighbor f with the largest error variable: Wr = 0.5 (w q + wf)' ? Insert edges connecting the new unit r with units q and f, and remove the original edge between q and f. ? Decrease the error variables of q and f by multiplying them with a constant 0:. Initialize the error variable of r with the new value of the error variable of q. 9. Decrease all error variables by multiplying them with a constant d. 10. If a stopping criterion (e.g., net size or some performance measure) is not yet fulfilled go to step 1. How does the described method work? The adaptation steps towards the input signals (5.) lead to a general movement of all units towards those areas of the input space where signals come from (P(~) > 0). The insertion of edges (6.) between the nearest and the second-nearest unit with respect to an input signal generates a single connection of the "induced Delaunay triangulation" (see fig. 1b) with respect to the current position of all units. The removal of edges (7.) is necessary to get rid of those edges which are no longer part of the "induced Delaunay triangulation" because their ending points have moved. This is achieved by local edge aging (3.) around the nearest unit combined with age re-setting of those edges (6.) which already exist between nearest and second-nearest units. With insertion and removal of edges the model tries to construct and then track the "induced Delaunay triangulation" which is a slowly moving target due to the adaptation of the reference vectors. The accumulation of squared distances (4.) during the adaptation helps to identify units lying in areas of the input space where the mapping from signals to units causes much error. To reduce this error, new units are inserted in such regions. 4 SIMULATION RESULTS We will now give some simulation results to demonstrate the general behavior of our model. The probability distribution in fig. 2 has been proposed by Martinetz and Schulten (1991) to demonstrate the non-incremental "neural gas" model. It can be seen that our model quickly learns the important topological relations in this rather complicated distribution by forming structures of different dimensionalities. The second example (fig. 3) illustrates the differences between the proposed model and the original NG network. Although the final topology is rather similar for both models, intermediate stages are quite different. Both models are able to identify the clusters in the given distribution. Only the "growing neural gas" model, however, Bernd Fritzke 630 Figure 2: The "growing neural gas" network adapts to a signal distribution which has different dimensionalities in different areas of the input space. Shown are the initial network consisting of two randomly placed units and the networks after 600, 1800, 5000, 15000 and 20000 input signals have been applied. The last network shown is not the necessarily the "final" one since the growth process could in principle be continued indefinitely. The parameters for this simulation were: A = 100, Eb = 0.2, En = 0.006, a = 0.5, a maz = 50, d = 0.995. could continue to grow to discover still smaller clusters (which are not present in this particular example, though). 5 DISCUSSION The "growing neural gas" network presented here is able to make explicit the important topological relations in a given distribution of input signals. An advantage over the NG method of Martinetz and Schulten is the incremental character of the model which eliminates the need to pre-specify a network size. Instead, the growth process can be continued until a user-defined performance criterion or network size is met. All parameters are constant over time in contrast to other models which heavily rely on decaying parameters (such as the NG method or the Kohonen feature map). pee) It should be noted that the topology generated by CHL is not an optional feature A Growing Neural Gas Network Learns Topologies "neural gas" and "competitive Hebbian learning" o 0 0 0 00 631 "growing neural gas" (uses "competitive Hebbian learning") 0 o~oo ?~~r~:;-.;. {]J'. 000 090- o V o 0 ~ 0 o j 00 co 00 000 8 0 Figure 3: The NG/CHL network of Martinetz and Schulten (1991) and the author's "growing neural gas" model adapt to a clustered probability distribution. Shown are the respective initial states (top row) and a number of intermediate stages. Both the number of units in the NG model and the final number of units in the "growing neural gas" model are 100. The bottom row shows the distribution of centers after 10000 adaptation steps (the edges are as in the previous row but not shown). The center distribution is rather similar for both models although the intermediate stages differ significantly. 632 Bernd Fritzke of our method (as it is for the NG model) but an essential component since it is used to direct the (completely local) adaptation as well as insertion of centers. It is probably the proper initialization of new units by interpolation from existing ones which makes it possible to have only constant parameters and local adaptations. Possible applications of our model are clustering (as shown) and vector quantization. The network should perform particularly well in situations where the neighborhood information (in the edges) is used to implement interpolation schemes between neighboring units. By using the error occuring in early phases it can be determined where to insert new units to generate a topological look-up table of different density and different dimensionality in particular areas of the input data space. Another promising direction of research is the combination with supervised learning. This has been done earlier with the "growing cell structures" (Fritzke, 1994c) and recently also with the "growing neural gas" described in this paper (Fritzke, 1994a). A crucial property for this kind of application is the possibility to choose an arbitrary insertion criterion. This is a feature not present, e.g., in the original "growing neural gas". The first results of this new supervised network model, an incremental radial basis function network, are very promising and we are further investigating this currently. References Fritzke, B. (1994a). Fast learning with incremental rbf networks. Neural Processing Letters, 1(1):2-5. Fritzke, B. (1994b). Growing cell structures - a self-organizing network for unsupervised and supervised learning. Neural Networks, 7(9):1441-1460. Fritzke, B. (1994c). Supervised learning with growing cell structures. In Cowan, J., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems 6, pages 255-262. Morgan Kaufmann Publishers, San Mateo, CA. Kohonen, T. (1982). Self-organized formation of topologically correct feature maps. Biological Cybernetics, 43:59-69. Martinetz, T. M. (1993). Competitive Hebbian learning rule forms perfectly topology preserving maps. In ICANN'93: International Conference on Artificial Neural Networks, pages 427-434, Amsterdam. Springer. Martinetz, T. M. and Schulten, K J. (1991). A "neural-gas" network learns topologies. In Kohonen, T., Makisara, K, Simula, 0., and Kangas, J., editors, Artificial Neural Networks, pages 397-402. North-Holland, Amsterdam. Martinetz, T. M. and Schulten, K J. (1994). Topology representing networks. Neural Networks, 7(3):507-522. Oja, E. (1982). A simplified neuron model as a principal component analyzer. Journal of Mathematical Biology, 15:267-273. Sanger, T. D. (1989). An optimality principle for unsupervised learning. In Touretzky, D. S., editor, Advances in Neural Information Processing Systems 1, pages 11-19. Morgan Kaufmann, San Mateo, CA. 

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JPMAX: Learning to Recognize Moving Objects as a Model-fitting Problem Suzanna Becker Department of Psychology, McMaster University Hamilton, Onto L8S 4K1 Abstract Unsupervised learning procedures have been successful at low-level feature extraction and preprocessing of raw sensor data. So far, however, they have had limited success in learning higher-order representations, e.g., of objects in visual images. A promising approach is to maximize some measure of agreement between the outputs of two groups of units which receive inputs physically separated in space, time or modality, as in (Becker and Hinton, 1992; Becker, 1993; de Sa, 1993). Using the same approach, a much simpler learning procedure is proposed here which discovers features in a single-layer network consisting of several populations of units, and can be applied to multi-layer networks trained one layer at a time. When trained with this algorithm on image sequences of moving geometric objects a two-layer network can learn to perform accurate position-invariant object classification. 1 LEARNING COHERENT CLASSIFICATIONS A powerful constraint in sensory data is coherence over time, in space, and across different sensory modalities. An unsupervised learning procedure which can capitalize on these constraints may be able to explain much of perceptual self-organization in the mammalian brain. The problem is to derive an appropriate cost function for unsupervised learning which will capture coherence constraints in sensory signals; we would also like it to be applicable to multi-layer nets to train hidden as well as output layers. Our ultimate goal is for the network to discover natural object classes based on these coherence assumptions. 934 1.1 Suzanna Becker PREVIOUS WORK Successive images in continuous visual input are usually views of the same object; thus, although the image pixels may change considerably from frame to frame, the image usually can be described by a small set of consistent object descriptors, or lower-level feature descriptors. We refer to this type of continuity as temporal coherence. This sort of structure is ubiquitous in sensory signals, from vision as well as other senses, and can be used by a neural network to derive temporally coherent classifications. This idea has been used, for example, in temporal versions of the Hebbian learning rule to associate items over time (Weinshall, Edelman and Biilt hoff, 1990; FOldiak, 1991). To capitalize on temporal coherence for higher-order feature extraction and classification, we need a more powerful learning principle. A promising approach is to maximize some measure of agreement between the outputs of two groups of units which receive inputs physically separated in space, time or modality, as in (Becker and Hinton, 1992; Becker, 1993; de Sa, 1993). This forces the units to extract features which are coherent across the different input sources. Becker and Hinton's (1992) Imax algorithm maximizes the mutual information between the outputs of two modules, y~ and Yb, connected to different parts of the input, a and b. Becker (1993) extended this idea to the problem of classifying temporally varying patterns by applying the discrete case of the mutual information cost function to the outputs of a single module at successive time steps, y-;'(t) and y-;'(t + 1). However, the success of this method relied upon the back-propagation of derivatives to train the hidden layer and it was found to be extremely susceptible to local optima. de Sa's method (1993) is closely related, and minimizes the probability of disagreement between output classifications, y~(t) and y1(t), produced by two modules having different inputs, e.g., from different sensory modalities. The success of this method hinges upon bootstrapping the first layer by initializing the weights to randomly selected training patterns, so this method too is susceptible to the problem of local optima. If we had a more flexible cost function that could be applied to a multi-layer network, first to each hidden layer in turn, and finally to the~utput layer for classification, so that the two layers could discover genuinely different structure, we might be able to overcome the problem of getting trapped in local optima, yielding a more powerful and efficient learning procedure. We can analyze the optimal solutions for both de Sa's and Becker's cost functions (see Figure 1 a) and see that both cost functions are maximized by having perfect one-to-one agreement between the two groups of units over all cases, using a one-of-n encoding, i.e., having only a single output unit on for each case. A major limitation of these methods is that they strive for perfect classifications by the units. While this is desirable at the top layer of a network, it is an unsuitable goal for training intermediate layers to detect low-level features. For example, features like oriented edges would not be perfect predictors across spatially or temporally nearby image patches in images of translating and rotating objects. Instead, we might expect that an oriented edge at one location would predict a small range of similar orientations at nearby locations. So we would prefer a cost function whose optimal solution was more like those shown in Figure 1 b) or c). This would allow a feature i in group a to agree with any of several nearby features, e.g. i - 1, i, or i + 1 in group b. JPMAX 935 b) a) ?? === ?.!. iii === ;;; ? ?11 II , ???? =???- ? I!!!!! ..... . c) , ?? == -== ?? ?? ? .1. II' ill!! II ? 11,? ? ? ????? ? .i. I' I' ??? = ? ? ? = ? 11 I!!!!! I ?? ;;; ? .. ?.'. .. ??.i. ??? ? II ? II II' I? ? 1' ? J[. , ? ??? .. :== I!!III'J!!'? ? ii ??? II "'-= , n ;;; . == ? Ilii Figure 1: Three possible joint distributions for the probability that the i th and j th units in two sets of m classification units are both on. White is high density, black is low density. The optimal joint distribution for Becker's and de Sa's algorithms is a matrix with all its density either in the diagonal as in a), or any subset of the diagonal entries for de Sa's method, or a permutation of the diagonal matrix for Becker's algorithm. Alternative distributions are shown in b) and c). 1.2 THE JPMAX ALGORITHM One way to achieve an arbitrary configuration of agreement over time between two groups of units (as in Figure 1 b) or c? is to treat the desired configuration as a prior joint probability distribution over their outputs. We can obtain the actual distribution by observing the temporal correlations between pairs of units' outputs in the two groups over an ensemble of patterns. We can then optimize the actual distribution to fit the prior. We now derive two different cost functions which achieve this result. Interestingly, they result in very similar learning rules. Suppose we have two groups of m units as shown in Figure 2 a), receiving inputs, x"7t and xi" from the same or nearby parts of the input image. Let Ca(t) and Cb(t) be the classifications of the two input patches produced by the network at time step t; the outputs of the two groups of units, y"7t(t) and yi,(t), represent these classification probabilities: enetoi(t) Yai(t) - P(Ca(t) = i) = Lj eneto;(t) Ybi(t) = enetbi (t) P(Cb(t) = i) = Lj enetb;(t) (1) (the usual "soft max" output function) where netai(t) and netbj(t) are the weighted net inputs to units. We could now observe the expected joint probability distribution qij = E [Yai(t)Ybj(t + 1)]t = E (P(Ca(t) = i, Cb(t + 1) = j)]t by computing the temporal covariances between the classification probabilities, averaged over the ensemble of training patterns; this joint probability is an m 2 -valued random variable. Given the above statistics, one possible cost function we could minimize is the -log probability of the observed temporal covariance between the two sets of units' outputs under some prior distribution (e.g. Figure 1 b) or c?. If we knew the actual frequency counts for each (joint) classification k = kll ,?? ., kIm, k21 , ... ,kmm, 936 Suzanna Becker b) At b'~-#--#----------"'-""""-"''''''''''''''''''''':iIo. <t (I) t Figure 2: a) Two groups of 15 units receive inputs from a 2D retina. The groups are able to observe each other's outputs across lateral links with unit time delays. b) A second layer of two groups of 3 units is added to the architecture in a). rather than just the observed joint probabilities, qij = E [~] , then given our prior model, pu, ... ,Pmm, we could compute the probability of the observations under a multinomial distribution: (2) Using the de Moivre-Laplace approximation leads to the following: P(k) ~ 1 exp v(27rn)m 2 -1 IL,j Pij (_! L 2 (k ij i,j - n Pij )2) (3) npij Taking the derivative of the - log probability wrt k ij leads to a very simple learning rule which depends only on the observed probabilities qij and priors Pij: fJ -logP(k) fJk ij ~ npij - k ij npij Pij - qij (4) Pij %!i: To obtain the final weight update rule, we just multiply this by n l . One problem with the above formulation is that the priors Pij must not be too close to zero for the de Moivre-Laplace approximation to hold. In practice, this cost function works well if we simply ignore the derivative terms where the priors are zero. An alternative cost function (as suggested by Peter Dayan) which works equally well is the Kullback-Liebler divergence or G-error between the desired joint probabilities Pij and the observed probabilities qij: G(p,q) = - LL j (Pij logpij - Pij lOgqij) (5) JPMAX 937 Figure 3: 10 of the 1500 tmining patterns: geometric objects centered in 36 possible locations on a 12-by-12 pixel grid. Object location varied mndomly between patterns. The derivative of G wrt qij is: aG Pij aqij qij (6) subject to Llij qij = 1 (enforced by the softmax output function). Note the similarity between the learning rules given by equations 4 and 6. 2 EXPERIMENTS The network shown in Figure 2 a) was trained to minimize equation 5 on an ensemble of pattern trajectories of circles, squares and triangles (see Figure 3) for five runs starting from different random initial weights, using a gradient-based learning method. For ten successive frames, the same object would appear, but with the centre varying randomly within the central six-by-six patch of pixels. In the last two frames, another randomly selected object would appear, so that object trajectories overlapped by two frames. These images are meant to be a crude approximation to what a moving observer would see while watching multiple moving objects in a scene; at any given time a single object would be approximately centered on the retina but its exact location would always be jittering from moment to moment. In these simulations, the prior distribution for the temporal covariances between the two groups of units' outputs was a block-diagonal configuration as in Figure 1 c), but with three five-by-five blocks along the diagonal. Our choice of a blockdiagonal prior distribution with three blocks encodes the constraint that units in a given block in one group should try to agree with units in the same block in the other group; so each group should discover three classes of features. The number of units within a block was varied in preliminary experiments, and five units was found to be a reasonable number to capture all instances of each object class (although the performance of the algorithm seemed to be robust with respect to the number of units per block). The learning took about 3000 iterations of steepest descent with 938 Suzanna Becker .... .. - - .. .. . ... ... . ... .... ?.... ... .. .. -. -.... .....-,..?. .......... .. . . -i?" .. . . . . . . .? ....?? ~~;. ? ? .. ? 1 ??. . . ? .. .. . . . .. .. . .... ...?? . . I'....... ..?. .... .? -1-?... . -.1 .... :1" . ....... ..... .' . ..... . ,'.=::::.: . ... ;~. :~:" i1 ? ? ? 1: . . . . . ' ~:. I' 'a_" _" ?? .. ... ? I .. ? 1 0; .... . ?? .. ? ??? _ _ _ ,. .. . . . ?? .. ? .a . . 1 ' 1 ? .. :" ~ . . ~i~" ._ , ' . ..? ::: ") ;: .. ?? ':. ?? " __ . ??? ? ?? ' .. .... .. . ? . ?1 ?????????? ~ .??..????? :.. ? ....... . ' .. .. ' e o_ .1. .... .... ?? . ? ? . ..... . .. ...... :: .. ? ?i . . . . . . . . . . . ???...? ~.~.: i ??? . _ _ .. ... .. I - ?1 ~::.... .; . ....... ????? II ........ ?????? ? . .. i " ' ;' :': ? ? ?:... a ? . :- .~:~ji----.. .........?:.......... ::: .;~ I .. . .; .. , ...... ?..... .... .. ?? a ???? .... ... . ..... .. ? .. a ? ? .... i??. .. .?. ??.. I?i....... I .. " ? . ?? :::' ":: . \~. ..... ... .. . ...... ... .. .. ......... . ....... .... .... ..... ? ? ? . .... .. ? .. ? ....... . Figure 4: Weights learned by one of the two groups of 15 units in the first layer. White weights are positive and black are negative. momentum to converge, but after about 1000 iterations only very small weight changes were made. Weights learned on a typical run for one of the two groups of fifteen units are shown in Figure 4. The units' weights are displayed in three rows corresponding to units in the three blocks in the block-diagonal joint prior matrix. Units within the same block each learned different instances of the same pattern class. For example, on this run units in the first block learned to detect circles in specific positions. Units in the second block tended to learn combinations of either horizontal or vertical lines, or sometimes mixtures of the two. In the third block, units learned blurred, roughly triangular shape detectors, which for this training set were adequate to respond specifically to triangles. In all five runs the network converged to equivalent solutions (only the groups' particular shape preferences varied across runs). Varying the number of units per block from three to five (Le. three three-by-three blocks versus three five-by-five blocks of units) produced similar results, except that with fewer units per block, each unit tended to capture multiple instances of a particular object class in different positions. A second layer of two groups of three units was added to the network, as shown in Figure 2 b). While keeping the first layer of weights frozen, this network was trained using exactly the same cost function as the first layer for about 30 iterations using a gradient-based learning method. This time the prior joint distribution for the two classifications was a three-by-three matrix with 80% of the density along the diagonal and 20% evenly distributed across the remainder of the distribution. Units in this layer learned to discriminate fairly well between the three object classes, as shown in Figure 5 a). On a test set with the ambiguous patterns removed (Le., patterns containing multiple objects), units in the second layer achieved very JPMAX 939 Loyer 2 Unit Responses on Trai ning Set a) I 1. ~. Circles Squares Triangles 0.80 .~ 0. 50 c ~ 1 0.40 '0 c i? 0.20 0.00 Loyer 2 Unit Responses on Test Set (no overlap s) b) 1.00 c . 0 . :;; 0.80 0 .; 0.60 i '0 0. 40 c 0 0 1?,20 0.00 LIllI ? ? UM3 .... ----. Unit 6 Figure 5: Response probabilities for the six output units to each of the three shapes. accurate object discrimination as shown in Figure 5 b). On ambiguous patterns containing multiple objects, the network's performance was disappointing. The output units would sometimes produce the "correct" response, i.e., all the units representing the shapes present in the image would be partially active. Most often, however, only one of the correct shapes would be detected, and occasionally the network's response indicated the wrong shape altogether. It was hoped? that the diagonally dominant prior mixed with a uniform density would allow units to occasionally disagree, and they would therefore be able to represent cases of multiple objects. It may have helped to use a similar prior for the hidden layer; however, this would increase the complexity of the learning considerably. 3 DISCUSSION We have shown that the algorithm can learn 2D-translation-invariant shape recognition, but it should handle equally well other types of transformations, such as rotation, scaling or even non-linear transformations. In principle, the algorithm should be applicable to real moving images; this is currently being investigated. Although we have focused here on the temporal coherence constraint, the algorithm could be applied equally well using other types of coherence, such as coherence 940 Suzanna Becker across space or across different sensory modalities. Note that the units in the first layer of the network did not learn anything about the geometric transformations between translated versions of the same object; they simply learned to associate different views together. In this respect, the representation learned at the hidden layer is similar to that predicted by the "privileged views" theory of viewpoint-invariant object recognition advocated by Weinshall et al. (1990) (and others). Their algorithm learns a similar representation in a single layer of competing units with temporal Hebbian learning applied to the lateral connections between these units. However, the algorithm proposed here goes further in that it can be applied to subsequent stages of learning to discover higher-order object classes. Yuille et al. (1994) have previously proposed an algorithm based on similar principles, which also involves maximizing the log probability of the network outputs under a prior; in one special case it is equivalent to Becker and Hinton's Imax algorithm. The algorithm proposed here differs substantially, in that we are dealing with the ensemble-averaged joint probabilities of two populations of units, and fitting this quantity to a prior; further, Yuille et aI's scheme employs back-propagation. One challenge for future work is to train a network with smaller receptive fields for the first layer units, on images of objects with common low-level features, such as squares and rectangles. At least three layers of weights would be required to solve this task: units in the first layer would have to learn local object parts such as corners, while units in the next layer could group parts into viewpoint-specific whole objects and in the top layer viewpoint-invariance, in principle, could be achieved. Acknowledgements Helpful comments from Geoff Hinton, Peter Dayan, Tony Plate and Chris Williams are gratefully acknowledged. References Becker, S. (1993). Learning to categorize objects using temporal coherence. In Advances in Neural Information Processing Systems 5, pages 361- 368. Morgan Kaufmann. Becker, S. and Hinton, G. E. (1992). A self-organizing neural network that discovers surfaces in random-dot stereograms. Nature, 355: 161-163. de Sa, V. R. (1993). Minimizing disagreement for self-supervised classification. In Proceedings of the 1993 Connectionist Models Summer School, pages 300-307. Lawrence Erlbaum associates. F51diak, P. (1991). Learning invariance from transformation sequences. Neural Computation, 3(2):194-200. Weinshall, D., Edelman, S., and Biilthoff, H. H. (1990). A self-organizing multiple-view representation of 3D objects. In Advances in Neural Information Processing Systems 2, pages 274-282. Morgan Kaufmann. Yuille, A. L., Stelios, M. S., and Xu, L. (1994). Bayesian Self-Organization. Technical Report No. 92-10, Harvard Robotics Laboratory. 

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Factorial Learning and the EM Algorithm Zoubin Ghahramani zoubin@psyche.mit.edu Department of Brain & Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract Many real world learning problems are best characterized by an interaction of multiple independent causes or factors. Discovering such causal structure from the data is the focus of this paper. Based on Zemel and Hinton's cooperative vector quantizer (CVQ) architecture, an unsupervised learning algorithm is derived from the Expectation-Maximization (EM) framework. Due to the combinatorial nature of the data generation process, the exact E-step is computationally intractable. Two alternative methods for computing the E-step are proposed: Gibbs sampling and mean-field approximation, and some promising empirical results are presented. 1 Introduction Many unsupervised learning problems fall under the rubric of factorial learning-that is, the goal of the learning algorithm is to discover multiple independent causes, or factors, that can well characterize the observed data (Barlow, 1989; Redlich, 1993; Hinton and Zemel, 1994; Saund, 1995). Such learning problems often arise naturally in response to the actual process by which the data have been generated. For instance, images may be generated by combining multiple objects , or varying colors, locations, and poses, with different light sources. Similarly, speech signals may result from an interaction of factors such as the tongue position, lip aperture, glottal state, communication line, and background noises. The goal of factorial learning is to invert this data generation process, discovering a representation that will both parsimoniously describe the data and reflect its underlying causes. A recent approach to factorial learning uses the Minimum Description Length (MDL) principle (Rissanen, 1989) to extract a compact representation of the input (Zemel, 1993; Hinton and Zemel, 1994). This has resulted in a learning architecture 618 Zoubin Ghahramani called Cooperative Vector Quantization (CVQ), in which a set of vector quantizers cooperates to reproduce the input . Within each vector quantizer a competitive learning mechanism operates to select an appropriate vector code to describe the input. The CVQ is related to algorithms based on mixture models, such as soft competitive clustering, mixtures of experts (Jordan and Jacobs, 1994), and hidden Markov models (Baum et al., 1970), in that each vector quantizer in the CVQ is itself a mixture model. However, it generalizes this notion by allowing the mixture models to cooperate in describing features in the data set, thereby creating a distributed representations of the mixture components. The learning algorithm for the CVQ uses MDL to derive a cost function composed of a reconstruction cost (e.g. sum squared error), representation cost (negative entropy of the vector code), and model complexity (description length of the network weights), which is minimized by gradient descent. In this paper we first formulate the factorial learning problem in the framework of statistical physics (section 2). Through this formalism, we derive a novel learning algorithm for the CVQ based on the Expectation-Maximization (EM) algorithm (Dempster et al., 1977) (section 3). The exact EM algorithm is intractable for this and related factorial learning problems-however, a tractable mean-field approximation can be derived. Empirical results on Gibbs sampling and the mean-field approximation are presented in section 4. 2 Statistical Physics Formulation The CVQ architecture, shown in Figure 1, is composed of hidden and observable units, where the observable units, y, are real-valued, and the hidden units are discrete and organized into vectors Si, i = 1, ... , d. The network models a data generation process which is assumed to proceed in two stages. First, a factor is independently sampled from each hidden unit vector, Sj, according to its prior probability density, ?ri. Within each vector the factors are mutually exclusive, i.e. if Sij = 1 for some j, then Sik = 0, Vk -# j. The observable is then generated from a Gaussian distribution with mean 2:1=1 WiSi. Notation: 0 0 0 0 0 VOl 51 0 0 0 0 0 52 ??? V0 2 0 0 0 0 0 d k p N Sd Sij Si Wi VOd Y number of vectors number of hidden units per vector number of outputs number of patterns hidden unit j in vector i vector i of units (Si = [Si1, ... , Sik]) weight matrix from Si to output network output (observable) Figure 1. The factorial learning architecture. Defining the energy of a particular configuration of hidden states and outputs as 1 H(s, y) = "2 lly - d L i=l d W i s ill 2 - k LL i=l j=l Sij log 7rij, (1) Factorial Learning and the EM Algorithm 619 the Boltzmann distribution p(s, y) 1 = -free Z exp{-11.(s,y)}, (2) exactly recovers the probability model for the CVQ. The causes or factors are represented in the multinomial variables Si and the observable in the multivariate Gaussian y. The undamped partition function, Zjree, can be evaluated by summing and integrating over all the possible configurations of the system to obtain Zjree =~ $ 1 exp{ -11.(s, y)}dy = (21l")P/2, (3) Y which is constant, independent of the weights. This constant partition function results in desirable properties, such as the lack of a Boltzmann machine-like sleep phase (Neal, 1992), which we will exploit in the learning algorithm. The system described by equation (1)1 can be thought of as a special form of the Boltzmann machine (Ackley et al., 1985). Expanding out the quadratic term we see that there are pairwise interaction terms between every unit. The evaluation of the partition function (3) tells us that when y is unclamped the quadratic term can be integrated out and therefore all Si are independent. However, when y is clamped all the Si become dependent. 3 The EM Algorithm Given a set of observable vectors, the goal of the unsupervised learning algorithm is to find weight matrices such that the network is most likely to have generated the data. If the hidden causes for each observable where known, then the weight matrices could be easily estimated . However, the hidden causes cannot be inferred unless these weight matrices are known. This chicken-and-egg problem can be solved by iterating between computing the expectation of the hidden causes given the current weights and maximizing the likelihood of the weights given these expected causes-the two steps forming the basis of the Expectation-Maximization (EM) algorithm (Dempster et al., 1977). Formally, from (2) we obtain the expected log likelihood of the parameters ?/: = (-11.(s,y) -logZjree)c,q, where ? denotes the current parameters, ? = {Wi}?=1, and (-)c,q, Q(?,?/) (4) denotes expectation given ? and the damped observables. The E-step of EM consists of computing this expected log likelihood. As the only random variables are the hidden causes, this simplifies to computing the (Si)c and (SiS])c terms appearing in the quadratic expansion of 11.. Once these terms have been computed, the M-step consists of maximizing Q with respect to the parameters. Setting the derivatives to zero we obtain a linear system, 1 For the remainder of the paper we will ignore the second term in (1), thereby assuming equal priors on the hidden states. Relaxing this assumption and estimating priors from the data is straightforward. 620 Zoubin Ghahramani which can be solved via the normal equations, where s is the vector of concatenated Si and the subscripts denote matrix size. For models in which the observable is a monotonic differentiable function ofLi WiSi, i.e. generalized linear models, least squares estimates of the weights for the M-step can be obtained iteratively by the method of scoring (McCullagh and NeIder, 1989). 3.1 E-step: Exact The difficulty arises in the E-step of the algorithm. The expectation of hidden unit j in vector i given pattern y is: P(Sij = 11Y; W) Ii; <X Ii; <x P(ylSij = 1;W)1l"ij Ii; L?? L .. LP(ylsij = 1, slh = 1, .. . ,Sdjd = 1; W)1l"ij jl=l ih,ti=l jd=l To compute this expectation it is necessary to sum over all possible configurations of the other hidden units. If each vector quantizer has k hidden units, each expectation has time complexity of O( k d - l ), i.e. O( N k d ) for a full E-step. The exponential time is due inherently to the cooperative nature of the model-the setting of one vector only determines the observable if all the other vectors are fixed. 3.2 E-step: Gibbs sampling Rather than summing over all possible hidden unit patterns to compute the exact expectations, a natural approach is to approximate them through a Monte Carlo method. As with Boltzmann machines, the CVQ architecture lends itself well to Gibbs sampling (Geman and Geman, 1984). Starting from a clamped observable y and a random setting of the hidden units {Sj}, the setting of each vector is updated in turn stochastically according to its conditional distribution Si '" p( sdY, {Sj h;ti; W). Each conditional distribution calculation requires k forward passes through the network, one for each possible state of the vector being updated, and k Gaussian distance calculations between the resulting predicted and clamped observables. If all the probabilities are bounded away from zero this process is guaranteed to converge to the equilibrium distribution of the hidden units given the observable. The first and second-order statistics, for (Si)c and (SiS])c respectively, can be collected using the Sij'S visited and p( Si Iy, {Sj h;ti; W) calculated during this sampling process. These estimated expectations are then used in the E-step. 3.3 E-step: Mean-field approximation Although Gibbs sampling is generally much more efficient than exact calculations, it too can be computationally demanding . A more promising approach is to approximate the intractable system with a tractable mean-field approximation (Parisi , 1988), and perform the E-step calculation on this approximation . We can write the Factorial Learning and the EM Algorithm 621 negative log likelihood minimized by the original system as a difference between the damped and undamped free energies: Cost -logp(y/W) = -log ~p(y, s/W) s -log~exp{-'J-l(y,s)} + log~ s Fcl - FJree s [ exp{-'J-l(y,s)}dy Jy The mean-field approximation allows us to replace each free energy in this cost with an upper bound approximation Cost M F F:t F - Ffrt'e. Unfortunately, a difference of two upper bounds is not generally an upper bound, and therefore minimizing Cost M F in, for example, mean-field Boltzmann machines does not guarantee that we are minimizing an upper bound on Cost. However, for the factorial learning architectures described in this paper we have the property that FJree is constant, and therefore the mean-field approximation of the cost is an upper bound on the exact cost. = The mean-field approximation can be obtained by approximating the probability density given by (1) and (2) by a completely factorized probability density: p(s, y) = (21r~P/2 exp{ -~/IY -1L/12} nm:y ',J In this approximation all units are independent: the observables are Gaussian distributed with mean IL and each hidden unit is binomially distributed with mean mij. To obtain the mean-field approximation we solve for the mean values that minimize the Kullback-Leibler divergence KC(p,p) == Ep[logp] - Ep[logp]. Noting that: Ep[Sij] = mij, Ep[S[j] = mij, Ep[SijSkd = mijmkl, and Ep[SijSik] = 0, we obtain the mean-field fixed point equations (5) where y == l:i WiIDi. The softmax function is the exponential normalized over the k hidden units in each IDi vector. The first term inside the softmax has an intuitive interpretation as the projection of the error in the observable onto the weights of the hidden unit vector i. The more a hidden unit can reduce this error, the higher its mean. The second term arises from the fact that Ep[s[j] = mij and not Ep[s[j] m[j. The means obtained by iterating equation (5) are used in the E-step by substituting mi for (Si)c and IDiID] for (sisJk = 4 Empirical Results Two methods, Gibbs sampling and mean-field, have been provided for computing the E-step of the factorial learning algorithm. There is a key empirical question that needs to be answered to determine the efficiency and accuracy of each method. For Gibbs sampling it is important to know how many samples will provide robust estimates of the expectations required for the E-step. It is well known that for stochastic Boltzmann machines the number of samples needed to obtain good 622 Zoubin Ghahramani estimates of the gradients is generally large and renders the learning algorithm prohibitively slow. Will this architecture suffer from the same problem? For mean-field it is important to know the loss incurred by approximating the true likelihood. We explore these questions by presenting empirical results on two small unsupervised learning problems. The first benchmark problem consists of a data set of 4 x 4 greyscale images generated by a combination of two factors: one producing a single horizontal line and the other, a vertical line (Figure 2a; cf. Zemel, 1993). Using a network with 2 vectors of 4 hidden units each, both the Gibbs sampling and mean-field EM algorithms converge on a solution after about a dozen steps (Figure 2b). The solutions found resemble the generative model of the data (Figure 2c & d). b) a) 70 .0 5o :'i 0 ." ~ 40 ~ l O ; i 2 0 _= s ~.r-r--- _--- ~-="-~- 10 oo-~_~_~_~_~ 10 J'i 20 Iteration c) d) Figure 2. Lines Problem. a) Complete data set of 160 patterns. b) Learning curves for Gibbs (solid) and mean-field (dashed) forms of the algorithm. c) A sample output weight matrix after learning (MSE=1.20). The top vector of hidden units has come to represent horizontal lines, and the bottom, vertical lines. d) Another typical output weight matrix (MSE=1.78). The second problem consists of a data set of 6 x 6 images generated by a combination of three shapes-a cross, a diagonal line, and an empty square-each of which can appear in one of 16 locations (Figure 3a). The data set of 300 out of 4096 possible images was presented to a network with the architecture shown in Figure 3b. After 30 steps of EM, each consisting of 5 Gibbs samples of each hidden unit, the network reconstructed a representation that approximated the three underlying causes of the data-dedicating one vector mostly to diagonal lines, one to hollow squares, and one to crosses (Figure 3c). To assess how many Gibbs samples are required to obtain accurate estimates of the expectations for the E-step we repeated the lines problem varying the number of samples. Clearly, as the number of samples becomes large the Gibbs E-step becomes exact. Therefore we expect performance to asymptote at the performance of the exact E-step. The results indicate that, for this problem, 3 samples are sufficient to achieve ceiling performance (Figure 4). Surprisingly, a single iteration of the Factorial Learning and the EM Algorithm 623 a) b) ~_- lSi- -~ , -:JI t rIl 1f --.~ ? .p- rj- ~_- ~~ 36 ;. ~+ .,. a:,. :'I,," D.... If :m' ~1r- ~.t ti .q. at- ..:.p If !ill1I- . ........ :11:1 ti+ "t:I:I .ttt~ ... 16 16 oo 16 0 0 c) Figure 3. Shapes Problem. a) Sample images from the data set. b) Learning architecture used. c) Output weight matrix after learning. mean-field equations also performs quite well. 5 Discussion The factorial learning problem for cooperative vector quantizers has been formulated in the EM framework, and two learning algorithms, based on Gibbs sampling and mean-field approximation, have been derived. Unlike the Boltzmann machine, Gibbs sampling for this architecture seems to require very few samples for adequate performance. This may be due to the fact that, whereas the Boltzmann machine relies on differences of noisy estimates for its weight changes, due to the constant partition function the factorial learning algorithm does not. The mean-field approximation also seems to perform quite well on all problems tested to date . This may also be a consequence of the constant partition function which guarantees that the mean-field cost is an upper bound on the exact cost. The framework can be extended to hidden Markov models (HMMs), showing that simple HMMs are a special case of dynamical CVQs, with the general case corresponding to parallel, factorial HMMs. The two principal advantages of such architectures are (1) unlike the traditional HMM, the state space can be represented as a combination of features, and (2) time series generated by multiple sources can be modeled. Simulation results on the Gibbs and mean-field EM algorithms for factorial HMMs are also promising (Ghahramani, 1995). 624 Zoubin Ghahramani 2.8 1 2 3 4 Gibbs samples or mean-field lIerarions Figure 4. Comparison of the Gibbs and mean-field EM algorithms for the lines data. Each data point shows the mean squared training error averaged over 10 runs of 20 EM steps, with standard error bars. For the Gibbs curve the abscissa is the number of samples per vector of hidden units; for the mean-field curve it is the number of iterations of equation (5). Acknowledgements The author wishes to thank Lawrence Saul and Michael Jordan for invaluable discussions. This project was supported in part by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, by a grant from Siemens Corporation, and by grant N00014-94-1-0777 from the Office of Naval Research. References Ackley, D., Hinton, G., and Sejnowski, T. (1985). A learning algorithm for Boltzmann machines. Cognitive Science, 9:147-169. Barlow, H. (1989). Unsupervised learning. Neural Computation, 1:295-31l. Baum, L., Petrie, T., Soules, G., and Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. The Annals of Mathematical Statistics, 41:164-17l. Dempster, A., Laird, N., and Rubin, D. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Royal Statistical Society Series B, 39:1-38. Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence,6:721-74l. Ghahramani, Z. (1995). Factorial learning and the EM algorithm. MIT Computational Cognitive Science TR 9501. Hinton, G. and Zemel, R. (1994) . Autoencoders, minimum description length, and Helmholtz free energy. In Cowan, J., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems 6. Morgan Kaufmanm Publishers, San Francisco, CA. Jordan, M. and Jacobs, R. (1994). Hierarchical mixtures of experts and the EM algorithm. Neural Computation, 6:181-214. McCullagh, P ..and NeIder, J. (1989). Generalized Linear Models. Chapman & Hall, London. Neal, R. (1992). Connectionist learning of belief networks. Artificial Intelligence, 56:71113. Parisi, G. (1988). Statistical Field Theory. Addison-Wesley, Redwood City, CA. Redlich, A. (1993). Supervised factorial learning. Neural Computation, 5:750-766. Rissanen, J. (1989). Stochastic Complexity in Statistical Inquiry. World Scientific, Singapore. Saund, E. (1995). A multiple cause mixture model for unsupervised learning. Neural Computation,7(1):51-7l. Zemel, R. (1993). A minimum description length framework for unsupervised learning. Ph.D. Thesis, Dept. of Computer Science, University of Toronto, Toronto, Canada. 

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Learning Local Error Bars for Nonlinear Regression David A.Nix Department of Computer Science and Institute of Cognitive Science University of Colorado Boulder, CO 80309-0430 dnix@cs.colorado.edu Andreas S. Weigend Department of Computer Science and Institute of Cognitive Science University of Colorado Boulder, CO 80309-0430 andreas@cs.colorado.edu? Abstract We present a new method for obtaining local error bars for nonlinear regression, i.e., estimates of the confidence in predicted values that depend on the input. We approach this problem by applying a maximumlikelihood framework to an assumed distribution of errors. We demonstrate our method first on computer-generated data with locally varying, normally distributed target noise. We then apply it to laser data from the Santa Fe Time Series Competition where the underlying system noise is known quantization error and the error bars give local estimates of model misspecification. In both cases, the method also provides a weightedregression effect that improves generalization performance. 1 Learning Local Error Bars Using a Maximum Likelihood Framework: Motivation, Concept, and Mechanics Feed-forward artificial neural networks used for nonlinear regression can be interpreted as predicting the mean of the target distribution as a function of (conditioned on) the input pattern (e.g., Buntine & Weigend, 1991; Bishop, 1994), typically using one linear output unit per output variable. If parameterized, this conditional target distribution (CID) may also be ?http://www.cs.colorado.edu/~andreas/Home.html. This paper is available with figures in colors as ftp://ftp.cs.colorado.edu/pub/ Time-Series/MyPapers/nix.weigenCLnips7.ps.Z . 490 David A. Nix, Andreas S. Weigend viewed as an error model (Rumelhart et al., 1995). Here, we present a simple method that provides higher-order information about the cm than simply the mean. Such additional information could come from attempting to estimate the entire cm with connectionist methods (e.g., "Mixture Density Networks," Bishop, 1994; "fractional binning, "Srivastava & Weigend, 1994) or with non-connectionist methods such as a Monte Carlo on a hidden Markov model (Fraser & Dimitriadis, 1994). While non-parametric estimates of the shape of a C1D require large quantities of data, our less data-hungry method (Weigend & Nix, 1994) assumes a specific parameterized form of the C1D (e.g., Gaussian) and gives us the value of the error bar (e.g., the width of the Gaussian) by finding those parameters which maximize the likelihood that the target data was generated by a particular network model. In this paper we derive the specific update rules for the Gaussian case. We would like to emphasize, however, that any parameterized unimodal distribution can be used for the em in the method presented here. j------------, I------T-------, I I A o y(x) I A2,. ) 0 cr IX I I /\ i O'OOh k : , '. \ I I I I I I I : ,-----------_.l Figure 1: Architecture of the network for estimating error bars using an auxiliary output unit. All weight layers have full connectivity. This architecture allows the conditional variance ~2 -unit access to both information in the input pattern itself and in the hidden unit representation formed while learning the conditional mean, y(x). We model the desired observed target value d as d(x) = y(x) + n(x), where y(x) is the underlying function we wish to approximate and n(x) is noise drawn from the assumed cm. Just as the conditional mean of this cm, y(x), is a function of the input, the variance (j2 of the em, the noise level, may also vary as a function of the input x (noise heterogeneity). Therefore, not only do we want the network to learn a function y(x) that estimates the conditional mean y(x) of the cm, but we also want it to learn a function a- 2 (x) that estimates the conditional variance (j2(x). We simply add an auxiliary output unit, the a- 2-unit, to compute our estimate of (j2(x). Since (j2(x) must be positive, we choose an exponential activation function to naturally impose this bound: a-2 (x) = exp [Lk Wq2khk (x) + ,8], where,8 is the offset (or "bias"), and Wq2k is the weight between hidden unit k and the a- 2 -unit. The particular connectivity of our architecture (Figure 1), in which the a- 2-unit has a hidden layer of its own that receives connections from both the y-unit's hidden layer and the input pattern itself, allows great flexibility in learning a- 2 (x). In contrast, if the a- 2 -unit has no hidden layer of its own, the a- 2 -unit is constrained to approximate (j2 (x) using only the exponential of a linear combination of basis functions (hidden units) already tailored to represent y(x) (since learning the conditional variance a- 2 (x) before learning the conditional mean y(x) is troublesome at best). Such limited connectivity can be too constraining on the functional forms for a- 2 ( x) and, in our experience, I The case of a single Gaussian to represent a unimodal distribution can also been generalized to a mixture of several Gaussians that allows the modeling of multimodal distributions (Bishop, 1994). Learning Local Error Bars for Nonlinear Regression 491 produce inferior results. This is a significant difference compared to Bishop's (1994) Gaussian mixture approach in which all output units are directly connected to one set of hidden units. The other extreme would be not to share any hidden units at all, i.e., to employ two completely separate sets of hidden units, one to the y(x)-unit, the other one to the a- 2(x)-unit. This is the right thing to do if there is indeed no overlap in the mapping from the inputs to y and from the inputs to cr2 ? The two examples discussed in this paper are between these two extremes; this justifies the mixed architecture we use. Further discussion on shared vs. separate hidden units for the second example of the laser data is given by Kazlas & Weigend (1995, this volume). For one of our network outputs, the y-unit, the target is easily available-it is simply given by d. But what is the target for the a- 2-unit? By maximizing the likelihood of our network ' model N given the data, P(Nlx, d), a target is "invented" as follows. Applying Bayes' rule and assuming statistical independence of the errors, we equivalently do gradient descent in the negative log likelihood of the targets d given the inputs and the network model, summed over all patterns i (see Rumelhart et at., 1995): C = - Li In P(dilxi, N). Traditionally, the resulting form of this cost function involves only the estimate Y(Xi) of the conditional mean; the variance of the CID is assumed to be constant for all Xi, and the constant terms drop out after differentiation. In contrast, we allow the conditional variance to depend on x and explicitly keep these terms in C, approximating the conditional variance for Xi by a- 2(Xi). Given any network architecture and any parametric form for the ern (Le., any error model), the appropriate weight-update equations for gradient decent learning can be straightforwardly derived. Assuming normally distributed errors around y(x) corresponds to a em density function of P(dilxj) = [27rcr2(Xi)t 1/ 2 exp {- d~:Y~.) Using the network output Y(Xi) ~ y(Xi) to estimate the conditional mean and using the auxiliary output a- 2(Xi) ~ cr2(xd to estimate the conditional variance, we obtain the monotonically related negative log . over all patterns likelib00, d - I n P(di IXi, nAf\J -- 2"IIn 2 7rcrA2() Xi + [di-y(Xi)]2 2".2(X.) . Summatlon 2}. gives the total cost: C = ! ,,{ [di =-2 y(xd] 2 + Ina-2(Xi) + In27r} 2~ , cr (Xi) (1) To write explicit weight-update equations, we must specify the network unit transfer functions. Here we choose a linear activation function for the y-unit, tanh functions for the hidden units, and an exponential function for the a- 2 -unit. We can then take derivatives of the cost C with respect to the network weights. To update weights connected to the Yand a- 2-units we have: 11 a-2~i) [di - 11 2a-2~Xi) {[di - (2) Y(Xi)] hj(Xi) y(Xi)f - a-2 (Xi) } hk (Xi) (3) where 11 is the learning rate. For weights not connected to the output, the weight-update equations are derived using the chain rule in the same way as in standard backpropagation. Note that Eq. (3) is equivalent to training a separate function-approximation network for a- 2(x) where the targets are the squared errors [d i - y(Xi)]2]. Note also that if a- 2(Xj) is 492 David A. Nix, Andreas S. Weigend constant, Eqs. (1)-(2) reduce to their familiar forms for standard backpropagation with a sum-squared error cost function. The 1/&2(X) term in Eqs. (2)-(3) can be interpreted as a form of "weighted regression," increasing the effective learning rate in low-noise regions and reducing it in high-noise regions. As a result, the network emphasizes obtaining small errors on those patterns where it can (low &2); it discounts learning patterns for which the expected error is going to be large anyway (large &2). This weighted-regression term can itself be highly beneficial where outliers (i.e., samples from high-noise regions) would ordinarily pull network resources away from fitting low-noise regions which would otherwise be well approximated. For simplicity, we use simple gradient descent learning for training. Other nonlinear minimization techniques could be applied, however, but only if the following problem is avoided. If the weighted-regression term described above is allowed a significant influence early in learning, local minima frequently result. This is because input patterns for which low errors are initially obtained are interpreted as "low noise" in Eqs. (2)-(3) and overemphasized in learning. Conversely, patterns for which large errors are initially obtained (because significant learning of y has not yet taken place) are erroneously discounted as being in "high-noise" regions and little subsequent learning takes place for these patterns, leading to highly-suboptimal solutions. This problem can be avoided if we separate training into the following three phases: Phase I (Initial estimate of the conditional mean): Randomly split the available data into equal halves, sets A and 8. Assuming u 2 (x) is constant, learn the estimate of the conditional mean y(x) using set A as the training set. This corresponds to "traditional" training using gradient descent on a simple squared-error cost function, i.e., Eqs. (1)-(2) without the 1/&2(X) terms. To reduce overfitting, training is considered complete at the minimum of the squared error on the cross-validation set 8, monitored at the end of each complete pass through the training data. Phase II (Initial estimate of the conditional variance): Attach a layer of hidden units connected to both the inputs and the hidden units of the network from Phase I (see Figure 1). Freeze the weights trained in Phase I, and train the &2-unit to predict the squared errors (see Eq. (3?, again using simple gradient descent as in Phase I. The training set for this phase is set 8, with set A used for cross-validation. If set A were used as the training set in this phase as well, any overfitting in Phase I could result in seriously underestimating u 2 (x). To avoid this risk, we interchange the data sets. The initial value for the offset (3 of the &2-unit is the natural logarithm of the mean squared error (from Phase I) of set 8. Phase II stops when the squared error on set A levels off or starts to increase. Phase ill (Weighted regression): Re-split the available data into two new halves, A' and 8'. Unfreeze all weights and train all network parameters to minimize the full cost function C on set A'. Training is considered complete when C has reached its minimum on set 8'. 2 Examples Example #1: To demonstrate this method, we construct a one-dimensional example problem where y(x) and u 2 (x) are known. We take the equation y(x) = sin(wax) sin(w,Bx) withw a = 3 andw,B = 5. We then generate (x, d) pairs by picking x uniformly from the interval [0, 7r /2] and obtaining the corresponding target d by adding normally distributed noise n(x) = N[0,u 2 (x)] totheunderlyingy(x), whereu 2(x) = 0.02+0.25 x [1-sin(w,Bx)j2. 493 Learnillg Local Error Bars for Nonlinear Regression Table 1: Results for Example #1. ENMS denotes the mean squared error divided by the overall variance of the target; "Mean cost" represents the cost function (Eq. (1)) averaged over all patterns . Row 4 lists these values for the ideal model (true y(x) and a 2 (x)) given the data generated. Row 5 gives the correlation coefficient between the network's predictions for the standard error (i.e., the square root of the &2 -unit's activation) and the actually occurring L1 residual errors, Id(Xi) - y(x;) I. Row 6 gives the correlation between the true a(x) and these residual errors . Rows 7-9 give the percentage of residuals smaller than one and two standard deviations for the obtained and ideal models as well as for an exact Gaussian. 1 Training 1 Phase I 2 3 4 Phase " Phase III n( x ) (exact additive noise) 5 6 ~ ~~ ~ \: residual errors) 7 8 9 % of errors % of errors I a( x ,reridual errors) < .,.~ xl; 2"'~ xl < a-(x); 2a-(x) (aact Gaussian) ENM ."< 0.576 0.576 0.552 0.545 (N -- 103 ) Er.. IU .,,< 0.593 0.593 0.570 0.563 0.853 0.542 0.440 0.430 I P 0.564 0.602 I sId 64.8 66.6 68.3 1 Evaluation Mean cost 2 sId 95.4 96.0 95.4 (N -- 105 ) 1 Mean cost 0.882 0.566 0.462 0.441 J p 0.548 0.584 I sId 67.0 68.4 68.3 2 sId 94.6 95.4 95.4 We generate 1000 patterns for training and an additional 105 patterns for post-training evaluation. Training follows exactly the three phases described above with the following details: 2 Phase I uses a network with one hidden layer of 10 tanh units and TJ = 10- 2 ? For Phase II we add an auxiliary layer of 10 tanh hidden units connected to the a.2-unit (see Figure 1) and use the same TJ. Finally, in Phase III the composite network is trained with TJ 10- 4. = At the end of Phase I (Figure 2a), the only available estimate of (1'2 (x ) is the global root-mean-squared error on the available data, and the model misspecification is roughly uniform over x-a typical solution were we training with only the traditional squared-error cost function. The corresponding error measures are listed in Table 1. At the end of Phase II, however, we have obtained an initial estimate of (1'2 (x ) (since the weights to the i)-unit are frozen during this phase, no modification of i) is made). Finally, at the end of Phase III, we have better estimates of both y{ x) and (1'2 (x). First we note that the correlations between the predicted errors and actual errors listed in Table 1 underscore the near-optimal prediction of local errors. We also see that these errors correspond, as expected, to the assumed Gaussian error model. Second, we note that not only has the value of the cost function dropped from Phase II to Phase III, but the generalization error has also dropped, indicating an improved estimate of y( x ). By comparing Phases I and III we see that the quality of i)(x) has improved significantly in the low-noise regions (roughly x < 0.6) at a minor sacrifice of accuracy in the high-noise region. Example #2: We now apply our method to a set of observed data, the 1000-point laser 2Purther details: all inputs are scaled to zero mean and unit variance. All initial weights feeding into hidden units are drawn from a uniform distribution between -1 j i and 1 j i where i is the number of incoming connections. All initial weights feeding into y or &2 are drawn from a uniform distribution between -sji and sji where s is the standard deviation of the (overall) target distribution. No momentum is used, and all weight updates are averaged over the forward passes of 20 patterns. 494 David A. Nix. Andreas S. Weigend (bl Phaao 1 Phaoolll Phalol _II _"I x x x 1~ 0.5 ~ Phaloll : 0 . -0.5 .? -1 o . ?.\ x 1 x x 11(1-11 Figure 2: (a) Example #1: Results after each phase of training. The top row gives the true y(x) (solid line) and network estimate y(x) (dotted line); the bottom row gives the true oo2(x) (solid line) and network estimate o-2(x) (dotted line). (b) Example #2: state-space embedding of laser data (evaluation set) using linear grey-scaling of 0.50 (lightest) < o-(Xt) < 6.92 (darkest). See text for details. intensity series from the Santa Fe competition. 3 Since our method is based on the network's observed errors, the predicted error a- 2 (x) actually represents the sum of the underlying system noise, characterized by 00 2 (x), and the model misspecification. Here, since we know the system noise is roughly uniform 8-bit sampling resolution quantization error, we can apply our method to evaluate the local quality of the manifold approximation. 4 The prediction task is easier if we have more points that lie on the manifold, thus better constraining its shape. In the competition, Sauer (1994) upsampled the 1000 available data points with an FFf method by a factor of 32. This does not change the effective sampling rate, but it "fills in" more points, more precisely defining the manifold. We use the same upsampling trick (without filtered embedding), and obtain 31200 full (x, d) patterns for learning. We apply the three-phase approach described above for the simple network of Figure 1 with 25 inputs (corresponding to 25 past values), 12 hidden units feeding the y-unit, and a libera130 hidden units feeding the a- 2 -unit (since we are uncertain as to the complexity of (j2(x) for this dataset). We use 11 = 10- 7 for Phase I and 11 = 10- 10 for Phases II and III. Since we know the quantization error is ?O. 5, error estimates less than this are meaningless. Therefore, we enforce a minimum value of (j2(x) = 0.25 (the quantization error squared) on the squared errors in Phases II and III. 3The data set and several predictions and characterizations are described in the volume edited by Weigend & Gershenfeld (1994). The data is available by anonymous ftp at ftp.cs.colorado.edu in /pub/Time-Series/SantaFe as A.dat. See also http://www . cs. colorado. edu/Time-Series/TSWelcome. html for further analyses of this and other time series data sets. 4When we make a single-step prediction where the manifold approximation is poor, we have little confidence making iterated predictions based on that predicted value. However, if we know we are in a low-error region, we can have increased confidence in iterated predictions that involve our current prediction. 495 Learning Local Error Bars for Nonlinear Regression Table 2: Results for Example #2 (See Table 1 caption for definitions) . I Training (N -- 975) I Evaluation (N - 23 950) row I . EMMC: I 3 Phase I Phase II Phase III 4 P( a( x ) . residual errors) p 0.557 % of errors < a( x); 2a( x) (exacl Gaussian) 1 sid 69A 68.3 2 0.00125 0.00125 0.00132 Mean <XlSt 1.941 1.939 1.725 E1"Juc: 0.0156 0.0156 0.0139 Mean cost 7.213 5.628 5.564 P 0.366 2std 94.9 95.4 1 sid 63.1 68.3 2 std 88.0 95.4 Results are given in Table 2 for patterns generated from the available 1000 points and 24,000 additional points used for evaluation. Even though we have used a Gaussian error model, we know the distribution of errors is not Gaussian. This is reflected in rows 5 and 6, where the training data is modeled as having a Gaussian em but the evaluation values become considerably distorted from an exact Gaussian. Again, however, not only do we obtain significant predictability of the errors, but the method also reduces the squared-error measure obtained in Phase I. We can use the estimated error to characterize the quality of the manifold approximation on 24,000 post-training evaluation points, as illustrated in Figure 2b. The manifold approximation is poorer (darker) for higher predicted values of Xt and for values nearer the edge of the manifold. Note the dark-grey (high-error) vertical streak leaving the origin and dark points to its left which represent patterns involving sudden changes in oscillation intensity. 3 Discussion Since we are in effect approximating two functions simultaneously, we can apply many of the existing variations for improving function approximation designed for networks learning only y(x). For example, when using limited amounts of data, especially if it is noisy, the particular split of data into training and cross-validation sets we use introduces significant variation in the resulting y(x) due to overfitting, as demonstrated on financial data by Weigend & LeBaron (1994). If we want to estimate local error bars, not only must we fear overfitting y(x), but we must also be concerned with overfitting &.2 (x). If the standard method of stopping at the minimum of an appropriate cross-validation set does not suffice for a given problem, it is straightforward to employ the usual anti-overfitting weaponry (smooth &2 as a function of x, pruning, weight-elimination, etc.). Furthermore, we can bootstrap over our available dataset and create multiple composite networks, averaging their predictions for both y(x) and &2(X) . Additionally, to incorporate prior information in a Bayesian framework as a form of regularization, Wolpert (personal communication, 1994) suggests finding the maximum a posteriori (instead of maximum-likelihood) conditional mean and variance using the same interpretation of the network outputs. In summary, we start with the maximum-likelihood principle and arrive at error estimates that vary with location in the input space. These local error estimates incorporate both underlying system noise and model misspecification. We have provided a computergenerated example to demonstrate the ease with which accurate error bars can be learned. We have also provided an example with real-world data in which the underlying system noise is small, uniform quantization error to demonstrate how the method can be used 496 David A. Nix, Andreas S. Weigend to characterize the local quality of the regression model. A significant feature of this method is its weighted-regression effect, which complicates learning by introducing local minima but can be potentially beneficial in constructing a more robust model with improved generalization abilities. In the framework presented, for any problem we must assume a specific parameterized cm then add one auxiliary output unit for each higher moment of the cm we wish to estimate locally. Here we have demonstrated the Gaussian case with a location parameter (conditional mean) and a scale parameter (local error bar) for a scalar output variable. The extension to multiple output variables is clear and allows a full covariance matrix to be used for weighted regression, including the cross-correlation between multiple targets. Acknowledgments This work is supported by the National Science Foundation under Grant No. RIA ECS-9309786 and by a Graduate Fellowship from the Office of Naval Research. We would like to thank Chris Bishop, Wray BUntine, Don Hush, Steve Nowlan, Barak Pearlmutter, Dave Rumelhart, and Dave Wolpert for helpful discussions. References C. Bishop. (1994) "Mixture Density Networks." Neural Computing Research Group Report NCRG/4288, Department of Computer Science, Aston University, Birmingham, UK. w.L. Buntine and A.S. Weigend. (1991) "Bayesian Backpropagation." Complex Systems, 5: 603--643 . A.M. Fraser and A. Dimitriadis. (1994) "Forecasting Probability Densities Using Hidden Markov Models with Mixed States." In Time Series Prediction: Forecasting the Future and Understanding the Past, A.S. Weigend and N.A. Gershenfeld. eds .. Addison-Wesley, pp. 265- 282. P.T. Kazlas and A.S. Weigend. (1995) "Direct Multi-Step TIme Series Prediction Using TD(>.)." In Advances in Neural Infonnation Processing Systems 7 (NIPS*94, this volume). San Francisco, CA: Morgan Kaufmann. D.E. Rumelhart, R. Durbin. R. Golden. and Y. Chauvin. (1995) "Backpropagation: The Basic Theory." In Backpropagation: Theory, Architectures and Applications, Y. Chauvin and D.E. Rumelhart, eds., Lawrence Erlbaum, pp. 1- 34. T. Sauer. (1994) 'T1l11.e Series Prediction by Using Delay Coordinate Embedding." In Time Series Prediction: Forecasting the Future and Understanding the Past, A.S. Weigend and N.A. Gershenfeld. eds., Addison-Wesley, pp. 175-193. A.N. Srivastava and A.S. Weigend. (1994) "Computing the Probability Density in Connectionist Regression." In Proceedings of the IEEE International Conference on Neural Networks (IEEEICNN'94), Orlando, FL, p. 3786--3789. IEEE-Press. A.S. Weigend and N.A. Gershenfeld, eds. (1994) Time Series Prediction: Forecasting the Future and Understanding the Past. Addison-Wesley. A.S. Weigend and B. LeBaron. (1994) "Evaluating Neural Network Predictors by Bootstrapping." In Proceedings of the International Conference on Neural Infonnation Processing (ICONIP'94), Seoul, }(orea,pp.1207-1212. A.S. Weigend and D.A. Nix. (1994) "Predictions with Confidence Intervals (Local Error Bars)." In Proceedings of the International Conference on Neural Information Processing (ICONIP'94), Seoul, }(orea, p. 847- 852. 

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PREDICTIVE CODING WITH NEURAL NETS: APPLICATION TO TEXT COMPRESSION J iirgen Schmidhuber Stefan Heil Fakultat fiir Informatik Technische Universitat Miinchen 80290 Miinchen, Germany Abstract To compress text files, a neural predictor network P is used to approximate the conditional probability distribution of possible "next characters", given n previous characters. P's outputs are fed into standard coding algorithms that generate short codes for characters with high predicted probability and long codes for highly unpredictable characters. Tested on short German newspaper articles, our method outperforms widely used Lempel-Ziv algorithms (used in UNIX functions such as "compress" and "gzip"). 1048 1 liirgen Schmidhuber, Stefan Heil INTRODUCTION The method presented in this paper is an instance of a strategy known as "predictive coding" or "model-based coding". To compress text files, a neural predictor network P approximates the conditional probability distribution of possible "next characters", given n previous characters. P's outputs are fed into algorithms that generate short codes for characters with low information content (characters with high predicted probability) and long codes for characters conveying a lot of information (highly unpredictable characters) [5]. Two such standard coding algorithms are employed: Huffman Coding (see e.g. [1]) and Arithmetic Coding (see e.g. [7]). With the off-line variant of the approach, P's training phase is based on a set F of training files. After training, the weights are frozen. Copies of P are installed at all machines functioning as message receivers or senders. From then on, P is used to encode and decode unknown files without being changed any more. The weights become part of the code of the compression algorithm. Note that the storage occupied by the network weights does not have to be taken into account to measure the performance on unknown files - just like the code for a conventional data compression algorithm does not have to be taken into account. The more sophisticated on-line variant of our approach will be addressed later. 2 A PREDICTOR OF CONDITIONAL PROBABILITIES Assume that the alphabet contains k possible characters Zl, Z2, ?.? , Z1c. The (local) representation of Zi is a binary k-dimensional vector r( Zi) with exactly one non-zero component (at the i-th position). P has nk input units and k output units. n is called the "time-window size". We insert n default characters Zo at the beginning of each file. The representation of the default character, r(zo), is the k-dimensional zero-vector. The m-th character of file f (starting from the first default character) is called efn. For all f E F and all possible m > n, P receives as an input r(e;"_n) 0 r(e;"_n+l) 0 ... 0 r(c!n_l), (1) where 0 is the concatenation operator, for vectors. P produces as an output Pin, a k-dimensional output vector. Using back-propagation [6][2][3][4], P is trained to mmlIDlze ~L L II r(c!n) - pin 112 . (2) jEFm>n Expression (2) is minimal if pin always equals E(r(efn) I e;"_n,?? ?,c!n-l), (3) the conditional expectation of r( efn), given r( e;"_n) ore e;"_n+1)o . . . or( c!n-l). Due to the local character representation, this is equivalent to (Pin); being equal to the 1049 Predictive Coding with Neural Nets conditional probability (4) for all / and for all appropriate m> n, where (P,{Jj denotes the j-th component of the vector P/n. In general, the (P/n)i will not quite match the corresponding conditional probabilities. For normalization purposes, we define _ PmI (.) 1 - (P/n)i f? L:j=I(Pm)j j: (5) No normalization is used during training, however. 3 HOW TO USE THE PREDICTOR FOR COMPRESSION We use a standard procedure for predictive coding. With the help of a copy of P, an unknown file / can be compressed as follows: Again, n default characters are inserted at the beginning. For each character cfn (m> n), the predictor emits its output P/n based on the n previous characters. There will be a k such that cfn = Zj:. The estimate of P(cfn = Zj: I Crn-l) is given by P/n(k). The code of cfn, code( cfn), is generated by feeding P/n (k) into the Huffman Coding algorithm (see below), or, alternatively, into the Arithmetic Coding algorithm (see below). code(cfn) is written into the compressed file. The basic ideas of both coding algorithms are described next. c!n-n, ... , 3.1 HUFFMAN CODING With a given probability distribution on a set of possible characters, Huffman Coding (e.g. [1]) encodes characters by bitstrings as follows. Characters are terminal nodes of a binary tree to be built in an incremental fashion. The probability of a terminal node is defined as the probability of the corresponding character. The probability of a non-terminal node is defined as the sum of the probabilities of its sons. Starting from the terminal nodes, a binary tree is built as follows: Repeat as long as possible: Among those nodes that are not children 0/ any non-terminal nodes created earlier, pick two with lowest associated probabilities. Make them the two sons 0/ a newly generated non-terminal node. The branch to the "left" son of each non-terminal node is labeled by a O. The branch to its "right" son is labeled by a 1. The code of a character c, code(c), is the bitstring obtained by following the path from the root to the corresponding node. Obviously, if c #- d, then code(c) cannot be the prefix of code(d). This makes the code uniquely decipherable. 1050 Jurgen Schmidhuber, Stefan Heil Characters with high associated probability are encoded by short bitstrings. Characters with low associated probability are encoded by long bitstrings. Huffman Coding guarantees minimal expected code length, provided all character probabilities are integer powers of ~. 3.2 ARlTHMETIC CODING In general, Arithmetic Coding works slightly better than Huffman Coding. For sufficiently long messages, Arithmetic Coding achieves expected code lenghts arbitrarily close to the information-theoretic lower bound. This is true even if the character probabilities are not powers of ~ (see e.g. [7]) . The basic idea of Arithmetic Coding is: a message is encoded by an interval of real numbers from the unit interval [0,1[. The output of Arithmetic Coding is a binary representation of the boundaries of the corresponding interval. This binary representation is incrementally generated during message processing. Starting with the unit interval, for each observed character the interval is made smaller, essentially in proportion to the probability of the character. A message with low information content (and high corresponding probability) is encoded by a comparatively large interval whose precise boundaries can be specified with comparatively few bits. A message with a lot of information content (and low corresponding probability) is encoded by a comparatively small interval whose boundaries require comparatively many bits to be specified. Although the basic idea is elegant and simple, additional technical considerations are necessary to make Arithmetic Coding practicable. See [7] for details. Neither Huffman Coding nor Arithmetic Coding requires that the probability distribution on the characters remains fixed. This allows for using "time-varying" conditional probability distributions as generated by the neural predictor. 3.3 HOW TO "UNCOMPRESS" DATA The information in the compressed file is sufficient to reconstruct the original file without loss of information. This is done with the "uncompress" algorithm, which works as follows: Again, for each character efn (m > n), the predictor (sequentially) emits its output based on the n previous characters, where the e{ with n < I < m were gained sequentially by feeding the approximations (k) of the probabilities P(e{ ZIc I e{-n,???, e{-l) into the inverse Huffman Coding procedure (see e.g. [1]), or, alternatively (depending on which coding procedure was used), into the inverse Arithmetic Coding procedure (e.g. [7]). Both variants allow for correct decoding of c{ from eode(c{) . With both variants, to correctly decode some character, we first need to decode all previous characters. Both variants are guaranteed to restore the original file from the compressed file. pin = p/ WHY NOT USE A LOOK-UP TABLE INSTEAD OF A NETWORK? Because a look-up table would be extremely inefficient. A look-up table requires k n +1 entries for all the conditional probabilities corresponding to all possible com- Predictive Coding with Neural Nets 1051 binations of n previous characters and possible next characters. In addition, a special procedure is required for dealing with previously unseen combinations of input characters. In contrast, the size of a neural net typically grows in proportion to n 2 (assuming the number of hidden units grows in proportion to the number of input units), and its inherent "generalization capability" is going to take care of previously unseen combinations of input characters (hopefully by coming up with good predicted probabilities). 4 SIMULATIONS We implemented both alternative variants of the encoding and decoding procedure described above. Our current computing environment prohibits extensive experimental evaluations of the method. The predictor updates turn out to be quite time consuming, which makes special neural net hardware recommendable. The limited software simulations presented in this section, however, will show that the "neural" compression technique can achieve "excellent" compression ratios. Here the term "excellent" is defined by a statement from [1]: "In general, good algorithms can be expected to achieve an average compression ratio of 1.5, while excellent algorithms based upon sophisticated processing techniques will achieve an average compression ratio exceeding 2.0." Here the average compression ratio is the average ratio between the lengths of original and compressed files. The method was applied to German newspaper articles. The results were compared to those obtained with standard encoding techniques provided by the operating system UNIX, namely "pack", "compress", and "gzip" . The corresponding decoding algorithms are "unpack", "uncompress", and "gunzip", respectively. ''pack'' is based on Huffman-Coding (e.g. [1]), while "compress" and "gzip" are based on techniques developed by Lempel and Ziv (e.g. [9]). As the file size goes to infinity, Lempel-Ziv becomes asymptotically optimal in a certain information theoretic sense [8]. This does not necessarily mean, however, that Lempel-Ziv is optimal for finite file sizes. The training set for the predictor was given by a set of 40 articles from the newspaper Miinchner M erkur, each containing between 10000 and 20000 characters. The alphabet consisted of k 80 possible characters, including upper case and lower case letters, digits, interpunction symbols, and special German letters like "0", "ii", "a.". P had 430 hidden units. A "true" unit with constant activation 1.0 was connected to all hidden and output units. The learning rate was 0.2. The training phase consisted of 25 sweeps through the training set. = The test set consisted of newspaper articles excluded from the training set, each containing between 10000 and 20000 characters. Table 1 lists the average compression ratios. The "neural" method outperformed the strongest conventional competitor, the UNIX "gzip" function based on a Lempel-Ziv algorithm. 1052 Jurgen Schmidhuber, Stefan Heil Method Huffman Coding (UNIX: pack) Lempel-Ziv Coding (UNIX: compress) Improved Lempel-Ziv ( UNIX: gzip -9) Neural predictor + Huffman Coding, n 5 Neural predictor + Arithmetic Coding, n 5 = = Compression Ratio 1.74 1.99 2.29 2.70 2.72 I Table 1: Compression ratios of various compression algorithms for short German 20000 Bytes) from the unknown test set. text files ? Method Huffman Coding (UNIX: pack) Lempel-Ziv Coding (UNIX: compress) Improved Lempel-Zlv ( UNIX: gzip -9) Neural predictor + Huffman Coding, n 5 Neural predictor + Arithmetic Coding, n 5 = = Compression Ratio 1.67 1.71 2.03 2.25 2.20 I Table 2: Compression ratios for articles from a different newspaper. The neural predictor was not retrained. How does a neural net trained on articles from Miinchner Merkurperform on articles from other sources? Without retraining the neural predictor, we applied all competing methods to 10 articles from another German newspaper (the Frankenpost). The results are given in table 2. The Frankenpost articles were harder to compress for all algorithms. But relative performance remained comparable. Note that the time-window was quite small (n = 5). In general, larger time windows will make more information available to the predictor. In turn, this will improve the prediction quality and increase the compression ratio. Therefore we expect to obtain even better results for n > 5 and for recurrent predictor networks. 5 DISCUSSION / OUTLOOK Our results show that neural networks are promising tools for loss-free data compression. It was demonstrated that even off-line methods based on small time windows can lead to excellent compression ratios - at least with small text files, they can outperform conventional standard algorithms. We have hardly begun, however, to exhaust the potential of the basic approach. 5.1 ON-LINE METHODS A disadvantage of the off-line technique above is that it is off-line: The predictor does not adapt to the specific text file it sees. This limitation is not essential, however. It is straight-forward to construct an on-line variant of the approach. Predictive Coding with Neural Nets 1053 With the on-line variant, the predictor continues to learn during compression. The on-line variant proceeds like this: Both the sender and the receiver start with exactly the same initial predictor. Whenever the sender sees a new character, it encodes it using its current predictor. The code is sent to the receiver who decodes it. Both the sender and the receiver use exactly the same learning protocol to modify their weights. This implies that the modified weights need not be sent from the sender to the receiver and do not have to be taken into account to compute the average compression ratio. Of course, the on-line method promises much higher compression ratios than the off-line method. 5.2 LIMITATIONS The main disadvantage of both on-line and off-line variants is their computational complexity. The current off-line implementation is clearly slower than conventional standard techniques, by about three orders of magnitude (but no attempt was made to optimize the code with respect to speed). And the complexity of the on-line method is even worse (the exact slow-down factor depends on the precise nature of the learning protocol, of course). For this reason, especially the promising on-line variants can be recommended only if special neural net hardware is available. Note, however, that there are many commercial data compression applications which rely on specialized electronic chips. 5.3 ONGOING RESEARCH There are a few obvious directions for ongoing experimental research: (1) Use larger time windows - they seem to be promising even for off-line methods (see the last paragraph of section 4). (2) Thoroughly test the potential of on-line methods. Both (1) and (2) should greatly benefit from fast hardware. (3) Compare performance of predictive coding based on neural predictors to the performance of predictive coding based on different kinds of predictors. 6 ACKNOWLEDGEMENTS Thanks to David MacKay for directing our attention towards Arithmetic Coding. Thanks to Margit Kinder, Martin Eldracher, and Gerhard Weiss for useful comments. 1054 Jiirgen Schmidhuber, Stefan Heil References [1] G. Held. Data Compression. Wiley and Sons LTD, New York, 1991. [2] Y. LeCun. Une procedure d'apprentissage pour reseau a. seuil asymetrique. Proceedings of Cognitiva 85, Paris, pages 599-604, 1985. [3] D. B. Parker. Learning-logic. Technical Report TR-47, Center for Compo Research in Economics and Management Sci., MIT, 1985. [4] D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representations by error propagation. In Parallel Distributed Processing, volume 1, pages 318-362. MIT Press, 1986. [5] J. H. Schmidhuber and S. Heil. Sequential neural text compression. IEEE Transactions on Neural Networks, 1994. Accepted for publication. [6] P. J. Werbos. Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences. PhD thesis, Harvard University, 1974. [7] I. H. Witten, R . M. Neal , and J. G. Cleary. Arithmetic coding for data compression. Communications of the ACM, 30(6):520-540, 1987. [8] A. Wyner and J . Ziv. Fixed data base version of the Lempel-Ziv data compression algorithm. IEEE Transactions In/ormation Theory, 37:878-880, 1991. [9] J. Ziv and A. Lempel. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory, IT-23(5):337-343, 1977. 

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Direction Selectivity In Primary Visual Cortex Using Massive Intracortical Connections Christof Koch CNS Program 216-76 Caltech Pasadena, CA 91125 Humbert Suarez CNS Program 216-76 Caltech Pasadena, CA 91125 Rodney Douglas MRC Anatomical Neuropharmacology Unit University of Oxford Oxford UK Abstract Almost all models of orientation and direction selectivity in visual cortex are based on feedforward connection schemes, where geniculate input provides all excitation to both pyramidal and inhibitory neurons. The latter neurons then suppress the response of the former for non-optimal stimuli. However, anatomical studies show that up to 90 % of the excitatory synaptic input onto any cortical cell is provided by other cortical cells. The massive excitatory feedback nature of cortical circuits is embedded in the canonical microcircuit of Douglas &. Martin (1991). We here investigate analytically and through biologically realistic simulations the functioning of a detailed model of this circuitry, operating in a hysteretic mode. In the model, weak geniculate input is dramatically amplified by intracortical excitation, while inhibition has a dual role: (i) to prevent the early geniculate-induced excitation in the null direction and (ii) to restrain excitation and ensure that the neurons fire only when the stimulus is in their receptive-field. Among the Humbert Suarez, Christo! Koch, Rodney Douglas 4 insights gained are the possibility that hysteresis underlies visual cortical function, paralleling proposals for short-term memory, and strong limitations on linearity tests that use gratings. Properties of visual cortical neurons are compared in detail to this model and to a classical model of direction selectivity that does not include excitatory corti co-cortical connections. The model explain a number of puzzling features of direction-selective simple cells, including the small somatic input conductance changes that have been measured experimentally during stimulation in the null direction. The model also allows us to understand why the velocity-response curve of area 17 neurons is different from that of their LG N afferents, and the origin of expansive and compressive nonlinearities in the contrast-response curve of striate cortical neurons. 1 INTRODUCTION Direction selectivity is the property of neurons that fire more strongly for one direction of motion of a bar (the preferred direction) than the other (nUll direction). It is one of the fundamental properties of neurons in visual cort~x and is intimately related to the processing of motion by the visual system. LGN neurons that provide input to visual cortex respond approximately symmetrically to motion in different directions; so cortical neurons must generate that direction specificity. Models of direction selectivity in primary visual. cortex generally overlook two important constraints. 1. at least 80% of excitatory synapses on cortical pyramidal cells originate from other pyramidal cells, and less than 10 % are thalamic afferents (Peters & Payne, 1993). 2. Intracellular in vivo recordings in cat simple cells by Douglas, et al. (1988) failed to detect significant changes in somatic input conductance during stimulation in the null direction, indicating that there is very little synaptic input to direction-selective neurons in that condition, including no massive "shunting" inhibition. One very attractive solution incorporating these two constraints was proposed by Douglas & Martin (1991) in the form of their canonical microcircuit: for motion of a visual stimulus in the preferred direction, weak geniculate excitation excites cortical pyramidal cells to respond moderately. This relatively small amount of cortical excitation is amplified via excitatory cortico-cortical connections. For motion in the null direction, the weak geniculate excitation is vetoed by weak inhibition (mediated via an interneuron) and the cortical loop is never activated. In order to test quantitatively this circuit against the large body of experimental data, it is imperative to model its operation through mathematical analysis and detailed simulations. 2 MODEL DESCRIPTION AND ANALYSIS The model consists of a retino-geniculate and a cortical stage (Fig. 1). The former includes a center-surround receptive field and bandpass temporal filtering (Victor, 1987). We simulate a I-D array of ON LGN neurons, with 208 LGN neurons at each of 6 spatial positions in this array. The output ofthe LGN-action potentials-feeds Direction Selectivity in Primary Visual Cortex 5 into the cortical module consisting of 640 pyramidal (excitatory) and 160 inhibitory neurons. Each neuron is modeled using 3-4 compartments whose parameters reflect, in a simplified way, the biophysics and morphology of cortical neurons; the somatic compartment produces action potentials in response to current injection. There are excitatory connections among all pyramidal neurons, and inhibitory connections from the inhibitory population to itself and to the pyramidal population. The receptive field of the geniculate input to the inhibitory cortical neurons is displaced in space from the geniculate input to the pyramidal neurons, so that in the null direction inhibition overlaps with geniculate excitation in the pyramidal neurons, resulting in direction selectivity in the pyramidal neurons. The time courses of the post-synaptic potentials (PSP's) are consistent with physiologically recorded PSP's. The EPSP's in our model arise exclusively from non-NMDA synapses, and the IPSP's originate from both GABAA and GABAB synapses. There are two operating modes of this cortical amplifier circuit, depending on parameter values. In the first mode, the pyramidal neurons' response increases proportionally to the stimulus strength over a substantial range of input values, before saturating. In the second mode, the response increases much faster over a narrow range of stimulus strengths, then saturates. Analytically, one can define a steadystate transfer function for the network of pyramidal neurons; then, in the first mode, the transfer function has a slope that is less than 1, so that the network's firing rate at equilibrium increases proportionally to the input strength and the network dynamics are rather slow. In the second mode the initial slope is larger than 1, so that the network can discharge briskly at equilibrium even without any input, show hysteresis, and has rather fast dynamics. The network does not fire without any input, because of the neuron's threshold. In this paper, we will show responses in that second, hysteretic mode of operation. We will compare the cortical amplifier model's response properties to a pure feedforward model, that has no excitatory connections between pyramidal neurons. In order to maintain strong responses in that model, the LGN was connected more strongly to the pyramidal neurons, and in order to maintain direction selectivity, the weights of the connections of the inhibitory neurons to the pyramids were increased as well. 3 AMPLIFICATION AND CONDUCTANCE CHANGE IN THE NULL DIRECTION The input conductance of pyramidal neurons, a measure of total synaptic input, changes by only 50 % in the cortical amplifier model versus 400 % in the feedforward model, and so is more consistent with physiology (Fig. 2). Indeed, most of the current causing firing in the preferred direction originates from other pyramidal neurons, so the connection weight from the LGN is small. Consequently, the inhibitory weight is also small, since it needs only be large enough to balance out the LG N current in the null direction. Since there is little firing in other pyramidal neurons in the null direction, there is also little total synaptic input to the fiducial cell. The cortical amplifier circuit provides substantial amplification of the LGN input. In the preferred direction, excitatory intracortical connections amplify the LGN input, providing a feedback current that is about 2.2 times larger than the 6 Humbert Suarez, Christo! Koch, Rodney Douglas 640 160 smooth cells AREA 17 208 neurons at each position LGN RETINA 111I11111111111111 ==-1111111111111111111 125 pixels Figure 1: Wiring diagram of the direction selectivity model. Input to LGN neurons comes from a one-dimensional array of retinal pixels. There are 208 LGN neurons at each of six spatial positions. The LG N neurons connect slightly differently with the two populations of cortical neurons (pyramidal and inhibitory) so that as a group the LGN inputs to the pyramids are displaced spatially by 5' from those to the inhibitory neurons. The open triangle symbols denote excitatory connections, the filled triangles inhibitory GABAA-mediated synapses, and the filled circles inhibitory GABAB-mediated synapses. The capital sigma symbol indicates convergence of inputs from many LGN neurons onto cortical neurons. Direction Selectivity in Primary Visual Cortex 7 0.5 b 04 ~ .s E ~ :; 1 0.3 r25 tOO ~:~I ~:':-~_.'~~::'>:~ :: ~"~:'~:':'.~:..~ .,)0.1:' 75 0.2 0 - 2sec 0 .1 0 0 0.1 0.2 0.3 0.5 0.4 Cumal 0 ? 2iec Tesl 0 .6 4()0 c 150 d 300 125 j ..... c 200 ~: 100 - :.....,...,..-..,.?? 100 o\V_,MJ _ ...: . .. . ; . :"-:~ .... ---..,;" '. ?o\...-ohl 75 501 - - - - - - - - - - ' o Time (seconds) Figure 2: (a) Total synaptic current in a simulated pyramidal cell from the LGN (with symbols) and from other pyramidal cells (continuous without symbols), during stimulation by a bar moving in the preferred direction . For the same stimulus moving in the null direction, somatic input conductance as a function of time, (b) for a direction-selective pyramidal neuron (data from Douglas et ai., 1988); (c) feedforward model; (d) cortical amplifier model; LGN current at 60 % contrast (Fig. 2). 4 CONTRAST-RESPONSE CURVES Contrast-response curves plot the peak firing rate to a grating moving in the preferred direction as function of its contrast, or stimulus amplitude (Albrecht & Hamilton, 1982). The cortical amplifier's contrast-response curve is very different from the LGN inputs' and is similar to those that have been described experimentally in cortex (Albrecht & Geisler 1991), having a steep power-function portion followed by abrupt saturation (Fig. 3). The network firing saturates at the fixed point of the transfer function mentioned above (see section 2) and the steep portion results from the fast rise to that fixed point when the stimulus has exceeded the neurons' threshold . In contrast, the feedforward model's contrast-response curve is similar to the LGN inputs' and does not match physiology. The response is very small in the null direction, resulting in very good direction selectivity at all contrasts (i.e., the average direction index is above 0.9). 5 VELOCITY-RESPONSE CURVES 8 Humbert Suarez, Christo! Koch, Rodney Douglas KlO a u 100 b S! '" <II .:a. L .. ~ 10 .. :< ~ 2' :,.; u. 1 1 100 100 10 Contrast('X.) 100 100 U- c S! . !.. 10 Contrast ('X.) <II ~ 10 ~ go :~ u. 1 1 U- ~ 10 Contrast ('X.) 100 d S! . <II 1 10 ~ 2' J; IL 1 1 Figure 3: Peak firing rate versus contrast during stimulation by a moving grating. (a) Visual cortical neuron. Data from Albrecht &, Hamilton, 1982. (b) LGN model. ( c) Feedforward model. (d) Cortical amplifier model. Velocity-response curves plot the peak response to a bar moving in the preferred direction as a function of its velocity. Again, the cortical amplifier's velocityresponse curve is very different from the LGN inputs' and is similar to physiology (Orban, 1984; Fig. 4). The LGN model is firing strongly at high velocities but the model pyramidal neurons are totally silent, due to a combination of GABAA inhibition, neuron threshold, and membrane low-pass filtering. At low velocities, the LGN model does not fire much, while the cortical neurons respond strongly. Indeed, the network firing has enough time to reach the fixed point of the transfer function and will reach it as long as the input is suprathreshold. In contrast, the feedforward model's velocity-response curve is again similar to the LGN input's and does not match physiology. Also shown in Fig. 4 is the response in the null direction for the cortical amplifier model. There is very good direction selectivity at all velocities, consistent with physiological data (Orban, 1984). The persistence of direction selectivity down to low velocities depends critically on the time constant of GABAB and the presence of a very small displacement between the LGN inputs to the pyramidal and inhibitory neurons. 6 OTHER PROPERTIES Recently, direction-selective cortical neurons have been tested for linearity using an intracellular grating superposition test and found to be quite linear (Jagadeesh et al., 1993). Despite that amplification in the present model is so nonlinear, the model is also linear according to that superposition test. An analysis of the test in the context of the model shows that such a test has limited usefulness and suggests improvements. Given that the network transfer function has a fixed point at high firing without any Direction Selectivity in Primary Visual Cortex U 9 400 ~ ~ lit .- ~ ~'OO .~ . LGN X a !! 200 .. .E .. iL D 1n0 GO 0 0.1 10 100 1000 Stimulus velocity (e/sec) 100 120 'S c 51100 CiJ Q) '! --... CIl Q) Cii CI c :~ u.. 'S 80 60 40 20 0 0.1 d 51 80 CiJ Q) '! ~ GN-::J 60 Cii ... 40 ell c: FF :~ 20 u.. 1 P Q) 10 Velocity (deg/sec) 100 0 0.1 NP 1 10 100 Velocity (deg/sec) Figure 4: Peak firing rate versus velocity during stimulation by a bar. (a) LGN neuron in cat (Frishman et aJ., 1983). (b) area 17 visual cortical neuron in cat (Orban, 1984). (c) LGN model neuron (curve labelled "LGN") and feedforward model (FF). (d) Cortical amplifier model in the preferred (curve labelled "P") and null directions (NP). 10 Humbert Suarez, Christo! Koch, Rodney Douglas input (see section 2), hysteresis may occur, whereby the network's discharge persists after the initial stimulus is withdrawn. Because of hysteresis, it is imperative to reset the network by presenting a negative, or inhibitory, input. A parallel can be drawn to several recent proposals for the mechanisms underlying short-term memory. 7 CONCLUSIONS From early on, neurophysiologists have proposed that the LG N provides most of the input to visual cortical neurons and shapes their receptive properties (Hubel and Wiesel, 1962). However, direction selectivity and several other stimulus selectivities are not present in LGN neurons, and other important discrepancies have appeared between the receptive field properties of cortical neurons and those of their LG N afferents. Anatomically, synaptic inputs from the LGN account for less than 10 % of synapses to pyramidal neurons in visual cortex; the remaining 90 % could clearly provide a substrate for these receptive field transformations. Although intracortical inhibition has often been invoked to explain various cortical properties, excitation is usually not mentioned, despite that most corti co-cortical synapses are excitatory. In this paper, we show that intracortical excitation can better account for several key properties of cortical neurons than a purely feedforward model, including the magnitude of the conductance change in the null direction, contrast-response curves, and velocity-response curves. Furthermore, surprinsingly, other key cell properties that are appear to point to feedforward models, such as linearity measured by superposition tests, are also properties of a model based on intracortical excitation. Acknowledgements This research was supported by the Office of Naval Research, the National Science Foundation, the National Eye Institute, and the McDonnell Foundation. References Albrecht, D.G., and Geisler, W.S . (1991) Visual Neuroscience 7, 531-546. Albrecht, D.B., and Hamilton, D.B. (1982) J. Neurophysiol. 48,217-237. Douglas, R.J., and Martin, K.A.C. (1991) J. Physiol. 440,735-769. Douglas, R.J ., Martin, K.A.C., and Whitteridge, D. (1988) Nature 332,642-644. Frishman, L.J., Schweitzer-Tong, D.E., and Goldstein, E.B. (1983) J. Neurophysiol. 50, 1393-1414. Hubel, D.H., and Wiesel, T.N. (1962) J. Physiol. 165,559-568. Jagadeesh, B., Wheat, H.S. and Ferster, D. (1993) Science 262, 1901-1904 Orban, G.A. (1984) Neuronal operations in the visual cortex. Springer, Berlin. Peters, A. and Payne, B. R. (1993) Cerebral Cortex 3, 69-78. Victor, J.D. (1987) J.Physiol. 386, 219-246. 

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SIMPLIFYING NEURAL NETS BY DISCOVERING FLAT MINIMA Sepp Hochreiter" Jiirgen Schmidhuber t Fakultat fiir Informatik, H2 Technische Universitat Miinchen 80290 Miinchen, Germany Abstract We present a new algorithm for finding low complexity networks with high generalization capability. The algorithm searches for large connected regions of so-called ''fiat'' minima of the error function. In the weight-space environment of a "flat" minimum, the error remains approximately constant. Using an MDL-based argument, flat minima can be shown to correspond to low expected overfitting. Although our algorithm requires the computation of second order derivatives, it has backprop's order of complexity. Experiments with feedforward and recurrent nets are described. In an application to stock market prediction, the method outperforms conventional backprop, weight decay, and "optimal brain surgeon" . 1 INTRODUCTION Previous algorithms for finding low complexity networks with high generalization capability are based on significant prior assumptions. They can be broadly classified as follows: (1) Assumptions about the prior weight distribution. Hinton and van Camp [3] and Williams [17] assume that pushing the posterior distribution (after learning) close to the prior leads to "good" generalization. Weight decay can be derived e.g. from Gaussian priors. Nowlan and Hinton [10] assume that networks with many similar weights generated by Gaussian mixtures are "better" a priori. MacKay's priors [6] are implicit in additional penalty terms, which embody the "hochreit@informatik. tu-muenchen .de t schmidhu@informatik.tu-muenchen.de 530 Sepp Hochreiter. Jurgen Schmidhuber assumptions made. (2) Prior assumptions about how theoretical results on early stopping and network complexity carryover to practical applications. Examples are methods based on validation sets (see [8]), Vapnik's "structural risk minimization" [1] [14], and the methods of Holden [5] and Wang et al. [15]. Our approach requires less prior assumptions than most other approaches (see appendix A.l). Basic idea of flat minima search. Our algorithm finds a large region in weight space with the property that each weight vector from that region has similar small error. Such regions are called "flat minima". To get an intuitive feeling for why ''flat'' minima are interesting, consider this (see also Wolpert [18]): a "sharp" minimum corresponds to weights which have to be specified with high precision. A ''flat'' minimum corresponds to weights many of which can be given with low precision. In the terminology of the theory of minimum description length (MDL), fewer bits of information are required to pick a ''flat'' minimum (corresponding to a "simple" or low complexity-network). The MDL principle suggests that low network complexity corresponds to high generalization performance (see e.g. [4, 13]). Unlike Hinton and van Camp's method [3] (see appendix A.3), our approach does not depend on explicitly choosing a "good" prior. Our algorithm finds "flat" minima by searching for weights that minimize both training error and weight precision. This requires the computation of the Hessian. However, by using Pearlmutter's and M~ner's efficient second order method [11, 7], we obtain the same order of complexity as with conventional backprop. A utomatically, the method effectively reduces numbers of units, weigths, and input lines, as well as the sensitivity of outputs with respect to remaining weights and units. Excellent experimental generalization results will be reported in section 4. 2 TASK / ARCHITECTURE / BOXES Generalization task. The task is to approximate an unknown relation [) c X x Z between a set of inputs X C RN and a set of outputs Z C RK. [) is taken to be a function. A relation D is obtained from [) by adding noise to the outputs. All training information is given by a finite relation Do C D. Do is called the training set. The pth element of Do is denoted by an input/target pair (xp, dp ). Architecture. For simplicity, we will focus on a standard feedforward net (but in the experiments, we will use recurrent nets as well). The net has N input units, K output units, W weights, and differentiable activation functions. It maps input vectors xp E RN to output vectors op E RK. The weight from unit j to i is denoted by Wij. The W -dimensional weight vector is denoted by w. Training error. Mean squared error Eq(w, Do) := l.riol E(xp,dp)ED o II dp - op 112 is used, where II . II denotes the Euclidian norm, and 1.1 denotes the cardinality of a set. To define regions in weight space with the property that each weight vector from that region has "similar small error", we introduce the tolerable error Etal, a positive constant. "Small" error is defined as being smaller than Etal. Eq(w, Do) > E tal implies "underfitting" . Boxes. Each weight W satisfying Eq(w, Do) ~ E tal defines an "acceptable minimum". We are interested in large regions of connected acceptable minima. 531 Simplifying Neural Nets by Discovering Flat Minima Such regions are called fiat minima. They are associated with low expected generalization error (see [4]). To simplify the algorithm for finding large connected regions (see below), we do not consider maximal connected regions but focus on so-called "boxes" within regions: for each acceptable minimum w, its box Mw in weight space is a W-dimensional hypercuboid with center w . For simplicity, each edge of the box is taken to be parallel to one weight axis. Half the length of the box edge in direction of the axis corresponding to weight Wij is denoted by ~wii ' which is the maximal (positive) value such that for all i, j, all positive K.ii :5 ~Wij can be added to or subtracted from the corresponding component of W simultaneously without violating Eq(. , Do) :5 Etol (~Wij gives the precision of Wij). Mw's box volume is defined by ~w := 2w ni,j ~Wij. 3 THE ALGORITHM The algorithm is designed to find a W defining a box Mw with maximal box volume ~w. This is equivalent to finding a box Mw with minimal B( w, Do) := -log(~w/2W) = Li,j -log ~Wi.j. Note the relationship to MDL (B is the number of bits required to describe the weights). In appendix A.2, we derive the following algorithm. It minimizes E(w, Do) = Eq(w, Do) + >'B(w, Do), where B= ~ (-WIOg{+ ~logL(:~~j)2 + WlogL ' ,3 " " (L: /~ ,)2) . (1) L"(8w;) ' ,3 0" Here is the activation of the kth output unit, { is a constant, and >. is a positive variable ensuring either Eq(w, Do) :5 Etol, or ensuring an expected decrease of E q (., Do) during learning (see [16] for adjusting >.). E(w, Do) is minimized by gradient descent. To minimize B(w, Do), we compute 8B(w, Do) 8 wuv 2 8B(w, Do) 8 0" = '"" L8 " 8Wij 8 wuv 8(~) I. .? "',' ,) ~ II lor a u, v . (2) vw., It can be shown (see [4]) that by using Pearlmutter's and M~ller's efficient second order method [11, 7], the gradient of B( w, Do) can be computed in O(W) time (see details in [4]) . Therefore, our algorithm has the same order of complexity as standard backprop. 4 EXPERIMENTAL RESULTS (see [4] for details) EXPERIMENT 1 - noisy classification. The first experiment is taken from Pearlmutter and Rosenfeld [12]. The task is to decide whether the x-coordinate of a point in 2-dimensional space exceeds zero (class 1) or does not (class 2). Noisy training examples are generated as follows: data points are obtained from a Gaussian with zero mean and stdev 1.0, bounded in the interval [-3.0,3.0]. The data points are misclassified with a probability of 0.05. Final input data is obtained by adding a zero mean Gaussian with stdev 0.15 to the data points. In a test with 2,000,000 data points, it was found that the procedure above leads to 9.27 per cent 532 Sepp Hochreiter, Jurgen Schmidhuber 1 2 3 4 5 Backprop MSE dto 0.220 1.35 0.223 1.16 0.222 1.37 0.213 1.18 0.222 1.24 New approach dto MSE 0.193 0.00 0.189 0.09 0.13 0.186 0.181 0.01 0.195 0.25 6 7 8 9 10 Backprop MSE dto 0.219 1.24 0.215 1.14 0.214 1.10 0.218 1.21 0.214 1.21 New approach dto MSE 0.187 0.04 0.187 0.07 0.01 0.185 0.190 0.09 0.188 0.07 Table 1: 10 comparisons of conventional backprop (BP) and our new method (FMS) . The second row (labeled "MSE") shows mean squared error on the test set. The third row ("dto") shows the difference between the fraction (in per cent) of misclassifications and the optimal fraction (9.27). The remaining rows provide the analoguous information for the new approach, which clearly outperforms backprop. misclassified data. No method will misclassify less than 9.27 per cent, due to the inherent noise in the data. The training set is based on 200 fixed data points. The test set is based on 120,000 data points. Results. 10 conventional backprop (BP) nets were tested against 10 equally initialized networks based on our new method ("flat minima search", FMS). After 1,000 epochs, the weights of our nets essentially stopped changing (automatic "early stopping"), while backprop kept changing weights to learn the outliers in the data set and overfit. In the end, our approach left a single hidden unit h with a maximal weight of 30.0 or -30.0 from the x-axis input. Unlike with backprop, the other hidden units were effectively pruned away (outputs near zero). So was the y-axis input (zero weight to h) . It can be shown that this corresponds to an "optimal" net with minimal numbers of units and weights. Table 1 illustrates the superior performance of our approach. EXPERIMENT 2 - recurrent nets. The method works for continually running fully recurrent nets as well. At every time step, a recurrent net with sigmoid activations in [0,1] sees an input vector from a stream of randomly chosen input vectors from the set {(0,0), (0,1),(1,0),(1,1)}. The task is to switch on the first output unit whenever an input (1,0) had occurred two time steps ago, and to switch on the second output unit without delay in response to any input (0,1). The task can be solved by a single hidden unit. Results. With conventional recurrent net algorithms, after training, both hidden units were used to store the input vector . Not so with our new approach. We trained 20 networks. All of them learned perfect solutions. Like with weight decay, most weights to the output decayed to zero. But unlike with weight decay, strong inhibitory connections (-30.0) switched off one of the hidden units, effectively pruning it away. EXPERIMENT 3 - stock market prediction. We predict the DAX (German stock market index) based on fundamental (experiments 3.1 and 3.2) and technical (experiment 3.3) indicators. We use strictly layered feedforward nets with sigmoid units active in [-1,1]' and the following performance measures: Confidence: output 0 > a - positive tendency, 0 < -a - negative tendency. Performance: Sum of confidently, incorrectly predicted DAX changes is subtracted Simplifying Neural Nets by Discovering Flat Minima 533 from sum of confidently, correctly predicted ones. The result is divided by the sum of absolute changes. EXPERIMENT 3.1: Fundamental inputs: (a) German interest rate ("Umlaufsrendite"), (b) industrial production divided by money supply, (c) business sentiments ("IFO Geschiiftsklimaindex"). 24 training examples, 68 test examples, quarterly prediction, confidence: 0: = 0.0/0.6/0.9, architecture: (3-8-1) . EXPERIMENT 3.2: Fundamental inputs: (a), (b) , (c) as in expo 3.1, (d) dividend rate, (e) foreign orders in manufacturing industry. 228 training examples, 100 test examples, monthly prediction, confidence: 0: = 0.0/0.6/0.8, architecture: (5-8-1). EXPERIMENT 3.3: Technical inputs: (a) 8 most recent DAX-changes, (b) DAX, (c) change of 24-week relative strength index ("RSI"), (d) difference of "5 week statistic", (e) "MACD" (difference of exponentially weighted 6 week and 24 week DAX). 320 training examples, 100 test examples, weekly predictions, confidence: 0: 0.0/0.2/0.4, architecture: (12-9-1). The following methods are tested: (1) Conventional backprop (BP), (2) optimal brain surgeon (OBS [2]), (3) weight decay (WD [16]), (4) flat minima search (FMS). = Results. Our method clearly outperforms the other methods . FMS is up to 63 per cent better than the best competitor (see [4] for details) . APPENDIX - THEORETICAL JUSTIFICATION A.t. OVERFITTING ERROR In analogy to [15] and [1], we decompose the generalization error into an "overfitting" error and an "underfitting" error. There is no significant underfitting error (corresponding to Vapnik's empirical risk) if Eq(w, Do) ~ E tol . Some thought is required, however, to define the "overfitting" error. We do this in a novel way. Since we do not know the relation D, we cannot know p(o: I D), the "optimal" posterior weight distribution we would obtain by training the net on D (- "sure thing hypothesis") . But, for theoretical purposes, suppose we did know p(o: I D) . Then we could use p(o: I D) to initialize weights before learning the training set Do . Using the Kullback-Leibler distance, we measure the information (due to noise) conveyed by Do, but not by D. In conjunction with the initialization above, this provides the conceptual setting for defining an overfitting error measure. But, the initialization does not really matter, because it does not heavily influence the posterior (see [4]). The overfittin~ error is the Kullback-Leibler distance of the posteriors: Eo(D , Do) = J p(o: I Do) log (p(o: I Do)/p(o: I D?) do:. Eo(D, Do) is the expectation of log (p(o: I Do)/p(o: I D)) (the expected difference of the minimal description of 0: with respect to D and Do, after learning Do). Now we measure the expected overfitting error relative to Mw (see section 2) by computing the expectation of log (p( 0: I Do) / p( 0: I D? in the range Mw: Ero(w) = f3 (1M", PM", (0: I Do)Eq(O:, D)do: - Eq(Do, M w ?) . (3) Here PM",(O: I Do) := p(o: I Do)/ IM", p(a: I Do)da: is the posterior of Do scaled to obtain a distribution within Mw, and Eq(Do, Mw) := IM", PM", (a I Do)Eq(a, Do)do: is the mean error in Mw with respect to Do. 534 Sepp Hochreiter. JiJrgen Schmidhuber Clearly, we would like to pick W such that Ero( w) is minimized. Towards this purpose, we need two additional prior assumptions, which are actually implicit in most previous approaches (which make additional stronger assumptions, see section 1): (1) "Closeness assumption": Every minimum of E q (., Do) is "close" to a maximum of p(aID) (see formal definition in [4]). Intuitively, "closeness" ensures that Do can indeed tell us something about D, such that training on Do may indeed reduce the error on D. (2) "Flatness assumption": The peaks of p(aID)'s maxima are not sharp. This MDL-like assumption holds if not all weights have to be known exactly to model D. It ensures that there are regions with low error on D. A.2. HOW TO FLATTEN THE NETWORK OUTPUT To find nets with flat outputs, two conditions will be defined to specify B(w, Do) (see section 3). The first condition ensures flatness. The second condition enforces "equal flatness" in all weight space ? directions. In both cases, linear approximations will be made (to be justified in [4]). We are looking for weights (causing tolerable error) that can be perturbed without causing significant output changes. Perturbing the weights w by 6w (with components 6Wij), we obtain ED(w,6w) := L,,(o"(w + 6w) - o"(w?)2, where o"(w) expresses o"'s dependence on w (in what follows, however, w often will be suppressed for convenience). Linear approximation (justified in [4]) gives us "Flatness Condition 1": ED(w, 6w) ~ 00" 00" L(L -;;;-:-:6Wij)2 :$ L(L 1-;;;-:-:1I 6wij1)2 :$ ( , .. .. uW'J .. .. uW'J .. '.J .. (4) '.J where ( > 0 defines tolerable output changes within a box and is small enough to allow for linear approximation (it does not appear in B(w, Do)'s gradient, see section 3). Many Mw satisfy flatness condition 1. To select a particular, very flat M w, the following "Flatness Condition 2" uses up degrees of freedom left by (4): . . ( 2 " ( 00")2 'VZ,),U,V: 6Wij) L-~ " 00" 2 = (6w uv ) 2L-(-~-)' (5) " uW'J " uWuv Flatness Condition 2 enforces equal "directed errors" EDij(W,6wij) L,,(O"(Wij + 6Wij) - O"(Wij)? ~ L,,(:;:;6wij?, where Ok(Wij) has the obvious meaning. It can be shown (see [4]) that with given box volume, we need flatness condition 2 to minimize the expected description length of the box center. Flatness condition 2 influences the algorithm as follows: (1) The algorithm prefers to increase the 6Wij'S of weights which currently are not important to generate the target output. (2) The algorithm enforces equal sensitivity of all output units with respect to the weights. Hence, the algorithm tends to group hidden units according to their relevance for groups of output units. Flatness condition 2 is essential: flatness condition 1 by itself corresponds to nothing more but first order derivative reduction (ordinary sensitivity reduction, e.g. [9]). Linear approximation is justified by the choice of f in equation (4). = We first solve equation (5) for 16w,; I = 16W., I ( 2:, (a':::.) 2/ 2:, U.::J 2) Simplifying Neural Nets by Discovering Flat Minima 535 (fixing u, v for all i, j) . Then we insert 16wij I into equation (4) (replacing the second "$" in (4) by ";;;:"). This gives us an equation for the 16wijl (which depend on w, but this is notationally suppressed): 16wi?1 J = Vii ook ( "(_)2 LowIJ.. k (6) The 16wijl approximate the ~Wij from section 2. Thus, B(w,Do) (see section 3) can be approximated by B(w, Do) :;;;: Ei,j -log 16wijl. This immediately leads to the algorithm given by equation (1) . How can this approximation be justified? The learning process itself enforces its validity (see justification in [4]). Initially, the conditions above are valid only in a very small environment of an "initial" acceptable minimum. But during search for new acceptable minima with more associated box volume, the corresponding environments are enlarged, which implies that the absolute values of the entries in the Hessian decrease . It can be shown (see [4]) that the algorithm tends to suppress the following values: (1) unit activations, (2) first order activation derivatives, (3) the sum of all contributions of an arbitary unit activation to the net output. Since weights, inputs, activation functions, and their first and second order derivatives are bounded, it can be shown (see [4]) that the entries in the Hessian decrease where the corresponding 16wij I increase. A.3. RELATION TO HINTON AND VAN CAMP Hinton and van Camp [3] minimize the sum of two terms: the first is conventional enor plus variance, the other is the distance f p( a I Do) log (p( a I Do) I p( a? da between posterior pea I Do) and prior pea). The problem is to choose a "good" prior. In contrast to their approach, our approach does not require a "good" prior given in advance. Furthermore, Hinton and van Camp have to compute variances of weights and units, which (in general) cannot be done using linear approximation. Intuitively speaking, their weight variances are related to our ~Wij. Our approach, however, does justify linear approximation. References [1] I. Guyon, V. Vapnik, B. Boser, L. Bottou, and S. A. Solla. Structural risk minimization for character recognition. In J. E. Moody, S. J. Hanson, and R. P. Lippman, editors, Advances in Neural Information Processing Systems 4, pages 471-479. San Mateo, CA: Morgan Kaufmann, 1992. [2] B. Hassibi and D. G. Stork. Second order derivatives for network pruning: Optimal bra.i.n surgeon. In J. D. Cowan S. J . Hanson and C. L. Giles, editors, Advances in Neural Information Processing Systems 5, pages 164-171 . San Mateo, CA: Morgan Kaufmann , 1993. [3] G. E. Hinton and D. van Camp. Keeping neural networks simple. In Proceedings of the International Conference on Artificial Neural Networks, Amsterdam, pa.ges 11-18. Springer, 1993. 536 Sepp Hoc/zreiter, Jiirgen Schmidhuber [4] S. Hochreiter and J. Schmidhuber. Flat Inllllma search for discovering simple nets. Technical Report FKI-200-94, Fakultiit fiir Informatik, Technische Universitiit Munchen, 1994. [5] S. B. Holden. On the Theory of Generalization and Self-Structuring in Linearly Weighted Connectionist Networks. PhD thesis, Cambridge University, Engineering Department, 1994. [6] D. J. C . MacKay. A practical Bayesian framework for backprop networks. Neural Computation, 4:448-472, 1992. [7] M. F. M~ller. Exact calculation of the product of the Hessian matrix offeed-forward network error functions and a vector in O(N) time. Technical Report PB-432, Computer Science Department, Aarhus University, Denmark, 1993. [8] J. E. Moody and J . Utans. Architecture selection strategies for neural networks: Application to corporate bond rating prediction. In A. N. Refenes, editor, Neural Networks in the Capital Markets. John Wiley & Sons, 1994. [9] A. F. Murray and P. J. Edwards. Synaptic weight noise during MLP learning enhances fault-tolerance, generalisation and learning trajectory. In J. D. Cowan S. J. Hanson and C. L. Giles, editors, Advances in Neural Information Processing Systems 5, pages 491-498. San Mateo, CA: Morgan Kaufmann, 1993. [10] S. J. Nowlan and G. E. Hinton. Simplifying neural networks by soft weight sharing. Neural Computation, 4:173-193, 1992. [11] B. A. Pearlmutter. Fast exact multiplication by the Hessian. Neural Computation, 1994. [12] B. A. Pearlmutter and R. Rosenfeld. Chaitin-Kolmogorov complexity and generalization in neural networks. In R. P. Lippmann, J. E. Moody, and D. S. Touretzky, editors, Advances in Neural Information Processing Systems 3, pages 925-931. San Mateo, CA: Morgan Kaufmann, 1991. [13] J. H. Schmid huber. Discovering problem solutions with low Kolmogorov complexity and high generalization capability. Technical Report FKI-194-94, Fakultiit fUr Informatik, Technische U niversitiit Munchen, 1994. [14] V. Vapnik. Principles of risk minimization for learning theory. In J. E . Moody, S. J. Hanson, and R. P. Lippman, editors, Advances in Neural Information Processing Systems 4, pages 831-838. San Mateo, CA: Morgan Kaufmann, 1992. [15] C . Wang, S. S. Venkatesh, and J. S. Judd. Optimal stopping and effective machine complexity in learning. In J . D. Cowan, G. Tesauro, and J. Alspector, editors, Advances in Neural Information Processing Systems 6, pages 303-310. Morgan Kaufmann, San Mateo, CA, 1994. [16] A. S. Weigend, D. E. Rumelhart, and B. A. Huberman. Generalization by weightelimination with application to forecasting. In R. P. Lippmann, J. E. Moody, and D. S. Touretzky, editors, Advances in Neural Information Processing Systems 3, pages 875-882. San Mateo, CA: Morgan Kaufmann, 1991. [17] P. M. Williams. Bayesian regularisation and pruning using a Laplace prior. Technical report, School of Cognitive and Computing Sciences, University of Sussex, Falmer, Brighton, 1994. [18] D. H. Wolpert. Bayesian backpropagation over i-o functions rather than weights. In J. D. Cowan, G. Tesauro, and J. Alspector, editors, Advances in Neural Information Processing Systems 6, pages 200-207. San Mateo, CA: Morgan Kaufmann, 1994. 

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22 LEARNING ON A GENERAL NETWORK Amir F. Atiya Department of Electrical Engineering California Institute of Technology Ca 91125 Abstract This paper generalizes the backpropagation method to a general network containing feedback t;onnections. The network model considered consists of interconnected groups of neurons, where each group could be fully interconnected (it could have feedback connections, with possibly asymmetric weights), but no loops between the groups are allowed. A stochastic descent algorithm is applied, under a certain inequality constraint on each intra-group weight matrix which ensures for the network to possess a unique equilibrium state for every input. Introduction It has been shown in the last few years that large networks of interconnected "neuron" -like elemp.nts are quite suitable for performing a variety of computational and pattern recognition tasks. One of the well-known neural network models is the backpropagation model [1]-[4]. It is an elegant way for teaching a layered feedforward network by a set of given input/output examples. Neural network models having feedback connections, on the other hand, have also been devised (for example the Hopfield network [5]), and are shown to be quite successful in performing some computational tasks. It is important, though, to have a method for learning by examples for a feedback network, since this is a general way of design, and thus one can avoid using an ad hoc design method for each different computational task. The existence of feedback is expected to improve the computational abilities of a given network. This is because in feedback networks the state iterates until a stable state is reached. Thus processing is perforrr:.ed on several steps or recursions. This, in general allows more processing abilities than the "single step" feedforward case (note also the fact that a feedforward network is a special case of a feedback network). Therefore, in this work we consider the problem of developing a general learning algorithm for feedback networks. In developing a learning algorithm for feedback networks, one has to pay attention to the following (see Fig. 1 for an example of a configuration of a feedback network). The state of the network evolves in time until it goes to equilibrium, or possibly other types of behavior such as periodic or chaotic motion could occur. However, we are interested in having a steady and and fixed output for every input applied to the network. Therefore, we have the following two important requirements for the network. Beginning in any initial condition, the state should ultimately go to equilibrium. The other requirement is that we have to have a unique ? American Institute of Physics 1988 23 equilibrium state. It is in fact that equilibrium state that determines the final output. The objective of the learning algorithm is to adjust the parameters (weights) of the network in small steps, so as to move the unique equilibrium state in a way that will result finally in an output as close as possible to the required one (for each given input). The existence of more than op.e equilibrium state for a given input causes the following problems. In some iterations one might be updating the weights so as to move one of the equilibrium states in a sought direction, while in other iterations (especially with different input examples) a different equilibrium state is moved. Another important point is that when implementing the network (after the completion oflearning), for a fixed input there can be more than one possible output. Independently, other work appeared recently on training a feedback network [6],[7],[8]. Learning algorithms were developed, but solving the problem of ensuring a unique equilibrium was not considered. This problem is addressed in this paper and an appropriate network and a learning algorithm are proposed. neuron 1 outputs inputs Fig . 1 A recurrent network The Feedback Network Consider a group of n neurons which could be fully inter-connected (see Fig. 1 for an example). The weight matrix W can be asymmetric (as opposed to the Hopfield network). The inputs are also weighted before entering into the network (let V be the weight matrix). Let x and y be the input and output vectors respectively. In our model y is governed by the following set of differential equations, proposed by Hopfield [5]: du Tdj = Wf(u) - u + Vx, y = f(u) (1) 24 where f(u) = (J(ud, ... , f(un)f, T denotes the transpose operator, f is a bounded and differentiable function, and.,. is a positive constant. For a given input, we would like the network after a short transient period to give a steady and fixed output, no matter what the initial network state was. This means that beginning any initial condition, the state is to be attracted towards a unique equilibrium. This leads to looking for a condition on the matrix W. Theorem: A network (not necessarily symmetric) satisfying L L w'fi i < l/max(J')2, i exhibits no other behavior except going to a unique equilibrium for a given input. Proof : Let udt) and U2(t) be two solutions of (1). Let where " II is the two-norm. Differentiating J with respect to time, one obtains Using (1) , the expression becomes dJ(t) = --lluI(t) 2 -d- u2(t))11 2 t 1" 2 + -(uI(t) - U2(t)) T W [f( uI(t) ) - f ( uz(t) )] . .,. Using Schwarz's Inequality, we obtain Again, by Schwarz's Inequality, i = 1, ... ,n where Wi denotes the ith row of W. Using the mean value theorem, we get Ilf(udt)) - f(U2(t))II ~ (maxl!'I)IIUl(t) - uz(t)ll. Using (2),(3), and the expression for J(t), we get d~~t) ~ where -aJ(t) (4) (3) (2) 25 By hypothesis of the Theorem, a is strictly positive. Multiplying both sides of (4) by exp( at), the inequality results, from which we obtain J(t) ~ J(O)e- at . From that and from the fact that J is non-negative, it follows that J(t) goes to zero as t -+ <Xl. Therefore, any two solutions corresponding to any two initial conditions ultimately approach each other. To show that this asymptotic solution is in fact an equilibrium, one simply takes U2(t) = Ul(t + T), where T is a constant, and applies the above argument (that J(t) -+ 0 as t -+ <Xl), and hence Ul(t + T) -+ udt) as t -+ <Xl for any T, and this completes the proof. For example, if the function I is of the following widely used sigmoid-shaped form, 1 I(u) = l+e- u ' then the sum of the square of the weights should be less than 16. Note that for any function I, scaling does not have an effect on the overall results. We have to work in our updating scheme subject to the constraint given in the Theorem. In many cases where a large network is necessary, this constraint might be too restrictive. Therefore we propose a general network, which is explained in the next Section. The General Network We propose the following network (for an example refer to Fig. 2). The neurons are partitioned into several groups. Within each group there are no restrictions on the connections and therefore the group could be fully interconnected (i.e. it could have feedback connections) . The groups are connected to each other, but in a way that there are no loops. The inputs to the whole network can be connected to the inputs of any of the groups (each input can have several connections to several groups). The outputs of the whole network are taken to be the outputs (or part of the outputs) of a certain group, say group I. The constraint given in the Theorem is applied on each intra-group weight matrix separately. Let (qa, s"), a = 1, .. . , N be the input/output vector pairs of the function to be implemented. We would like to minimize the sum of the square error, given by a=l where M e" = I)y{ - si}2, i=l and yf is the output vector of group f upon giving input qa, and M is the dimension of vector s". The learning process is performed by feeding the input examples qU sequentially to the network, each time updating the weights in an attempt to minimize the error. 26 inputs J---V outputs Fig. 2 An example of a general network (each group represents a recurrent network) Now, consider a single group l. Let Wi be the intra-group weight matrix of group l, vrl be the matrix of weights between the outputs of group,. and the inputs of group l, and yl be the output vector of group I. Let the respective elements be w~i' V[~., and y~. Furthermore, let be the number of neurons of group l. Assume that the time constant l' is sufficiently small so as to allow the network to settle quickly to the equilibrium state, which is given by the solution of the equation n, yl = f(W'yl + L vrlyr) . (5) r?A I where A, is the set of the indices of the groups whose outputs a.re connected to the inputs of group ,. We would like each iteration to update the weight matrices Wi and vrl so as to move the equilibrium in a direction to decrease the error. We need therefore to know the change in the error produced by a small change in the weight matrices. Let .:;';, , and aa~~, denote the matrices whose (i, j)th element are :~'.' and ::~ respectively. Let ~ be the column vector '1 '1 :r whose ith element is ~. We obtain the following relations: uy. 8e a = 8W' 8e a 8V tl [A' _ (W')T] -1 8ea ( 8yl Y ')T , a ( r)T = [A' _ (W')T] -1 8e 8yl y , where A' is the diagonal matrix whose ith diagonal element is l/f'(Lk w!kY~ + LrLktJ[kyk) for a derivation refer to Appendix). The vector ~ associated with groUp l can be obtained in terms of the vectors ~, fEB" where B, is the set of the indices of the groups whose inputs are connected to the outputs of group ,. We get (refer to Appendix) 8e a 8yl = '" (V'i)T[Ai _ (Wi{r 1 8e".. ~ 8y3 JlBI (6) The matrix A' ~ (W')T for any group l can never be singular, so we will not face any problem in the updating process. To prove that, let z be a vector satisfying [A' - (W'f]z = o. 27 We can write zdmaxlf' I ~ LW~.Zk' i = I, ... , nl k where Zi is the ,"th element of z. Using Schwarz's Inequality, we obtain i = I, ... ,nl Squaring both sides and adding the inequalities for i = I, ... , nl, we get L/; ~ max(J')2(Lz~) LL(w~i)2. i k Since the condition (7) k LL(W!k)2 < I/max(J')2), k is enforced, it follows that (7) cannot be satisfied unless z is the zero vector. Thus, the matrix A' - (W')T cannot be singular. For each iteration we begin by updating the weights of group f (the group contammg the final outputs). For that group ~ equals simply 2(y{ - SI, ... , yf.t - SM, 0, ... , O)T). Then we move backwards to the groups connected to that group and obtain their corresponding vectors using (6), update the weights, and proceed in the same manner until we complete updating all the groups. Updating the weights is performed using the following stochastic descent algorithm for each group, !!J: 8e a t:. V = -a3 8V + a4 ea R , where R is a noise matrix whose elements are characterized by independent zero-mean unityvariance Gaussian densities, and the a's are parameters. The purpose of adding noise is to allow escaping local minima if one gets stuck in any of them. Note that the control parameter is taken to be ea. Hence the variance of the added noise tends to decrease the more we approach the ideal zero-error solution. This makes sense because for a large error, i.e. for an unsatisfactory solution, it pays more to add noise to the weight matrices in order to escape local minima. On the other hand, if the error is small, then we are possibly near the global minimum or to an acceptable solution, and hence we do not want too much noise in order not to be thrown out of that basin. Note that once we reach the ideal zero-error solution the added noise as well as the gradient of ea become zero for all a and hence the increments of the weight matrices become zero. If after a certain iteration W happens to violate the constraint Liiwlj ~ constant < I/max(J')2, then its elements are scaled so as to project it back onto the surface of the hypershere. Implementation Example A pattern recognition example is considered. Fig. 3 shows a set of two-dimensional training patterns from three classes. It is required to design a neural network recognizer with 28 three output neurons. Each of the neurons should be on if a sample of the corresponding class is presented, and off otherwise, i.e. we would like to design a "winner-take-all" network. A singlelayer three neuron feedback network is implemented. We obtained 3.3% error. Performing the same experiment on a feedforward single-layer network with three neurons, we obtained 20% error. For satisfactory results, a feedforward network should be two-layer. With one neuron in the first layer and three in the second layer, we got 36.7% error. Finally, with two neurons in the first layer and three in the second layer, we got a match with the feedback case, with 3.3% error . z z z z z z z z z z z z z z z z zil 1 33 3 3 3 1 3 3 33 3 3 3 ~ 3 3 3 3 3 3 Fig. 3 A pattern recognition example Conclusion A way to extend the backpropagation method to feedback networks has been proposed . A condition on the weight matrix is obtained, to insure having only one fixed point, so as to prevent having more than one possible output for a fixed input. A general structure for networks is presented, in which the network consists of a number of feedback groups connected to each other in a feedforward manner. A stochastic descent rule is used to update the weights. The lJ!ethod is applied to a pattern recognition example. With a single-layer feedback network it obtained good results. On the other hand, the feedforward backpropagation method achieved good resuls only for the case of more than one layer, hence also with a larger number of neurons than the feedback case. 29 Acknow ledgement The author would like to gratefully acknowledge Dr . Y. Abu-Mostafa for the useful discussions. This work is supported by Air Force Office of Scientific Research under Grant AFOSR-86-0296. Appendix Differentiating (5), one obtains aI - Yj a I = w kp aI k,p = 1, ... ,n, I Ym p jk , f '(')(,,", Zj L..,Wjm-a I +Y'6) w kp m where if j = k otherwise, and We can write a~' = (A' _ Wi) -lbkz> (A - 1) aw kp where b kp is the nt-dimensional vector whose b~l> = {y~ ? 0 ith component is given by ifi = k otherwise. By the chain rule, aea _ ""' ae a ay; -a I -L..,-aI - a I' w kp j Yj w kp which, upon substituting from (A - 1), can be put in the form y!,gk~' where gk is the column of (A' - Wt)-l. Finally, we obtain the required expression, which is ae" = [At _ (WI)T] aw' -1 ae" ( ,)T ayl y . Regarding a()~~I' it is obtained by differentiating (5) with respect to vr~,. We get similarly where C kl' is the nt-dimensional vector whose ith component is given by if i = k otherwise. kth 30 A derivation very similar to the case of :~l results in the following required expression: Be a = BVrl 8 8 [A' _ (w,)T] -1 Be a ( r)T. By' y 8yJ j Now, finally consider ~. Let ~, jf.B, be the matrix whose (k,p)th element is ~. The elements of ~ can be obtained by differentiating the equation for the fixed point for group . uy J, as follows, Hence, :~~. = (Ai - Wi) -IV'i. (A - 2) Using the chain rule, one can write Be a By' ?T = ~(ByJ) ~ Byl JEEr Be a By;' We substitute from (A - 2) into the previous equation to complete the derivation by obtaining References 111 P. Werbos, "Beyond regression: New tools for prediction and analysis in behavioral sciences", Harvard University dissertation, 1974. [21 D. Parker, "Learning logic", MIT Tech Report TR-47, Center for Computational Research in Economics and Management Science, 1985. [31 Y. Le Cun, "A learning scheme for asymmetric threshold network", Proceedings of Cognitiva, Paris, June 1985. [41 D. Rumelhart, G.Hinton, and R. Williams, "Learning internal representations by error propagation", in D. Rumelhart, J. McLelland and the PDP research group (Eds.), Parallel distributed processing: Explorations in the microstructure of cognition, Vol. 1, MIT Press, Cambridge, MA, 1986. 151 J. Hopfield, "Neurons with graded response have collective computational properties like those of two-state neurons", Proc. N atl. Acad. Sci. USA, May 1984. [61 L. Ahneida, " A learning rule for asynchronous perceptrons with feedback in a combinatorial environment", Proc. of the First Int. Annual Conf. on Neural Networks, San Diego, June 1987. [71 R. Rohwer, and B. Forrest, "Training time-dependence in neural networks", Proc. of the First Int. Annual Conf. on Neural Networks, San Diego, June 1987. [81 F. Pineda, "Generalization of back-propagation to recurrent neural networks", Phys. Rev. Lett., vol. 59, no. 19, 9 Nov. 1987. 

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584 PHASOR NEURAL NETVORKS Andr~ J. Noest, N.I.B.R., NL-ll0S AZ Amsterdam, The Netherlands. ABSTRACT A novel network type is introduced which uses unit-length 2-vectors for local variables. As an example of its applications, associative memory nets are defined and their performance analyzed. Real systems corresponding to such 'phasor' models can be e.g. (neuro)biological networks of limit-cycle oscillators or optical resonators that have a hologram in their feedback path. INTRODUCTION Most neural network models use either binary local variables or scalars combined with sigmoidal nonlinearities. Rather awkward coding schemes have to be invoked if one wants to maintain linear relations between the local signals being processed in e.g. associative memory networks, since the nonlinearities necessary for any nontrivial computation act directly on the range of values assumed by the local variables. In addition, there is the problem of representing signals that take values from a space with a different topology, e.g. that of the circle, sphere, torus, etc. Practical examples of such a signal are the orientations of edges or the directions of local optic flow in images, or ~he phase of a set of (sound or EM) waves as they arrive on an array of detectors. Apart from the fact that 'circular' signals occur in technical as well as biological systems, there are indications that some parts of the brain (e.g. olfactory bulb, cf. Dr.B.Baird's contribution to these proceedings) can use limit-cycle oscillators formed by local feedback circuits as functional building blocks, even for signals without circular symmetry. Vith respect to technical implementations, I had speculated before the conference whether it could be useful to code information in the phase of the beams of optical neurocomputers, avoiding slow optical switching elements and using only (saturating) optical amplification and a ? American Institute of Physics 1988 585 hologram encoding the (complex) 'synaptic' weight factors. At the conference, I learnt that Prof. Dana Anderson had independently developed an optical device (cf. these proceedings) that basically works this way, at least in the slow-evolution limit of the dynamic hologram. Hopefully, some of the theory that I present here can be applied to his experiment. In turn, such implementations call for interesting extensions of the present models. BASIC ELEMENTS OF GENERAL PHASOR NETVORKS Here I study the perhaps simplest non-scalar network by using unitlength 2-vectors (phasors) as continuous local variables. The signals processed by the network are represented in the relative phaseangles. Thus, the nonlinearities (unit-length 'clipping') act orthogonally to the range of the variables coding the information. The behavior of the network is invariant under any rigid rotation of the complete set of phasors, representing an arbitrary choice of a global reference I phase. Statistical physicists will recognize the phasor model as a generalization of 02-spin models to include vector-valued couplings. All 2-vectors are treated algebraically as complex numbers, writing x for Ixl for the length, Ixl for the phase-angle, and the complex conjugate of a 2-vector x. A phasor network then consists of N?l phasors s. , with Is.l=l, 1 interacting via couplings c .. , with 1J C .. 11 1 = O. The c 1J .. are allowed to be complex-valued quantities. For optical implementations this is clearly a natural choice, but it may seem less so for biological systems. However, if the coupling between two limitcycle oscillators with frequency f is mediated via a path having propagationdelay d, then that coupling in fact acquires a phaseshift of f.d.2~ radians. Thus, complex couplings can represent such systems more faithfully than the usual models which neglect propagationdelays altogether. Only 2-point couplings are treated here, but multi-point couplings c.1)'k' etc., can be treated similarly. The dynamics of each phasor depends only on its local field h.= 1 !z:4~ c 1J .. s. J J + n. 1 where z is the number of inputs 586 c .. ~O per cell and n. is a local noise term (complex and Gaussian). 1J 1 Various dynamics are possible, and yield largely similar results: Continuous-time, parallel evolution: ("type A") d (/s./) = Ih. l.sin(/h.1 - Is./) (IT 1 1 1 1 s.(t+dt)= h.1 Ih. I , either serially in Discrete-time updating: 1 1 1 random i-sequence ("type B"), or in parallel for all i ("type C"). The natural time scale for type-B dynamics is obtained by scaling the discrete time-interval eft as ,.., liN ; type-C dynamics has cl't=l. LYAPUNOV FUNCTION (alias "ENERGY", or "HAMILTONIAN" ) If one limits the attention temporarily to purely deterministic (n.=O) models, then the question suggests itself whether a class of 1 couplings exists for which one can easily find a Lyapunov function i.e. a function of the network variables that is monotonic under the dynamics. A well-known example 1 is the 'energy' of the binary and scalar Hopfield models with symmetric interactions. It turns out that a very similar function exists for phasor networks with type-A or B dynamics and a Hermitian matrix of couplings. -H = L ? 1 Hermiticity (c .. 1J 5.1 h. 1 = =c .. ) makes J 1 (lIz) L 5.1 c 1J .. s. J ? . 1,J H real-valued and non-increasing in time. This can be shown as follows, e.g. for the serial dynamics (type B). Suppose, without loss of generality, that phasor i=l is updated. Then -z H Ls. c ' l sl 1>1 + sl' 2: c ' 1 5. i>l = + 1 z 51 h1 + 1 1 I.I: i ,j>l -s. 1 c .. s. 1J J + constant. 1 Vith Hermitian couplings, H becomes real-valued, and one also has l:c 1 ? i>l I:c' 5. i>1 1 l 1 Thus, - H - constant 1 s. 1 = z h1 . 51 h1 + sl h1 = 2 Re(sl h 1 ) Clearly, H is minimized with respect to sl by sl(t+1) = hll Ih11 ? Type-A dynamics has the same Lyapunovian, but type C is more complex. = The existence of Hermitian interactions and the corresponding energy function simplifies greatly the understanding and design of phasor networks, although non-Hermitian networks can still have a Lyapunov- 587 function, and even networks for which such a function is not readily found can be useful, as will be illustrated later. AN APPLICATION: ASSOCIATIVE MEMORY. A large class of collective computations, such as optimisations and content-addressable memory, can be realised with networks having an energy function. The basic idea is to define the relevant penalty function over the solution-space in the form of the generic 'energy' of the net, and simply let the network relax to minima of this energy. As a simple example, consider an associative memory built within the framework of Hermitian phasor networks. In order to store a set of patterns in the network, i.e. to make a set of special states (at least approximatively) into attractive fixed points of the dynamics, one needs to choose an appropriate set of couplings. One particularly simple way of doing this is via the phasor-analog of "Hebb's rule" (note the Hermiticity) p s(.k). s-(.k), h c .. = were s.(k).IS phasor 1. .In I earne d pattern k . IJ rk 1 J 1 The rule is understood to apply only to the input-sets 'i of each i. Such couplings should be realisable as holograms in optical networks, but they may seem unrealistic in the context of biological networks of oscillators since the phase-shift (e.g. corresponding to a delay) of a connection may not be changeable at will. However, the required coupling can still be implemented naturally if e.g. a few paths with different fixed delays exist between pairs of cells. The synaps in each path then simply becomes the projection of the complex coupling on the direction given by the phase of its path, i.e. it is just a classical Hebb-synapse that computes the correlation of its pre- and post-synaptic (imposed) signals, which now are phase-shifted versions of the phasors s~~)The required complex c .. are then realised as the 1 IJ vector sum over at least two signals arriving via distinct paths with corresponding phase-shift and real-valued synaps. Two paths suffice if they have orthogonal phase-shifts, but random phases will do as well if there are a reasonable number of paths. Ve need to have a concise way of expressing how 'near' any state of the net is to one or more of the stored patterns. A natural way 588 of doing this is via a set of p order parameters called "overlaps" N -(k) 1 s 1.. s.1 ? N 11: 1 I ; 1 Note the constraint on the p overlaps P I < k -< p - ? 2 Mk ~ 1 if all the patterns k are orthogonal, or merely random in the limit N-.QO. This will be assumed from now on. Also, one sees at once that the whole behaviour of the network does not depend on any rigid rotation of all phasors over some angle since H, Mk , c .. and the dynamics are invariant under 1J multiplication of all s. by a fixed phasor : s~ = S.s. with ISI=1. I I I Let us find the performance at low loading: N,p,z .. oo, with p/z.. O and zero local noise. Also assume an initial overlap m)O with only one pattern, say with k=1. Then the local field is hi 1 s~1~ hP~ Z 1 1 and h.* 1 f s~k) ~s .. k s~k~ 1 J J j' i 1 = -z I: sP~s. jl'i = J J (1) = m1 . si ? S ~ fs~k). L: s~k~s. z k=2 1 j(~i J J h(1) i + + O(1//Z) h71 , with O( ./( p-l) Iz') where S~f(i);ISI=1, . Thus, perfect recall (M 1=1) occurs in one 'pass' at loadings p/z ... O. EXACTLY SOLVABLE CASE: SPARSE and ASYMMETRIC couplings Although it would be interesting to develop the full thermodynamics of Hermitian phasor networks with p and z of order N (analogous to the analysis of the finite-T Hopfield model by the teams of Amit 2 and van Hemmen 3 ), I will analyse here instead a model with sparse, asymmetric connectivity, which has the great advantages of being exactly solvable with relative ease, and of being arguably more realistic biologically and more easily scalable technologically. In neurobiological networks a cell has up to z;10 4 asymmetric connections, whereas N;101~ This probably has the same reason as applies to most VLSI chips, namely to alleviate wiring problems. For my present purposes, the theoretical advantage of getting some exact results is of primary interest 4 Suppose each cell has z incoming connections from randomly selected other cells. The state of each cell at time t depends on at most zt . cells at time t=O. Thus, If z t ?N 112 and N large, then the respective 589 4 trees of 'ancestors' of any pair cells have no cells in common. In x particular, if z_ (logN) , for any finite x, then there are no common ancestors for any finite time t in the limit N-.OO. For fundamental information-theoretic reasons, one can hope to be able to store p patterns with p at most of order z for any sort of 2-point couplings. Important questions to be settled are: Yhat are the accuracy and speed of the recall process, and how large are the basins of the attractors representing recalled patterns? Take again initial conditions (t=O) with, say, m(t)= Hl > H>l = O. Allowing again local random Gaussian (complex) noise n., the local ? ld s become, In . now f amI'1'Iar notatIon, . h .= h(l) 1 n .? f Ie . + h*. + 1 1 1 1 As in the previous section, the h~l)term consists of the 'signal' 1 m(t).s. (modulo the rigid rotation S) and a random term of variance * 1 at most liz. For p _ z, the h. term becomes important. Being sums of * 1 z(p-1) phasors oriented randomly relative to the signal, the h. are 1 independent Gaussian zero-mean 2-vectors with variance (p-1)/z , as p,z and N.. oo . Finally, let the local noises n.1 have variance r2. Then the distribution of the s.(t+l) phasors can be found in terms of 1 * .? 2 the signal met) and the total variance a=(p/z)+r of the random h.+n 1 1 After somewhat tedious algebraic manipulations (to be reported in detail elsewhere) one obtains the dynamic behaviour of met) m(t+1) = F(m(t),a) for discrete parallel (type-C) dynamics, and d met) = F(m(t),a) - met) for type-A or type-B dynamics , Tt where the function m F(m,a) = +" 2 Idx.(1+cos2x).expl-(m.sinx) la].(l+erfl(m.cosx)/~) -1'C The attractive fixed points H* (a)= F(H * ,a) represent the retrieval accuracy when the loading-pIus-noise factor equals a. See figure 1. For a?l one obtains the expansion 1-H* (a) = a/4 + 3a 2 132 + O(a 3 ). The recall solutions vanish continuously as H*_(a -a) 112 at a =tc/4. c c One also obtains (at any t) the distribution of the phase scatter of the phasors around the ideal values occurring in the stored pattern. 590 P(/u./) 1 = (1/2n).exp(-m 2 /a).(1+I1t.L.exp(L 2 ).(1+erf(L? -(k) , and L = (m/la).cos(/u./) 1 where u.= s. s. 111 , (modulo S). Useful approximations for the high, respectively low M regimes are: M ?ra: PUu./) 1 (MIl'a1l).exp[-(M./u./)2 /a ] 1 ; I/u./1 ?"XI2 1 M ? f i : PUu./) = (1I21t).(1+L ?./;l) 1 Figure ~ RETRIEVAL-ERROR and BASIN OF ATTRACTION versus LOADING + NOISE. Q Q Q en Q Q .,; Q ".,; Q I: UI ..,) Q -C0 a. 1:) CD x - c- Q Q . Q Q Q '"Q 0 Q 0 Q 0 c: "'0.00 0.10 0.20 O. 30 0 ? 40 a 0 ? 50 = p/z 0 ? 60 + r-r O. 70 0 ? 80 0 ? 90 1. 00 591 DISCUSSION It has been shown that the usual binary or scalar neural networks can be generalized to phasor networks, and that the general structure of the theoretical analysis for their use as associative memories can be extended accordingly. This suggests that many of the other useful applications of neural nets (back-prop, etcJ can also be generalized to a phasor setting. This may be of interest both from the point of view of solving problems naturally posed in such a setting, as well as from that of enabling a wider range of physical implementations, such as networks of limit-cycle oscillators, phase-encoded optics, or maybe even Josephson-junctions. The performance of phasor networks turns out to be roughly similar to that of the scalar systems; the maximum capacity p/z=~/4 for phasor nets is slightly larger than its value 2/n for binary nets, but there is a seemingly faster growth of the recall error 1-M at small a (linear for phasors, against exp(-1/(2a? for binary nets). However, the latter measures cannot be compared directly since they stem from quite different order parameters. If one reduces recalled phasor patterns to binary information, performance is again similar. Finally, the present methods and results suggest several roads to further generalizations, some of which may be relevant with respect to natural or technical implementations. The first class of these involves local variables ranging over the k-sphere with k>l. The other generalizations involve breaking the O(n) (here n=2) symmetry of the system, either by forcing the variables to discrete positions on the circle (k-sphere), and/or by taking the interactions between two variables to be a more general function of the angular distance between them. Such models are now under development. REFERENCES 1. J.J.Hopfield, Proc.Nat.Acad.Sci.USA 79, 2554 (1982) and idem, Proc.Nat.Acad.Sci.USA 81, 3088 (1984). 2. D.J.Amit, H.Gutfreund and H.Sompolinski, Ann.Phys. 173, 30 (1987). 3. D.Grensing, R.Kuhn and J.L. van Hemmen, J.Phys.A 20, 2935 (1987). 4. B.Derrida, E.Gardner and A.Zippelius, Europhys.Lett. 4, 167 (1987) 

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Active Learning for Function Approximation Kah Kay Sung (sung@ai.mit.edu) Massachusetts Institute of Technology Artificial Intelligence Laboratory 545 Technology Square Cambridge, MA 02139 Partha Niyogi (pn@ai.mit.edu) Massachusetts Institute of Technology Artificial Intelligence Laboratory 545 Technology Square Cambridge, MA 02139 Abstract We develop a principled strategy to sample a function optimally for function approximation tasks within a Bayesian framework. Using ideas from optimal experiment design, we introduce an objective function (incorporating both bias and variance) to measure the degree of approximation, and the potential utility of the data points towards optimizing this objective. We show how the general strategy can be used to derive precise algorithms to select data for two cases: learning unit step functions and polynomial functions. In particular, we investigate whether such active algorithms can learn the target with fewer examples. We obtain theoretical and empirical results to suggest that this is the case. 1 INTRODUCTION AND MOTIVATION Learning from examples is a common supervised learning paradigm that hypothe- sizes a target concept given a stream of training examples that describes the concept. In function approximation, example-based learning can be formulated as synthesizing an approximation function for data sampled from an unknown target function (Poggio and Girosi, 1990). Active learning describes a class of example-based learning paradigms that seeks out new training examples from specific regions of the input space, instead of passively accepting examples from some data generating source. By judiciously selecting ex- 594 Kah Kay Sung, Parlha Niyogi amples instead of allowing for possible random sampling, active learning techniques can conceivably have faster learning rates and better approximation results than passive learning methods. This paper presents a Bayesian formulation for active learning within the function approximation framework. Specifically, here is the problem we want to address: Let Dn {(Xi, Yi)li 1, ... , n} be a set of n data points sampled from an unknown target function g, possibly in the presence of noise. Given an approximation function concept class, :F, where each f E :F has prior probability P;:-[J], one can use regularization techniques to approximate 9 from Dn (in the Bayes optimal sense) by means of a function 9 E:F. We want a strategy to determine at what input location one should sample the next data point, (XN+l, YN+d, in order to obtain the "best" possible Bayes optimal approximation of the unknown target function 9 with our concept class :F. = = The data sampling problem consists of two parts: 1) Defining what we mean by the "best" possible Bayes optimal approximation of an unknown target function. In this paper, we propose an optimality criterion for evaluating the "goodness" of a solution with respect to an unknown target function. 2) Formalizing precisely the task of determining where in input space to sample the next data point. We express the above mentioned optimality criterion as a cost function to be minimized, and the task of choosing the next sample as one of minimizing the cost function with respect to the input space location of the next sample point. Earlier work (Cohn, 1991; MacKay, 1992) have tried to use similar optimal experiment design (Fedorov, 1972) techniques to collect data that would provide maximum information about the target function. Our work differs from theirs in several respects. First, we use a different, and perhaps more general, optimality criterion for evaluating solutions to an unknown target function, based on a measure of function uncertainty that incorporates both bias and variance components of the total output generalization error. In contrast, MacKay and Cohn use only variance components in model parameter space. Second, we address the important sample complexity question, i.e., does the active strategy require fewer examples to learn the target to the same degree of uncertainty? Our results are stated in PAC-style (Valiant, 1984). After completion of this work, we learnt that Sollich (1994) had also recently developed a similar formulation to ours. His analysis is conducted in a statistical physics framework. The rest of the paper is organized as follows: Section 2, develops our active sampling paradigm. In Sections 3 and 4, we consider two classes offunctions for which active strategies are obtained, and investigate their performance both theoretically and empirically. 2 THE MATHEMATICAL FRAMEWORK In order to optimally select examples for a learning task, one should first have a clear notion of what an "ideal" learning goal is for the task. We can then measure an example's utility in terms of how well the example helps the learner achieve the Active Learning for Function Approximation 595 goal, and devise an active sampling strategy that selects examples with maximum potential utility. In this section, we propose one such learning goal - to find an approximation function g E :F that "best" estimates the unknown target function g. We then derive an example utility cost function for the goal and finally present a general procedure for selecting examples. 2.1 EVALUATING A SOLUTION TO AN UNKNOWN TARGET THE EXPECTED INTEGRATED SQUARED DIFFERENCE Let 9 be the target function that we want to estimate by means of an approximation function 9 E :F. If the target function 9 were known, then one natural measure of how well (or badly) g approximates 9 would be the Integrated Squared Difference (ISD) of the two functions: 8(g,g) = 1x", (g(x) - g(x))2dx. (1) Xlo In most function approximation tasks, the target 9 is unknown, so we clearly cannot express the quality of a learning result, g, in terms of g. We can, however, obtain an expected integrated squared difference (EISD) between the unknown target, g, and its estimate, g, by treating the unknown target 9 as a random variable from the approximation function concept class:F. Taking into account the n data points, D n , seen so far, we have the following a-posteriori likelihood for g: P(gIDn) ex PF[g]P(Dnlg). The expected integrated squared difference (EISD) between an unknown target, g, and its estimate, g, given D n , is thus: EF[8(g,g)IDn] = 2.2 f P(gIDn)8(g,g)dg 19EF =f PF[g]P(D n lg)8(y,g)dg. (2) 19EF SELECTING THE NEXT SAMPLE LOCATION We can now express our learning goal as minimizing the expected integrated squared difference (EISD) between the unknown target 9 and its estimate g. A reasonable sampling strategy would be to choose the next example from the input location that minimizes the EISD between g and the new estimate gn+l. How does one predict the new EISD that results from sampling the next data point at location Xn+l ? Suppose we also know the target output value (possibly noisy), Yn+l, at Xn+l. The EISD between 9 and its new estimate gn+l would then be EF[8(gn+l, g)IDn U (x n+l,Yn+d]' where gn+l can be recovered from Dn U (X n+l,Yn+l) via regularization. In reality, we do not know Yn+l, but we can derive for it the following conditional probability distribution: P(Yn+llxn+l,Dn) ex f P(Dn U(Xn+l,Yn+dlf)PF[J]df. (3) liEF This leads to the following expected value for the new EISD, if we sample our next data point at X n +l: U(Yn+lIDn , xn+1) = 1: P(Yn+lIXn+1' Dn)EF[8(gn+1' g)IDn U (Xn+b Yn+l)]dYn+1' ~l Clearly, the optimal input location to sample next is the location that minimizes cost function in Equation 4 (henceforth referred to as the total output uncertainty), l.e., (5) 596 Kah Kay Sung, Partha Niyogi 2.3 SUMMARY OF ACTIVE LEARNING PROCEDURE We summarize the key steps involved in finding the optimal next sample location: 1) Compute P(gIDn). This is the a-posteriori likelihood of the different functions 9 given Dn , the n data points seen so far. 2) Fix a new point Xn+l to sample. 3) Assume a value Yn+l for this Xn+l . One can compute P(glDn U (Xn+l' Yn+d) and hence the expected integrated squared difference between the target and its new estimate. This is given by EF[8(Un+l, g)IDn U (Xn+l' Yn+t>]. See also Equation 2. 4) At the given Xn+l, Yn+l has a probability distribution given by Equation 3. Averaging over all Yn+l 's, we obtain the total output uncertainty for Xn+l, given by U(Un+lIDn, xn+t> in Equation 4. 5) Sample at the input location that minimizes the total output uncertainty cost function. 3 EXAMPLE 1: UNIT STEP FUNCTIONS To demonstrate the usefulness of the above procedure, let us first consider the following simple class of indicator functions parameterized by a single parameter a which takes values in [0,1]. Thus = :F {I[a,l]IO ~ a ~ I} We obtain a prior peg = l[a,l]? by assuming that a has an a-priori uniform distribution on [0,1]. Assume that data, Dn = {(Xi;Yi); i 1, .. n} consistent with some unknown target function l[at,l] (which the learner is to approximate) has been obtained. We are interested in choosing a point x E [0,1] to sample which will provide us with maximal information. Following the general procedure outlined above we go through the following steps. = For ease of notation, let x R be the right most point belonging to Dn whose Y value maXi=l, ..n{xdYi = OJ. Similarly let XL millj=l, .. n{Xs!Yi I} and is 0, i.e., XR let XL - XR = W. = = = 1) We first need to get P(gIDn ). It is easy to show that P(g = l[a,l]IDn) =~ if a E [XR' XL]; 0 otherwise 2) Suppose we sample next at a particular distribution P(y = OIDn , x) = (XL - x) XL - XR = (XL - X E [0,1]' we would obtain Y with the x) if x E [XR' XL]; 1 if x ~ XR; 0 otherwise W = For a particular y, the new data set would be Dn+l Dn U (x, y) and the corresponding EISD can be easily obtained using the distribution P(gIDn+d. Averaging this over P(YIDn, x) as in step 4 of the general procedure, we obtain if x ~ x R or x otherwise ~ XL 597 Active Learning for Function Approximation Clearly the point which minimizes the expected total output uncertainty is the midpoint of XL and XR. . Xn+1 arg mm U(gIDn, x) (XL + xR)/2 = = XE[O,l] Thus applying the general procedure to this special case reduces to a binary search learning algorithm which queries the midpoint of XR and XL. An interesting question at this stage is whether such a strategy provably reduces the sample complexity; and if so, by how much. It is possible to. prove the following theorem which shows that for a certain pre-decided total output uncertainty value, the active learning algorithm takes fewer examples to learn the target to the same degree of total output uncertainty than a random drawing of examples according to a uniform distribution. Theorem 1 Suppose we want to collect examples so that we are guaranteed with high probability (i. e. probability> 1 - 6) that the total output uncertainty is less than f. Then a passive learner would require at least ~ In(I/6) examples while y(48f) the active strategy described earlier would require at most (1/2) In(1 / 12f) examples. 4 . EXAMPLE 2: THE CASE OF POLYNOMIALS In this section we turn our attention to a class of univariate polynomials (from [-5,5] to ~) of maximum degree K, i.e., K :F = {g(ao, ... , aK) = L: aixi} i:=O As before, the prior on :F is obtained here by assuming a prior on the parameters; in particular we assume that a = (ao, aI, ... , aK) has a multivariate normal distribution N(O, S). For simplicity, it is assumed that the parameters are independent, i.e., S is a diagonal matrix with Si,i = (7;' In this example we also incorporate noise (distributed normally according to N(O, (72)). As before, there is a target gt E G which the learner is to approximate on the basis of data. Suppose the learner is in possession of a data set Dn = {(Xi, Yi = gt(Xi) + 1]); i = 1 ... n} and is to receive another data point. The two options are 1) to sample the function at a point X according to a uniform distribution on the domain [-5,5] (passive learning) and 2) follow our principled active learning strategy to select the next point to be sampled. 4.1 ACTIVE STRATEGY Here we derive an exact expression for Xn+1 (the next query point) by applying the general procedure described earlier. Going through the steps as before, 1)It is possible to show that p(g(a)IDn ) = P(aIDn ) is again a multivariate normal distribution N(J1., 1: n ) where J1. L~l Yixi, Xi (1, Xi, xl, . .. , xf)T and 1 n T ) 1 1:S-l + _ " (x.x. n 2(72 L..J 1 1 = = = i:=l 2)Computation of the total output uncertainty U(gn+1IDn, x) requires several steps. Taking advantage of the Gaussian distribution on both the parameters a and the noise, we obtain (see Niyogi and Sung, 1995 for details): U(gIDn, x) = l1:n+1AI Kah Kay Sung. Partha Niyogi 598 20 20 i\ Noise=0.1 15 to/ I~J \ A "\ 10 15 0\ I'\\J\l r\ \ \.j\ 10 '\ "'- 5 0 Noise=1.0 \ J 50 10". 'v.J\ 5 0 50 Figure 1: Comparing active and passive learning average error rates at different sample noise levels. The two graphs above plot log error rates against number of samples. See text for detailed explanation. The solid and dashed curves are the active and passive learning error rates respectively. where A is a matrix of numbers whose i,j element is J~5 t(i+i- 2)dt . En+l has the same form as En and depends on the previous data, the priors, noise and Xn+l. When minimized over Xn+l, we get xn +! as the maximum utility location where the active learner should next sample the unknown target function. 4.2 SIMULATIONS We have performed several simulations to compare the performance of the active strategy developed in the previous section to that of a passive learner (who receives examples according to a uniform random distribution on the domain [-5,5]). The following issues have been investigated. 1) Average error rate as a function of the number of examples: Is it indeed the case that the active strategy has superior error performance for the same number of examples? To investigate this we generated 1000 test target polynomial functions (of maximum degree 9) according ~o the following Gaussian prior on the parameters: for each ai,P(ai) = N(0,0.9'). For each target polynomial, we collected data according to the active strategy as well as the passive (random) strategy for varying number of data points. Figure 1 shows the average error rate (i .e., the integrated squared difference between the actual target function and its estimate, averaged over the 1000 different target polynomials) as a function of the number of data points. Notice that the active strategy has a lower error rate than the passive for the same number of examples and is particularly true for small number of data. The active strategy uses the same priors that generate the test target functions. We show results of the same simulation performed at two noise levels (noise standard deviation 0.1 and 1.0). In both cases the active strategy outperforms the passive learner indicating robustness in the face of noise. 2) Incorrect priors: How sensitive is the active learner to possible differences between its prior assumptions on the class :F and the true priors? We repeated the function learning task of the earlier case with the test targets generated in the same way as before. The active learner assumes a slightly different Gaussian prior and polynomial degree from the target (Std(ai) 0.7i and K 7 for the active learner versus Std(ad = O.Si and K = S for the target). Despite its inaccurate priors, the = = 599 Active Learning for Function Approximation 8.5 r--------r----r---~--...---___, OJ 8 1tS ~7.5 o Y' '- Different Gauss Prior & Deg. \ \ "- ~ ~ w 7 C) o -6.5 -- '\ ""' ....... ........ --... ....... -- -- 6~--~~--~----~----~----~ o 10 20 30 40 50 Figure 2: Active lea.rning results with different Gaussian priors for coefficients, and a lower a priori polynomial degree K. See text for detailed explanation. The solid and dashed curves a.re the active a.nd passive learning error rates respectively. active learner outperforms the passive case. 3) The distribution of points: How does the active learner choose to sample the domain for maximally reducing uncertainty? There are a few sampling trends which are noteworthy here. First, the learner does not simply sample the domain on a uniform grid. Instead it chooses to cluster its samples typically around K + 1 locations for concept classes with maximum degree K as borne out by simulations where K varies from 5 to 9. One possible explanation for this is it takes only K + 1 points to determine the target in the absence of noise. Second, as the noise increases, although the number of clusters remains fixed, they tend to be distributed away from the origin. It seems that for higher noise levels, there is less pressure to fit the data closely; consequently the prior assumption of lower order polynomials dominates. For such lower order polynomials, it is profitable to sample away from the origin as it reduces the variance of the resulting fit. (Note the case of linear regression) . Remarks 1) Notice that because the class of polynomials is linear in its model parameters, a, the new sample location (xn+d does not depend on the y values actually observed but only on the x values sampled . Thus if the learner is to collect n data points, it can pre-compute the n points at which to sample from the start. In this sense the active algorithm is not really adaptive. This behavior has also been observed by MacKay (1992) and Sollich (1994) . 2) Needless to say, the general framework from optimal design can be used for any function class within a Bayesian framework. We are currently investigating the possibility of developing active strategies for Radial Basis Function networks. While it is possible to compute exact expressions for Xn+l for such RBF networks with fixed centers, for the case of moving centers, one has to resort to numerical minimization. For lack of space we do not include those results in this paper. 600 Kah Kay Sung, Partha Niyogi 10 (DO 0 00 (D CJX:? 0 o 2 8 o 00 (I) 0 0 00 6 1 0 0 o~~~~----~~~~ -5 o 5 4 -5 0 0 00 0 0 0 5 Figure 3: Distribution of active learning sample points as a function of (i) noise strength and (ii) a-priori polynomial degree. The horizontal axis of both graphs represents the input space [-5,5]. Each circle indicates a sample location. The Left graph shows the distribution of sample locations (on x axis) for different noise level (indicated on y-axis). The Right graph shows the distribution of sample locations (on x-axis) for different assumptions on the maximum polynomial degree K (indicated on y-axis) . 5 CONCLUSIONS We have developed a Bayesian framework for active learning using ideas from optimal experiment design. Our focus has been to investigate the possibility of improved sample complexity using such active learning schemes. For a simple case of unit step functions, we are able to derive a binary search algorithm from a completely different standpoint. Such an algorithm then provably requires fewer examples for the same error rate. We then show how to derive specific algorithms for the case of polynomials and carry out extensive simulations to compare their performance against the benchmark of a passive learner with encouraging results. This is an application of the optimal design paradigm to function learning and seems to bear promise for the design of more efficient learning algorithms. References D. Cohn. (1991) A Local Approach to Optimal Queries. In D. Touretzky (ed.), Proc. of 1990 Connectionist Summer School, San Mateo, CA , 1991. Morgan Kaufmann Publishers. V. Fedorov. (1972) Theory of Optimal Experiments. Academic Press, New York, 1972. D. MacKay. (1992) Bayesian Methods for Adaptive Models. PhD thesis, CalTech, 1992. P. Niyogi and K. Sung. (1995) Active Learning for Function Approximation: Paradigms from Optimal Experiment Design. Tech Report AIM-1483, AI Lab. , MIT, In Preparation. M. Plutowski and H. White. (1991) Active Selection of Training Examples for Network Learning in Noiseless Environments. Tech Report CS91-180, Dept. of Computer Science and Engineering, University of California, San Diego, 1991. T. Poggio and F. Girosi. (1990) Regularization Algorithms for Learning that are Equivalent to Multilayer Networks. Science, 247:978-982, 1990. P. Sollich. (1994) Query Construction, Entropy, Generalization in Neural Network Models. Physical Review E, 49 :4637-4651, 1994. 1. Valiant. (1984) A Theory of Learnable. Proc. of the 1984 STOC, p436-445, 1984. 

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Higher Order Statistical Decorrelation without Information Loss Gustavo Deco SiemensAG Central Research Otto-Hahn-Ring 6 81739 Munich GeIIDany Wilfried Brauer Technische UniversiUit MUnchen Institut fur InfoIIDatik Arcisstr. 21 80290 Munich GeIIDany Abstract A neural network learning paradigm based on information theory is proposed as a way to perform in an unsupervised fashion, redundancy reduction among the elements of the output layer without loss of information from the sensory input. The model developed performs nonlinear decorrelation up to higher orders of the cumulant tensors and results in probabilistic ally independent components of the output layer. This means that we don't need to assume Gaussian distribution neither at the input nor at the output. The theory presented is related to the unsupervised-learning theory of Barlow, which proposes redundancy reduction as the goal of cognition. When nonlinear units are used nonlinear principal component analysis is obtained. In this case nonlinear manifolds can be reduced to minimum dimension manifolds. If such units are used the network performs a generalized principal component analysis in the sense that non-Gaussian distributions can be linearly decorrelated and higher orders of the correlation tensors are also taken into account. The basic structure of the architecture involves a general transfOlmation that is volume conserving and therefore the entropy, yielding a map without loss of infoIIDation. Minimization of the mutual infoIIDation among the output neurons eliminates the redundancy between the outputs and results in statistical decorrelation of the extracted features. This is known as factorialleaming. 248 Gustavo Deco, Wilfried Brauer 1 INTRODUCTION One of the most important theories of feature extraction is the one proposed by Barlow (1989). Barlow describes the process of cognition as a preprocessing of the sensorial information performed by the nervous system in order to extract the statistically relevant and independent features of the inputs without loosing information. This means that the brain should statistically decorrelate the extracted information. As a learning strategy Barlow (1989) formulated the principle of redundancy reduction. This kind of learning is called factorial learning. Recently Atick and Redlich (1992) and Redlich (1993) concentrate on the original idea of Barlow yielding a very interesting formulation of early visual processing and factorial learning. Redlich (1993) reduces redundancy at the input by using a network structure which is a reversible cellular automaton and therefore guarantees the conservation of information in the transformation between input and output. Some nonlinear extensions of PCA for decorrelation of sensorial input signals were recently introduced. These follow very closely Barlow's original ideas of unsupervised learning. Redlich (1993) use similar information theoretic concepts and reversible cellular automata architectures in order to define how nonlinear decorrelation can be performed. The aim of our work is to formulate a neural network architecture and a novel learning paradigm that performs Barlow's unsupervised learning in the most general fashion. The basic idea is to define an architecture that assures perfect transmission without loss of information. Consequently the nonlinear transformation defined by the neural architecture is always bijective. The architecture performs a volume-conserving transformation (determinant of the Jacobian matrix is equal one). As a particular case we can derive the reversible cellular automata architecture proposed by Redlich (1993). The learning paradigm is defined so that the components of the output signal are statistically decorrelated. Due to the fact that the output distribution is not necessarily Gaussian, even if the input is Gaussian, we perform a cumulant expansion of the output distribution and find the rules that should be satisfied by the higher order correlation tensors in order to be decorrelated. 2 THEORETICAL FORMALISM Let us consider an input vector x of dimensionality d with components distributed according to the probability distribution P (X) , which is not factorial, i.e. the components of x are correlated. The goal of Barlow's unsupervised learning rule is to find a transformation ,. ->,. y = F (x) (2.1) such that the components of the output vector d -dimensional yare statistically decorrelated. This means that the probability distributions of the components Y j are independent and therefore, d pO) = IT P (y) . (2.2) j The objl?,.ctive of factorial learning is to find a neural network, which performs the transformation F ( ) such that the joint probability distribution P 0) of the output signals is factorized as in eq. (2.2). In order to implement factorial learning, the information contained in the input should be transferred to the output neurons without loss but, the probability distribution of the output neurons should be statistically decorrelated. Let us now define Higher Order Statistical Decorrelation without /n/onl1atioll Loss 249 these facts from the information theory perspective. The first aspect is to assure the entropy is conselVed, i.e. H (x) = H (y) (2.3) where the symbol !! (a) denotes the entropy of a and H (a/b) the conditional entropy of given b. One way to achieve this goal is to construct an architecture that independently of its synaptic parameters satisfies always eq. (2.3). Thus the architecture will conselVe information or entropy. The transmitted entropy satisfies a H (J) ~ H (x) + JP (x) In (det (~r? dx (2.4) -" where equality holds only if F is bijective, i.e. reversible. ConselVation of information and bijectivity is assured if the neural transformation conselVes the volume, which mathematically can be expressed by the fact that the Jacobian of the transformation should have determinant unity. In section 3 we fonnulate an architecture that always conselVes the entropy. Let us now concentrate on the main aspect of factorial learning, namely the decorrelation of the output components. Here the problem is to find a volume-conselVing transformation that satisfies eq. (2.2). The major problem is that the distribution of the output signal will not necessarily be Gaussian. Therefore it is impossible to use the technique of minimizing the mutual information between the components of the output as done by Redlich (1993). The only way to decorrelate non-Gaussian distributions is to expand the distribution in higher orders of the correlation matrix and impose the independence condition of eq. (2.2). In order to achieve this we propose to use a cumulant expansion of the output distribution. Let us define the Fourier transform of the output distribution, ~(K) = JdY ei(K?h P(J;~(Ki) = fdYi ei(Ki?Yi) P(y) (2.5) The cumulant expansion of a distribution is (Papoulis, 1991) (2.6) In the Fourier space the independence condition is given by (papoulis, 1991) (2.7) which is equivalent to ...>. In(~(K? = Incrl~(Ki? = Eln(~(Ki? I (2.8) I Putting eq. (2.8) and the cumulant expansions of eq. (2.6) together, we obtain that in the case of independence the following equality is satisfied (2.9) In both expansions we will only consider the first four cumulants. Mter an extra transformation ~ .. -,.- y' .. y - (y) (2.10) Gustavo Deco, Wilfried Brauer 250 to remove the bias <y). we can rewrite eq. (2.9) using the cumulants expression derived in the Papoulius (1991): 1 -21: KjKj {Cjj - Cj (2) '.J 1 + 24 i 5jj} -"6 E KiKjKd Cijl.: - Cj 5jjk } (3) (2.11) '.J.k E KjKjKkKd (Cjjkl-3CijCkl) - (4) (Cj (2) -3 (C; 2 ? 5jjkl } =0 i ,j, t , I Equation (2.11) should be satisfied for all values of K. The multidimensional correlation tensors C j . .. j and the one-dimensional higher order moments C?) are given by ~j ... j = Jdy' P(y') y'j ... y'j ; C?) = Jdy'j P(y'j) (y'i) (2.12) The 5. . denotes Kroenecker's delta. Due to the fact that eq. (2.11) should be satisfied for all K,'iiU coefficients in each summation of eq. (2.11) must be zero. This means that C jj = 0, if(i oF j) (2.13) C jjk C jjkl = 0, - 0, if(i oF j v i oF k) (2.l4) if( {ioFjvioFkvioFl} I\-,L) C Ujj - CijCjj = 0, if(i oF j) . (2.15) (2.16) In eq. (2.15) L is the logical expression L - {(i .. j 1\ k .. 11\ j oF k) v (i = k 1\ j = I 1\ i oF j) v (i - 11\ j .. k 1\ i oF j) }, (2.17) which excludes the cases considered in eq. (2.16). The conditions of independence given by eqs. (2. 13-2.l6) can be achieved by minimization of the cost function E - aE Cfj+ 13i<j~k E Cfjk+Yi<j~';'1 , Ctkl+ 5E (C;;jj - CjjCjj) 2 ;<j (2.18) i<j where a, 13, Y, 5 are the inverse of the number of elements in each summation respectively. In conclusion, minimizing the cost given by eq. (2.18) with a volume-conserving network, we achieve nonlinear decorrelation of non-Gaussian distributions. It is very easy to test wether a factorized probability distribution (eq. 2.2) satisfies the eqs. (2.13-2.16). As a particular case if only second order terms are used in the cumulant expansion, the learning rule reduces to eq. (2.13), which expresses nothing more than the diagonalization of the second order covariance matrix. In this case, by anti-transforming the cumulant expansion of the Fourier transform of the distribution,we obtain a Gaussian distribution. Diagonalization of the covariance matrix decorrelates statistically the components of the output only if we assume a Gaussian distribution of the outputs. In general the distribution of the output is not Gaussian and therefore higher orders of the cumulant expansion should be taken into account. yielding the learning rule conditions eqs. (2.13-2.l6) (up to fourth order, generalization to higher orders is straightforward). In the case of Gaussian distribution, minimization of the sum of the variances at each output leads to statistically decorrelation. This fact has a nice information theoretic background namely the minimization of the mutual information between the output components. Statistical independence as expressed in eq. (2.2) is equivalent to (Atick and Redlich. 1992) MH = EH(Yj) J -H(~) =0 (2.19) Higher Order Statistical Decorrelatioll without Illformation Loss 251 This means that in order to minimize the redundancy at the output we minimize the mutual information between the different components of the output vector. Due to the fact that the volume-conserving structure of the neural network conserves the entropy, the minimizaH (Yj) . tion of MH reduces to the minimization of E J 3 VOLUME-CONSERVING ARCHITECTURE AND LEARNING RULE In this section we define a neural network architecture that is volume-conserving and therefore can be used for the implementation of the learning rules described in the last section. Figure l.a shows the basic architecture of one layer. The dimensionality of input and output layer is the same and equal to d. A similar architecture was proposed by Redlich (1993b) using the theory of reversible cellular automata. ~ x ~ x ~ x ~ Y (c) (b) (a) Figure 1: Volume-conserving Neural Architectures. The analytical definition of the transformation defined by this architecture can be written as, )'i = xi +1, (x o' ... , Xj' Wi)' with j < i (3.1) where Wi represents a set of parameters of the function h' Note that independently of the functions fi the network is always volume-conserving. In particular h can be calculated by another neural network, by a sigmoid neuron, by polynomials (higher order neurons), etc. Due to the asymmetric dependence on the input variables and the direct connections with weights equal to 1 between corresponding components of input and output neurons, the Jacobian matrix of the transformation defined in eq. (3.1) is an upper triangular matrix with diagonal elements all equal to one, yielding a determinant equal to one. A network with inverted asymmetry also can be defined as )'i = Xi + gi (xj' ... , Xd' 8;>, with i <j (3.2) corresponding to a lower triangular Jacobian matrix 'fith diagonal elements all equal to one, being therefore volume-conserving. The vectors 8 j represent the parameters of functions gj. In order to yield a general nonlinear transformation from inputs to outputs (without asymmetric dependences) it is possible to build a multilayer architecture like the one shown in Fig. 1.b, which involves mixed versions of networks described by eq. (3.1) and eq. (3.2), respectively. Due to the fact that successive application of volume-conserving transformation is also volume-conserving, the multilayer architecture is also volume-con- 252 Gustavo Deco, Wilfried Brauer serving. In the two-layer case (Fig. I.c) the second layer can be interpreted as asymmetric lateral connections between the neurons of the first layer. However, in our case the feedfOlWard connections between input layer and first layer are also asymmetric. As demonstrated in the last section, we minimize a cost function E to decorrelate nonlinearly correlated non-Gaussian inputs. Let us analyze for simplicity a two-layer architecture (Fig. l.c) with the first layer given by eq. (3.l) and the second layer by eq. (3.2). Let us denote the output of the hidden layer by h and use it as input of eq. (3.2) with output y. The extension to multilayer architectures is straightfolWard. The learning rule can be easily expressed by gradient descent method: aE e.=e.-T\ae; .>. .>. I I (3.3) ..>. In order to calculate the derivative of the cost functions we need a _ _ Ci ... j 1 == N-~ {",,\"::"(y.-y .) ... (y .-y.) o i..J P 00 I I ] ] _ a _. ayi 1 a + (y.-y.) ... ""\"::"(y . -y, -a = -~ {,\::"(y .)}(3.4) I I 00 ] 0 N i..J 00 I p J e e where represents the parameters j and Wi. The sums in both equations extend over the N training patterns. The gradients of the different outputs are a _ a .j _ a a a W; Wi ---;:-Yi - - -g; .. ,~Yt - (ahgt) (a ..... !;)&i>k+ (a ..... !;) ae; ae; [OJ I (3.5) where &i> k is equal to I if i > k and 0 othelWise. In this paper we choose a polynomial fonn for the functions I and g. This model involves higher order neurons. In this case each function Ii or g; is a product of polynomial functions of the inputs. The update equations are given by (3.6) where R is the order of the polynomial used. In this case the two-layer architecture is a higher order network with a general volume-conserving structure. The derivatives involved in the learning rule are given by a _ -a-Yk 0) ? . ljr (3.7) 253 Higher Order Statistical Decorrelation without Information Loss 4 RESULTS AND SIMULATIONS We will present herein two different experiments using the architecture defined in this paper. The input space in all experiments is two-dimensional in order to show graphically the results and effects of the presented model. The experiments aim at learning noisy nonlinear polynomial and rational curves. Figure 2.a and 2.b plot the input and output space of the second experiment after training is finished, respectively. In this case the noisy logistic map was used to generate the input: (4.1) where'\) introduces I % Gaussian noise. In this case a one-layer polynomial network with R - 2 was used. The learning constant was 11 - 0.01 and 20000 iterations of training were performed. The result of Fig. 2.b is remarkable. The volume-conserving network decorrelated the output space extracting the strong nonlinear correlation that generated the curve in the input space. This means that after training only one coordinate is important to describe the curve. (b) (a) ?? ?? o? os ?? 04 02 ' ?1'=" .2 ---.,....---:.:'-:-.---:.:':-.---,.,':-.-----::':: ..-----7----,,' .. .g7.-~-~.:'-:-2---,O~.--0~.-~ ??~~-~ Figure 2: Input and Output space distribution after training with a one-layer polynomial volume-conseIVing network of order for the logistic map. (a) input space; (b) output space. The whole information was compressed into the first coordinate of the output. This is the generalization of data compression normally performed by using linear peA (also called Karhunen-Loewe transformation). The next experiment is similar, but in this case a twolayer network of order R .. 4 was used. The input space is given by the rational function 3 x2 = O.2x I + Xl 2 (1 +x I ) + '\l (4.2) where Xl and '\) are as in the last case. The results are shown in Fig. 4.a (input space) and Fig. 4.b (output space). Fig. 4.c shows the evolution of the four summands of eq (2.18) during learning. It is important to remark that at the beginning the tensors of second and third order are equally important. During learning all summands are simultaneously minimized, resulting in a statistically decorrelated output. The training was performed during 20000 iterations and the learning constant was 11 = 0.005 . 254 Gustavo Deco, Wilfried Brauer (b) (a) " (c) " etrIlIt' II. " .. .... " " ?? .- /' ~st2 II ? .l I. I ~. _,,, ...... H"'t . I. , . . . . . . .1? .. -"' - ' ... . . t. ,'" ..I ... ...t.. I .. 'I. .. . " '1. " " " cost4b .~ .. II. M.. ,- Figure 4: Input and Output space distribution after training with a two-layer polynomial volume-conselVing network of order for the noisy CUIVe of~. (4.2). (a) input space; (b) output space (c) Development of the four summands of the cost function (~ 2.18) during learning: (cost2) fiist summand (second order COIrelation tensor); (cost 3) second summand (tliird correlation order tensor); (cost 4a) third summand (fourth order correlation tensor); (cost4b) fourth summand (fourth order correlation tensor). 5 CONCLUSIONS We proposed a unsupervised neural paradigm, which is based on Infonnation Theory. The algorithm perfonns redundancy reduction among the elements of the output layer without loosing infonnation, as the data is sent through the network. The model developed perfonns a generalization of Barlow's unsupervised learning, which consists in nonlinear decorrelation up to higher orders of the cumulant tensors. After training the components of the output layer are statistically independent. Due to the use of higher order cumulant expansion arbitrary non-Gaussian distributions can be rigorously handled. When nonlinear units are used nonlinear principal component analysis is obtained. In this case nonlinear manifolds can be reduced to a minimum dimension manifolds. When linear units are used, the network performs a generalized principal component analysis in the sense that non-Gaussian distribution can be linearly decorrelated.This paper generalizes previous works on factorial learning in two ways: the architecture performs a general nonlinear transformation without loss of information and the decorrelation is perfonned without assuming Gaussian distributions. References: H. Barlow. (1989) Unsupervised Learning. Neural Computation, 1,295-311. A. Papoulis. (1991) Probability, Random Variables, and Stochastic Processes. 3. Edition, McGraw-Hill, New York. A. N. Redlich. (1993) Supervised Factorial Learning. Neural Computation, 5, 750-766. 

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A Neural Model of Delusions and Hallucinations in Schizophrenia Eytan Ruppin and James A. Reggia Department of Computer Science University of Maryland, College Park, MD 20742 ruppin@cs.umd .edu reggia@cs.umd.edu David Horn School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel horn@vm.tau.ac.il Abstract We implement and study a computational model of Stevens' [19921 theory of the pathogenesis of schizophrenia. This theory hypothesizes that the onset of schizophrenia is associated with reactive synaptic regeneration occurring in brain regions receiving degenerating temporal lobe projections. Concentrating on one such area, the frontal cortex, we model a frontal module as an associative memory neural network whose input synapses represent incoming temporal projections. We analyze how, in the face of weakened external input projections, compensatory strengthening of internal synaptic connections and increased noise levels can maintain memory capacities (which are generally preserved in schizophrenia) . However, These compensatory changes adversely lead to spontaneous, biased retrieval of stored memories, which corresponds to the occurrence of schizophrenic delusions and hallucinations without any apparent external trigger, and for their tendency to concentrate on just few central themes. Our results explain why these symptoms tend to wane as schizophrenia progresses, and why delayed therapeutical intervention leads to a much slower response. 150 1 Eytan Ruppin, James A. Reggia, David Hom Introduction There has been a growing interest in recent years in the use of neural models to investigate various brain pathologies and their cognitive and behavioral effects. Recent published examples of such studies include models of cortical plasticity following stroke, Alzheimer's disease and schizophrenia, and cognitive and behavioral explorations of aphasia, acquired dyslexia and affective disorders (reviewed in [1, 2]). Continuing this line of study, we present a computational account linking specific pathological synaptic changes that are postulated to occur in schizophrenia, and the emergence of schizophrenic delusions and hallucinations. The latter symptoms denote persistent, unrealistic, psychotic thoughts (delusions) or percepts (hallucinations) that may at times flood the patient in an overwhelming, stressful manner. The wealth of data gathered concerning the pathophysiology of schizophrenia supports the involvement of both the frontal and the temporal lobes. On the one hand, there are atrophic changes in the hippocampus and parahippocampal areas including neuronal loss and gliosis. On the other hand, neurochemical and morphometric studies testify to an expansion of various receptor binding sites and increased dendritic branching in the frontal cortex of schizophrenics. Stevens has recently presented a theory linking these temporal and frontal findings, claiming that the onset of schizophrenia is associated with reactive anomalous sprouting and synaptic reorganization taking place in the projection sites of degenerating temporal neurons, including (among various cortical and subcortical structures) the frontal lobes [3]. This paper presents a computational study of Stevens' theory. Within the framework of a memory model of hippocampal-frontal interaction, we show that the introduction of the 'microscopic' synaptic changes that underlie Stevens' hypothesis can help preserve memory function but results in specific 'pathological' changes in the 'macroscopic' behavior of the network. A small subset of the patterns stored in the network are now spontaneously retrieved at times, without being cued by any specific input pattern. This emergent behavior shares some of the important characteristics of schizophrenic delusions and hallucinations, which frequently appear in the absence of any apparent external trigger, and tend to concentrate on a limited set of recurrent themes [4]. Memory capacities are fairly preserved in schizophrenics, until late stages of the disease [5]. In Section 2 we present our model. The analytical and numerical results obtained are described in Section 3, followed by our conclusions in Section 4. 2 The Model As illustrated in Figure 1, we model a frontal module as an associative memory attractor neural network, receiving its input memory cues from decaying external input fibers (representing the degenerating temporal projections). The network's internal connections, which store the memorized patterns, undergo synaptic strengthening changes that model the reactive synaptic regeneration within the frontal module. The effect of other diffuse external projections is modeled as background noise. A frontal module represents a macro-columnar unit that has been suggested as a basic functional building block of the neocortex [6]. The assumption that memory retrieval from the frontal cortex is invoked by the firing of incoming A Neural Model of Delusions and Hallucinations in Schizophrenia 151 temporal projections is based on the notion that temporal structures have an important role in establishing long-term memory in the neocortex and in the retrieval of facts and events (e.g., [7]). ,,- --- " Other cortical : modules I T I I ....... : (Influence nxxJeled as,' .... .. ! noise) ," ......... .. \ ,I" .... .. Figure 1: A schematic illustration of the model. A frontal module is modeled as an attractor neural network whose neurons receive inputs via three kinds of connections: internal connections from other frontal neurons, external connections from temporal lobe neurons, and diffuse external connections from other cortical modules, modeled as noise. The attractor network we use is a biologically-motivated variant of Hopfield's ANN model, proposed by Tsodyks & Feigel'man [8]. Each neuron i is described by a binary variable Si = {1,0} denoting an active (firing) or passive (quiescent) state, respectively. M = aN distributed memory patterns IJ are stored in the network. The elements of each memory pattern are chosen to be 1 (0) with probability p (1- p) respectively, with p ~ 1. All N neurons in the network have a fixed uniform threshold O. e In its initial, undamaged state, the weights of the internal synaptic connections are M Wij =N CO" L....t(e IJ i - p)(e j - p) , (1) IJ=I where Co = 1. The post-synaptic potential (input field) hi of neuron i is the sum of internal contributions from other neurons and external projections Fi e hi(t) = L WijSj(t - 1) + Fie. (2) j The updating rule for neuron i at time t is given by S .(t) = 1 {I,0, with p~ob . G(hi(t) - 0) otherwIse (3) 152 Eytan Ruppin, James A. Reggia, David Hom where G is the sigmoid function G(x) = 1/(1+exp( -x/T)), and T denotes the noise level. The activation level of the stored memories is measured by their overlaps mil with the current state of the network, defined by 1 mll(t) = N (1 _ )N P P L)er - p)Si(t) . (4) i=l Stimulus-dependent retrieval is modeled by orienting the field Fe with one of the memorized patterns (the cued pattern, say e), such that F/ = e . e\ , (e > 0) . (5) Following the presentation of an external input cue, the network state evolves until it converges to a stable state. The network parameters are tuned such that in its initial, undamaged state it correctly retrieves the cued patterns (eo = 0.035 , Co = 1, T = 0.005). We also examine the network's behavior in the absence of any specific stimulus. The network may either continue to wander around in a state of random low baseline activity, or it may converge onto a stored memory state. We refer to the latter process as spontaneous retrieval. Our investigation of Stevens' work proceeds in two stages. First we examine and analyze the behavior of the network when it undergoes uniform synaptic changes that represent the pathological changes occurring in accordance with Stevens' theory. These include the weakening of external input projections (e !) and the increase in the internal projections (c i) and noise levels (T i). In the second stage, we add the assumption that the internal synaptic compensatory changes have an additional Hebbian activity-dependent component, and examine the effect of the rule (6) where Sk is 1 (0) only if neuron k has been consecutively firing (quiescent) for the last r iterations, and 'Y is a constant. 3 Results We now show some simulation and analytic results, examining the effects of the 'microscopic' pathological changes, taking place in accordance with Stevens' theory, on the 'macroscopic' behavior of the network . The analytical results presented have been derived by calculating the magnitude of randomly formed initial 'biases', and comparing their effect on the network 's dynamics versus the effect of externally presented input cues. This comparison is performed by formulating a corresponding overlap master equation , whose fixed point dynamics are investigated via phaseplane analysis, as described in [9]. First, we study whether the reactive synaptic changes (occurring in both internal and external, diffuse synapses) are really compensatory, i.e., to what extent can they help maintaining memory capacities in the face of degenerating external input synapses. As illustrated in Figure 2, we find that increased noise levels can (up to some degree) preserve memory retrieval in the face of decreased external input strength. Increased synaptic strengthening preserves 153 A Neural Model of Delusions and Hallucinations in Schizophrenia (a) (b) 1.0 1.0 ,/ i' , - . _ ,! O.O35 ------ ?? iO.025 O.B ---- ?? '0.015 - -? ? 10.014 G---o ?? ; 0.013 <)--0._ 10.012 - - - ?? ; 0.005 O.B 0.6 0.8 . t ! 6 6 0.( . ~_.__ ? _ 0.025 - --- ? ? 0.015 - - - ? ? 0.005 '- -- 0.2 0.2 0.0 0.000 0.( : - . _ 0.035 I - :... ---- 0.010 T 0.020 0?8. 0.020 T Figure 2: Stimulus-dependent retrieval performance, measured by the average final overlap m, as a function of the noise level T. Each curve displays this relation at a different magnitude of external input projections e. (a) Simulation results. (b) Analytic approximation. memory retrieval in a similar manner, and the combined effect of these synaptic compensatory measures is synergistic. Second, although the compensatory synaptic changes help maintain memory retrieval capacities, they necessarily have adverse effects, leading eventually to the emergence of spontaneous activation of non-cued memory patterns; the network converges to some of its memory patterns in a pathological, autonomous manner , in the absence of any external input stimuli. This emergence of pathological spontaneous retrieval , when either the noise level or the internal synaptic strength (or both) are increased beyond some point, is demonstrated in Figure 3. Third, when the compensatory regeneration of internal synapses has an additional Hebbian component (representing a period of increased activity-dependent plasticity due to the regenerative synaptic changes), a biased spontaneous retrieval distribution is obtained. That is, as time evolves (measured in time units of 'trials'), the distribution of patterns spontaneously retrieved by the network in a pathological manner tends to concentrate only on one or two of all the memory patterns stored in the network, as is shown in Figure 4a. This highly peaked distribution is maintained for a few hundred additional trials until memory retrieval sharply collapses to zero as a global mixed-state attractor is formed. Such a mixed attractor state does not have very high overlap with any memorized pattern, and thus does not represent any well-defined cognitive or perceptual item. It is an end state of the Hebbian, activity-dependent evolution of the network. Yet, even after activitydependent changes ensue , if spontaneous activity does not emerge the distribution of retrieved memories remains homogeneous (see Figure 4b). Eventually, a global 154 Eytan Ruppin. James A. Reggia. David Hom (b) (a) 1.0 , ,--- - . 1.0 , ........... . ~ .. .. .. ,1' " , , O.B , O.B , 0.6 0.6 , t ! f AnalytIc _Imollon - - - - Slmulallon 0.4 0.4 0.2 0.2 ,, 0.0 0.005 ' , 0.010 0.015 0.020 0.0 ':--~-'-:-'-::--~~--~----:' 2.0 2.5 3.0 3.5 4.0 T Figure 3: (a) Spontaneous retrieval, measured as the highest final overlap m achieved with any of the stored memory patterns, displayed as a function of the noise level T. c = 1. (b) Spontaneous retrieval as a function of internal synaptic compensation factor c. T 0.009. = mixed-state attractor is formed, and the network looses its retrieval capacities, but during this process no memory pattern gets to dominate the retrieval output. Our results remain qualitatively similar even when bounds are placed on the absolute magnitude of the synaptic weights. 4 Conclusions Our results suggest that the formation of biased spontaneous retrieval requires the concomitant occurrence of both degenerative changes in the external input fibers, and regenerative Hebbian changes in the intra-modular synaptic connections. They add support to the plausibility of Stevens' theory by showing that it may be realized within a neural model, and account for a few characteristics of schizophrenic symptoms: ? The emergence of spontaneous, non-homogeneous retrieval is a self-limiting phenomenon (as eventually a cognitively meaningless global attractor is formed) - this parallels the clinical finding that as schizophrenia progresses both delusions and hallucinations tend to wane, while negative symptoms are enhanced [10]. ? Once converged to, the network has a much larger tendency to remain in a biased memory state than in a non biased one - this is in accordance with the persistent characteristic of schizophrenic florid symptoms. ? As more spontaneous retrieval trials occur the frequency of spontaneous retrieval increases - indeed, while early treatment in young psychotic adults A Neural Model of Delusions and Hallucinations in Schizophrenia 155 (b) (a) 1.0 ~-~-~~-:----~-----. 0.100 0.8 0.080 G---f] ~ 0.6 t - v Aft... 800triala Q .,. 1 ~ ? ( ? Aft ... 400tr1a1a ~ 0.4 >' ., .,~ ~. ; , ~ 1 0.000 t 1 l .~ 0.2 ~AftOf200trlals Aft.r 200 tria_ {3 - -(:~Aft.r500trials ~ ~----"-~~--~-----. 0.040 0.020 Figure 4: (a) The distribution of memory patterns spontaneously retrieved . The xaxis enumerates the memories stored, and the y-axis denotes the retrieval frequency of each memory. 'Y = 0.0025. (b) The distribution of stimulus-dependent retrieval of memories. 'Y = 0.0025 . leads to early response within days, late, delayed intervention leads to a much slower response during one or more months [11]. The current model generates some testable predictions: ? On the neuroanatomical level, the model can be tested quantitatively by searching for a positive correlation between a recent history of florid psychotic symptoms and postmortem neuropathological findings of synaptic compensation. (For example, this kind of correlation, between indices of synaptic area and cognitive functioning was found in Alzheimer patients [12]). ? On the physiological level, the increased compensatory noise should manifest itself in increased spontaneous neural activity. While this prediction is obviously difficult to examine directly, EEG studies in schizophrenics show significant increase in slow-wave delta activity which may reflect increased spontaneous activity [13]. ? On the clinical level, due to the formation of a large and deep basin of attraction around the memory pattern which is at the focus of spontaneous retrieval, the proposed model predicts that its retrieval (and the elucidation of the corresponding delusions or hallucinations) may be frequently triggered by various environmental cues. A recent study points in this direction [14]. 156 Eytan Ruppin, James A. Reggia, David Hom Acknowledgements This research has been supported by a Rothschild Fellowship to Dr. Ruppin. References [1] J. Reggia, R. Berndt, and L. D' Autrechy. Connectionist models in neuropsychology. In Handbook of Neuropsychology, volume 9. 1994, in press. [2] E. Ruppin. Neural modeling of psychiatric disorders. Network: Computation in Neural Systems, 1995. Invited review paper, to appear. [3] J .R. Stevens. Abnormal reinnervation as a basis for schizophrenia: A hypothesis. Arch. Gen . Psychiatry, 49 :238-243, 1992. [4] S.K. Chaturvedi and V.D. Sinha. Recurrence of hallucinations in consecutive episodes of schizophrenia and affective disorder. Schizophrenia Research, 3: 103106, 1990. [5] M. Marsel Mesulam. Schizophrenia and the brain. New England Journal of Medicine, 322(12) :842-845, 1990. [6] P.S. Goldman and W.J .H . Nauta. Columnar distribution of cortica-cortical fibers in the frontal, association , limbic and motor cortex of the developing rhesus monkey. Brain Res., 122:393-413,1977. [7] L. R. Squire. Memory and the hippocampus: A synthesis from findings with rats, monkeys, and humans . Psychological Review, 99 :195-231 , 1992. [8] M.V. Tsodyks and M.V. Feigel'man. The enhanced storage capacity in neural networks with low activity level. Europhys. Lett., 6:101 - 105, 1988. [9] D. Horn and E. Ruppin . Synaptic compensation in attractor neural networks: Modeling neuropathological findings in schizophrenia. Neural Computation, page To appear, 1994. [10] W .T. Carpenter and R.W. Buchanan. Schizophrenia. New England Journal of Medicine, 330:10, 1994. [11] P. Seeman. Schizophrenia as a brain disease: The dopamine receptor story. Arch . Neurol., 50:1093-1095, 1993. [12] S. T. DeKosky and S.W. Scheff. Synapse loss in frontal cortex biopsies in alzheimer 's disease: Correlation with cognitive severity. Ann. Neurology, 27(5):457-464, 1990. [13] Y. Jin, S.G. Potkin, D. Rice, and J. Sramek et. al. Abnormal EEG responses to photic stimulation in schizophrenic patients. Schizophrenia Bulletin, 16(4):627634, 1990. [14] R.E. Hoffman and J .A. Rapaport. A psycholoinguistic study of auditory /verbal hallucinations: Preliminary findings. In David A. and Cutting J. , editors, The Neuropsychology of Schizophrenia. Erlbaum, 1993. 

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A Silicon Axon Bradley A. Minch, Paul Hasler, Chris Diorio, Carver Mead Physics of Computation Laboratory California Institute of Technology Pasadena, CA 91125 bminch, paul, chris, carver@pcmp.caltech.edu Abstract We present a silicon model of an axon which shows promise as a building block for pulse-based neural computations involving correlations of pulses across both space and time. The circuit shares a number of features with its biological counterpart including an excitation threshold, a brief refractory period after pulse completion, pulse amplitude restoration , and pulse width restoration. We provide a simple explanation of circuit operation and present data from a chip fabricated in a standard 2Jlm CMOS process through the MOS Implementation Service (MOSIS). We emphasize the necessity of the restoration of the width of the pulse in time for stable propagation in axons. 1 INTRODUCTION It is well known that axons are neural processes specialized for transmitting information over relatively long distances in the nervous system. Impulsive electrical disturbances known as action potentials are normally initiated near the cell body of a neuron when the voltage across the cell membrane crosses a threshold. These pulses are then propagated with a fairly stereotypical shape at a more or less constant velocity down the length of the axon . Consequently, axons excel at precisely preserving the relative timing of threshold crossing events but do not preserve any of the initial signal shape. Information , then , is presumably encoded in the relative timing of action potentials. 740 Bradley A. Minch, Paul Hasler, Chris Diorio, Carver Mead The biophysical mechanisms underlying the initiation and propagation of action potentials in axons have been well studied since the seminal work of Hodgkin and Huxley on the giant axon of Loligo. (Hodgkin & Huxley, 1952) Briefly, when the voltage across a small patch of the cell membrane increases to a certain level, a population of ion channels permeable to sodium opens, allowing an influx of sodium ions, which, in turn, causes the membrane voltage to increase further and a pulse to be initiated. This population of channels rapidly inactivates, preventing the passage of additional ions. Another population of channels permeable to potassium opens after a brief delay causing an efflux of potassium ions, restoring the membrane to a more negative potential and terminating the pulse. This cycle of ion migration is coupled to neighboring sections of the axon, causing the action potential to propagate. The sodium channels remain inactivated for a brief interval of time during which the affected patch of membrane will not be able to support another action potential. This period of time is known as the refractory period. The axon circuit which we present in this paper does not attempt to model the detailed dynamics of the various populations of ion channels, although such detailed neuromimes are both possible (Lewis, 1968; Mahowald & Douglas, 1991) and useful for learning about natural neural systems. Nonetheless, it shares a number of important features with its biological counterpart including having a threshold for excitation and a refractory period. It is well accepted that the amplitude of the action potential must be restored as it propagates. It is not as universally understood is that the width of the action potential must be restored in time if it is to propagate over any appreciable distance. Otherwise, the pulse would smear out in time resulting in a loss of precise timing information, or it would shrink down to nothing and cease to propagate altogether. In biological axons, restoration of the pulse width is accomplished through the dynamics of sodium channel inactivation and potassium channel activation. In our silicon model, the pulse width is restored through feedback from the successive stage. This feedback provides an inactivation which is similar to that of the sodium channels in biological axons and is also the underlying cause of refractoriness in our circuit . In the following section we provide a simple description of how the circuit behaves. Following this, data from a chip fabricated in a standard 2p.m CMOS process through MOSIS are presented and discussed. 2 THE SILICON AXON CmCUIT An axon circuit which is to be used as a building block in large-scale computational systems should be made as simple and low-power as possible , since it would be replicated many times in any such system. Each stage of the axon circuit described below consists of five transistors and two small capacitors, making the axon circuit very compact. The axon circuit uses the delay through a stage to time the signal which is fed back to restore the pulse width, thus avoiding the need for an additional delay circuit for each section. Additonally, the circuit operates with low power; during typical operation (a pulse of width 2ms travelling at 10 3 stages/s), pulse propagation costs about 4pJ / stage of energy. Under these circumstances, the circuit consumes about 2nW/stage of static power. A Silicon Axon 741 Figure 1: Three sections of the axon circuit. Three stages of the axon circuit are depicted in Figure 1. A single stage consists of two capacitors and what would be considered a pseudo-nMOS NAND gate and a pseudo-nMOS inverter in digital logic design. These simple circuits are characterized by a threshold voltage for switching and a slew rate for recharging. Consider the inverter circuit. If the input is held low for a sufficiently long time, the pull-up transistor will have charged the output voltage almost completely to the positive rail. If the input voltage is ramped up toward the positive rail, the current in the pull-down transistor will increase rapidly. At some input voltage level, the current in the pull-down transistor will equal the saturation current of the pull-up transistor; this voltage is known as the threshold. The output voltage will begin to discharge at a rapidly increasing rate as the input voltage is increased further . After a very short time, the output will have discharged almost all the way to the negative rail. Now, if the input were decreased rapidly, the output voltage would ramp linearly in time (slew) up toward the positive rail at a rate set by the saturation current in the pull-up transistor and the capacitor on the output node . The NAND gate is similar except both inputs must be (roughly speaking) above the threshold in order for the output to go low. If either input goes low , the output will charge toward the positive rail. Note also that if one input of the NAND gate is held high, the circuit behaves exactly as an inverter. The axon circuit is formed by cascading multiple copies of this simple five transistor circuit in series. Let the voltage on the first capacitor of the nth stage be denoted by Un and the voltage on the second capacitor by v n . Note that there is feedback from U n +1 to the lower input of the NAND gate of the nth stage. Under quiescent conditions, the input to the first stage is low (at the negative rail) , the U nodes of all stages are high (at the positive rail), and all of the v nodes are held low (at the negative rail). The feedback signal to the final stage in the line would be tied to the positive rail. The level of the bias voltages 71 and 72 determine whether or not a narrow pulse fed into the input of the first stage will propagate and, if so, the width and velocity with which it does. In order to obtain a semi-quantitative understanding of how the axon circuit behaves, we will first consider the dynamics of a cascade of simple inverters (three sections of which are depicted in Figure 2) and then consider the addition of feedback. Under most circumstances, discharges will occur on a much faster time scale Bradley A. Minch, Paul Hasler, Chris Diorio, Carver Mead 742 than the recharges, so we make the simplifying assumption that when the input of an inverter reaches the threshold voltage, the output discharges instantaneously. Additionally, we assume that saturated transistors behave as ideal current sources (i.e., we neglect the Early effect) so that the recharges are linear ramps in time. TTT TTT TTT TTT TTT TTT Figure 2: A cascade of pseudo-nMOS inverters. Let hand h be the saturation currents in the pull-up transistors with bias voltages and 1"2, respectively. Let 6 1 and 62 be the threshold voltages of the first and second inverters in a single stage, respectively. Also, let u = IdC and v = I2/C be the slew rates for the U and v nodes, respectively. Let a 1 = 6dv and a 2 = 6 2 /u be the time required for Un to charge from the negative rail up to E>1 and for Vn to charge from the negative rail up to 6 2 , respectively. Finally, let v~ denote the peak value attained by the v signal of the nth stage. 1"1 V? v?71 vn IE ~IE 671 IE ~I a2 (a) II > h Figure 3: Geometry of the idealized ~IE 671 Un and Vn ~I a2 (b) h < 12 signals under the bias conditions (a) h > 12 and (b) II < 12, Consider what would happen if V n - l exceeded 6 1 for a time 671 , In this case, Un would be held low for 671 and then released. Meanwhile , Vn would ramp up to v6 n . Then , Un would begin to charge toward the positive rail while Vn continues to charge. This continues for a time a2 at which point Un will have reached 6 2 causing A Silicon Axon 743 to be discharged to the negative rail. Now, Un+l is held low while Vn exceeds 8 1 this interval of time is precisely On+!. Figures 3a and 3b depict the geometry of the Un and Vn signals in this scenario under the bias conditions h > 12 and h < 12, respectively. Simple Euclidean geometry implies that the evolution of On will be governed by the first-order difference equation Vn which is trivially solved by Thus, the quantity a2-al determines what happens to the width of the pulse as it propagates. In the event that a 2 < aI, the pulse will shrink down to nothing from its initial width. If a 2 > aI, the pulse width will grow without bound from its initial width. The pulse width is preserved only if a 2 = a l . This last case, however, is unrealistic. There will always be component mismatches (with both systematic and random parts), which will cause the width of the pulse to grow and shrink as it propagates down the line, perhaps cancelling on average. Any systematic offsets will cause the pulse to shrink to nothing or to grow without bound as it propagates. In any event, information about the detailed timing of the initial pulse will have been completely lost. Now, consider the action of the feedback in the axon circuit (Figure 1). If Un were to be held low for a time longer than a l (i.e., the time it takes Vn to charge up to 8d, Un+1 would come back and release Un, regardless of the state of the input. Thus, the feedback enforces the condition On ::; a l . If 11 > h (i.e., a 2 < ad, a pulse whose initial width is larger than a 1 will be clipped to a 1 and then shrink down to nothing and disappear. In the event that II < h (i.e., a 2 > ad, a pulse whose initial duration is too small will grow up until its width is limited by the feedback. The axon circuit normally operates under the latter bias condition. The dynamics of the simple inverter chain cause a pulse which is to narrow to grow and the feedback loop serves to limit the pulse width; thus, the width of the pulse is restored in time . The feedback is also the source of the refractoriness in the axon; that is, until U n +l charges up to (roughly) 8 1 , V n -l can have no effect on Un. 3 EXPERIMENTAL DATA In this section, data from a twenty-five stage axon will be shown. The chip was fabricated in a standard 2f.lm p-well (Orbit) CMOS process through MOSIS. Uniform Axon A full space-time picture of pulse (taken at the v nodes of the circuit) propagation down a uniform axon is depicted on the left in Figure 4. The graph on the right in Figure 4 shows the same data from a different perspective. The lower sloped curve represents the time of the initial rapid discharge of the U node at each successive stage-this time marks the leading edge of the pulse taken at the v node of that stage. 744 Bradley A. Minch. Paul Hasler. Chris Diorio. Carver Mead The upper sloped curve marks the time of the final rapid discharge of the v node of each stage-this time is the end of the pulse taken at the v node of that stage. The propagation velocity of the pulse is given (in units of stages/s) by the reciprocal of the slope of the lower inclined curve. The third curve is the difference of the other two and represents the pulse width as a function of position along the axon. The graph on the left of Figure 5 shows propagation velocity as a function of the 72 bias voltage- so long as the pulse propagates, the velocity is nearly independent of 71. Two orders of magnitude of velocity are shown in the plot; these are especially well matched to the time scales of motion in auditory and visual sensory data. The circuit is tunable over a much wider range of velocities (from about one stage per second to well in excess of 10 4 stages/ s). The graph on the right of Figure 5 shows pulse width as a function of 71 for various values of 72-the pulse width is mainly determined by 71 with 72 setting a lower limit. Tapered Axon In biological axons, the propagation velocity of an action potential is related to the diameter of the axon- the bigger the diameter, the greater the velocity. If the axon were tapered, the velocity of the action potential would change as it propagated. If the bias transistors in the axon circuit are operated in their subthreshold region, the effect of an exponentially tapered axon can be simulated by applying a small voltage difference to the ends of each ofthe 71 and 72 bias lines. (Lyon & Mead , 1989) These narrow wires are made with a relatively resistive layer (polysilicon); hence, putting a voltage difference across the ends will linearly interpolate the bias voltages for each stage along the line. In subthreshold, the bias currents are exponentially related to the bias voltages. Since the pulse width and velocity are related to the bias currents , we expect that a pulse will either speed up and get narrower or slow down and get wider (depending on the sign of the applied voltage) exponentially as a function of position along the line. The graph on the left of Figure 6 depicts the boundaries of a pulse as it propagates along of the axon circuit for a positive (*'s) and negative (x 's) voltage difference applied to the 7 lines. The graph on the right of Figure 6 shows the corresponding pulse width for each applied voltage difference. Note that in each case, the width changes by more than an order of magnitude, but the pulse maintains its integrity. That is, the pulse does not disappear nor does it split into multiple pulses-this behavior would not be possible if the pulse width were not restored in time. 4 CONCLUSIONS In this paper we have presented a low-power, compact axon circuit, explained its operation , and presented data from a working chip fabricated through MOSIS. The circuit shares a number of features with its biological counterpart including an excitation threshold , a brief refractory period after pulse completion, pulse amplitude restoration , and pulse width restoration. It is tunable over orders of magnitude in pulse propagation velocity-including those well matched to the time scales of auditory and visual signals- and shows promise for use in synthetic neural systems which perform computations by correlating events which occur over both space and time such as those presented in (Horiuchi et ai, 1991) and (Lazzaro & Mead, 1989). A Silicon Axon 745 Acknowledgements This material is based upon work supported in part under a National Science Foundation Graduate Research Fellowship, the Office of Naval Research, DARPA, and the Beckman Foundation. References A. 1. Hodgkin and A. K. Huxley, (1952). A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve. Journal of Physiology, 117:6, 500-544. T. Horiuchi, J. Lazzaro , A. Moore , and C. Koch, (1991) . A Delay-Line Based Motion Detection Chip. Advances in Neural Information Processing Systems 3. San Mateo, CA: Morgan Kaufmann Publishers, Inc . 406-412 . J. Lazzaro and C. Mead, (1989). A Silicon Model of Auditory Localization. Neural Computation, 1:1 , 47-57. R . Lyon and C. Mead, (1989). Electronic Cochlea. Analog VLSI and Neural Systems. Reading, MA: Addison-Wesley Publishing Company, Inc. 279-302. E. R. Lewis , (1968). Using Electronic Circuits to Model Simple Neuroelectric Interactions. Proceedings of the IEEE, 56:6, 931-949. M. Mahowald and R. Douglas, (1991). A Silicon Neuron. Nature , 354:19 , 515-518. o.o35r-----.,-----.,--------, 0.03 2 0.025 1.5 ~ ., .& '0 0 .02 ~ f= 0.015 0 .5 > 0 -0 . 5 30 0.06 Stage o 0 TilTIe (s) ?0~--~1~0---~2~0---~30 Stage Figure 4: Pulse propagation along a uniform axon. (Left) Perspective view. (Right) Overhead view . *: pulse boundaries , x: pulse width. T1 = 0.720V, T2 = 0.780V . Velocity = 1, 100stages/s , Width = 3.8ms Bradley A. Minch, Paul Hasler, Chris Diorio, Carver Mead 746 10" r----~--~--~--____, II< o o "" """" "" """" 00 II< 'ttl" 3: "" ..."""" Tau2 = 0.700 V ~Xx 10.2 II< II< ~ "" 0 .740 Tau2 = 0.7 0 V ~o T a u2 = ~, 10-3 """" = +d> l!l ? II< Tau2 ~+ ~ II< ~ Tau2 = 0 .660 V ~ 110 11<... .820 V Ta\lp = 0 .860 V 10~~.5-----0~.6-----0~.~7-----0~.8~---0~.9 10~~ . 6-------0~.~7-------0~ . 8-------0~. 9 Tau2 (V) Taul (V) Figure 5: Uniform axon. (Left) Pulse velocity as a function of r2. (Right) Pulse width as a function of rl for various values of r2. 10" r------~--------~------___, 0 . 14r-------~-------~----~ 0 . 12 )0( II< 0 .1 il:! 0 .08 E::: II< )0( )O()O( x 0.06 x )0( )0( x )0( )0( )0( x 0.0 4#. . x )0( .... )0( )0( x )O()O( )0( )0( ><"XxX )0()0( X X X 1 O?3~------,'---------_'_------__=' 0 . 020~------~1~0~------2~0~------370 Stage o 10 20 30 Stage Figure 6: (Left) Pulse propagation along a tapered axon. (Right) Pulse width as 0.770V, r;ight 0.600V , a function of position along a tapered axon. *: ri eft eft r~eft 0.B20V , r;ight 0.650. x : ri 0.600V , r;ight = 0.770V, r~eft = 0.650V , r;ight = 0.B20V. = = = = = 

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A solvable connectionist model of immediate recall of ordered lists Neil Burgess Department of Anatomy, University College London London WC1E 6BT, England (e-mail: n.burgessOucl.ac.uk) Abstract A model of short-term memory for serially ordered lists of verbal stimuli is proposed as an implementation of the 'articulatory loop' thought to mediate this type of memory (Baddeley, 1986). The model predicts the presence of a repeatable time-varying 'context' signal coding the timing of items' presentation in addition to a store of phonological information and a process of serial rehearsal. Items are associated with context nodes and phonemes by Hebbian connections showing both short and long term plasticity. Items are activated by phonemic input during presentation and reactivated by context and phonemic feedback during output. Serial selection of items occurs via a winner-take-all interaction amongst items, with the winner subsequently receiving decaying inhibition. An approximate analysis of error probabilities due to Gaussian noise during output is presented. The model provides an explanatory account of the probability of error as a function of serial position, list length, word length, phonemic similarity, temporal grouping, item and list familiarity, and is proposed as the starting point for a model of rehearsal and vocabulary acquisition. 1 Introduction Short-term memory for serially ordered lists of pronounceable stimuli is well described, at a crude level, by the idea of an 'articulatory loop' (AL). This postulates that information is phonologically encoded and decays within 2 seconds unless refreshed by serial rehearsal, see (Baddeley, 1986). It successfully accounts for (i) 52 Neil Burgess the linear relationship between memory span s (the number of items s such that 50% of lists of s items are correctly recalled) and articulation rate r (the number of items that can be said per second) in which s ~ 2r + c, where r varies as a function of the items, language and development; (ii) the fact that span is lower for lists of phonemically similar items than phonemically distinct ones; (iii) unattended speech and articulatory distract or tasks (e.g. saying blah-blah-blah ... ) both reduce memory span. Recent evidence suggests that the AL plays a role in the learning of new words both during development and during recovery after brain traumas, see e.g. (Gathercole & Baddeley, 1993). Positron emission tomography studies indicate that the phonological store is localised in the left supramarginal gyrus, whereas subvocal rehearsal involves Broca's area and some of the motor areas involved in speech planning and production (Paulesu et al., 1993). However, the detail of the types of errors committed is not addressed by the AL idea. Principally: (iv) the majority of errors are 'order errors' rather than 'item errors', and tend to involve transpositions of neighbouring or phonemically similar items; (v) the probability of correctly recalling a list as a function of list length is a sigmoid; (vi) the probability of correctly recalling an item as a function of its serial position in the list (the 'serial position curve') has a bowed shape; (vii) span increases with the familiarity of the items used, specifically the c in s ~ 2r + c can increase from 0 to 2.5 (see (Hulme et al., 1991)), and also increases if a list has been previously presented (the 'Hebb effect'); (viii) 'position specific intrusions' occur, in which an item from a previous list is recalled at the same position in the current list. Taken together, these data impose strong functional constraints on any neural mechanism implementing the AL. Most models showing serial behaviour rely on some form of 'chaining' mechanism which associates previous states to successive states, via recurrent connections of various types. Chaining of item or phoneme representations generates errors that are incompatible with human data, particularly (iv) above, see (Burgess & Hitch, 1992, Henson, 1994). Here items are maintained in serial order by association to a repeatable time-varying signal (which is suggested by position specific intrusions and is referred to below as 'context'), and by the recovery from suppression involved in the selection process - a modification of the 'competitive queuing' model for speech production (Houghton, 1990). The characteristics of STM for serially ordered items arise due to the way that context and phoneme information prompts the selection of each item. 2 The model The model consists of 3 layers of artificial neurons representing context, phonemes and items respectively, connected by Hebbian connections with long and short term plasticity, see Fig. 1. There is a winner-take-all (WTA) interaction between item nodes: at each time step the item with the greatest input is given activation 1, and the others o. The winner at the end of each time step receives a decaying inhibition that prevents it from being selected twice consecutively. During presentation, phoneme nodes are activated by acoustic or (translated) visual input, activation in the context layer follows the pattern shown in Fig. 1, item nodes receive input from phoneme nodes via connections Wij. Connections A Solvable Connectionist Model of Immediate Recall of Ordered Lists A) .....1 - - - - nc 53 ---I.~ ? ? ? ? ? ? 00000 0 0 ? ? ? ? ? ? 0000 0 00 ? ? ? ? ? ? 0 0 0 0 B) context phonemes 0000000 Wij(t) items (WTA + suppression) ~ t=l t=2 t=3 . translated ..,/ visual input 0000000 \ acoustic input buffer [0 output Figure 1: A) Context states as a function of serial position t; filled circles are active nodes, empty circles are inactive nodes. B) The architecture of the model. Full lines are connections with short and long term plasticity; dashed lines are routes by which information enters the model. Wij (t) learn the association between the context state and the winning item, and Wij and Wij learn the association with the active phonemes. During recall, the context layer is re-activated as in presentation, activation spreads to the item layer (via Wij(t)) where one item wins and activates its phonemes (via Wij(t?. The item that now wins, given both context and phoneme inputs, is output, and then suppressed. As described so far, the model makes no errors. Errors occur when Gaussian noise is added to items' activations during the selection of the winning item to be output. Errors are likely when there are many items with similar activation levels due to decay of connection weights and inhibition since presentation. Items may then be selected in the wrong order, and performance will decrease with the time taken to present or recall a list. 2.1 Learning and familiarity Connection weights have both long and short term plasticity: W ij (t) (similarly Wij(t) and Wij(t)) have an incremental long term component Wi~(t), and a oneshot short term component Wl,(t) which decays by a factor b.. per second. The net weight of the connection is the sum of the two components: Wij(t) = Wi~(t)+W/i(t). Learning occurs according to: if Cj(t)ai(t) otherwise, > Wij(t)j 54 Neil Burgess { Wil(t) + eCj(t)Uoi(t) if Cj(t)Uoi(t) > 0; Wij (t) otherwise, (1) where Cj(t) and Uoi(t) are the pre- and post-connection activations, and e decreases with IW/i(t)1 so that the long term component saturates at some maximum value. These modifiable connection weights are never negative. An item's cfamiliarity' is reflected by the size of the long term components wfj and wfj of the weights storing the association with its phonemes. These components increase with each (error-free) presentation or recall of the item. For lists of totally unfamiliar items, the item nodes are completely interchangeable having only the short-term connections w!j to phoneme nodes that are learned at presentation. Whereas the presentation of a familiar item leads to the selection of a particular item node (due to the weights wfj) and, during output, this item will activate its phonemes more strongly due to the weights w! '. Unfamiliar items that are phonemically similar to a familiar item will tend to be represented by the familiar item node, and can take advantage of its long-term item-phoneme weights wfj. Presentation of a list leads to an increase in the long term component of the contextitem association. Thus, if the same list is presented more than once its recall improves, and position specific intrusions from previous lists may also occur. Notice that only weights to or from an item winning at presentation or output are increased. 3 Details There are nw items per list, np phonemes per item, and a phoneme takes time lp seconds to present or recall. At time t, item node i has activation Uoi(t) , context node i has activation Ci(t), Ct is the set of nc context nodes active at time t, phoneme node i has activation bi(t) and Pi is the set of np phonemes comprising item i. Context nodes have activation 0 or J3/2n c , phonemes take activation 0 or 1/ y'n;, so Wij(t) ~ J3/2n c and wlj(t) = Wji(t) ~ 1/h' see (1). This sets the relative effect that context and phoneme layers have on items' activation, and ensures that items of neither few nor many phonemes are favoured, see (Burgess & Hitch, 1992). The long-term components of phoneme-item weights wfj(t) and wji(t) are 0.45/ y'n; for familiar items, and 0.15/ y'n; for unfamiliar items (chosen to match the data in Fig. 3B). The long-term components of context-item weights Wi~(t) increase by 0.15/.Jn; for each ofthe first few presentations or recalls of a list. Apart from the WTA interaction, each item node i has input: (2) where Ii(t) < 0 is a decaying inhibition imposed following an item's selection at presentation or output (see below), TJi is a (0, u) Gaussian random variable added at output only, and Ei(t) is the excitatory input to the item from the phoneme layer during presentation and the context and phoneme layers during recall: during presentation; during recall. During recall phoneme nodes are activated according to bi(t) = 2:j Wij(t)aj(t). (3) A Solvable Connectionist Model of Immediate Recall of Ordered Lists 55 One time step refers to the presentation or recall of an item and has duration nplp. The variable t increases by 1 per time step, and refers to both time and serial position. Short term connection weights and inhibition Ii(t) decay by a factor .6. per second, or .6. nplp per time step. The algorithm is as follows; rehearsal corresponds to repeating the recall phase. Presentation o. Set activations, inhibitions and short term weights to zero, t = 1. 1. Set the context layer to state Ct : Ci(t) = J3/2n c if i E Ct ; Ci(t) = 0 otherwise. 2. Input items, i.e. set the phoneme layer to state 1't : bi(t) = 1/..;n; if i E 1't; bi(t) 0 otherwise. 3. Select the winning item, i.e. ak(t) = 1 where hk(t) = maJC.i{hi(t)}; ai(t) = 0, for i =1= k. 4. Learning, i.e. increment all connection weights according to (1). 5. Decay, i.e. multiply short-term connection weights Wl;(t), w[j(t) and w[j(t), and inhibitions Ii(t) by a factor .6.n plp. 6. Inhibit winner, i.e. set Ik(t) = -2, where k is the item selected in 3. 7. t ---+ t + 1, go to 1. Recall o. t = 1. 1. Set the context layer to state Ct , as above. 2. Set all phoneme activations to zero. 3. Select the winning item, as above. 4. Output. Activate phonemes via Wji(t), select the winning item (in the presence of noise) . 5. Learning, as above. 6. Decay, as above. 7. Inhibit winner, i.e. set Ik(t) = -2, where k is the item selected in 4. 8. t ---+ t + 1, go to 1. = 4 Analysis The output of the model, averaged over many trials, depends on (i) the activation values of all items at the output step for each time t and, (ii) given these activations and the noise level, the probability of each item being the winner. Estimation is necessary since there is no simple exact expression for (ii), and (i) depends on which items were output prior to time t. I define "Y(t, i) to be the time elapsed, by output at time t, since item i was last selected (at presentation or output), i.e. in the absence of errors: . {(t-i)lpnp ifi<t; "Y(t, l) = (nw - (i - t))lpnp if i 2: t. (4) If there have been no prior errors, then at time t the inhibition of item l IS Ii(t) = -2(.6.)7(t,i+l), and short term weights to and from item i have decayed by a factor .6.7(t,i). For a novel list of familiar items, the excitatory input to item i during output at time t is, see (3): Ei(t) = 3.6.7(t,i)IICi n Ct 11/2nc + (0.45 + .6.7(t,i))21I1'i n 1't II/np, (5) 56 Neil Burgess A) 0.90 0.85 0.80 s 0.6 0.75 2 6 4 2 4 6 Figure 2: Serial position curves. Full lines show the estimation, extra markers are error bars at one standard deviation of 5 simulations of 1,000 trials each, see ?5 for parameter values. A) Rehearsal. Four consecutive recalls of a list of 7 digits ('1', .. ,'4'). B) Phonemic similarity. SPCs are shown for lists of dissimilar letters ('d'), similar letters ('s'), and alternating similar and dissimilar letters with the similar ones in odd ('0') and even ('e') positions. C.f. (Baddeley, 1968, expt. V). where IIX II is the number of elements in set X. The probability p(t, i) that item i wins at time t function(Brindle, 1990): .) '" p (t, 1, '" IS estimated by the softmax exp (TrI.i (t)/ 0") (), , Lj=1 exp (mj t /0' ) ntu (6) where TrI.i(t) is hi(t) without the noise term, see (2-3), and 0" = 0.750'. For 0' = 0.5 (the value used below), the r.m.s. difference between p(t, i) estimated by simulation (500 trials) and by (6) is always less than 0.035 for -1 < TrI.i(t) < 1 with 2 to 6 items. Which items have been selected prior to time t affects Ii(t) in hi(t) via "I(t, i). p(t, i) is estimated for all combinations of up to two prior errors using (6) with appropriate values of TrI.i(t), and the average, weighted by the probability of each error combination, is used. The 'missing' probability corresponding to more than two prior errors is corrected for by normalising p(t, i) so that Li p(t, i) = 1 for t = 1, .. , nw' This overestimates the recency effect, especially in super-span lists. 5 Performance The parameter values used are Do = 0.75, nc = 6, 0' = 0.5. Different types of item are modelled by varying (np,lp) : 'digits' correspond to (2,0.15), 'letters' to (2,0.2), and 'words' to (5,0.15-0.3). 'Similar' items all have 1 phoneme in common, dissimilar items have none. Unless indicated otherwise, items are dissimilar and familiar, see ?3 for how familiarity is modelled. The size of 0' relative to Do is set so that digit span ~ 7. np and lp are such that approximately 7 digits can be said in 2 seconds. The model's performance is shown in Figs. 2 and 3. Fig. 2A: the increase in the long-term component of context-item connections during rehearsal brings stability after a small number of rehearsals, i.e. no further errors are committed. Fig. 2B: serial position curves show the correct effect of phonemic similarity among items. 57 A Solvable Connectionist Model of Immediate Recall of Ordered Lists B) r w 4 T 3 u n - 2 0 5 10 0.0 0.5 1.0 1.5 Figure 3: Item span. Full lines show the estimation, extra markers (A only) are error bars at one standard deviation of 3 simulations of 1,000 trials each, see ?5 and ?3 for parameter values. A) The probability of correctly recalling a whole list versus list length. Lists of digits ('d'), unfamiliar items (of the same length, 'u'), and experimental data on digits (adapted from (Guildford & Dallenbach, 1925), 'x') are shown. B) Span versus articulation rate (rate= 1/ipnp, with np = 5 and ip =0.15,0.2, and 0.3). Calculated curves are shown for novel lists of familiar ('f') and unfamiliar ('u') words and lists of familiar words after 5 repetitions ('r'). Data on recall of words ('w') and non-words ('n') are also shown, adapted from (Hulme et al., 1991). Fig. 3A: the probability of recalling a list correctly as a function of list length shows the correct sigmoidal relationship. Fig. 3B: item span shows the correct, approximately linear, relationship to articulation rate, with span for unfamiliar items below that for familiar items. Span increases with repeated presentations of a list in accordance with the 'Hebb effect'. Note that span is slightly overestimated for short lists of very long words. 5.1 Discussion and relation to previous work This model is an extension of (Burgess & Hitch, 1992), primarily to model effects of rehearsal and item and list familiarity by allowing connection weights to show plasticity over different timescales, and secondly to show recency and phonemic similarity effects simultaneously by changing the way phoneme nodes are activated during recall. Note that the 'context' timing signal varies with serial position: reflecting the rhythm of presentation rather than absolute time (indeed the effect of temporal grouping can be modelled by modifying the context representations to reflect the presence of pauses during presentation (Hitch et al., 1995)), so presentation and recall rates cannot be varied. The decaying inhibition that follows an items selection increases the locality of errors, i.e. if item i + 1 replaces item i, then item i is most likely to replace item i + 1 in turn (rather than e.g. item i+ 2). The model has two remaining problems: (i) selecting an item node to form the long term representation of a new item, without taking over existing item nodes, and (ii) learning the correct order of the phonemes within an item - a possible extension to address this problem is presented in (Hartley & Houghton, 1995). The mechanism for selecting items is a modification of competitive queuing 58 Neil Burgess (Houghton, 1990) in that the WTA interaction occurs at the item layer, rather than in an extra layer, so that only the winner is active and gets associated to context and phoneme nodes (this avoids partial associations of a context state to all items similar to the winner, which would prevent the zig-zag curves in Fig. 2B). The basic selection mechanism is sufficient to store serial order in itself, since items recover from suppression in the same order in which they were selected at presentation. The model ma.ps onto the articulatory loop idea in that the selection mechanism corresponds to part of the speech production ('articulation') system and the phoneme layer corresponds to the 'phonological store', and predicts that a 'context' timing signal is also present. Both the phoneme and context inputs to the item layer serve to increase span, and in addition, the former causes phonemic similarity effects and the latter causes recency, position specific intrusions and temporal grouping effects. 6 Conclusion I have proposed a simple mechanism for the storage and recall of serially ordered lists of items. The distribution of errors predicted by the model can be estimated mathematically and models a very wide variety of experimental data. By virtue of long and short term plasticity of connection weights, the model begins to address familiarity and the role of rehearsal in vocabulary acquisition. Many of the predicted error probabilities have not yet been checked experimentally: they are predictions. However, the major prediction of this model, and of (Burgess & Hitch, 1992), is that, in addition to a short-term store of phonological information and a process of sub-vocal rehearsal, STM for ordered lists of verbal items involves a third component which provides a repeatable time-varying signal reflecting the rhythm of the items' presentation. Acknowledgements: I am grateful for discussions with Rik Henson and Graham Hitch regarding data, and with Tom Hartley and George Houghton regarding error probabilities, and to Mike Page for suggesting the use of the softmax function. This work was supported by a Royal Society University Research Fellowship. References Baddeley AD (1968) Quarterly Journal of Ezperimental Pllychology 20 249-264. Baddeley AD (1986) Working Memory, Clarendon Press. Brindle, J S (1990) in: D S Tourebky (ed.) Advancell in Neural Information ProcelJlling Syatemll ! . San Mateo, CA: Morgan Kaufmann. Burgess N & Hitch G J (1992) J. Memory and Language 31 429-460. Gathercole S E & Baddeley A D (1993) Working memory and language, Erlbaum. Guildford J P & Dallenbach K M (1925) American J. of Pllychology 36 621-628. Hartley T & Houghton G (1995) J. Memory and Language to be published. Henson R (1994) Tech. Report, M.R.C. Applied Psychology Unit, Cambridge, U.K. Hitch G, Burgess N, Towse J & Culpin V (1995) Quart. J. of Ezp. Pllychology, submitted. Houghton G (1990) in: R Dale, C Mellish & M Zock (eds.), Current Rellearch in Natural Language Generation 287-319. London: Academic Press. Hulme C, Maughan S & Brown G D A (1991) J. Memory and Language 30685-701. Paulesu E, Frith C D & Frackowiak R S J (1993) Nature 362 342-344. PART II NEUROSCIENCE 

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An Alternative Model for Mixtures of Experts Lei Xu Dept. of Computer Science, The Chinese University of Hong Kong Shatin, Hong Kong, Emaillxu@cs.cuhk.hk Michael I. Jordan Dept. of Brain and Cognitive Sciences MIT Cambridge, MA 02139 Geoffrey E. Hinton Dept. of Computer Science University of Toronto Toronto, M5S lA4, Canada Abstract We propose an alternative model for mixtures of experts which uses a different parametric form for the gating network. The modified model is trained by the EM algorithm. In comparison with earlier models-trained by either EM or gradient ascent-there is no need to select a learning stepsize. We report simulation experiments which show that the new architecture yields faster convergence. We also apply the new model to two problem domains: piecewise nonlinear function approximation and the combination of multiple previously trained classifiers. 1 INTRODUCTION For the mixtures of experts architecture (Jacobs, Jordan, Nowlan & Hinton, 1991), the EM algorithm decouples the learning process in a manner that fits well with the modular structure and yields a considerably improved rate of convergence (Jordan & Jacobs, 1994). The favorable properties of EM have also been shown by theoretical analyses (Jordan & Xu, in press; Xu & Jordan, 1994). It is difficult to apply EM to some parts of the mixtures of experts architecture because of the nonlinearity of softmax gating network. This makes the maximiza- 634 Lei Xu, Michael!. Jordan, Geoffrey E. Hinton tion with respect to the parameters in gating network nonlinear and analytically unsolvable even for the simplest generalized linear case. Jordan and Jacobs (1994) suggested a double-loop approach in which an inner loop of iteratively-reweighted least squares (IRLS) is used to perform the nonlinear optimization. However, this requires extra computation and the stepsize must be chosen carefully to guarantee the convergence of the inner loop. We propose an alternative model for mixtures of experts which uses a different parametric form for the gating network. This form is chosen so that the maximization with respect to the parameters of the gating network can be handled analytically. Thus, a single-loop EM can be used, and no learning stepsize is required to guarantee convergence. We report simulation experiments which show that the new architecture yields faster convergence. We also apply the model to two problem domains. One is a piecewise nonlinear function approximation problem with smooth blending of pieces specified by polynomial, trigonometric, or other prespecified basis functions. The other is to combine classifiers developed previously-a general problem with a variety of applications (Xu, et al., 1991, 1992). Xu and Jordan (1993) proposed to solve the problem by using the mixtures of experts architecture and suggested an algorithm for bypassing the difficulty caused by the softmax gating networks. Here, we show that the algorithm of Xu and Jordan (1993) can be regarded as a special case of the single-loop EM given in this paper and that the single-loop EM also provides a further improvement. 2 MIXTURES OF EXPERTS AND EM LEARNING The mixtures of experts model is based on the following conditional mixture: K P(ylx,6) I: 9j(X, lI)P(ylx, OJ), P(ylx, OJ) where x ERn, and 6 consists of 1I,{Oj}f, and OJ consists of {wj}f,{rj}f. The vector Ij (x, Wj) ~s the output ofthe j-th expert net. The scalar 9j (x, 1I), j 1, ... , K is given by the softmax function: = 9j(X,1I) = e.B;(x,v)/I:e.B;(X,v). (2) In this equation, Pj (x, 1I), j = 1, ... ,K are the outputs of the gating network. The parameter 6 is estimated by Maximum Likelihood (ML), where the log likelihood is given by L = Lt In P(y(t) Ix(t), 6). The ML estimate can be found iteratively using the EM algorithm as follows. Given the current estimate 6(1~), each iteration consists of two steps. (1) E-step. For each pair {x(t),y(t)}, compute h~k)(y(t)lx(t?) = PUlx(t),y(t?), and then form a set of objective functions: Qj(Oj) I:h;k)(y(t)lx(t?) InP(y(t)lx(t), OJ), j = 1,,, ',K; An Alternative Model for Mixtures of Experts Qg(V) = 635 L L h)k) (y(t) Ix(t?) lngt) (x(t) ,v(k?). t (3) j (2). M-step. Find a new estimate e(k+l) = {{ot+l)}f=I,V(k+l)} with: OJk+l) = argmax Qj(Oj), j = 1, ... , K; ~ v(k+l) = arg max Qg(v). (4) v In certain cases, for example when I; (x, Wj) is linear in the parameters Wj, maXe j Qj(Oj) can be solved by solving 8Qj/80j = O. When l;(x,Wj) is nonlinear with respect to Wj, however, the maximization cannot be performed analytically. Moreover, due to the nonlinearity of softmax, maXv Qg(v) cannot be solved analytically in any case. There are two possibilities for attacking these nonlinear optimization problems. One is to use a conventional iterative optimization technique (e.g., gradient ascent) to perform one or more inner-loop iterations. The other is to simply find a new estimate such that Qj(ot+l?) ~ Qj(Ojk?), Qg(v(k+l?) ~ Qg(v(k?). Usually, the algorithms that perform a full maximization during the M step are referred as "EM" algorithms, and algorithms that simply increase the Q function during the M step as "GEM" algorithms. In this paper we will further distinguish between EM algorithms requiring and not requiring an iterative inner loop by designating them as double-loop EM and single-loop EM respectively. Jordan and Jacobs (1994) considered the case of linear {3j(x, v) = vJ[x,l] with v = [VI,???, VK] and semi-linear I; (wn x , 1]) with nonlinear 1;(.). They proposed a double-loop EM algorithm by using the IRLS method to implement the inner-loop iteration. For more general nonlinear {3j(x, v) and I;(x, OJ), Jordan and Xu (in press) showed that an extended IRLS can be used for this inner loop. It can be shown that IRLS and the extension are equivalent to solving eq. (3) by the so-called Fisher Scoring method. 3 A NEW GATING NET AND A SINGLE-LOOP EM To sidestep the need for a nonlinear optimization routine in the inner loop of the EM algorithm, we propose the following modified gating network: gj(x, v) = CkjP(xlvj)/ L:i CkiP(xIVi) , L:j Ckj = 1, Ckj ~ 0, P(xIVj) = aj(vj)-lbj(x) exp{cj(Vj)Ttj(x)} (5) where v = {Ckj,Vj,j = 1,???,K}, tj(x) is a vector of sufficient statistics, and the P(xlvj)'s are density functions from the exponential family. The most common example is the Gaussian: (6) In eq. (5), gj(x, v) is actually the posterior probability PUlx) that x is assigned to the partition corresponding to the j-th expert net, obtained from Bayes' rule: gj(x, v) = PUlx) = CkjP(xIVj)/ P(x, v), P(x, v) = L CkiP(xlvi). (7) 636 Lei Xu, Michael I. Jordan, Geoffrey E. Hinton Inserting this 9j(X, v) into the model eq. (1), we get P(ylx, 8) ?) = "L-. a?P(xlv ~( ) P(ylx, OJ). X,V (8) 3 If we do ML estimation directly on this P(ylx,8) and derive an EM algorithm, we again find that the maximization maXv Q9(v) cannot be solved analytically. To avoid this difficulty, we rewrite eq. (8) as: P(y, x) = P(ylx, 8)P(x, v) = L ajP(xlvj)P(ylx, OJ). (9) j This suggests an asymmetrical representation for the joint density. We accordingly perform ML estimation based on L' = 2:t In P(y(t), x(t?) to determine the parameters a j , Vj, OJ of the gating net and the expert nets. This can be done by the following EM algorithm: (1) E-step. Compute h(k)(y(t) Ix(t?) _ j - a\k) P( x(t) Iv~k ?)P(y(t) Ix(t) O<k?) 3 3 ' 3 . 2:i a~k) P(x(t)lv??)P(y(t)lx(t), 0)"?)' (10) Then let Qj(Oj),j = 1,? .. , K be the same as given in eq. (3), and decompose Q9(v) further into L h)k '(y(t) Ix(t?) In P( x(t) IVj), j = 1, ... , K; t LLh;k)(y(t)lx(t?)lnaj, with a= {al, .. ?,aK}. (11) j (2). M-step. Find a new estimate for j = 1,???, K O;k+l) = argmaXS j Qj(Oj), V]"+l) = argmaxllj QJ(Vj)' a(k+1) = arg maXa, QlX, s.t. 2:j aj = 1. (12) The maximization for the expert nets is the same as in eq. (4). However, for the gating net the maximization now becomes analytically solvable as long as P(xlvj) is from the exponential family. That is, we have: V~k+l) = 2:t h)")(y(t) Ix(t?)tj(x(t?) 3 2:t h)k)(y(t)lx(t?) , a;k+l) = ~ L h)")(y(t) Ix(t?). t In particular, when P(xIVj) is a Gaussian density, the update becomes: (13) An Alternative Model for Mixtures of Experts 637 Two issues deserve to be emphasized further: (1) The gating nets eq. (2) and eq. (5) become identical when f3j(x, v) = lnaj + In bj (x) +Cj (Vj)T tj (x) -In aj(vj) . In other words, the gating net in eq. (5) explicitly uses this function family instead of the function family defined by a multilayer feedforward network. (2) It follows from eq. (9) that max In P(y, xiS) = max [In P(ylx, S) + In P(xlv)]. So, the solution given by eqs. (10) through (14) is actually different from the one given by the original eqs. (3) and (4). The former tries to model both the mapping from x to y and the input x, while the latter only models the mapping from x and y. In fact, here we learn the parameters of the gating net and the expert nets via an asymmetrical representation eq. (9) of the joint density P(y, x) which includes P(ylx) implicitly. However, in the testing phase, the total output still follows eq. (8). 4 PIECEWISE NONLINEAR APPROXIMATION The simple form /j(x, Wj) = wJ[x , 1] is not the only case to which single-loop EM applies. Whenever /j(x, Wj) can be written in a form linear in the parameters: (15) where 4>i,j(X) are prespecified basis functions, maX8 j Qf!(Oj),j = 1"", K in eq. (3) is still a weighted least squares problem that can be soived analytically. One useful special case is when 4>i,j(X) are canonical polynomial terms X~l .. 'X~d, r j ~ 0. In this case, the mixture of experts model implements piecewise polynomial approximations. Another case is that 4>i,j(X) is TIi sini (jll'xt) cosi(jll'xt}, ri ~ 0, in which case the mixture of experts implements piecewise trigonometric approximations. 5 COMBINING MULTIPLE CLASSIFIERS Given pattern classes Ci, i = 1, ... , M, we consider classifiers ej that for each input x produce an output Pj(ylx): Pj(ylx) = [Pj(ll x ), ... ,pj(Mlx)), pj(ilx) ~ 0, LPj(ilx) = 1. (16) The problem of Combining Multiple Classifiers (CMC) is to combine these Pj(ylx)'s to give a combined estimate of P(ylx) . Xu and Jordan (1993) proposed to solve CMC problems by regarding the problem as a special example of the mixture density problem eq. (1) with the Pj(ylx)'s known and only the gating net 9j(X, v) to be learned. In Xu and Jordan (1993), one problem encountered was also the nonlinearity of softmax gating networks, and an algorithm was proposed to avoid the difficulty. Actually, the single-loop EM given by eq. (10) and eq. (13) can be directly used to solve the CMC problem. In particular, when P(xlvj) is Gaussian, eq. (13) becomes eq. (14). Assuming that al = . . . = aK in eq. (7), eq. (10) becomes 638 Lei Xu, Michaell. Jordan, Geoffrey E. Hinton h)k) (y(t) Ix(t)) = P(x(t)lvt))P(y(t)lx(t))/ L:i P(x(t)lvi(k))p(y(t)lx(t)). If we divide L:i P(x(t) Ivi(k)), we get ht)(y(t) Ix(t)) = gj(x, v)P(y(t)lx(t))/ L:i gj(x, v)P(y(t)lx(t)) . Comparing this equation with eq. (7a) in Xu and Jordan (1993), we can see that the two equations are actually the same. Despite the different notation, C?j(x) and Pj(y1.t) Ix(t)) in Xu and Jordan (1993) are the same as gj(x, v) and P(y(t)lx(t)) in Section 3. So the algorithm of Xu and Jordan (1993) is a special case of the single-loop EM given in Section 3. both the numerator and denominator by 6 SIMULATION RESULTS We compare the performance of the EM algorithm presented earlier with the model of mixtures of experts presented by Jordan and Jacobs (1994). As shown in Fig. l(a), we consider a mixture of experts model with K 2. For the expert nets, each P(ylx, OJ) is Gaussian given by eq. (1) with linear !;(x,Wj) = wJ[x, 1] . For the new gating net, each P(x, Vj) in eq. (5) is Gaussian given by eq. (6) . For the old gating net eq. (2), f31(X,V) 0 and f32(X,v) = vT[x, 1]. The learning speeds of the two are significantly different. The new algorithm takes k=15 iterations for the log-likelihood to converge to the value of -1271.8. These iterations require about 1,351,383 MATLAB flops. For the old algorithm, we use the IRLS algorithm given in Jordan and Jacobs (1994) for the inner loop iteration. In experiments, we found that it usually took a large number of iterations for the inner loop to converge. To save computations, we limit the maximum number of iterations by Tmaz = 10. We found that this saved computation without obviously influencing the overall performance. From Fig. 1(b), we see that the outer loop converges in about 16 iterations. Each inner loop takes 290498 flops and the entire process requires 5,312,695 flops. So, we see that the new algorithm yields a speedup of about 4,648,608/1,441,475 = 3.9. Moreover, no external adjustment is needed to ensure the convergence of the new algorithm. But for the old one the direct use of IRLS can make the inner loop diverge and we need to appropriately rescale the updating stepsize of IRLS. = = Figs. 2(a) and (b) show the results of a simulation of a piecewise polynomial approximation problem utilizing the approach described in Section 4. We consider a mixture of experts model with K = 2. For expert nets, each P(ylx, OJ) is Gaussian given by eq. (1) with !;(x,Wj) = W3,jX 3 +W2,jX 2 +Wl,jX+WD,j. In the new gating net eq. (5), each P(x, Vj) is again Gaussian given by eq. (6) . We see that the higher order nonlinear regression has been fit quite well. For multiple classifier combination, the problem and data are the same as in Xu and Jordan (1993). Table 1 shows the classification results. Com-old and Com-new denote the method given in in Xu and Jordan (1993) and in Section 5 respectively. We see that both improve the classification rate of each individual network considerably and that Com - new improves on Com - old. Training set Testing set Classifer el 89.9% 89.2% Classifer el 93.3% 92.7% Com- old Com- new 98.6% 99.4% 98.0% 99.0% Table 1 A comparison of the correct classification rates An Alternative Model for Mixtures of Experts o.s U (a) 2 U 3 U 639 4 (b) Figure 1: (a) 1000 samples from y = a1X + a2 + c, a1 = 0.8, a2 = 0.4, x E [-1,1.5] with prior a1 = 0.25 and y = ai x + a~ + c, ai = 0.8, a2 =' 2.4, x E [1,4] with prior a2 = 0.75, where x is uniform random variable and z is from Gaussian N(O, 0.3). The two lines through the clouds are the estimated models of two expert nets. The fits obtained by the two learning algorithms are almost the same. (b) The evolution of the log-likelihood. The solid line is for the modified learning algorithm. The dotted line is for the original learning algorithm (the outer loop iteration). 7 REMARKS Recently, Ghahramani and Jordan (1994) proposed solving function approximation problems by using a mixture of Gaussians to estimate the joint density of the input and output (see also Specht, 1991; Tresp, et al., 1994). In the special case of linear I;(x, Wj) = wnx,I] and Gaussian P(xlvj) with equal priors, the method given in Section 3 provides the same result as Ghahramani and Jordan (1994) although the parameterizations of the two methods are different. However, the method of this paper also applies to nonlinear l;(x,Wj) = Wn<pj(x) , 1] for piecewise nonlinear approximation or more generally I; (x, Wj) that is nonlinear with respect to Wj, and applies to cases in which P(y, xlvj, OJ) is not Gaussian, as well as the case of combining multiple classifiers. Furthermore, the methods proposed in Sections 3 and 4 can also be extended to the hierarchical mixture of experts architecture (Jacobs & Jordan, 1994) so that single-loop EM can be used to facilitate its training. References Ghahramani, Z., & Jordan, M.I. (1994). Function approximation via density estimation using the EM approach. In Cowan, J.D., Tesauro, G., and Alspector, J., (Eds.), Advances in NIPS 6. San Mateo, CA: Morgan Kaufmann. Jacobs, R.A., Jordan, M.L, Nowlan, S.J., & Hinton, G.E. (1991). Adaptive mixtures of local experts. Neural Computation, 9, 79-87. 640 Lei Xu, Michael!. Jordan, Geoffrey E. Hinton 110' 0 .as -\ I1 -\.5 -2 ~ -2.5 -2 -3 -3.5 -4 0 (a) (b) Figure 2: Piecewise 3rd polynomial approximation. (a) 1000 samples from y = alx3+a3x+a4+c, x E [-1,1.5] with prior 0'1 = 0.4 and y = a~x2+a~x2+a4+c, x E [1,4] with prior 0'2 = 0.6, where x is uniform random variable and z is from Gaussian N(0,0.15). The two curves through the clouds are the estimated models of two expert nets. (b) The evolution of the log-likelihood. Jordan, M.I., & Jacobs, R.A. (1994). Hierarchical mixtures of experts and the EM algorithm. Neural Computation, 6, 181-214. Jordan, M.I., & Xu, L. (in press). Convergence results for the EM approach to mixtures-of-experts architectures. Neural Networks. Specht, D. (1991). A general regression neural network. IEEE Trans. Networks, 2, 568-576. Neural Tresp, V., Ahmad, S., and Neuneier, R. (1994). Training neural networks with deficient data. In Cowan, J.D., Tesauro, G., & Alspector, J., (Eds.), Advances in NIPS 6, San Mateo, CA: Morgan Kaufmann. Xu, L., Krzyzak A., & Suen, C.Y. (1991). Associative switch for combining multiple classifiers. Proc. of 1991 HCNN, Vol. I. Seattle, 43-48. Xu, L., Krzyzak A., & Suen, C.Y. (1992). Several methods for combining multiple classifiers and their applications in handwritten character recognition. IEEE Thans. on SMC, Vol. SMC-22, 418-435. Xu, L., & Jordan, M.I. (1993). EM Learning on a generalized finite mixture model for combining multiple classifiers. Proceedings of World Congress on Neural Networks, Vol. IV. Portland, OR, 227-230. Xu, L., & Jordan, M.I. (1994). On convergence properties of the EM algorithm for Gaussian mixtures. Submitted to Neural Computation. 

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Classifying with Gaussian Mixtures and Clusters Nanda Kambhatla and Todd K. Leen Department of Computer Science and Engineering Oregon Graduate Institute of Science & Technology P.O. Box 91000 Portland, OR 97291-1000 nanda@cse.ogi.edu, tleen@cse.ogi.edu Abstract In this paper, we derive classifiers which are winner-take-all (WTA) approximations to a Bayes classifier with Gaussian mixtures for class conditional densities. The derived classifiers include clustering based algorithms like LVQ and k-Means. We propose a constrained rank Gaussian mixtures model and derive a WTA algorithm for it. Our experiments with two speech classification tasks indicate that the constrained rank model and the WTA approximations improve the performance over the unconstrained models. 1 Introduction A classifier assigns vectors from Rn (n dimensional feature space) to one of K classes, partitioning the feature space into a set of K disjoint regions. A Bayesian classifier builds the partition based on a model of the class conditional probability densities of the inputs (the partition is optimal for the given model). In this paper, we assume that the class conditional densities are modeled by mixtures of Gaussians. Based on Nowlan's work relating Gaussian mixtures and clustering (Nowlan 1991), we derive winner-take-all (WTA) algorithms which approximate a Gaussian mixtures Bayes classifier. We also show the relationship of these algorithms to non-Bayesian cluster-based techniques like LVQ and k-Means. The main problem with using Gaussian mixtures (or WTA algorithms thereof) is the explosion in the number of parameters with the input dimensionality. We propose 682 Nanda Kambhatla. Todd K. Leen a constrained rank Gaussian mixtures model for classification. Constraining the rank of the Gaussians reduces the effective number of model parameters thereby regularizing the model. We present the model and derive a WTA algorithm for it. Finally, we compare the performance of the different mixture models discussed in this paper for two speech classification tasks. 2 Gaussian Mixture Bayes (GMB) classifiers Let x denote the feature vector (x E 'Rn ), and {n I , I = 1, ... , K} denote the classes. Class priors are denoted p(nI) and the class-conditional densities are denoted p(x InI). The discriminant function for the Bayes classifier is (1) An input feature vector x is assigned to class I if 6 I(x) > 6J (x) 'VJ:F I . Given the class conditional densities, this choice minimizes the classification error rate (Duda and Hart 1973). We model each class conditional density by a mixture composed of QI component Gaussians. The Bayes discriminant function (see Figure 1) becomes QI I "j,foiTlexp[-~(X-~J{~J-\x-~J)], j'(x)=PW)L ;=1 I~fl (21r)n/2 where ~J and ~f are the mean and the covariance matrix of the ponent for nl. lh (2) mixture com- 0.12 0.1 0.08 0.06 0.04 0.02 5 10 --~)o 20 X 25 Fig. 1: Figure showing the decision rule of a GMB classifier for a two class problem with one input feature. The horizontal axis represents the feature and the vertical axis represents the Bayes discriminant functions. In this example, the class conditional densities are modelled as a mixture of two Gaussians and equal priors are assumed. To implement the Gaussian mixture Bayes classifier (GMB) we first separate the training data into the different classes. We then use the EM algorithm (Dempster Classifying with Gaussian Mixtures and Clusters 683 et al 1977, Nowlan 1991) to determine the parameters for the Gaussian mixture density for each class. 3 Winner-take-all approximations to G MB classifiers In this section, we derive winner-take-all (WTA) approximations to GMB classifiers. We also show the relationship of these algorithms to non-Bayesian cluster-based techniques like LVQ and k-Means. 3.1 The WTA model for GMB The WTA assumptions (relating hard clustering to Gaussian mixtures; see (Nowlan 1991)) are: ? p(x 10 1 ) are mixtures of Gaussians as in (2). ? The summation in (2) is dominated by the largest term. This is "equivalent to assigning all of the responsibility for an observation to the Gaussian with the highest probability of generating that observation" (Nowlan 1991). To draw the relation between GMB and cluster-based classifiers, we further assume that: ? The mixing proportions (oj) are equal for a given class. ? The number of mixture components QI is proportional to p(O/). Applying all the above assumptions to (2), taking logs and discarding the terms that are identical for each class, we get the discriminant function AI [1"2 log IEj!) + "21(x - ILj) 1 Q. (I -y (x) = - ~f I T I-I ~j I ] (x - ILj) . (3) The discriminant function (3) suggests an algorithm that approximates the Bayes classifier. We segregate the feature vectors by class and then train a separate vector quantizer (VQ) for each class. We then compute the means ILl and the covariance each Voronoi cell of each quantizer, and use (3) for classifying new matrices patterns. We call this algorithm VQ-Covariance. Note that this algorithm does not do a maximum likelihood estimation of its parameters based on the probability model used to derive (3). The probability model is only used to classify patterns. E1Jor 3.2 The relation to LVQ and k-Means Further assume that for each class, the mixture components are spherically sym= 0"2 I, with 0"2 identical for all classes. We obtain metric with covariance matrix the discriminant function, El A -y/(x) =- QI r?n )=1 II x - J.t)~ 2 II . (4) 684 Nanda Kamblwtla, Todd K. Leen This is exactly the discriminant function used by the learning vector quantizer (LVQ; Kohonen 1989) algorithm. Though LVQ employs a discriminatory training procedure (i.e it directly learns the class boundaries and does not explicitly build a separate model for each class), the implicit model of the class conditional densities used by LVQ corresponds to a GMB model under all the assumptions listed above. This is also the implicit model underlying any classifier which makes its classification decision based on the Euclidean distance measure between a feature vector and a set of prototype vectors (e.g. a k-Means clustering followed by classification based on (4)) . 4 Constrained rank GMB classifiers In the preceding sections, we have presented a GMB classifier and some WTA approximations to GMB. Mixture models such as GMB generally have too many parameters for small data sets. In this section, we propose a way of regularizing the mixture densities and derive a WTA classifier for the regularized model. 4.1 The constrained rank model In section 2, we assumed that the class conditional densities of the feature vectors x are mixtures of Gaussians (5) where p.J and EJ are the means and covariance matrices for the jth component Gaussian. eJi and AJi are the orthonormal eigenvectors and eigenvalues of I::Ji (ordered such that A}l ~ '" ~ AJn). In (5), we have written the Mahalanobis distance in terms of the eigenvectors. For a particular data point x, the Mahalanobis distance is very sensitive to changes in the squared projections onto the trailing eigen-directions, since the variances are very small in these directions. This is a potential problem with small data sets. When there are insufficient data points to estimate all the parameters of the mixture density accurately, the trailing eigen-directions and their associated eigenvalues are likely to be poorly estimated. Using the Mahalanobis distance in (5) can lead to erroneous results in such cases. We propose a method for regularizing Gaussian mixture classifiers based on the above ideas. We assume that the trailing n - m eigen-directions of each Gaussian component are inaccurate due to overfitting to the training set. We rewrite the class conditional densities (5) retaining only the leading m (0 < m ::; n) eigen-directions Classifying with Gaussian Mixtures and Clusters 685 in the determinants and the Mahalanobis distances p(x IIll) = f j=1 m 2)ft=1 \i I m (21T) / exp [-~(x - ILJ{ (f e;~fiT) (x - ILJ)] i=l Jl (6) We choose the value of m (the reduced rank) by cross-validation over a separate validation set. Thus, our model can be considered to be regularizing or constraining the class conditional mixture densities. If we apply the above model and derive the Bayes discriminant functions (1), we get, 8I (x) = p(nI) ~ j=1 m (21T) / 2 a} m I exp y'ft=l \i [-~(X - ILJ)T (f e}~f/) (x - ILJ)]. i=1 Jl (7) We can implement a constrained rank Gaussian mixture Bayes (GMB-Reduced) classifier based on (7) using the EM algorithm to determine the parameters of the mixture density for each class. We segregate the data into different classes and use the EM algorithm to determine the parameters of the full mixture density (5). We then use (7) to classify patterns. 4.2 A constrained rank WTA algorithm We now derive a winner-take-all (WTA) approximation for the constrained rank mixture model described above. We assume (similar to section 3.1) that ? p(x I nI) are constrained mixtures of Gaussians as in (6). ? The summation in (6) is dominated by the largest term (the WTA assumption). . ? The mixing proportions (a}) are equal for a given class and the number of components QI is proportional to p(nI). Applying these assumptions to (7), taking logs and discarding the terms that are identical for each class, we get the discriminant function (8) It is interesting to compare (8) with (3) . Our model postulates that the trailing m eigen-directions of each Gaussian represent overfitting to noise in the training set. The discriminant functions reflect this; (8) retains only those terms of (3) which are in the leading m eigen-directions of each Gaussian. n- We can generate an algorithm based on (8) that approximates the reduced rank Bayes classifier. We separate the data based on classes and train a separate vector the covariance quantizer (VQ) for each class. We then compute the means matrices ~} for each Voronoi cell of each quantizer and the orthonormal eigenvectors IL}, 686 Nanda Kambhatla. Todd K. Leen Table 1: The test set classification accuracies for the TIMIT vowels data for different algorithms. ALGORITHM MLP (40 nodes in hidden layer) GMB (1 component; full) GMB (1 component; diagonal) GMB-Reduced (1 component; 13-D) VQ-Covariance (1 component) VQ-Covariance-Reduced (1 component; 13-D) LVQ (48 cells) ACCURACY 46.8% 41.4% 46.3% 51.2% 41.4% 51.2% 41.4% eJi and eigenvalues AJ for each covariance matrix EJ. We use (8) for classifying new patterns. Notice that the algorithm described above is a reduced rank version of VQ-Covariance (described in section 3.1). We call this algorithm VQ-CovarianceReduced. 5 Experimental Results In this section we compare the different mixture models and a multi layer perceptron (MLP) for two speech phoneme classification tasks. The measure used is the classification accuracy. 5.1 TIMIT data The first task is the classification of 12 monothongal vowels from the TIMIT database (Fisher and Doddington 1986). Each feature vector consists of the lowest 32 DFT coefficients, time-averaged over the central third of the vowel. We partitioned the data into a training set (1200 vectors), a validation set (408 vectors) for model selection, and a test set (408 vectors). The training set contained 100 examples of each class. The values of the free parameters for the algorithms (the number of component densities, number of hidden nodes for the MLP etc.) were selected by maximizing the performance on the validation set. Table 1 shows the results obtained with different algorithms. The constrained rank models (GMB-Reduced and VQ-Covariance-Reduced 1 ) perform much better than all the unconstrained ones and even beat a MLP for this task. This data set consists of very few data points per class, and hence is particularly susceptible to over fitting by algorithms with a large number of parameters (like GMB). It is not surprising that constraining the number of model parameters is a big win for this task. INote that since the best validation set performance is obtained with only one component for each mixture density, the WTA algorithms are identical to the GMB algorithms (for these results). Classifying with Gaussian Mixtures and Clusters 687 Table 2: The test set classification accuracies for the CENSUS data for different algorithms. ALGORITHM MLP (80 nodes in hidden layer) GMB (1 component; full) GMB (8 components; diagonal) GMB-Reduced (2 components; 35-D) VQ-Covariance (3 components) VQ-Covariance-Reduced (4 components; 38-D) LVQ (55 cells) 5.2 ACCURACY 88.2% 77.2% 70.9% 82.5% 77.5% 84.2% 67.3% CENSUS data The next task we experimented with was the classification of 9 vowels (found in the utterances ofthe days of the week). The data was drawn from the CENSUS speech corpus (Cole et alI994). Each feature vector was 70 dimensional (perceptual linear prediction (PLP) coefficients (Hermansky 1990) over the vowel and surrounding context}. We partitioned the data into a training set (8997 vectors), a validation set (1362 vectors) for model selection, and a test set (1638 vectors). The training set had close to a 1000 vectors per class. The values of the free parameters for the different algorithms were selected by maximizing the validation set performance. Table 2 gives a summary of the classification accuracies obtained using the different algorithms. This data set has a lot more data points per class than the TIMIT data set. The best accuracy is obtained by a MLP, though the constrained rank mixture models still greatly outperform the unconstrained ones. 6 Discussion We have derived WTA approximations to GMB classifiers and shown their relation to LVQ and k-Means algorithms. The main problem with Gaussian mixture models is the explosion in the number of model parameters with input dimensionality, resulting in poor generalization performance. We propose constrained rank Gaussian mixture models for classification. This approach ignores some directions ( "noise") locally in the input space, and thus reduces the effective number of model parameters. This can be considered as a way of regularizing the mixture models. Our results with speech vowel classification indicate that this approach works better than using full mixture models, especially when the data set size is small. The WTA algorithms proposed in this paper do not perform a maximum likelihood estimation of their parameters. The probability model is only used to classify data. We can potentially improve the performance of these algorithms by doing maximum likelihood training with respect to the models presented here. 688 Nanda Kambhatla, Todd K. Leen Acknowledgments This work was supported by grants from the Air Force Office of Scientific Research (F49620-93-1-0253), Electric Power Research Institute (RP8015-2) and the Office of Naval Research (NOOOI4-91-J-1482). We would like to thank Joachim Utans, OGI for several useful discussions and Zoubin Ghahramani, MIT for providing MATLAB code for the EM algorithm. We also thank our colleagues in the Center for Spoken Language Understanding at OGI for providing speech data. References R.A. Cole, D.G. Novick, D. Burnett, B. Hansen, S. Sutton, M. Fanty. (1994) Towards Automatic Collection of the U.S. Census. Proceedings of the International Conference on Acoustics, Speech and Signal Processing 1994. A.P. Dempster, N.M. Laird, and D.B. Rubin. (1977) Maximum Likelihood from Incomplete Data via the EM Algorithm. J. Royal Statistical Society Series B, vol. 39, pp. 1-38. R.O. Duda and P.E. Hart. (1973) Pattern Classification and Scene Analysis. John Wiley and Sons Inc. W.M Fisher and G.R Doddington. (1986) The DARPA speech recognition database: specification and status. In Proceedings of the DARPA Speech Recognition Workshop, p93-99, Palo Alto CA. H. Hermansky. (1990) Perceptual Linear Predictive (PLP) analysis of speech. J. Acoust. Soc. Am., 87(4):1738-1752. T. Kohonen. (1989) Self-Organization and Associative Memory (3rd edition). Berlin: Springer-Verlag. S.J. Nowlan. (1991) Soft Competitive Adaptation: Neural Network Learning Algorithms based on Fitting Statistical Mixtures. CMU-CS-91-126 PhD thesis, School of Computer Science, Carnegie Mellon University. 

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The Use of Dynamic Writing Information in a Connectionist On-Line Cursive Handwriting Recognition System Stefan Manke Michael Finke University of Karlsruhe Computer Science Department D-76128 Karlsruhe, Germany mankeCO)ira. uka.de, finkem@ira.uka.de Alex Waibel Carnegie Mellon University School of Computer Science Pittsburgh, PA 15213-3890, U.S.A. wai bel CO) cs. cm u.ed u Abstract In this paper we present NPen ++, a connectionist system for writer independent, large vocabulary on-line cursive handwriting recognition. This system combines a robust input representation, which preserves the dynamic writing information, with a neural network architecture, a so called Multi-State Time Delay Neural Network (MS-TDNN), which integrates rec.ognition and segmentation in a single framework. Our preprocessing transforms the original coordinate sequence into a (still temporal) sequence offeature vectors, which combine strictly local features, like curvature or writing direction, with a bitmap-like representation of the coordinate's proximity. The MS-TDNN architecture is well suited for handling temporal sequences as provided by this input representation. Our system is tested both on writer dependent and writer independent tasks with vocabulary sizes ranging from 400 up to 20,000 words. For example, on a 20,000 word vocabulary we achieve word recognition rates up to 88.9% (writer dependent) and 84.1 % (writer independent) without using any language models. 1094 1 Stefan Manke, Michael Finke, Alex Waibel INTRODUCTION Several preprocessing and recognition approaches for on-line handwriting recognition have been developed during the past years . The main advantage of on-line handwriting recognition in comparison to optical character recognition (OCR) is the temporal information of handwriting, which can be recorded and used for recognition . In general this dynamic writing information (i.e. the time-ordered sequence of coordinates) is not available in OCR, where input consists of scanned text. In this paper we present the NPen++ system, which is designed to preserve the dynamic writing information as long as possible in the preprocessing and recognition process. During preprocessing a temporal sequence of N-dimensional feature vectors is computed from the original coordinate sequence, which is recorded on the digitizer. These feature vectors combine strictly local features, like curvature and writing direction [4], with so-called context bitmaps, which are bitmap-like representations of a coordinate's proximity. The recognition component of NPen++ is well suited for handling temporal sequences of patterns, as provided by this kind of input representation. The recognizer, a so-called Multi-State Time Delay Neural Network (MS-TDNN), integrates recognition and segmentation of words into a single network architecture. The MS-TDNN, which was originally proposed for continuous speech recognition tasks [6, 7], combines shift-invariant, high accuracy pattern recognition capabilities of a TDNN [8, 4] with a non-linear alignment procedure for aligning strokes into character sequences. Our system is applied both to different writer dependent and writer independent, large vocabulary handwriting recognition tasks with vocabulary sizes up to 20,000 words. Writer independent word recognition rates range from 92.9% with a 400 word vocabulary to 84.1% with a 20,000 word vocabulary. For the writer dependent system, word recognition rates for the same tasks range from 98.6% to 88.9% [1]. In the following section we give a description of our preprocessing performed on the raw coordinate sequence, provided by the digitizer. In section 3 the architecture and training of the recognizer is presented. A description of the experiments to evaluate the system and the results we have achieved on different tasks can be found in section 4. Conclusions and future work is described in section 5. 2 PREPROCESSING The dynamic writing information, i.e. the temporal order of the data points, is preserved throughout all preprocessing steps. The original coordinate sequence {(x(t), y(t?hE{O ...T'} recorded on the digitizer is transformed into a new temporal sequence X6 = Xo ... XT, where each frame Xt consists of an N-dimensional realvalued feature vector (h(t), . .. , fN(t? E [-1, l]N. Several normalization methods are applied to remove undesired variability from the original coordinate sequence. To compensate for different sampling rates and varying writing speeds the coordinates originally sampled to be equidistant in time are resampled yielding a new sequence {(x(t),y(t?hE{O ...T} which is equidistant in Dynamic Writing Information in Cursive Handwriting Recognition 1095 normalized coordinate (b) writing direction curvature , ,, , x(.?2),y(??2) 'f ~ ,';'1 x(t-l),y('?l) t~, " .... , x('),y(') ' 1" x(t+2),y(t+2) ? x('+1),y('+l) x( ??2),y('.2) , , ~ , : ' /1 ' x(.+2),Y(1+2) x(t-11y('?1). \. ~ " .. ...... x(t+l),y(t+l) x(.),y(') Figure I: Feature extraction for the normalized word "able". The final input representation is derived by c,alculating a 15-dimensional feature vector for each data point, which consists of a context bitmap (a) and information about the curvature and writing direction (b). space. This resampled trajectory is smoothed using a moving average window in order to remove sampling noise. In a final normalization step the goal is to find a representation of the trajectory that is reasonably invariant against rotation and scaling of the input. The idea is to determine the words' baseline using an EM approach similar to that described in [5] and rescale the word such that the center region of the word is assigned to a fixed size. From the normalized coordinate sequence {(x(t), y(t))hE{O .. .T} the temporal sequence 2::r; of N-dimensional feature vectors ~t = (!l(t), ... .IN(t)) is computed (Figure I). Currently the system uses N = 15 features for each data point. The first two features fl(t) = x(t)-x(t-I) and h(t) = y(t)-b describe the relative X movement and the Y position relative to the baseline b. The features /get) to f6(t) are used to desc,ribe the curvature and writing direction in the trajectory [4] (Figure I(b)). Since all these features are strictly local in the sense that they are local both in time and in space they were shown to be inadequate for modeling temporal long range context dependencies typically observed in pen trajectories [2]. Therefore, nine additional features h(t) to !J5(t) representing :3 x:3 bitmaps were included in each feature vector (Figure I(a?. These so-called context bitmaps are basically low resolution, bitmap-like descriptions of the coordinate's proximity, which were originally described in [2]. Thus, the input representation as shown in Figure I combines strictly local features like writing direction and curvature with the c,ontext bitmaps, which are still local 1096 Stefan Manke, Michael Finke, Alex Waibel in space but global in time. That means, each point of the trajectory is visible from each other point of the trajectory in a small neighbourhood. By using these context bitmaps in addition to the local features, important information about other parts of the trajectory, which are in a limited neighbourhood of a coordinate, are encoded. 3 THE NPen++ RECOGNIZER The NPell ++ recognizer integrates recognition and segmentation of words into a single network architecture, the Multi-State Time Delay Neural Network (MSTDNN) . The MS-TDNN, which was originally proposed for continuous speech recognition tasks [6 , 7], combines the high accuracy single character recognition capabilities of a TDNN [8, 4] with a non-linear time alignment algorithm (dynamic time warping) for finding stroke and character boundaries in isolated handwritten words. 3.1 MODELING ASSUMPTIONS Let W = {WI, .. . WK} be a vocabulary consisting of K words. Each of these words Wi is represented as a sequence of characters Wi == Ci l Ci 2 ? ?? Cik where each character Cj itself is modelled by a three state hidden markov model Cj == qj q] qJ. The idea of using three states per character is to model explicitly the imtial, middle and final section of the characters. Thus , Wi is modelled by a sequence of states Wi == qioqi l ? .. qhk . In these word HMMs the self-loop probabilities p(qij/qij) and the transition probabilities p(qij/qij_l) are both defined to be ~ while all other transition probabilities are set to zero. During recognition of an unknown sequence of feature vectors ~'(; = ~o . . . ~T we have to find the word Wi E W in the dictionary that maximizes the a-posteriori probability p( Wi /~a 10) given a fixed set of parameters 0 and the observed coordinate sequence. That means , a written word will be recognized such that Wj = argmaXw,EWp(Wi/Za,O). In our Multi-State Time Delay Neural Network approach the problem of modeling the word posterior probability p( wdz'{; , 0) is simplified by using Bayes' rule which expresses that probability as p(Z'{;/WilO)P(Wi/O) p(z'{;/O) Instead of approximating p( Wi /z'{;, 0) directly we define in the following section a network that is supposed to model the likelihood of the feature vector sequence p(z'{; /Wi, 0). 3.2 THE MS-TDNN ARCHITECTURE In Figure 2 the basic MS-TDNN architecture for handwriting recognition is shown. The first three layers constitute a standard TDNN with sliding input windows in each layer. In the current implementation of the system, a TDNN with 15 input Dynamic Writing InJonnation in Cursive Handwriting Recognition 0 -g :; 0 0 "E i::' '" ..'!l.D .D'"0 1i " mm.o .? ? '" 1097 ~ '" '" '" ::::::: :." .D ] '" '" ~.m.W.J ... -.......-.... E 0 0 N ??? 0 ~ r:'?~ """~L-,;~???i]- ... ~ [~~~::~~~::~ ~~~~~~~~:::~::~:~~~~:J- ... ? ? ? ;r?5_-m~;.?i~::~:~?:?:?:I ; .~ . ",. : ~ __ - _. _____ n ___ ___ . . . __ n n_. n ~. __ nn __ __ __ ~ t. ~-- -- n---:011- ~ 7-n ?? --- -- __ ..~.:~ __ ??-:.____ n --.~-- "_ ........ : ! I I : I I ,I 11 L" ~ X Y ?. --------------- ----.-- -- ~~~: ~ ~~~~~~~:~~~:~~~}:_ ~ z'---_ _ _ _ _ _-l::?-_ _ _ _ _- - ' --------------. time - - - - - - - - - - - - - - - - - - - Figure 2: The Multi-State TDNN architecture , consisting of a 3-layer TDNN to estimate the a posteriori probabilities of the character states combined with word units, whose scores are derived from the word models by a Viterbi approximation of the likelihoods p(x6'IWi). units , 40 units in the hidden layer , and 78 state output units is used . There are 7 time delays both in the input and hidden layer. The softmax normalized output of the states layer is interpreted as an estimate of the probabilities of the states qj given the input window x!~~ = Xt-d .. . Xt+d for each t ime frame t , i.e . exp(1/j (t)) 2::k exp(1/k(t)) ( 1) where 1lj (t) represents the weighted sum of inputs to state unit j at time t . Based on these estimates, the output of the word units is defined to be a Viterbi approximation of the log likelihoods of the feature vector sequence given the word model Stefan Manke, Michael Finke, Alex Waibel 1098 T logp(zrlwi) ~ m~ qo L 10gp(z;~~lqt, Wi) + logp(qtlqt-I, Wi) t=1 T ~ m~ Llogp(qtlz;~~) -logp(qt) + logp(qtlqt-1, Wi). qo t=1 (2) Here, the maximum is over all possible sequences of states q'{; = qo . .. qT given a word model, p(qtlz!~~) refers to the output of the states layer as defined in (1) and p(qt) is the prior probability of observing a state qt estimated on the training data. 3.3 TRAINING OF THE RECOGNIZER During training the goal is to determine a set of parameters 0 that will maximize the posterior probability p( wlzr, 0) for all training input sequences. But in order to make that maximization computationally feasible even for a large vocabulary system we had to simplify that maximum a posteriori approach to a maximum likelihood training procedure that maximizes p(zrlw, 0) for all words instead. The first step of our maximum likelihood training is to bootstrap the recognizer using a subset of approximately 2,000 words of the training set that were labeled manually with the character boundaries to adjust the paths in the word layer correctly. After training on this hand-labeled data, the recognizer is used to label another larger set of unlabeled training data. Each pattern in this training set is processed by the recognizer. The boundaries determined automatically by the Viterbi alignment in the target word unit serve as new labels for this pattern. Then, in the second phase, the recognizer is retrained on both data sets to achieve the final performance of the recognizer. 4 EXPERIMENTS AND RESULTS We have tested our system both on writer dependent and writer independent tasks with vocabulary sizes ranging from 400 up to 20,000 words. The word recognition results are shown in Table 1. The scaling of the recognition rates with respect to the vocabulary size is plotted in Figure 3b. T a bl e 1 W' rlter depen dent an d'In depen dent recogmtIOn resu ts Writer Dependent Writer Independent Vocabulary RecognitIon KecogmtIOn Test Test Task Size Patterns Patterns Rate Rate crtAOO 400 800 98.6% 92.9% 800 wsj_l,OOO 1,000 800 97.8% wsj_7,000 7,000 2,500 89.3% wsj_l0,000 10,000 1,600 92.1% 2,500 87.7% wsj..20,000 20,000 1,600 88.9% 2,500 84.f% Dynamic Writing Information in Cursive Handwriting Recognition 1099 100r-----.------.----~------., ... writer dependent ....-. writer independent .-+- ................... 95 .......... ......-----... 90 ................................... - .................... ............... ....................-.... ----- "'-"'-'. --.. 85 80 75~----~----~----~------~ 5000 10000 15000 20000 size of vocabulary (b) Figure 3: (a) Different writing styles in the database : cursive (top), hand-printed (middle) and a mixture of both (bottom) (b) Recognition results with respect to the vocabulary size For the writer dependent evaluation, the system was trained on 2,000 patterns from a 400 word vocabulary, written by a single writer, and tested on a disjunct set of patterns from the same writer. In the writer dependent case, the training set consisted of 4,000 patterns from a 7,000 word vocabulary, written by approximately 60 different writers. The test was performed on data from an independent set of 40 writers . All data used in these experiments was collected at the University of Karlsruhe, Germany. Only minimal instructions were given to the writers. The writers were asked to write as natural as they would normally do on paper, without any restrictions in writing style. The consequence is , that the database is characterized by a high variety of different writing styles, ranging from hand-printed to strictly cursive patterns or a mixture of both writing styles (for example see Figure 3a). Additionally the native language of the writers was german, but the language of the dictionary is english. Therefore , frequent hesitations and corrections can be observed in the patterns of the database. But since this sort of input is typical for real world applications , a robust recognizer should be able to process these distorted patterns, too . From each of the writers a set of 50-100 isolated words, choosen randomly from the 7,000 word vocabulary, was collected. The used vocabularies CRT (Conference Registration Task) and WSJ (ARPA Wall Street Journal Task) were originally defined for speech recognition evaluations. These vocabularies were chosen to take advantage of the synergy effects between handwriting recognition and speech recognition, since in our case the final goal is to integrate our speech recognizer JANUS [10] and the proposed NPen++ system into a multi-modal system. 1100 5 Stefan Manke, Michael Finke, Alex Waibel CONCLUSIONS In this paper we have presented the NPen++ system, a neural recognizer for writer dependent and writer independent on-line cursive handwriting recognition. This system combines a robust input representation, which preserves the dynamic writing information, with a neural network integrating recognition and segmentation in a single framework. This architecture has been shown to be well suited for handling temporal sequences as provided by this kind of input. Evaluation of the system on different tasks with vocabulary sizes ranging from 400 to 20,000 words has shown recognition rates from 92.9% to 84.1 % in the writer independent case and from 98.6% to 88.9% in the writer dependent case. These results are especially promising because they were achieved with a small training set compared to other systems (e.g. [3]). As can be seen in Table 1, the system has proved to be virtually independent of the vocabulary. Though the system was trained on rather small vocabularies (e.g. 400 words in the writer dependent system), it generalizes well to completely different and much larger vocabularies. References [1] S. Manke and U. Bodenhausen, "A Connectionist Recognizer for Cursive Handwriting Recognition", Proceedings of the ICASSP-94, Adelaide, April 1994. [2] S. Manke, M. Finke, and A. Waibel, "Combining Bitmaps with Dynamic Writing Information for On-Line Handwriting Recognition", Proceedings of the ICPR94, Jerusalem, October 1994. [3] M. Schenkel, I. Guyon, and D. Henderson, "On-Line Cursive Script Recognition Using Time Delay Neural Networks and Hidden Markov Models", Proceedings of the ICASSP-94, Adelaide, April 1994. [4] I. Guyon, P. Albrecht, Y. Le Cun, W. Denker, and W. Hubbard, "Design of a Neural Network Character Recognizer for a Touch Terminal", Pattern Recognition, 24(2), 1991. [5] Y. Bengio and Y. LeCun. "Word Normalization for On-Line Handwritten Word Recognition", Proceedings of the ICPR-94, Jerusalem, October 1994. [6] P. Haffner and A. Waibel, "Multi-State Time Delay Neural Networks for Continuous Speech Recognition", Advances in Neural Information Processing Systems (NIPS-4) , Morgan Kaufman, 1992. [7] C. Bregler, H. Hild, S. Manke, and A. Waibel, "Improving Connected Letter Recognition by Lipreading", Proceedings of the ICASSP-93, Minneapolis, April 1993. [8] A. Waibel, T. Hanazawa, G . Hinton, K. Shiano, and K. Lang, "Phoneme Recognition using Time-Delay Neural Networks", IEEE Transactions on Acoustics, Speech and Signal Processing, March 1989. [9] W. Guerfali and R. Plamondon, "Normalizing and Restoring On-Line Handwriting", Pattern Recognition, 16(5), 1993. [10] M. Woszczyna et aI., "Janus 94: Towards Spontaneous Speech Translation", Proceedings of the ICASSP-94, Adelaide, April 1994. 

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Forward dynamic models in human motor control: Psychophysical evidence Daniel M. Wolpert wolpert@psyche .mit .edu Zouhin Ghahramani zoubin@psyche.mit.edu Michael I. Jordan jordan@psyche.mit.edu Department of Brain & Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract Based on computational principles, with as yet no direct experimental validation, it has been proposed that the central nervous system (CNS) uses an internal model to simulate the dynamic behavior of the motor system in planning, control and learning (Sutton and Barto, 1981; Ito, 1984; Kawato et aI., 1987; Jordan and Rumelhart, 1992; Miall et aI., 1993). We present experimental results and simulations based on a novel approach that investigates the temporal propagation of errors in the sensorimotor integration process. Our results provide direct support for the existence of an internal model. 1 Introduction The notion of an internal model, a system which mimics the behavior of a natural process, has emerged as an important theoretical concept in motor control (Jordan, 1995). There are two varieties of internal models-"forward models," which mimic the causal flow of a process by predicting its next state given the current state and the motor command, and "inverse models," which are anticausal, estimating the motor command that causes a particular state transition. Forward modelsthe focus of this article-have been been shown to be of potential use for solving four fundamental problems in computational motor control. First, the delays in most sensorimotor loops are large, making feedback control infeasible for rapid 44 Daniel M. Wolpert, Zoubin Ghahramani, Michaell. Jordan movements. By using a forward model for internal feedback the outcome of an action can be estimated and used before sensory feedback is available (Ito, 1984; Miall et al., 1993). Second, a forward model is a key ingredient in a system that uses motor outflow ("efference copy") to anticipate and cancel the reafferent sensory effects of self-movement (Gallistel, 1980; Robinson et al., 1986). Third, a forward model can be used to transform errors between the desired and actual sensory outcome of a movement into the corresponding errors in the motor command, thereby providing appropriate signals for motor learning (Jordan and Rumelhart, 1992). Similarly by predicting the sensory outcome of the action, without actually performing it, a forward model can be used in mental practice to learn to select between possible actions (Sutton and Barto, 1981). Finally, a forward model can be used for state estimation in which the model's prediction of the next state is combined with a reafferent sensory correction (Goodwin and Sin, 1984). Although shown to be of theoretical importance, the existence and use of an internal forward model in the CNS is still a major topic of debate. When a subject moves his arm in the dark, he is able to estimate the visual location of his hand with some degree of accuracy. Observer models from engineering formalize the sources of information which the CNS could use to construct this estimate (Figure 1). This framework consists of a state estimation process (the observer) which is able to monitor both the inputs and outputs of the system. In particular, for the arm, the inputs are motor commands and the output is sensory feedback (e.g. vision and proprioception). There are three basic methods whereby the observer can estimate the current state (e.g. position and velocity) of the hand form these sources: It can make use of sensory inflow, it can make use of integrated motor outflow (dead reckoning), or it can combine these two sources of information via the use of a forward model. u(t) Input System X(t) Output y(t) S ensory Moto r Command Fe edback - Observer State estimate x(t) Figure 1. Observer model of state estimation . 45 Forward Dynamic Models in Human Motor Control 2 State Estimation Experiment To test between these possibilities, we carried out an experiment in which subjects made arm movements in the dark. The full details of the experiment are described in the Appendix. Three experimental conditions were studied, involving the use of null, assistive and resistive force fields. The subjects' internal estimate of hand location was assessed by asking them to localize visually the position of their hand at the end of the movement. The bias of this location estimate, plotted as a function of movement duration shows a consistent overestimation of the distance moved (Figure 2). This bias shows two distinct phases as a function of movement duration, an initial increase reaching a peak of 0.9 cm after one second followed by a sharp transition to a region of gradual decline. The variance of the estimate also shows an initial increase during the first second of movement after which it plateaus at about 2 cm 2 . External forces had distinct effects on the bias and variance propagation. Whereas the bias was increased by the assistive force and decreased by the resistive force (p < 0.05), the variance was unaffected. a - c - 1.0 1.5 E 1.0 0 0.5 (J) 0.0 E 0 (J) 0.5 <tI i:i5 CD <] -1 .0 0.0 0.5 E 0 Q) 1.0 1.5 2.0 - 2.5 (\I E 0 2.0 > - :~::- . -' 0.5 1.0 1.5 2.0 2.5 2.0 2.5 Time (5) 2 1 ..... . Q) 1.5 0 c:: c:: 1.0 <tI L... 0.0 d 0 <tI .:... -0.5 2.5 Time (5) b - ~ . -- -1.5 0.0 (\I -- <tI 0 <tI L... 0.5 <tI > <] 0.0 0.0 0.5 1.0 1.5 Time (5) 2.0 2.5 -1 -2 0.0 0.5 1.0 1.5 Time (5) Figure 2. The propagation of the (a) bias and (b) variance of the state estimate is shown, with standard error lines, against movement duration. The differential effects on (c) bias and (d) variance of the external force, assistive (dotted lines) and resistive (solid lines), are also shown relative to zero (dashed line). A positive bias represents an overestimation of the distance moved. 46 3 Daniel M. Wolpert, Zoubin Ghahramani, Michael I. Jordan Observer Model Simulation These experimental results can be fully accounted for only if we assume that the motor control system integrates the efferent outflow and the reafferent sensory inflow. To establish this conclusion we have developed an explicit model of the sensorimotor integration process which contains as special cases all three of the methods referred to above. The model-a Kalman filter (Kalman and Bucy, 1961)-is a linear dynamical system that produces an estimate of the location of the hand by monitoring both the motor outflow and the feedback as sensed, in the absence of vision, solely by proprioception. Based on these sources of information the model estimates the arm's state, integrating sensory and motor signals to reduce the overall uncertainty in its estimate. Representing the state of the hand at time t as x(t) (a 2 x 1 vector of position and velocity) acted on by a force u = [Uint, Uext]T, combining both internal motor commands and external forces, the system dynamic equations can be written in the general form of (1) x(t) = Ax(t) + Bu(t) + w(t), where A and B are matrices with appropriate dimension. The vector w(t) represents the process of white noise with an associated covariance matrix given by Q = E[w(t)w(t)T]. The system has an observable output y(t) which is linked to the actual hidden state x(t) by y(t) = Cx(t) + v(t), (2) where C is a matrix with appropriate dimension and the vector v(t) represents the output white noise which has the associated covariance matrix R = E[v(t)v(t)T]. In our paradigm, y(t) represents the proprioceptive signals (e.g. from muscle spindles and joint receptors). In particular, for the hand we approximate the system dynamics by a damped point mass moving in one dimension acted on by a force u(t). Equation 1 becomes (3) where the hand has mass m and damping coefficient {3. We assume that this system is fully observable and choose C to be the identity matrix. The parameters in the simulation, {3 = 3.9 N ,s/m, m = 4 kg and Uint = 1.5 N were chosen based on the mass of the arm and the observed relationship between time and distance traveled. The external force Uext was set to -0.3, 0 and 0.3 N for the resistive, null and assistive conditions respectively. To end the movement the sign of the motor command Uint was reversed until the arm was stationary. Noise covariance matrices of Q = 9.5 X 10- 5 [ and R = 3.3 x 1O- 4 [ were used representing a standard deviation of 1.0 cm for the position output noise and 1.8 cm s-l for the position component of the state noise. At time t = 0 the subject is given full view of his arm and, therefore, starts with an estimate X(O) = x(O) with zero bias and variance-we assume that vision calibrates the system. At this time the light is extinguished and the subject must rely on the inputs and outputs to estimate the system's state. The Kalman filter, using a Forward Dynamic Models in Human Motor Control 47 model of the system A, Band C, provides an optimal linear estimator of the state given by i(t) = ,Ax(t) + Bu(t).I + ,K(t)[y(t) - Cx(t)] I 'V' V forward model sensory correction where K(t) is the recursively updated gain matrix. The model is, therefore, a combination of two processes which together contribute to the state estimate. The first process uses the current state estimate and motor command to predict the next state by simulating the movement dynamics with a forward model. The second process uses the difference between actual and predicted reafferent sensory feedback to correct the state estimate resulting from the forward model. The relative contributions of the internal simulation and sensory correction processes to the final estimate are modulated by the Kalman gain matrix K(t) so as to provide optimal state estimates. We used this state update equation to model the bias and variance propagation and the effects of the external force. By making particular choices for the parameters of the Kalman filter, we are able to simulate dead reckoning, sensory inflow-based estimation, and forward modelbased sensorimotor integration. Moreover, to accommodate the observation that subjects generally tend to overestimate the distance that their arm has moved, we set the gain that couples force to state estimates to a value that is larger than its veridical value; B = ~ [1~4 1~6] while both A and C accurately reflected the true system. This is consistent with the independent data that subjects tend to under-reach in pointing tasks suggesting an overestimation of distance traveled (Soechting and Flanders, 1989). Simulations of the Kalman filter demonstrate the two distinct phases of bias propagation observed (Figure 3). By overestimating the force acting on the arm the forward model overestimates the distance traveled, an integrative process eventually balanced by the sensory correction. The model also captures the differential effects on bias of the externally imposed forces. By overestimating an increased force under the assistive condition, the bias in the forward model accrues more rapidly and is balanced by the sensory feedback at a higher level. The converse applies to the resistive force. In accord with the experimental results the model predicts no change in variance under the two force conditions. 4 Discussion We have shd-ttn that the Kalman filter is able to reproduce the propagation of the bias and variance of estimated position of the hand as a function of both movement duration and external forces . The Kalman filter also simulates the interesting and novel empirical result that while the variance asymptotes, the bias peaks after about one second and then gradually declines. This behavior is a consequence pf a trade off between the inaccuracies accumulating in the internal simulation of the arm's dynamics and the feedback of actual sensory information. Simple models which do not trade off the contributions of a forward model with sensory feedback, such as those based purely on sensory inflow or on motor outflow, are unable to reproduce the observed pattern of bias and variance propagation. The ability of the Kalman filter to parsimoniously model our data suggests that the processes embodied in the Daniel M. Wolpert, Zoubin Ghahramani, Michaell. Jordan 48 c a - - 1.0 E - E 0 0 en en 0.5 n1 [Ii n1 en <l 0.0 0.5 E 1.5 2.0 2.5 Time (s) b C\I 1.0 0.5 - 2.5 C\I 1.0 1.5 2.0 2.5 2.0 2.5 Time (s) d 2 E 2.0 Q) 1.5 - c:: n1 1.0 .... c:: n1 n1 0.5 n1 > 0.0 0 ~-------:.:...:.:...:,;.;.:.;.. 0.0 0 - 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 0 Q) 0 .... 0 -1 > <l -2 0.0 0.5 1.0 1.5 Time (s) 2.0 2.5 0.0 0.5 1.0 1.5 Time (s) Figure 3. Simulated bias and variance propagation, in the same representation and scale as Figure 2, from a Kalman filter model of the sensorimotor integration process. filter, namely internal simulation through a forward model together with sensory correction, are likely to be embodied in the sensorimotor integration process. We feel that the results of this state estimation study provide strong evidence that a forward model is used by the CNS in maintaining its estimate of the hand location. Furthermore, the state estimation paradigm provides a framework to study the sensorimotor integration process in both normal and patient populations. For example, the specific predictions of the sensorimotor integration model can be tested in both patients with sensory neuropathies, who lack proprioceptive reafference, and in patients with damage to the cerebellum, a proposed site for the forward model (Miall et al., 1993). Acknowledgements We thank Peter Dayan for suggestions about the manuscript. This project was supported by grants from the McDonnell-Pew Foundation, ATR Human Information Processing Research Laboratories, Siemens Corporation, and by grant N00014-94-10777 from the Office of Naval Research. Daniel M. Wolpert and Zoubin Ghahramani are McDonnell-Pew Fellows in Cognitive Neuroscience. Michael!. Jordan is a NSF Presidential Young Investigator. Forward Dynamic Models in Human Motor Control 49 Appendix: Experimental Paradigm To investigate the way in which errors in the state estimate change over time and with external forces we used a setup (Figure 4) consisting of a combination of planar virtual visual feedback with a two degree of freedom torque motor driven manipulandum (Faye, 1986). The subject held a planar manipulandum on which his thumb was mounted. The manipulandum was used both to accurately measure the position of the subject's thumb and also, using the torque motors, to constrain the hand to move along a line across the subject's body. A projector was used to create virtual images in the plane of the movement by projecting a computer VGA screen onto a horizontal rear projection screen suspended above the manipulandum. A horizontal semi-silvered mirror was placed midway between the screen and manipulandum. The subject viewed the reflected image of the rear projection screen by looking down at the mirror; all projected images, therefore, appeared to be in the plane of the thumb, independent of head position. Projector '0' \ ,', I I \ : , I' I' \ " .,,>1. '> It; .~ Cursor Image \ \ Screen Finger Semi?silvered mirror Bulb Manlpulandum ~orque motors Figure 4. Experimental Setup Eight subjects participated and performed 300 trials each. Each trial started with the subject visually placing his thumb at a target square projected randomly on the movement line. The arm was then illuminated for two seconds, thereby allowing the subject to perceive visually his initial arm configuration. The light was then extinguished leaving just the initial target. The subject was then required to move his hand either to the left or right, as indicated by an arrow in the initial starting square. This movement was made in the absence of visual feedback of arm configuration. The subject was instructed to move until he heard a tone at which point he stopped. The timing of the tone was controlled to produce a uniform distribution of path lengths from 0-30 cm. During this movement the subject either moved in a randomly selected null or constant assistive or resistive 0.3N force field generated by the torque motors. Although it is not possible to directly probe a subject's internal representation of the state of his arm, we can examine a function of this state-the estimated visual location of the thumb. (The relationship between the state of the arm and the visual coordinates of the hand is known as 50 Daniel M. Wolpert, Zoubin Ghahramani, Michaell. Jordan the kinematic transformation; Craig, 1986.) Therefore, once at rest the subject indicated the visual estimate of his unseen thumb position using a trackball, held in his other hand, to move a cursor projected in the plane of the thumb along the movement line. The discrepancy between the actual and visual estimate of thumb location was recorded as a measure of the state estimation error. The bias and variance propagation of the state estimate was analyzed as a function of movement duration and external forces. A generalized additive model (Hastie and Tibshirani, 1990) with smoothing splines (five effective degrees of freedom) was fit to the bias and variance as a function of final position, movement duration and the interaction of the two forces with movement duration, simultaneously for main effects and for each subject. This procedure factors out the additive effects specific to each subject and, through the final position factor, the position-dependent inaccuracies in the kinematic transformation. References Craig, J. (1986). Introduction to robotics. Addison-Wesley, Reading, MA. Faye, I. (1986). An impedence controlled manipulandum for human movement studies. MS Thesis, MIT Dept. Mechanical Engineering, Cambridge, MA. Gallistel, C. (1980). The organization of action: A new synthesis. Erlbaum, Hilladale, NJ. Goodwin, G. and Sin, K. (1984). Adaptive filtering prediction and control. PrenticeHall, Englewood Cliffs, NJ. Hastie, T. and Tibshirani, R. (1990). Generalized Additive Models. Chapman and Hall, London. Ito, M. (1984). The cerebellum and neural control. Raven Press, New York. Jordan, M. and Rumelhart, D. (1992). Forward models: Supervised learning with a distal teacher. Cognitive Science, 16:307-354. Jordan, M. I. (1995). Computational aspects of motor control and motor learning. In Heuer, H. and Keele, S., editors, Handbook of Perception and Action: Motor Skills. Academic Press, New York. Kalman, R. and Bucy, R. S. (1961). New results in linear filtering and prediction. Journal of Basic Engineering (ASME), 83D:95-108. Kawato, M., Furawaka, K., and Suzuki, R. (1987). A hierarchical neural network model for the control and learning of voluntary movements. Biol.Cybern., 56:117. Miall, R., Weir, D., Wolpert, D., and Stein, J. (1993). Is the cerebellum a Smith Predictor? Journal of Motor Behavior, 25(3):203-216. Robinson, D., Gordon, J., and Gordon, S. (1986). A model of the smooth pursuit eye movement system. Biol.Cybern., 55:43-57. Soechting, J. and Flanders, M. (1989). Sensorimotor representations for pointing to targets in three- dimensional space. J.Neurophysiol., 62:582-594. Sutton, R. and Barto, A. (1981). Toward a modern theory of adaptive networks: expettation and prediction. Psychol.Rev., 88:135-170. 

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652 Scaling Properties of Coarse-Coded Symbol Memories Ronald Rosenfeld David S. Touretzky Computer Science Department Carnegie Mellon University Pittsburgh, Pennsylvania 15213 Abstract: Coarse-coded symbol memories have appeared in several neural network symbol processing models. In order to determine how these models would scale, one must first have some understanding of the mathematics of coarse-coded representations. We define the general structure of coarse-coded symbol memories and derive mathematical relationships among their essential parameters: memory 8ize, 8ymbol-8et size and capacity. The computed capacity of one of the schemes agrees well with actual measurements oC tbe coarse-coded working memory of DCPS, Touretzky and Hinton's distributed connectionist production system. 1 Introduction A di8tributed repre8entation is a memory scheme in which each entity (concept, symbol) is represented by a pattern of activity over many units [3]. If each unit participates in the representation of many entities, it is said to be coar8ely tuned, and the memory itself is called a coar8e-coded memory. Coarse-coded memories have been used for storing symbols in several neural network symbol processing models, such as Touretzky and Hinton's distributed connectionist production system DCPS [8,9], Touretzky's distributed implementation of linked list structures on a Boltzmann machine, BoltzCONS [10], and St. John and McClelland's PDP model of case role defaults [6]. In all of these models, memory capacity was measured empirically and parameters were adjusted by trial and error to obtain the desired behavior. We are now able to give a mathematical foundation to these experiments by analyzing the relationships among the fundamental memory parameters. There are several paradigms for coarse-coded memories. In a feature-based repre8entation, each unit stands for some semantic feature. Binary units can code features with binary values, whereas more complicated units or groups of units are required to code more complicated features, such as multi-valued properties or numerical values from a continuous scale. The units that form the representation of a concept define an intersection of features that constitutes that concept. Similarity between concepts composed of binary Ceatures can be measured by the Hamming distance between their representations. In a neural network implementation, relationships between concepts are implemented via connections among the units forming their representations. Certain types of generalization phenomena thereby emerge automatically. A different paradigm is used when representing points in a multidimensional continuous space [2,3]. Each unit encodes values in some subset of the space. Typically the @ American Institute of Physics 1988 653 subsets are hypercubes or hyperspheres, but they may be more coarsely tuned along some dimensions than others [1]. The point to be represented is in the subspace formed by the intersection of all active units. AB more units are turned on, the accuracy of the representation improves. The density and degree of overlap of the units' receptive fields determines the system's resolution [7]. Yet another paradigm for coarse-coded memories, and the one we will deal with exclusively, does not involve features. Each concept, or symbol, is represented by an arbitrary subset of the units, called its pattern. Unlike in feature-based representations, the units in the pattern bear no relationship to the meaning of the symbol represented. A symbol is stored in memory by turning on all the units in its pattern. A symbol is deemed present if all the units in its pattern are active. l The receptive field of each unit is defined as the set of all symbols in whose pattern it participates. We call such memories coarsecoded symbol memories (CCSMs). We use the term "symbol" instead of "concept" to emphasize that the internal structure of the entity to be represented is not involved in its representation. In CCSMs, a short Hamming distance between two symbols does not imply semantic similarity, and is in general an undesirable phenomenon. The efficiency with which CCSMs handle sparse memories is the major reason they have been used in many connectionist systems, and hence the major reason for studying them here. The unit-sharing strategy that gives rise to efficient encoding in CCSMs is also the source of their major weakness. Symbols share units with other symbols. AB more symbols are stored, more and more of the units are turned on. At some point, some symbol may be deemed present in memory because all of its units are turned on, even though it was not explicitly stored: a "ghost" is born. Ghosts are an unwanted phenomenon arising out of the overlap among the representations of the various symbols. The emergence of ghosts marks the limits of the system's capacity: the number of symbols it can store simultaneously and reliably. 2 Definitions and Fundamental Parameters A coarse coded symbol memory in its most general form consists of: ? A set of N binary state units. ? An alphabet of Q symbols to be represented. Symbols in this context are atomic entities: they have no constituent structure. ? A memory scheme, which is a function that maps each symbol to a subset of the units - its pattern. The receptive field of a unit is defined as the set of all symbols to whose pattern it belongs (see Figure 1). The exact nature of the lThis criterion can be generalized by introducing a visibility threshold: a fraction of the pattern that should be on in order for a symbol to be considered present. Our analysis deals only with a visibility criterion of 100%, but can be generalized to accommodate nOise. 654 I I 81 I 82 I 88 I 8 I 85 I 86 I 87 I 88 I 4 Ul U2 U8 U4 U5 U6 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Figure 1: A memory scheme (N = 6, Q = 8) defined in terms of units Us and symbols 8;. The columns are the symbols' patterns. The rows are the units' receptive fieldB. memory scheme mapping determines the properties of the memory, and is the central target of our investigation. As symbols are stored, the memory fills up and ghosts eventually appear. It is not possible to detect a ghost simply by inspecting the contents of memory, since there is no general way of distinguishing a symbol that was stored from one that emerged out of overlaps with other symbols. (It is sometimes possible, however, to conclude that there are no ghosts.) Furthermore, a symbol that emerged as a ghost at one time may not be a ghost at a later time if it was subsequently stored into memory. Thus the definition of a ghost depends not only on the state of the memory but also on its history. Some memory schemes guarantee that no ghost will emerge as long as the number of symbols stored does not exceed some specified limit. In other schemes, the emergence of ghosts is an ever-present possibility, but its probability can be kept arbitrarily low by adjusting other parameters. We analyze systems of both types. First, two more bits of notation need to be introduced: Pghost: Probability of a ghost. The probability that at least one ghost will appear after some number of symbols have been stored. k: Capacity. The maximum number of symbols that can be stored simultaneously before the probability of a ghost exceeds a specified threshold. If the threshold is 0, we say that the capacity is guaranteed. A localist representation, where every symbol is represented by a single unit and every unit is dedicated to the representation of a single symbol, can now be viewed as a special case of coarse-coded memory, where k = N = Q and Pghost = o. Localist representations are well suited for memories that are not sparse. In these cases, coarsecoded memories are at a disadvantage. In designing coarse-coded symbol memories we are interested in cases where k ? N ? Q. The permissible probability for a ghost in these systems should be low enough so that its impact can be ignored. 655 3 3.1 Analysis of Four Memory Schemes Bounded Overlap (guaranteed capacity) If we want to construct the memory scheme with the largest possible a (given Nand k) while guaranteeing Pghost = 0, the problem can be stated formally as: Given a set of size N, find the largest collection of subsets of it such that no union of k such subsets subsumes any other subset in the collection. This is a well known problem in Coding Theory, in slight disguise. Unfortunately, no complete analytical solution is known. We therefore simplify our task and consider only systems in which all symbols are represented by the same number of units (i.e. all patterns are of the same size). In mathematical terms, we restrict ourselves to constant weight codes. The problem then becomes: Given a set of size N, find the largest collection of subsets of size exactly L such that no union of k such subsets subsumes any other subset in the collection. There are no known complete analytical solutions for the size of the largest collection of patterns even when the patterns are of a fixed size. Nor is any efficient procedure for constructing such a collection known. We therefore simplify the problem further. We now restrict our consideration to patterns whose pairwise overlap is bounded by a given number. For a given pattern size L and desired capacity k, we require that no two patterns overlap in more than m units, where: _lL -k 1J m- (1) -- Memory schemes that obey this constraint are guaranteed a capacity of at least k symbols, since any k symbols taken together can overlap at most L - 1 units in the pattern of any other symbol - one unit short of making it a ghost. Based on this constraint, our mathematical problem now becomes: Given a set of size N, find the largest collection of subsets of size exactly L such that the intersection of any two such subsets is of size ~ m (where m is given by equation 1.) Coding theory has yet to produce a complete solution to this problem, but several methods of deriving upper bounds have been proposed (see for example [4]). The simple formula we use here is a variant of the Johnson Bound. Let abo denote the maximum number of symbols attainable in memory schemes that use bounded overlap. Then (m~l) (m~l) (2) 656 The Johnson bound is known to be an exact solution asymptotically (that is, when N, L, m -+ 00 and their ratios remain finite). Since we are free to choose the pattern size, we optimize our memory scheme by maximizing the above expression over all possible values of L. For the parameter subspace we are interested in here (N < 1000, k < 50) we use numerical approximation to obtain: < max ( LeII,N] N L - m )m+l < (3) (Recall that m is a function of Land k.) Thus the upper bound we derived depicts a simple exponential relationship between Q and N/k. Next, we try to construct memory schemes of this type. A Common Lisp program using a modified depth-first search constructed memory schemes for various parameter values, whose Q'S came within 80% to 90% of the upper bound. These results are far from conclusive, however, since only a small portion of the parameter space was tested. In evaluating the viability of this approach, its apparent optimality should be contrasted with two major weaknesses. First, this type of memory scheme is hard to construct computationally. It took our program several minutes of CPU time on a Symbolics 3600 to produce reasonable solutions for cases like N = 200, k = 5, m = 1, with an exponential increase in computing time for larger values of m. Second, if CCSMs are used as models of memory in naturally evolving systems (such as the brain), this approach places too great a burden on developmental mechanisms. The importance of the bounded overlap approach lies mainly in its role as an upper bound for all possible memory schemes, subject to the simplifications made earlier. All schemes with guaranteed capacities can be measured relative to equation 3. 3.2 Random Fixed Size Patterns (a stochastic approach) Randomly produced memory schemes are easy to implement and are attractive because of their naturalness. However, if the patterns of two symbols coincide, the guaranteed capacity will be zero (storing one of these symbols will render the other a ghost). We therefore abandon the goal of guaranteeing a certain capacity, and instead establish a tolerance level for ghosts, Pghost. For large enough memories, where stochastic behavior is more robust, we may expect reasonable capacity even with very small Pghost. In the first stochastic approach we analyze, patterns are randomly selected subsets of a fixed size L. Unlike in the previous approach, choosing k does not bound Q. We may define as many symbols as we wish, although at the cost of increased probability of a ghost (or, alternatively, decreased capacity). The probability of a ghost appearing after k symbols have been stored is given by Equation 4: (4) 657 TN,L(k, e) is the probability that exactly e units will be active after k symbols have been stored. It is defined recursively by Equation 5": TN,L(O,O) = 1 TN,L(k, e) 0 for either k 0 and e 1= 0, or k > 0 and e < L TN,L(k, e) = E~=o T(k - 1, c - a) . (N-~-a)) . (~:~)/(~) = = (5) We have constructed various coarse-coded memories with random fixed-size receptive fields and measured their capacities. The experimental results show good agreement with the above equation. The optimal pattern size for fixed values of N, k, and a can be determined by binary search on Equation 4, since Pghost(L) has exactly one maximum in the interval [1, N]. However, this may be expensive for large N. A computational shortcut can be achieved by estimating the optimal L and searching in a small interval around it. A good initial estimate is derived by replacing the summation in Equation 4 with a single term involving E[e]: the expected value of the number of active units after k symbols have been stored. The latter can be expressed as: The estimated L is the one that maximizes the following expression: An alternative formula, developed by Joseph Tebelskis, produces very good approximations to Eq. 4 and is much more efficient to compute. After storing k symbols in memory, the probability Pz that a single arbitrary symbol x has become a ghost is given by: Pz(N,L,k,a) .(L) (N L_i)k / (N)k L L i = f.(-1)' (6) If we now assume that each symbol's Pz is independent of that of any other symbol, we obtain: (7) This assumption of independence is not strictly true, but the relative error was less than 0.1% for the parameter ranges we considered, when Pghost was no greater than 0.01. We have constructed the two-dimensional table TN,L(k, c) for a wide range of (N, L) values (70 ~ N ~ 1000, 7 ~ L ~ 43), and produced graphs of the relationships between N, k, a, and Pghost for optimum pattern sizes, as determined by Equation 4. The 658 results show an approximately exponential relationship between a and N /k [5]. Thus, for a fixed number of symbols, the capacity is proportional to the number of units. Let arl p denote the maximum number of symbols attainable in memory schemes that use random fixed-size patterns. Some typical relationships, derived from the data, are: ~ 0.0086. eO.46S f arlp(Pghost = 0.001) ~ O.OOOS. eO. 47S f arlP(Pghost 3.3 = 0.01) (8) Random Receptors (a stochastic approach) A second stochastic approach is to have each unit assigned to each symbol with an independent fixed probability s. This method lends itself to easy mathematical analysis, resulting in a closed-form analytical solution. After storing k symbols, the probability that a given unit is active is 1 - (1 - s)k (independent of any other unit). For a given symbol to be a ghost, every unit must either be active or else not belong to that symbol's pattern. That will happen with a probability [1 - s . (1 - s)k] N, and thus the probability of a ghost is: (9) Pghost(a, N, k,s) Assuming Pghost simplified to: ? 1 and k ? a (both hold in our case), the expression can be Pghost(a,N,k,s) a? [1- s. (1- s)k]N from which a can be extracted: arr(N, k, 8, Pghost) (10) We can now optimize by finding the value of s that maximizes a, given any desired upper bound on the expected value of Pghost. This is done straightforwardly by solving Ba/Bs = o. Note that 8? N corresponds to L in the previous approach. The solution is s = l/(k + 1), which yields, after some algebraic manipulation: (11) A comparison of the results using the two stochastic approaches reveals an interesting similarity. For large k, with Pghost = 0.01 the term 0.468/k of Equation 8 can be seen as a numerical approximation to the log term in Equation 11, and the multiplicative factor of 0.0086 in Equation 8 approximates Pghost in Equation 11. This is hardly surprising, since the Law of Large Numbers implies that in the limit (N, k -+ 00, with 8 fixed) the two methods are equivalent. 659 Finally, it should be. noted that the stochastic approaches we analyzed generate a family of memory schemes, with non-identical ghost-probabilities. Pghost in our formulas is therefore better understood as an expected value, averaged over the entire family. 3.4 Partitioned Binary Coding (a reference point) The last memory scheme we analyze is not strictly distributed. Rather, it is somewhere in between a distributed and a localist representation, and is presented for comparison with the previous results. For a given number of units N and desired capacity k, the units are partitioned into k equal-size "slots," each consisting of N / k units (for simplicity we assume that k divides N). Each slot is capable of storing exactly one symbol. The most efficient representation for all possible symbols that may be stored into a slot is to assign them binary codes, using the N / k units of each slot as bits. This would allow 2N Jic symbols to be represented. Using binary coding, however, will not give us the required capacity of 1 symbol, since binary patterns subsume one another. For example, storing the code '10110' into one of the slots will cause the codes '10010', '10100' and '00010' (as well as several other codes) to become ghosts. A possible solution is to use only half of the bits in each slot for a binary code, and set the other half to the binary complement of that code (we assume that N/k is even). This way, the codes are guaranteed not to subsume one another. Let Qpbc denote the number of symbols representable using a partitioned binary coding scheme. Then, '"pbc -_ ..... 2NJ2Ic -- eO.847 !:!-.. (12) Once again, Q is exponential in N /k. The form of the result closely resembles the estimated upper bound on the Bounded Overlap method given in Equation 3. There is also a strong resemblance to Equations 8 and 11, except that the fractional multiplier in front of the exponential, corresponding to Pghost, is missing. Pghost is 0 for the Partitioned Binary Coding method, but this is enforced by dividing the memory into disjoint sets of units rather than adjusting the patterns to reduce overlap among symbols. As mentioned previously, this memory scheme is not really distributed in the sense used in this paper, since there is no one pattern associated with a symbol. Instead, a symbol is represented by anyone of a set of k patterns, each N /k bits long, corresponding to its appearance in one of the k slots. To check whether a symbol is present, all k slots must be examined. To store a new symbol in memory, one must scan the k slots until an empty one is found. Equation 12 should therefore be used only as a point of reference. 4 Measurement of DCPS The three distributed schemes we have studied all use unstructured patterns, the only constraint being that patterns are at least roughly the same size. Imposing more complex structure on any of these schemes may is likely to reduce the capacity somewhat. In 660 Memory Scheme Bounded Overlap Random Fixed-size Patterns Random Receptors Partitioned Binary Coding Qbo(N, k) < Result eO.367 t r Q,,!p(Pghost = 0.01) ~ 0.0086. e?.468 Q,,!p(Pghost = 0.001) ~ 0.0008 . e?.473f Q _ P . e N .1og (k+1)"'Tl/((k+l)"'Tl_k"') ,.,. - ghost eO.347r Qpbc -- Table 1 Summary of results for various memory schemes. order to quantify this effect, we measured the memory capacity of DCPS (BoltzCONS uses the same memory scheme) and compared the results with the theoretical models analyzed above. DCPS' memory scheme is a modified version of the Random Receptors method [5]. The symbol space is the set of all triples over a 25 letter alphabet. Units have fixed-size receptive fields organized as 6 x 6 x 6 subspaces. Patterns are manipulated to minimize the variance in pattern size across symbols. The parameters for DCPS are: N = 2000, Q = 25 3 = 15625, and the mean pattern size is (6/25)3 x 2000 = 27.65 with a standard deviation of 1.5. When Pghost = 0.01 the measured capacity was k = 48 symbols. By substituting for N in Equation 11 we find that the highest k value for which Q,.,. ~ 15625 is 51. There does not appear to be a significant cost for maintaining structure in the receptive fields. 5 Summary and Discussion Table 1 summarizes the results obtained for the four methods analyzed. Some differences must be emphasiz'ed: and Qpbc deal with guaranteed capacity, whereas Q,.!p and Q,.,. are meaningful only for Pghost > O. ? Qbo ? Qbo is only an upper bound. ? Q,.!p is based on numerical estimates. ? Qpbc is based on a scheme which is not strictly coarse-coded. The similar functional form of all the results, although not surprising, is aesthetically pleasing. Some of the functional dependencies among the various parameters_ can be derived informally using qualitative arguments. Only a rigorous analysis, however, can provide the definite answers that are needed for a better understanding of these systems and their scaling properties. 661 Acknowledgments We thank Geoffrey Hinton, Noga Alon and Victor Wei for helpful comments, and Joseph Tebelskis for sharing with us his formula for approximating Pghost in the case of fixed pattern sizes. This work was supported by National Science Foundation grants IST-8516330 and EET-8716324, and by the Office of Naval Research under contract number NOOO14-86K-0678. The first author was supported by a National Science Foundation graduate fellowship. References [1] Ballard, D H. (1986) Cortical connections and parallel processing: structure and function. Behavioral and Brain Sciences 9(1). [2] Feldman, J. A., and Ballard, D. H. (1982) Connectionist models and their properties. Cognitive Science 6, pp. 205-254. [3] Hinton, G. E., McClelland, J. L., and Rumelhart, D. E. (1986) Distributed representations. In D. E. Rumelhart and J. L. McClelland (eds.), Parallel Distributed Processing: Explorations in the Microstructure of Cognition, volume 1. Cambridge, MA: MIT Press. [4] Macwilliams, F.J., and Sloane, N.J.A. (1978). The Theory of Error-Correcting Codes, North-Holland. [5] Rosenfeld, R. and Touretzky, D. S. (1987) Four capacity models for coarse-coded symbol memories. Technical report CMU-CS-87-182, Carnegie Mellon University Computer Science Department, Pittsburgh, PA. [6] St. John, M. F. and McClelland, J. L. (1986) Reconstructive memory for sentences: a PDP approach. Proceedings of the Ohio University Inference Conference. [7] Sullins, J. (1985) Value cell encoding strategies. Technical report TR-165, Computer Science Department, University of Rochester, Rochester, NY. [8] Touretzky, D. S., and Hinton, G. E. (1985) Symbols among the neurons: details of a connectionist inference architecture. Proceedings of IJCAI-85, Los Angeles, CA, pp. 238-243. [9] Touretzky, D. S., and Hinton, G. E. (1986) A distributed connectionist production system. Technical report CMU-CS-86-172, Computer Science Department, Carnegie Mellon University, Pittsburgh, PA. [10] Touretzky, D. S. (1986) BoltzCONS: reconciling connectionism with the recursive nature of stacks and trees. Proceedings of the Eighth A nnual Conference of the Cognitive Science Society, Amherst, MA, pp. 522-530. 

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Unsupervised Classification of 3D Objects from 2D Views Satoshi Suzuki Hiroshi Ando ATR Human Information Processing Research Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, Japan satoshi@hip.atr.co.jp, ando@hip.atr.co.jp Abstract This paper presents an unsupervised learning scheme for categorizing 3D objects from their 2D projected images. The scheme exploits an auto-associative network's ability to encode each view of a single object into a representation that indicates its view direction. We propose two models that employ different classification mechanisms; the first model selects an auto-associative network whose recovered view best matches the input view, and the second model is based on a modular architecture whose additional network classifies the views by splitting the input space nonlinearly. We demonstrate the effectiveness of the proposed classification models through simulations using 3D wire-frame objects. 1 INTRODUCTION The human visual system can recognize various 3D (three-dimensional) objects from their 2D (two-dimensional) retinal images although the images vary significantly as the viewpoint changes. Recent computational models have explored how to learn to recognize 3D objects from their projected views (Poggio & Edelman, 1990). Most existing models are, however, based on supervised learning, i.e., during training the teacher tells which object each view belongs to. The model proposed by Weinshall et al. (1990) also requires a signal that segregates different objects during training. This paper, on the other hand, discusses unsupervised aspects of 3D object recognition where the system discovers categories by itself. 950 Satoshi Suzuki, Hiroshi Ando This paper presents an unsupervised classification scheme for categorizing 3D objects from their 2D views. The scheme consists of a mixture of 5-layer auto-associative networks, each of which identifies an object by non-linearly encoding the views into a representation that describes transformation of a rigid object. A mixture model with linear networks was also studied by Williams et al. (1993) for classifying objects under affine transformations. We propose two models that employ different classification mechanisms. The first model classifies the given view by selecting an auto-associative network whose recovered view best matches the input view. The second model is based on the modular architecture proposed by Jacobs et al. (1991) in which an additional 3-layer network classifies the views by directly splitting the input space. The simulations using 3D wire-frame objects demonstrate that both models effectively learn to classify each view as a 3D object. This paper is organized as follows. Section 2 describes in detail the proposed models for unsupervised classification of 3D objects. Section 3 describes the simulation results using 3D wire-frame objects. In these simulations, we test the performance of the proposed models and examine what internal representations are acquired in the hidden layers. Finally, Section 4 concludes this paper. 2 THE NETWORK MODELS This section describes an unsupervised scheme that classifies 2D views into 3D objects. We initially examined classical unsupervised clustering schemes, such as the k-means method or the vector quantization method, to see whether such methods can solve this problem (Duda & Hart, 1973). Through simulations using the wire-frame objects described in the next section, we found that these methods do not yield satisfactory performance. We, therefore, propose a new unsupervised learning scheme for classifying 3D objects. The proposed scheme exploits an auto-associative network for identifying an object. An auto-associative network finds an identity mapping through a bottleneck in the hidden layer, i.e., the l network approximates functions . F and F- 1 such that R n ~Rm r ) R n where m < n. The network, thus, compresses the input into a low dimensional representation by eliminating redundancy. If we use a five-layer perceptron network, the network can perform nonlinear dimensionality reduction, which is a nonlinear analogue to the principal component analysis (Oja, 1991; DeMers & Cottrell, 1993). The proposed classification scheme consists of a mixture of five-layer auto-associative networks which we call the identification networks, or the I-Nets. In the case where the inputs are the projected views of a rigid object, the minimum dimension that constrains the input variation is the degree of freedom of the rigid object, which is six in the most general case, three for rotation and three for translation. Thus, a single I-Net can compress the views of an object into a representation whose dimension is its degree of freedom. The proposed scheme categorizes each view of a number of 3D objects into its class through selecting an appropriate I-Net. We present the following two models for different selection and learning methods. Model I: The model I selects an I-Net whose output best fits the input (see Fig. 1). SpecIfically, we assume a classifier whose output vector is given by the softmax function of a negative squared difference between the input and the output of the I-Nets, i.e., (1) Unsupervised CLassification of 3D Objects from 2D Views I-Net I-Net ??? 951 I-Net I-Net I-Net... I-Net 2D Projected Images of 3D Objects 2D Projected Images of 3D Objects Modell Model II Figure 1: Model I and Model II. Each I-Net (identification net) is a 5-layer auto-associative network and the C-Net (classification net) is a 3-layer network. where Y * and Yi denote the input and the output of the i th I-Net, respectively. Therefore, if only one of the I-Nets has an output that best matches the input, then the output value of the corresponding unit in the classifier becomes nearly one and the output values of the other units become nearly zero. For training the network, we maximize the following objective function: Lexp[-ally * _YiI1 2 ] In-'~'--~------~ exp[ Yi 112 ] L -Ily *- (2) i where a (>1) denotes a constant. This function forces the output of at least one I-Net to fit the input, and it also forces the rest of I-Nets to increase the error between the input and the output. Since it is difficult for a single I-Net to learn more than one object, we expect that the network will eventually converge to the state where each I-Net identifies only one object. Model II: The model II, on the other hand, employs an additional network which we call the classification network or the C-Net, as illustrated in Fig. 1. The C-Net classifies the given views by directly partitioning the input space. This type of modular architecture has been proposed by Jacobs et al. (1991) based on a stochastic model (see also Jordan & Jacobs, 1992). In this architecture, the final output, Y, is given by (3) where Yi denotes the output of the i th I-Net, and gi is given by the softmax function gi = eXP[SiVtexP[Sj] where Si is the weighted sum arriving at the (4) i th output unit of the C-Net. For the C-Net, we use three-layer perceptron, since a simple perceptron with two layers did not provide a good performance for the objects used for our simulations (see Section Satoshi Suzuki, Hiroshi Ando 952 3). The results suggest that classification of such objects is not a linearly separable problem. Instead of using MLP (multi-layer perceptron), we could use other types of networks for the C-Net, such as RBF (radial basis function) (Poggio & Edelman, 1990). We maximize the objective function In LgjO'-1 exp[-lly*-yJ /(20'2)] (5) j where 0'2 is the variance. This function forces the C-Net to select only one I-Net, and at the same time, the selected I-Net to encode and decode the input information. Note that the model I can be interpreted as a modified version of the model II, since maximizing (2) is essentially equivalent to maximizing (5) if we replace Sj of the C-Net in (4) with a ne&ative s~uared difference between the input and the output of the i th I-Net, i.e., Sj = -Ily * -yj Although the model I is a more direct classification method that exploits auto-associative networks, it is interesting to examine what information can be extracted from the input for classification in the model II (see Section 3.2). Ir . 3 SIMULATIONS We implemented the network models described in the previous section to evaluate their performance. The 3D objects that we used for our simulations are 5-segment wire-frame objects whose six vertices are randomly selected in a unit cube, as shown in Fig. 2 (a) (see also Poggio & Edelman, 1990). Various views of the objects are obtained by orthographically projecting the objects onto an image plane whose position covers a sphere around the object (see Fig. 2 (b?. The view position is defined by the two parameters, 8 and fj). In the simulations, we used x, y image coordinates of the six vertices of three wire-frame objects for the inputs to the network. The models contain three I-Nets, whose number is set equal to the number of the objects. The number of units in the third layer of the five-layer I-Nets is set equal to the number of the view parameters, which is two in our simulations. We used twenty units in the second and fourth layers. To train the network efficiently, we initially limited the ranges of 8 and fj) to 1r /8 and 1r /4 and gradually increased the range until it covered the whole sphere. During the training, objects were randomly selected among the three and their views were randomly selected within the view range. The steepest ascent method was used for maximizing the objective functions (2) and (5) in our simulations, but more efficient methods, such as the conjugate gradient method, can also be used. (a) (b) z View y Figure 2: (a) 3D wire-frame objects. (b) Viewpoint defined by two parameters, 8 and fj). 953 Unsupervised Classification of 3D Objects from 2D Views 3.1 SIMULATIONS USING THE MODEL I This section describes the simulation results using the model!. As described in Section 2, the classifier of this model selects an I-Net that produces minimum error between the output and the input. We test the classification performance of the model and examine internal representations of the I-Nets after training the networks. The constant a in the objective function (2) was set to 50 during the training. Fig. 3 shows the output of the classifier plotted over the view directions when the views of an object are used for the inputs. The output value of a unit is almost equal to one over the entire range of the view direction, and the outputs of the other two units are nearly zero. This indicates that the network effectively classifies a given view into an object regardless of the view directions. We obtained satisfactory results for classification if more than five units are used in the second and fourth layers of the I-Nets. Fig. 4 shows examples of the input views of an object and the views recovered by the corresponding I-Net. The recovered views are significantly similar to the input views, indicating that each auto-associative I-Net can successfully compress and recover the views of an object. In fact, as shown in Fig. 5, the squared error between the input and the output of an I-Net is nearly zero for only one of the objects. This indicates that each I-Net can be used for identifying an object for almost entire view range. UNIT 1 UNIT 2 UNIT 3 Figure 3: Outputs of the classifier in the model I. The output value of the second unit is almost equal to one over the full view range, and the outputs of the other two units are nearly zero for one of the 3D objects. Recovered views Input views Figure 4: Examples of the input and recovered views of an object. The recovered views are significantly similar to the input views. 954 Satoshi Suzuki, Hiroshi Ando We further analyzed what information is encoded in the third layer of the I-Nets. Fig. 6 (a) illustrates the outputs of the third layer units plotted as a function of the view direction ( (}, ?) of an object. Fig. 6 (b) shows the view direction ( (} , ?) plotted as a function of the outputs of the third layer units. Both figures exhibit single-valued functions, i.e. the view direction of the object uniquely determines the outputs of the hidden units, and at the same time the outputs of the hidden units uniquely determine the view direction. Thus, each I-Net encodes a given view of an object into a representation that has one-to-one correspondence with the view direction. This result is expected from the condition that the dimension of the third layer is set equal to the degree of freedom of a rigid object. Object 1 Object 2 Object 3 Figure 5: Error between the input view and the recovered view of an I-Net for each object. The figures show that the I-Net recovers only the views of Object 3. (a) unit! unit2 (b) a Wlit2 unit2 Figure 6: (a) Outputs of the third layer units of an I-Net plotted over the view direction ( (}, ?) of an object. (b) The view direction plotted over the outputs of the third layer units. Figure (b) was obtained by inversely replotting Figure (a). 3.2 SIMULATIONS USING THE MODEL n In this section, we show the simulation results using the model II. The C-Net in the model learns to classify the views by splitting the input space nonlinearly. We examine internal representations of the C-Net that lead to view invariant classification in its output. 955 Unsupervised Classification of 3D Objects from 2D Views In the simulations, we used the same 3 wire-frame objects used in the previous simulations. The C-Net contains 20 units in the hidden layer. The parameter cr in the objective function (5) was set to 0.1. Fig. 7 (a) illustrates the values of an output unit in the C-Net for an object. As in the case of the model I, the model correctly classified the views into their original object for almost entire view range. Fig. 7 (b) illustrates the outputs of two of the hidden units as examples, showing that each hidden unit has a limited view range where its output is nearly one. The C-Net, thus, combines these partially invariant representations in the hidden layer to achieve full view invariance at the output layer. To examine a generalization ability of the model, we limited the view range in the training period and tested the network using the images with the full view range. Fig. 8 (a) and (b) show the values of an output unit of the C-Net and the error of the corresponding I-Net plotted over the entire view range. The region surrounded by a rectangle indicates the range of view directions where the training was done. The figures show that the correct classification and the small recovery error are not restricted within the training range but spread across this range, suggesting that the network exhibits a satisfactory capability of generalization. We obtained similar generalization results for the model I as well. We also trained the networks with a sparse set of views rather than using randomly selected views. The results show that classification is nearly perfect regardless of the viewpoints if we use at least 16 training views evenly spaced within the full view range. Figure 7: (a) Output values of an output unit of the C-Net when the views of an object are given (cf. Fig.3). (b) Output values of two hidden units ofthe C-Net for the same object. OUTPUT ERROR Figure 8: (a) Output values of an output unit of the C-Net. (b) Errors between the input views and the recovered views of the corresponding I-Net. The region surrounded by a rectangle indicates the view range where the training was done. 956 Satoshi Suzuki, Hiroshi Ando 4 CONCLUSIONS We have presented an unsupervised classification scheme that classifies 3D objects from their 2D views. The scheme consists of a mixture of non-linear auto-associative networks each of which identifies an object by encoding an input view into a representation that indicates its view direction. The simulations using 3D wire-frame objects demonstrated that the scheme effectively clusters the given views into their original objects with no explicit identification of the object classes being provided to the networks. We presented two models that utilize different classification mechanisms. In particular, the model I employs a novel classification and learning strategy that forces only one network to reconstruct the input view, whereas the model II is based on a conventional modular architecture which requires training of a separate classification network. Although we assumed in the simulations that feature points are already identified in each view and that their correspondence between the views is also established, the scheme does not, in principle, require the identification and correspondence of features, because the scheme is based solely on the existence of non-linear mappings between a set of images of an object and its degree of freedom. Therefore, we are currently investigating how the proposed scheme can be used to classify real gray-level images of 3D objects. Acknowledgments We would like to thank Mitsuo Kawato for extensive discussions and continuous encouragement, and Hiroaki Gomi and Yasuharu Koike for helpful comments. We are also grateful to Tommy Poggio for insightful discussions. References DeMers, D. and Cottrell, G. (1993). Non-linear dimensionality reduction. In Hanson, S. 1., Cowan, 1. D. & Giles, C. L., (eds), Advances in Neural Information Processing Systems 5. Morgan Kaufmann Publishers, San Mateo, CA. 580-587. Duda, R. O. and Hart, P. E. (1973). Pattern Classification and Scene Analysis. John Wiley & Sons, NY. Jacobs, R. A., Jordan, M. I., Nowlan, S. 1. and Hinton, G. E. (1991). Adaptive mixtures of local experts. Neural Computation, 3,79-87. Jordan, M. I. and Jacobs, R. A. (1992). Hierarchies of adaptive experts. In Moody, J. E., Hanson, S. J. & Lippmann, R. P., (eds), Advances in Neural Information Processing Systems 4. Morgan Kaufmann Publishers, San Mateo, CA. 985-992. Oja, E. (1991). Data compression, Feature extraction, and autoassociation in feedforward neural networks. In Kohonen, K. et al. (eds), Anificial Neural Networks. Elsevier Science publishers B.V., North-Holland. Poggio, T. and Edelman, S. (1990). A network that learns to recognize three-dimensional objects. Nature, 343, 263. Weinshall, D., Edelman, S. and Btilthoff, H. H. (1990). A self-organizing multiple-view representation of 3D objects. In Touretzky, D. S., (eds), Advances in Neural Information Processing Systems 2. Morgan Kaufmann Publishers, San Mateo, CA. 274-281. Williams, C. K. I., Zemel, R. S. and Mozer, M. C. (1993). Unsupervised learning of object models. AAAI Fall 1993 Symposium on Machine Learning in Computer Vision. 

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A Study of Parallel Perturbative Gradient Descent D. Lippe? J. Alspector Bellcore Morristown, NJ 07960 Abstract We have continued our study of a parallel perturbative learning method [Alspector et al., 1993] and implications for its implementation in analog VLSI. Our new results indicate that, in most cases, a single parallel perturbation (per pattern presentation) of the function parameters (weights in a neural network) is theoretically the best course. This is not true, however, for certain problems and may not generally be true when faced with issues of implementation such as limited precision. In these cases, multiple parallel perturbations may be best as indicated in our previous results. 1 INTRODUCTION Motivated by difficulties in analog VLSI implementation of back-propagation [Rumelhart et al., 1986] and related algorithms that calculate gradients based on detailed knowledge of the neural network model, there were several similar recent papers proposing to use a parallel [Alspector et al., 1993, Cauwenberghs, 1993, Kirk et al., 1993] or a semi-parallel [Flower and Jabri, 1993] perturbative technique which has the property that it measures (with the physical neural network) rather than calculates the gradient. This technique is closely related to methods of stochastic approximation [Kushner and Clark, 1978] which have been investigated recently by workers in fields other than neural networks. [Spall, 1992] showed that averaging multiple parallel perturbations for each pattern presentation may be asymptotically preferable in the presence of noise. Our own results [Alspector et al., 1993] indicated ?Present address: Dept. of EECSj MITj Cambridge, MA 02139; dalippe@mit.edu 804 D. Lippe, 1. Alspector that multiple parallel perturbations are also preferable when only limited precision is available in the learning rate which is realistic for a physical implementation. In this work we have investigated whether multiple parallel perturbations for each pattern are non-asymptotically preferable theoretically (without noise). We have also studied this empirically, to the limited degree that simulations allow, by removing the precision constraints of our previous work. 2 GRADIENT ESTIMATION BY PARALLEL WEIGHT PERTURBATION Following our previous work, one can estimate the gradient of the error, E( w), with respect to any weight, Wi, by perturbing Wi by 6w 1 and measuring the change in the output error, 6E, as the entire weight vector, W, except for component Wi is held constant. 6E 6w1 E(w + 6;1) - E(w) 6Wi We now consider perturbing all weights simultaneously. However, we wish to have the perturbation vector, 6w, chosen uniformly on a hypercube. Note that this requires only a random sign multiplying a fixed perturbation and is natural for VLSI using a parallel noise generator [Alspector et al., 1991J. This leads to the approximation (ignoring higher order terms) w 6E - -+ 8E 2:(8E) -6w? -8w? (6Wi) -6w?? - 8w? , 'i?1 ] (1 ) , The last term has expectation value zero for random and independently distributed 6w1 ? The weight change rule where 1] is a learning rate, will follow the gradient on the average but with considerable noise. For each pattern, one can reduce the variance of the noise term in (1) by repeating the random parallel perturbation many times to improve the statistical estimate. If we average over P perturbations, we have where p indexes the perturbation number. A Study of Parallel Perturbative Gradient Descellf 3 3.1 805 THEORETICAL RELATIVE EFFICIENCY BACKGROUND Spall [Spall, 1992] shows in an asymptotic sense that multiple perturbations may be faster if only a noisy measurement of E( tV) is available, and that one perturbation is superior otherwise. His results are asymptotic in that they compare the rate of convergence to the local minimum if the algorithms run for infinite time. Thus, his results may only indicate that 1 perturbation is superior close to a local minimum. Furthermore, his result implicitly assumes that P perturbations per weight update takes P times as long as 1 perturbation per weight update. Experience shows that the time required to present patterns to the hardware is often the bottleneck in VLSI implementations of neural networks [Brown et al., 1992]. In a hardware implementation of a perturbative learning algorithm, a few perturbations might be performed with no time penalty while waiting for the next pattern presentation. The remainder of this section sketches an argument that multiple perturbations may be desirable for some problems in a non-asymptotic sense, even in a noise free environment and under the assumption of a multiplicative time penalty for performing multiple perturbations. On the other hand, the argument also shows that there is little reason to believe in practice that any given problem will be learned more quickly by multiple perturbations. Space limitations prevent us from reproducing the full argument and discussion of its relevance which can be found in [Lippe, 1994]. The argument fixes a point in weight space and considers the expectation value of the change in the error induced by one weight update under both the 1 perturbation case and the multiple perturbation case. [Cauwenberghs, 1994] contains a somewhat related analysis of the relative speed of one parallel perturbation and weight perturbation as described in [Jabri and Flower, 1991]. The analysis is only truly relevant far from a local minimum because close to a local minimum the variance of the change of the error is as important as the mean of the change of the error. 3.2 Calculations If P is the number of perturbations, then our learning rule is ~Wi -'TJ ~ 6E(p) =P If W is the number of weights, then ~E, 1 w 8E (2) L.J~. P=1 6wi calculated to second order in 'TJ, is W W 82E ~E = 'L.J " -8 W ? ~Wi + -2 '" L.J '" L.J 8 W? 8 W ? ~Wi~Wj. i=l' Expanding 6E(p) i=l j=l to second order in (j (where W 6E(p) 8E = ' " -6w~P) L.J 8w' j=l J J W 6Wi W ? = ?(j), we obtain 82 + !2 "'" E 6w~P)6w(P). L.J L.J 8w' k J k j=l k=l (3) J J 8w (4) 806 D. Lippe, 1. Alspector [Lippe, 1994] shows that combining (2)-(4), retaining only first and second order terms, and taking expectation values gives 2 < l:1E >= -TJX + ~ (Y + PZ) (5) where x w L i=l (8E)2 8w' ' ' Z Y Note that first term in (5) is strictly less than or equal to 0 since X is a sum of squares l . The second term, on the other hand, can be either positive or negative. Clearly then a sufficient condition for learning is that the first term dominates the second term. By making TJ small enough, we can guarantee that learning occurs. Strictly speaking, this is not a necessary condition for learning. However, it is important to keep in mind that we are only focusing on one point in weight space. If, at this point in weight space, < l:1E > is negative but the second term's magnitude is close to the first term's magnitude, it is not unlikely that at some other point in weight space < l:1E > will be positive. Thus, we will assume that for efficient learning to occur, it is necessary that TJ be small enough to make the first term dominate the second term. Assume that some problem can be successfully learned with one perturbation, at learning rate TJ(I). Then the first order term in (5) dominates the second order terms. Specifically, at any point in weight space we have, for some large constant J1., TJ(I)X ~ J1.TJ(I)2IY + ZI In order to learn with P perturbations, we apparently need TJ( P)X ~ J1. TJ(~)2 IY + P ZI (6) The assumption that the first order term of (5) dominates the second order terms implies that convergence time is proportional to Thus, learning is more efficient in the multiple perturbation case if ,.lp), J1.TJ(P) > J1.TJ(I) P (7) It turns out, as shown in [Lippe, 1994] that the conditions (6) and (7) can be met simultaneously with multiple perturbations if ~ 2. =f lIf we are at a stationary point then the first term in (5) is O. A Study of Parallel Perturbative Gradient Descent 807 It is shown in [Lippe, 1994], by using the fact that the Hessian of a quadratic function with a minimum is positive semi-definite, that if E is guadratic and has < 2). Any well a minimum, then Y and Z have the same sign (and hence behaved function acts quadratically sufficiently close to a stationary point. Thus, we can not get < flE > more than a factor of P larger by using P perturbations near local minima of well behaved functions. Although, as mentioned earlier, we are entirely ignoring the issue of the variance of flE, this may be some indication of the asymptotic superiority of 1 perturbation. =f 3.3 Discussion of Results -i The result that multiple perturbations are superior when ~ 2 may seem somewhat mysterious . It sheds some light on our answer to rewrite (5) as Y < flE >= -"IX + "I2(p + Z). For strict gradient descent, the corresponding equation is < flE >= flE = -"IX + "I2Z. The difference between strict gradient descent and perturbative gradient descent, on average, is the second order term "I2~. This is the term which results from not following the gradient exactly, and it Obviously goes down as P goes up and the gradient measurement becomes more accurate. Thus, if Z and Y have different signs, P can be used to make the second order term disappear. There is no way to know whether this situation will occur frequently. Furthermore, it is important to keep in mind that if Y is negative and Z is positive, then raising P may make the magnitude of the second order term smaller, but it makes the term itself larger. Thus, in general, there is little reason to believe that multiple perturbations will help with a randomly chosen problem. An example where multiple perturbations help is when we are at a point where the error surface is convex along the gradient direction, and concave in most other directions. Curvature due to second derivative terms in Y and Z help when the gradient direction is followed, but can hurt when we stray from the gradient. In this case, Z < 0 and possibly Y > 0, so multiple perturbations might be preferable in order to follow the gradient direction very closely. 4 4.1 SIMULATIONS OF SINGLE AND MULTIPLE PARALLEL PERTURBATION CONSTANT LEARNING RATES The second order terms in (5) can be reduced either by using a small learning rate, or by using more perturbations, as discussed briefly in [Cauwenberghs, 1993]. Thus, if "I is kept constant, we expect a minimum necessary number of perturbations in order to learn. This in itself might be of importance in a limited precision implementation. If there is a non-trivial lower bound on "I, then it might be necessary to use multiple perturbations in order to learn. This is the effect that was noticed in [Alspector et al., 1993]. At that time we thought that we had found empirically 808 P 1 1 1 1 1 7 7 7 7 7 D. Lippe, J. Alspector TJ .0005 .001 .002 .003 .004 .00625 .008 .0125 .025 .035 Table 1: Running times for the first initial weight vector Time for < . 1 Time for < .5 1,121,459 831 , 684 784, 768 4 94,029 1,695,974 707,840 583,654 922,880 1,010,355 Not tested 32,179 18,534 11,008 9,933 9,728 23,834 16,845 13,261 12,006 17,024 that multiple perturbations were necessary for learning. The problem was that we failed to decrease the learning rate with the number of perturbations. 4.2 EMPIRICAL RELATIVE EFFICIENCY OF SINGLE AND MULTIPLE PERTURBATION ALGORITHMS Section 3 showed that, in theory, multiple perturbations might be faster than 1 perturbation. We investigated whether or not this is the case for the 7 input hamming error correction problem as described in [Biggs, 1989]. This is basically a nearest neighbor problem. There exist 16 distinct 7 bit binary code words. When presented with an arbitrary 7 bit binary word, the network is to output the code word with the least hamming distance from the input. After preliminary tests with 50, 25, 7, and 1 perturbation, it seemed that 7 perturbations provided the fastest learning, so we concentrated on running simulations for both the 1 perturbation and the 7 perturbation case. Specifically, we chose two different (randomly generated) initial weight vectors, and five different seeds for the pseudo-random function used to generate the bWi. For each of these ten cases, we tested both 1 perturbation and 7 perturbations with various learning rates in order to obtain the fastest possible learning. The 128 possible input patterns were repeatedly presented in order. We investigated how many pattern presentations were necessary to drive the MSE below .1 and how many presentations were necessary to drive it below .5. Recalling the theory developed in section 3, we know that multiple perturbations can be helpful only far away from a stationary point. Thus, we expected that 7 perturbations might be quicker reaching .5 but would be slower reaching .1. The results are summarized in tables 1 and 2. Each table summarizes information for a different initial weight vector. All of the data presented are averaged over 5 runs, one with each of the different random seeds. The two columns labeled "Time for < .5" and "Time for < .1" are adjusted according to the assumption that one weight update at 7 perturbations takes 7 times as long as one weight update at 1 perturbation. In each table, the following four numbers appear in italics: the shortest time to reach .1 with 1 perturbation, the shortest time to reach .1 with 7 perturbations, the shortest time to reach .5 with 1 perturbation, and the shortest time to reach .5 with 7 perturbations. 7 perturbations were a loss in three out of four of the experiments. Surprisingly, A Study of Parallel Perturbative Gradient Descent l' 'T/ 1 1 1 1 1 1 1 1 1 .001 .002 .003 .004 .00625 .008 .0125 .025 .035 809 Table 2: Running times for the second initial weight vector TIme for < .1 TIme for < .5 928,236 719 , 078 154,139 1,603,354 629 , 530 611,610 912,333 1,580,442 Not tested 22,133 12,817 10,675 11,150 21,059 19,112 15,949 14,515 11,141 the one time that multiple perturbations helped was in reaching .1 from the second initial weight vector. There are several possible explanations for this. To begin with, these learning times are averages over only five simulations each, which makes their statistical significance somewhat dubious. Unfortunately, it was impractical to perform too many experiments as the data obtained required 180 computer simulations, each of which sometimes took more than a day to complete. Another possible explanation is that .1 may not be "asymptotic enough." The numbers .5 and .1 were chosen somewhat arbitrarily to represent non-asymptotic and asymptotic results. However, there is no way of predicting from the theory how close the error must be to its minimum before asymptotic results become relevant. The fact that 1 perturbation outperformed 7 perturbations in three out of four cases is not surprising. As explained in section 3, there is in general no reason to believe that multiple perturbations will help on a randomly chosen problem. 5 CONCLUSION Our results show that, under ideal computational conditions, where the learning rate can be adjusted to proper size, that a single parallel perturbation is, except for unusual problems, superior to multiple parallel perturbations. However, under the precision constraints imposed by analog VLSI implementation, where learning rates may not be adjustable and presenting a pattern takes longer than performing a perturbation, multiple parallel perturbations are likely to be the best choice. Acknowledgment We thank Gert Cauwenberghs and James Spall for valuable and insightful discussIons. References [Alspector et al., 1991] Alspector, J., Gannett, J. W ., Haber, S., Parker, M. B., and Chu, R. (1991). A VLSI-efficient technique for generating multiple uncorrelated noise sources and its application to stochastic neural networks. IEEE Transactions on Circuits and Systems, 38:109-123 . [Alspector et al., 1993] Alspector, J., Meir, R., Yuhas, B., Jayakumar, A., and Lippe, D. (1993). A parallel gradient descent method for learning in analog 810 D. Lippe, J. A/spector VLSI neural networks. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems 5, pages 836-844, San Mateo, California. Morgan Kaufmann Publishers. [Biggs, 1989] Biggs, N. L. (1989). Discrete Math. Oxford University Press. [Brown et al., 1992] Brown, T. X., Tran, M. D., Duong, T., and Thakoor, A. P. (1992). Cascaded VLSI neural network chips: Hardware learning for pattern recognition and classification. Simulation, 58(5):340-347. [Cauwenberghs, 1993] Cauwenberghs, G. (1993). A fast stochastic error-descent algorithm for supervised learning and optimization. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems 5, pages 244-251, San Mateo, California. Morgan Kaufmann Publishers. [Cauwenberghs, 1994] Cauwenberghs, G. (1994). Analog VLSI Autonomous Systems for Learning and Optimization. PhD thesis, California Institute of Technology. [Flower and Jabri, 1993J Flower, B. and Jabri, M. (1993). Summed weight neuron perturbation: An o(n) improvement over weight perturbation. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems 5, pages 212-219, San Mateo, California. Morgan Kaufmann Publishers. [Jabri and Flower, 1991] Jabri, M. and Flower, B. (1991). Weight perturbation: An optimal architecture and learning technique for analog VLSI feedforward and recurrent multilayer networks. In Neural Computation 3, pages 546-565. [Kirk et al., 1993] Kirk, D., Kerns, D., Fleischer, K., and Barr, A. (1993). Analog VLSI implementation of gradient descent. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems 5, pages 789-796, San Mateo, California. Morgan Kaufmann Publishers. [Kushner and Clark, 1978] Kushner, H. and Clark, D. (1978). Stochastic Approzimation Methods for Constrained and Unconstrained Systems. Springer-Verlag, New York. [Lippe, 1994] Lippe, D. A. (1994). Parallel, perturbative gradient descent methods for learning in analog VLSI neural networks. Master's thesis, Massachusetts Institute of Technology. [Rumelhart et al., 1986] Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986). Learning internal representations by error propogation. In Rumelhart, D. E. and McClelland, J. L., editors, Parallel Distributed Processing: Ezplorations in the Microstructure of Cognition, page 318. MIT Press, Cambridge, MA. [Spall, 1992] Spall, J. C. (1992). Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Transactions on A utomatic Control, 37(3):332-341. 

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Advantage Updating Applied to a Differential Game Mance E. Harmon Wright Laboratory WL/AAAT Bldg. 635 2185 Avionics Circle Wright-Patterson Air Force Base, OH 45433-7301 harmonme@aa.wpafb.mil Leemon C. Baird III? Wright Laboratory baird@cs.usafa.af.mil A. Harry Klopr Wright Laboratory klopfah@aa.wpafb.mil Category: Control, Navigation, and Planning Keywords: Reinforcement Learning, Advantage Updating, Dynamic Programming, Differential Games Abstract An application of reinforcement learning to a linear-quadratic, differential game is presented. The reinforcement learning system uses a recently developed algorithm, the residual gradient form of advantage updating. The game is a Markov Decision Process (MDP) with continuous time, states, and actions, linear dynamics, and a quadratic cost function. The game consists of two players, a missile and a plane; the missile pursues the plane and the plane evades the missile. The reinforcement learning algorithm for optimal control is modified for differential games in order to find the minimax point, rather than the maximum. Simulation results are compared to the optimal solution, demonstrating that the simulated reinforcement learning system converges to the optimal answer. The performance of both the residual gradient and non-residual gradient forms of advantage updating and Q-learning are compared. The results show that advantage updating converges faster than Q-learning in all simulations. The results also show advantage updating converges regardless of the time step duration; Q-learning is unable to converge as the time step duration ~rows small. U.S .A.F. Academy, 2354 Fairchild Dr. Suite 6K4l, USAFA, CO 80840-6234 Mance E. Hannon, Leemon C. Baird ll/, A. Harry Klopf 354 1 ADVANTAGE UPDATING The advantage updating algorithm (Baird, 1993) is a reinforcement learning algorithm in which two types of information are stored. For each state x, the value V(x) is stored, representing an estimate of the total discounted return expected when starting in state x and performing optimal actions. For each state x and action u, the advantage, A(x,u), is stored, representing an estimate of the degree to which the expected total discounted reinforcement is increased by performing action u rather than the action currently considered best. The optimal value function V* (x) represents the true value of each state. The optimal advantage function A *(x,u) will be zero if u is the optimal action (because u confers no advantage relative to itself) and A*(x,u) will be negative for any suboptimal u (because a suboptimal action has a negative advantage relative to the best action). The optimal advantage function A * can be defined in terms of the optimal value function v*: A*(x,u) = ~[RN(X,U)- V*(x)+ rNV*(x')] bat (1) The definition of an advantage includes a l/flt term to ensure that, for small time step duration flt, the advantages will not all go to zero. Both the value function and the advantage function are needed during learning, but after convergence to optimality, the policy can be extracted from the advantage function alone. The optimal policy for state x is any u that maximizes A *(x,u). The notation ~ax (x) = max A(x,u) (2) " defines Amax(x). If Amax converges to zero in every state, the advantage function is said to be normalized. Advantage updating has been shown to learn faster than Q-Iearning (Watkins, 1989), especially for continuous-time problems (Baird, 1993). If advantage updating (Baird, 1993) is used to control a deterministic system, there are two equations that are the equivalent of the Bellman equation in value iteration (Bertsekas, 1987). These are a pair of two simultaneous equations (Baird, 1993): A(x,u)-maxA(x,u') =(R+ y l1l V(x')- V(X?)_l (3) maxA(x,u)=O (4) w ~t " where a time step is of duration L1t, and performing action u in state x results in a reinforcement of R and a transition to state Xt+flt. The optimal advantage and value functions will satisfy these equations. For a given A and V function, the Bellman residual errors, E, as used in Williams and Baird (1993) and defined here as equations (5) and (6).are the degrees to which the two equations are not satisfied: E1 (xl,u) =(R(x"u)+ y l1l V(xt+l1I)- V(XI?)~- A(x"u)+ max A(x,,u' ) (5) E2 (x,u)=-maxA(x,u) (6) ~t " w Advantage Updating Applied to a Differential Game 2 355 RESIDUAL GRADIENT ALGORITHMS Dynamic programming algorithms can be guaranteed to converge to optimality when used with look-up tables, yet be completely unstable when combined with functionapproximation systems (Baird & Harmon, In preparation). It is possible to derive an algorithm that has guaranteed convergence for a quadratic function approximation system (Bradtke, 1993), but that algorithm is specific to quadratic systems. One solution to this problem is to derive a learning algorithm to perfonn gradient descent on the mean squared Bellman residuals given in (5) and (6). This is called the residual gradient form of an algorithm. There are two Bellman residuals, (5) and (6), so the residual gradient algorithm must perfonn gradient descent on the sum of the two squared Bellman residuals. It has been found to be useful to combine reinforcement learning algorithms with function approximation systems (Tesauro, 1990 & 1992). If function approximation systems are used for the advantage and value functions, and if the function approximation systems are parameterized by a set of adjustable weights, and if the system being controlled is deterministic, then, for incremental learning, a given weight W in the functionapproximation system could be changed according to equation (7) on each time step: dW = _ a a[E;(x"u,) + E;(x"u,)] 2 aw __ E( ) aE1 (x, ,u,) _ E ( ) aE2 (x" u, ) - a 1 x"u, aw a 2 x"u, aw = _a(_l (R + yMV(X'+M) - V (x) ) - A(x"u,) + max A(X"U)) dt u _(_I (yfJJ aV(x,+fJJ) _ av(x,))_ aA(x"u,) + am~XA(xt'U)J dt aw -amaxA(x"u) U aw aw (7) aw amaxA(x"u) U a W As a simple, gradient-descent algorithm, equation (7) is guaranteed to converge to the correct answer for a deterministic system, in the same sense that backpropagation (Rumelhart, Hinton, Williams, 1986) is guaranteed to converge. However, if the system is nondetenninistic, then it is necessary to independently generate two different possible "next states" Xt+L1t for a given action Ut perfonned in a given state Xt. One Xt+L1t must be used to evaluate V(Xt+L1t), and the other must be used to evaluate %w V(xt+fJJ)' This ensures that the weight change is an unbiased estimator of the true Bellman-residual gradient, but requires a system such as in Dyna (Sutton, 1990) to generate the second Xt+L1t. The differential game in this paper was detenninistic, so this was not needed here. 356 Mance E. Harmon, Leemon C. Baird /11, A. Harry KLopf 3 THE SIMULATION 3.1 GAME DEFINITION We employed a linear-quadratic, differential game (Isaacs, 1965) for comparing Q-learning to advantage updating, and for comparing the algorithms in their residual gradient forms. The game has two players, a missile and a plane, as in games described by Rajan, Prasad, and Rao (1980) and Millington (1991). The state x is a vector (xm,xp) composed of the state of the missile and the state of the plane, each of which are composed of the poSition and velocity of the player in two-dimensional space. The action u is a vector (um,up) composed of the action performed by the missile and the action performed by the plane, each of which are the acceleration of the player in two-dimensional space. The dynamics of the system are linear; the next state xt+ 1 is a linear function of the current state Xl and action Ul. The reinforcement function R is a quadratic function of the accelerations and the distance between the players. R(x,u)= [distance2 + (missile acceleration)2 - 2(plane acceleration)2]6t (8) R(X,U)=[(X m -X p)2 +U~-2U!]llt (9) In equation (9), squaring a vector is equivalent to taking the dot product of the vector with itself. The missile seeks to minimize the reinforcement, and the plane seeks to maximize reinforcement. The plane receives twice as much punishment for acceleration as does the missile, thus allowing the missile to accelerate twice as easily as the plane. The value function V is a quadratic function of the state. In equation (10), Dm and Dp are weight matrices that change during learning. (10) The advantage function A is a quadratic function of the state X and action u. The actions are accelerations of the missile and plane in two dimensions. A(x,u)=x~Amxm +x~BmCmum +u~Cmum + (11) x~Apxp +x~BpCpup +u~Cpup The matrices A, B, and C are the adjustable weights that change during learning. Equation (11) is the sum of two general quadratic functions. This would still be true if the second and fifth terms were xBu instead of xBCu. The latter form was used to simplify the calculation of the policy. Using the xBu form, the gradient is zero when u=-C-lBx!2. Using the xBCu form, the gradient of A(x,u) with respect to u is zero when u=-Bx!2, which avoids the need to invert a matrix while calculating the policy. 3.2 THE BELLMAN RESIDUAL AND UPDATE EQUATIONS Equations (5) and (6) define the Bellman residuals when maximizing the total discounted reinforcement for an optimal control problem; equations (12) and (13) modify the algorithm to solve differential games rather than optimal control problems. Advantage Updating Applied to a Differential Game 357 E1(x"u,) = (R(x"u,)+ r6tV(xl+6t)- V(X,?)..!...- A(x"u,)+ minimax A(x,) tl.t E 2 (x"u,)=-minimax A(x,) (12) (13) The resulting weight update equation is: tl.W = -aU .((rt:., R+ r 6tV (X'M')- V(x,?) 1t - A(x"u,)+minimax A(X,?) aV~6t) aV(X,?)_1 _ aA(x"u,) + aminimax A(X,?) aw aw aW tl.t (14) " A() aminimax A(x,) -amzmmax x, aw For Q-leaming, the residual-gradient form of the weight update equation is: tl.W =-a( R+ r 6t minimax Q(Xl+dt)-Q(x"u,?) (15) .( r 6t -kminimax Q(x,+6t)--kQ(x"u,?) 4 RESULTS 4.1 RESIDUAL GRADIENT ADVANTAGE UPDATING RESULTS The optimal weight matrices A *, B *, C *, and D* were calculated numerically with Mathematica for comparison. The residual gradient form of advantage updating learned the correct policy weights, B, to three significant digits after extensive training. Very interesting behavior was exhibited by the plane under certain initial conditions. The plane learned that in some cases it is better to turn toward the missile in the short term to increase the distance between the two in the long term. A tactic sometimes used by pilots. Figure 1 gives an example. 1 0 r - - - - - - -...... ....................... ~/ ............................~ ......... ....... .' .'...... i ...?? I \. \. GO V C ....til? 0.01 .001 .0001 0 0.04 0.08 0.12 Time Figure 1: Simulation of a missile (dotted line) pursuing a plane (solid line), each having learned optimal behavior. The graph of distance vs. time show the effects of the plane's maneuver in turning toward the missile. Mance E. Harmon. Leemon C. Baird III. A. Harry Klopf 358 4.2 COMPARATIVE RESULTS . The error in the policy of a learning system was defined to be the sum of the squared errors in the B matrix weights. The optimal policy weights in this problem are the same for both advantage updating and Q-learning, so this metric can be used to compare results for both algorithms. Four different learning algorithms were compared: advantage updating, Q-Iearning, Residual Gradient advantage updating, and Residual Gradient Qlearning. Advantage updating in the non-residual-gradient form was unstable to the point that no meaningful results could be obtained, so simulation results cannot be given for it. 4.2.1 Experiment Set 1 The learning rates for both forms of Q-Iearning were optimized to one significant digit for each simulation. A single learning rate was used for residual-gradient advantage updating in all four simulations. It is possible that advantage updating would have performed better with different learning rates. For each algorithm, the error was calculated after learning for 40,000 iterations. The process was repeated 10 times using different random number seeds and the results were averaged. This experiment was performed for four different time step durations, 0.05, 0.005, 0.0005, and 0.00005. The non-residualgradient form of Q-Iearning appeared to work better when the weights were initialized to small numbers. Therefore, the initial weights were chosen randomly between 0 and 1 for the residual-gradient forms of the algorithms, and between 0 and 10-8 for the non-residualgradient form of Q-learning. For small time steps, nonresidual-gradient Q-Iearning performed so poorly that the error was lower for a learning rate of zero (no learning) than it was for a learning rate of 10- 8 . Table 1 gives the learning rates used for each simulation, and figure 2 shows the resulting error after learning. ? 8 6 Final Error ---0 0-- [J -D--FQ 4 2 ? ? -?-RAU 0 0.05 0.005 0.0005 0.00005 TIme Step Duration Figure 2: Error vs. time step size comparison for Q-Learning (Q), residual-gradient Q-Learning(RQ), and residual-gradient advantage updating(RAU) using rates optimal to one significant figure for both forms of Q-Iearning, and not optimized for advantage updating. The final error is the sum of squared errors in the B matrix weights after 40,000 time steps of learning. The final error for advantage updating was lower than both forms of Q-learning in every case. The errors increased for Qlearning as the time step size decreased. 359 Advantage Updating Applied to a Differential Game Time step duration, III 5.10-2 5.10- 3 5.10-4 5.10-5 Q 0.02 0.06 0.2 0.4 RQ 0.08 0.09 0 0 RAU 0.005 0.005 0.005 0.005 Table 1: Learning rates used for each simulation. Learning rates are optimal to one significant figure for both forms of Q-learning, but are not necessarily optimal for advantage updating. 4.2.2 Experiment Set 2 Figure 3 shows a comparison of the three algorithms' ability to converge to the correct policy. The figure shows the total squared error in each algorithms' policy weights as a function of learning time. This simulation ran for a much longer period than the simulations in table 1 and figure 2. The learning rates used for this simulation were identical to the rates that were found to be optimal for the shorter run. The weights for the non-Residual gradient form of Q-Iearning grew without bound in all of the long experiments, even after the learning rate was reduced by an order of magnitude. Residual gradient advantage updating was able to learn the correct policy, while Q-learning was unable to learn a policy that was better than the initial, random weights. Leorning Ability Comporison 10~------------------, Error ---RAU .1 - - - - -, RO, ,01 ,001 0 2 3 4 5 Time Steps in millions Figure 3 5 Conclusion The experimental data shows residual-gradient advantage updating to be superior to the three other algorithms in all cases. As the time step grows small, Q-learning is unable to learn the correct policy. Future research will include the use of more general networks and implementation of the wire fitting algorithm proposed by Baird and Klopf (1994) to calculate the policy from a continuous choice of actions in more general networks. 360 Mance E. Hannon. Leemon C. Baird Ill. A. Harry Klopf Acknowledgments This research was supported under Task 2312Rl by the Life and Environmental Sciences Directorate of the United States Air Force Office of Scientific Research. References Baird, L.C. (1993). Advantage updating Wright-Patterson Air Force Base, OH. (Wright Laboratory Technical Report WL-TR-93-1146, available from the Defense Technical information Center, Cameron Station, Alexandria, VA 22304-6145). Baird, L.C., & Harmon, M. E. (In preparation). Residual gradient algorithms WrightPatterson Air Force Base, OH. (Wright Laboratory Technical report). Baird, L.C., & Klopf, A. H. (1993). Reinforcement learning with high-dimensional. continuous actions Wright-Patterson Air Force Base, OH. (Wright Laboratory technical report WL-TR-93-1147, available from the Defense Technical information Center, Cameron Station, Alexandria, VA 22304-6145). Bertsekas, D. P. (1987). Dynamic programming : Deterministic and stochastic models. Englewood Cliffs, NJ: Prentice-Hall. Bradtke, S. J. (1993). Reinforcement Learning Applied to Linear Quadratic Regulation. Proceedings of the 5th annual Conference on Neural Information Processing Systems. Isaacs, Rufus (1965). Differential games. New York: John Wiley and Sons, Inc. Millington, P. J. (1991). Associative reinforcement learning for optimal control. Unpublished master's thesis, Massachusetts Institute of Technology, Cambridge, MA. Rajan, N., Prasad, U. R., and Rao, N. J. (1980). Pursuit-evasion of two aircraft in a horizontal plane. Journal of Guidance and Control. 3(3). May-June, 261-267. Rumelhart, D., Hinton, G., & Williams, R. (1986). Learning representations by backpropagating errors. Nature. 323.. 9 October, 533-536. Sutton, R. S. (1990). Integrated architectures for learning, planning, and reacting based on approximating dynamic programming. Proceedings of the Seventh International Conference on Machine Learning. Tesauro, G. (1990). Neurogammon: A neural-network backgammon program. Proceedings of the International Joint Conference on Neural Networks . 3.. (pp. 33-40). San Diego, CA. Tesauro, G. (1992). Practical issues in temporal difference learning. Machine Learning, 8(3/4), 279-292. Watkins, C. J. C. H. (1989). Learningfrom delayed rewards. Doctoral thesis, University, Cambridge, England. Cambr~dge 

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Generalisation in Feedforward Networks Adam Kowalczyk and Herman Ferra Telecom Australia, Research Laboratories 770 Blackburn Road, Clayton, Vic. 3168, Australia (a.kowalczyk@trl.oz.au, h.ferra@trl.oz.au) Abstract We discuss a model of consistent learning with an additional restriction on the probability distribution of training samples, the target concept and hypothesis class. We show that the model provides a significant improvement on the upper bounds of sample complexity, i.e. the minimal number of random training samples allowing a selection of the hypothesis with a predefined accuracy and confidence. Further, we show that the model has the potential for providing a finite sample complexity even in the case of infinite VC-dimension as well as for a sample complexity below VC-dimension. This is achieved by linking sample complexity to an "average" number of implement able dichotomies of a training sample rather than the maximal size of a shattered sample, i.e. VC-dimension. 1 Introduction A number offundamental results in computational learning theory [1, 2, 11] links the generalisation error achievable by a set of hypotheses with its Vapnik-Chervonenkis dimension (VC-dimension, for short) which is a sort of capacity measure. They provide in particular some theoretical bounds on the sample complexity, i.e. a minimal number of training samples assuring the desired accuracy with the desired confidence. However there are a few obvious deficiencies in these results: (i) the sample complexity bounds are unrealistically high (c.f. Section 4.), and (ii) for some networks they do not hold at all since VC-dimension is infinite, e.g. some radial basis networks [7]. 216 Adam Kowalczyk, Hemzan Ferra One may expect that there are at least three main reasons for this state of affairs: (a) that the VC-dimension is too crude a measure of capacity, (b) since the bounds are universal they may be forced too high by some malicious distributions, (c) that particular estimates themselves are too crude, and so might be improved with time. In this paper we will attack the problem along the lines of (a) and (b) since this is most promising. Indeed, even a rough analysis of some proofs of lower bound (e.g. [I)) shows that some of these estimates were determined by clever constructions of discrete, malicious distributions on sets of "shattered samples" (of the size of VC-dimension). Thus this does not necessarily imply that such bounds on the sample complexity are really tight in more realistic cases, e.g. continuous distributions and "non-malicious" target concepts, the point eagerly made by critics of the formalism. The problem is to find such restrictions on target concepts and probability distributions which will produce a significant improvement. The current paper discusses such a proposition which significantly improves the upper bounds on sample complexity. 2 A Restricted Model of Consistent Learning First we introduce a few necessary concepts and some basic notation. We assume we are given a space of samples X with a probability measure J.L, a set H of binary functions X t-+ {O, I} called the hypothesis space and a target concept t E H. For an n-sample i = (Z1, ... , zn) E xn and h : H -+ {O, I} the vector (h(Z1), ... , h(zn)) E {O, l}n will be denoted by h(i). We define two projections 7r:e and 7rt,:e of H onto {0,1 } as follows 7r:e(h) = h z) = (h(Z1), ... , h(zn)) and 7rt,:e(h) = 7r:e(lt - hi) = It - hl(i) for every h E H . Below we shall use the notation 151 for the cardinality of the set 5 . The average density of the sets of projections 7r:e( H) or 7rt ,:e( H) in {O, l}n is defined as n de! (_ de! de! PrH(i) d~ 17r:e(H)I/2n = l7rt,:e(H)I/2n (equivalently, this is the probability of a random vector in {O, l}n belonging to the set 7r:e(H)). Now we define two associated quantities: d.:J PrH'Il(n) J PrH(i)J.Ln(di) de! = = 2- n J 17r:e(H) lJ.Ln (di), (1) = We recall that dH max{n ; 3:eEX .. I7r:e(H)1 2n} is called the VapnikChervonenkis dimension (VC-dimension) of H [1, 11]. If d H ~ 00 then Sauer's lemma implies the estimates (c.f. [1, 2, 10)) PrH'Il(n) ~ PrH,max(n) ~ 2- n cf?(dH, n) ~ 2- n (en/dH )dH where cf?(d,n) d;J 2::1=0 (~) (we assume , (2) C) d;J ?ifi > n). Now we are ready to formulate our main assumption in the model. We say that the space of hypotheses His (J.Ln , C)-uniform around t E 2x iffor every set 5 C {O, l}n Jl 7r t,:e(H) n 51J.Ln(dx) ~ CI 5 IPrH'Il(n). (3) Generalisation in Feedforward Networks 217 The meaning of this condition is obvious: we postulate that on average the number of different projections 7rt,ar(h) of hypothesis h E H falling into S has a bound proportional to the probability PrH,~(n) of random vector in {O, l}n belonging to the set 7rt,ar(H). Another heuristic interpretation of (3) is as follows. Imagine that elements of 7rt,ar(H) are almost uniformly distributed in {O, 1}n, i.e. with average density Par ~ CI7rt,ar(H)1/2n. Thus the "mass" of the volume lSI is l7rt,ar(H) n SI ~ PariSI and so its average f l7rt,ar(H) n SIJ.tn(di) has the estimate ~ lSI f parJ.tn(di) ~ CISiPrH,~(n). Of special interest is the particular case of con8istent learning [1], i.e. when the target concept and the hypothesis fully agree on the training sample. In this case, for any E > 0 we introduce the notation QE(m) d;J de! m {i E Xm ; 3hEH ert,ar(h) = 0 & ert,~(h) ~ E}, de! f where ert,ar(h) = Ei=llt-hl(zi)/mand ert,~(h) = It-hl(z)J.t(dz) denote error rates on the training sample i = (ZlJ ... , zm) and X, respectively. Thus QE(m) is the set of all m-samples for which there exists a hypothesis in H with no error on the sample and the error at least E on X. Theorem 1 If the hypothesis space H is (J.t 2m , C)-uniform around t E H then for anyE > 8/m (4) (5) Proof of the theorem is given in the Appendix. Given E,6 > O. The integer mL(6, E) d;J minim > 0 j J.tm(QE) ~ 6} will be called the 8ample complezity following the terminology of computational learning theory (c.f. [1]). Note that in our case the sample complexity depends also (implicitly) on the target concept t, the hypothesis space H and the probability measure J.t. Corollary 2 If the hypothesis space H is (J.t n , C)-uniform around t E H for any n> 0, then mL(6, E) < maz{8/E, minim j 2CPrH,~(2m)(3/2r < 6}} (6) C < maz{8/E, 6.9 dH + 2.4log2 6}' 0 (7) The estimate (7) of Corollary 2 reduces to the estimate mL(6, E) ~ 6.9 dH independent of 6 and E. This predicts that under the assumption of the corollary a transition to perfect generalisation occurs for training samples of size ~ 6.9 d H , which is in agreement with some statistical physics predictions showing such transition occurring below ~ 1.5dH for some simple neural networks (c.f. [5]). Proof outline. Estimate (6) follows from the first estimate (5). Estimate (7) can be derived from the second bound in (5) (virtually by repeating the proof of [1, Theorem 8.4.1] with substitution of 4log2(4/3) for E and 6/C for 6). Q.E.D. 2I8 Adam Kowalczyk, Herman Ferra 10-1 fie 10-2 10-300 '---'-..J....J............L.U.L_.L-J'--'--<u..LJw......L...L!-O.!.-'-'-u.uJ 10 102 m 103 104 :; . 10-3 10 m Figure 1: Plots of estimates on PrH,J.I(m) and f(m, ?)/C ~ f(m,O)/C = (3/2) 3m PrH,J.I (2m) (Fig_ a) for the analytic threshold neuron (PrH'J.I(m) = <P(dH' 2m)(3/4)3~ and (Fig_ b) for the abstract perceptron according to the estimate (11) on PrH,J.I(m) for m1 = 50, m3 = 1000 and p = 0.015_ The upper bound on the sample complexity, mL(?, 6), corresponds to the abscissa value of the intersection of the curve f(m,O) with the level 6/C (c.f. point A in Fig. b). In this manner for 6/C 0.01 we obtain estimates mL ~ 602 and mL < 1795 in the case of Fig. a for d H = 100 and d H = 300 and mL ~ 697 in the case of Fig. b (d H = ~ = 1000), respectively. = 3 An application to feedforward networks In this section we shall discuss the problem of estimation of PrH'J.I(m) which is crucial for application of the above formalism. First we discuss an example of an analytic threshold neuron on R n [6] when H is the family of all functions R n -+ {O, 1}, x 1-+ 9(~:=1 Wiai(X)), where ai : R n -+ R are fixed real analytic functions and 9 the ordinary hard threshold. In this case d H equals the number of linearly independent functions among ab a3, ... , all. For any continuous probability distribution J.L on R n we have: PrH(i) = <p(dH , m)/2m (Vm and 'Vi E (Rnr with probability 1), (8) and consequently PrH'J.I(m) = PrH,max(m) = <P(dH' m)/2m for every m. (9) Note that this class of neural networks includes as particular cases, the linear threshold neuron (if Ie = n+ 1 and a1, ... , a n +1 are chosen as 1, Xl, ... , xn) and higher order networks (if ai(x) are polynomials); in the former case (8) follows also from the classical result of T. Cover [3]. Now we discuss the more complex case of a linear threshold multilayer perceptron on X = R n , with H defined as the family of all functions R n -+ {0,1} that such an architecture may implement and J.L is any continuous probability measure Generalisation in Feedforward Networks 219 on Rn. In this case there exist two functions, Blow (m) and Bup(m), such that Blow(m) ~ PrH(x) ~ Bup(m)foranyxE (Rn)mwithprobabilityl. In other words, PrH(x) takes values within a "bifurcation region" similar to the shaded region in Fig. l.b. Further, it is known that Blow(m) = 1 for m ~ nh l + 1, Bup(m) = 1 for m ~ dH and, in general, Blow (m) ~ cI.i(nh l + 1, m) and Bup(m) ~ cI.i(dH , m), where hl is the number of neurons in the first hidden layer. Given this we can say that PrH'I'(m) takes values somewhere within the "bifurcation region". It is worth noting that the width of the "bifurcation region" , which approximately equals 2( dH - nh l ) (since it is known that values on the boundaries have a positive probability of being randomly attained) increases with increasing hl since [9] O(nhllog:z(h l )) = maxp(n - p)(hd2 - 2P ) ~ d H ? (10) p Estimate (6) is better than (7) in general, and in particular, if PrH,I'(m) "drops" to 0 much quicker than Bup = cI.i(dH , m), we may expect that it will provide an estimate of sample complexity mL even below dHi if PrH'I'(m) is close to Bup , then the difference between both estimates will be negligible. In order to clarify this issue we shall consider now a third, abstract example. We introduce the abstract perceptron defined as the set of hypotheses H on a probabilistic space (X, J.?) with the following property for a random m + 1-tuple (x,z) E xm x X: dvc(x, z) = dvc(x) + 1 with probability = 1 if dvc(x) < ml and with probability pifml ~ dvc(x) < m:z, and dvc(x,z) = dvc(X), otherwise. Here 0 ~ ml ~ m2 ~ 00 are two (integer) constants, 0 ~ p ~ 1 is another constant and dvc(z1J ... , zm) is the maximal n such that 17r(:Z:'l, ...,:z:,,,)(H)1 = 2n for some 1 ~ i l < ... < in ~ m. It can easily be seen that d H = m:z in this case and that the threshold analytic neuron is a particular example of abstract percept ron (with de! = 0 and ml = m:z = dH, Blow(m) = Bup = cI.i(mlJ m)J2m ) . Note further, that if 0 < p < 1, then for any m ~ ml, dvc(x) = ml with probability> 0, and for any m ~ m:z, dvc(x) = dH = m:z with probability> o. In this regard the abstract percept ron resembles the linear threshold multilayer percept ron (with ml and m:z corresponding to nhl + 1 and d H , respectively). However, the main advantage of this model is that we can derive the following estimate: p PrH,.( m) <:; 2- m ~~' (m ~ m, ) pm-m, -'( 1 - p)'<J> (min( m - i, m,), m) (11) Using this estimate we find that for sufficiently low p (and sufficiently large m:z) the sample complexity upper bound (6) is determined by ml and can even be lower than m:z = dH (c.f. Figure l.b). In particular, the sample complexity determined by Eqns. (6) and (11) can be finite even if dH = m:z = 00 (c.f. the curve E(m, 0) for p = .05 in Fig. l.b which is the same for m:z = 1000 and m:z = (0). 4 Discussion The paper strongly depends on the postulate (3) of (J.?n, C)-uniformity. We admit that this is an ad hoc assumption here as we do not give examples when it is 220 Adam Kowalczyk, Herman Ferra satisfied nor a method to determine the constant C. From this point of view our results at the current stage have no predictive power, perchaps only explanatory one. The paper should be viewed as an attempt in the direction to explain within VC-formalism some known generalisation properties of neural networks which are out of the reach of the formalism to date, such as the empirically observed peak generalisation for backpropagation network trained with samples of the size well below VC-dimension [8] or the phase transitions to perfect generalisation below 1.5. xVC-dimension [5]. We see the formalism in this paper as one of a number of possible approaches in this direction. There are other possibilities here as well (e.g. [5, 12]) and in particular other, weaker versions of (p.fI., C)-uniformity can be used leading to similar results. For instance in Theorem 1 and Corollary 2 it was c.f. the enough to assume (p.fI. , C)-uniformity for a special class of sets S (S = Appendix); we intend to discuss other options in this regard on another occasion. s;;:;m, Now we relate this research to some previous results (e.g. [2, 4]) which imply the following estimates on sample complexity (c.f. [1, Theorems 8.6.1-2]): - 1 ,-In(<<5)/?)~mL(<<5,?)~ .. max ( dH32? r-; 4 ( 12 2)1 ' dH log:l-;-+log:l6" (12) where the lower bound is proved for all ? ~ 1/8 and ?5 ~ 1/100; here mL(<<5, ?) is the "universal" sample complexity, i.e. for all target concepts t and all probability distributions p.. For ? = ?5 = 0.01 and dH ? 1 this estimate yields 3dH < mi.(.Ol, .01) < 4000dH' These bounds should be compared against estimates of Corollary 2 of which (7) provides a much tighter upper bound, mL(.Ol, .01) ~ 6.9dH , if the assumption on (p.m, C)-uniformity of the hypothesis space around the target concept t is satisfied. 5 Conclusions We have shown that und~r appropriate restriction on the probability distribution and target concept, the upper bound on sample complexity (and "perfect generalisation") can be lowered to ~ 6.9x VC-dimension, and in some cases even below VC-dimension (with a strong possibility that multilayer perceptron could be such). We showed that there are other parameters than VC-dimension potentially impacting on generalisation capabilities of neural networks. In particular we showed by example (abstract perceptron) that a system may have finite sample complexity and infinite VC dimension at the same time. The formalism of this paper predicts transition to perfect generalisation at relatively low training sample sizes but it is too crude to predict scaling laws for learning curves (c.f. [5, 12] and references in there). Acknowledgement. The permission of Managing Director, Research and Information Technology, Telecom Australia, to publish this paper is gratefully acknowledged. References [1] M. Anthony and N. Biggs. Computational Learning Theory. Cambridge Uni- 221 Generalisation in Feedfom'ard Networks versity Press, 1992. [2] A. Blumer, A. Ehrenfeucht, D. Haussler, and M.K. Warmuth. Learnability and the Vapnik-Chervonenkis dimensions. Journal of the ACM, 36:929-965, (Oct. 1989). [3] T.M. Cover. Geometrical and statistical properties of linear inequalities with applications to pattern recognition. IEEE Trans. Elec. Comp., EC-14:326334, 1965. [4] A. Ehrenfeucht, D. Haussler, M. Kearns, and L. Valiant. A general lower bound on the number of examples needed for learning. Information and Computation, 82:247-261, 1989. [5] D. Hausler, M. Kearns, H.S. Seung, and N. Tishby. Rigorous learning curve bounds from statistical mechanics. Technical report, 1994. [6] A. Kowalczyk. Separating capacity of analytic neuron. In Proc. ICNN'94 , Orlando, 1994. [7] A. Macintyre and E. Sontag. Finiteness results for sigmoidal "neural" networks. In Proc. of the 25th Annual ACM Symp. Theory of Comp., pages 325-334, 1993. [8] G.L. Martin and J .A. Pitman. Recognizing handprinted letters and digits using backpropagation learning. Neural Comput., 3:258-267, 1991. [9] A. Sakurai. Tighter bounds of the VC-dimension of three-layer networks. In Proceedings of the 1999 World Congress on Neural Networks, 1993. [10] N. Sauer. On the density of family of sets. Journal of Combinatorial Theory (Series A), 13:145-147, 1972). [11] V. Vapnik. Estimation of Dependences Based on Empirical Data. SpringerVerlag, 1982. [12] V. Vapnik, E. Levin, and Y. Le Cun. Measuring the vc-dimension of a learning machine. Neural Computation, 6 (5):851-876, 1994). 6 Appendix: Sketch of the proof of Theorem 1 The proof is a modification of the proof of [1, Theorem 8.3.1]. We divide it into three stages. Stage 1. Let . de! 'RJ = {(x,y) E Xm X Xm ~ X 2 m ; 3hEHer:ch = 0 & eryh = jim} (13) for j E {O, 1, ... , m}. Using a Chernoff bound on the "tail" of binomial distribution it can be shown [1, Lemma 8.3.2] that for m ~ 8/? m Jr(Qf(m)) ::; 2 L j.L2m(nj) (14) j~rmf/21 Stage 2. Now we use a combinatorial argument to estimate j.L2m(1~j). We consider the 2m -element commutative group G m of transformations of xm x xm ~ X 2m generated by all "co-ordinate swaps" of the form (Xb ... , Xm, Y1I ... , Ym) 1-+ (Xl, ... , Xi-1, Yi, Xi+1I ... , Xm, Yb ... , Yi-1, Xi, Yi+b ... , Ym), 222 Adam Kowalczyk, Herman Ferra for 1 ~ i ~ m. We assume also that Gm transforms {O, l}m x {O, l}m ~ {O, 1}2m in a similar fashion. Note that (15) As transformation u E Gm preserve the measure p.2m on xm 2mp.2m(ni) = IGmlp.2m(ni) = I: X xm we obtain J p.2m(d?dY)X'RAu(?, Y? uEG ... J p.2m(didY) I: X'R.i(u(i, Y?. (16) uEG ... Let s;;:;m = de! {h- = (hI, h2) E {O,l}m x {O, l}m = ?& - - hI IIh211 = j} and de! = {h- E {O,l} 2m ; Ilhll = j}, where Ilhll = hI + ... + hm for any h = (h1, ... , hm) E {O, l}m. Then 181ml = (2j), u(s;;:;m) C 81m for any u E Gm and 8j2m de! ni = {(i, YJ E xm x xm ; 3h E 7rt,(z,y)(H) n s;;:;m}, (17) Thus from Eqn. (16) we obtain 2mp.2m(ni) ~ J J J J p.2m(didY) I: xs;:-;-(uh) uEG ... hEWt,(.,f)(H)nS;- I: p.2m(didY) I: xs;:-;",(uh) hEwt,{.,f)(H)ns;'" uEG ... p.2m(didY) _ = I: I: _... I{u E Gm j uh E 81m}1 hEw t,(.,f)(H)nS o,; p.2m (didY) I{7rt,(:ii,y) (H) n 8;m}1 2m- j . Applying now the condition of (p.2m, C)-uniformity (Eqn. 3), Eqn. 17 and dividing by 2m we get Stage 3. On substitution of the above estimate into (14) we obtain estimate (4). To derive (5) let us observe that 2:7=rmE / 21 (2j) 2- j ~ (1 + 1/2)2m. Q.E.D. 

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A Rigorous Analysis Of Linsker-type Hebbian Learning J. Feng Mathematical Department University of Rome "La Sapienza? P. Ie A. Moro, 00185 Rome, Italy H. Pan V. P. Roychowdhury School of Electrical Engineering Purdue University West Lafayette, IN 47907 feng~at.uniroma1.it hpan~ecn.purdue.edu vwani~drum.ecn.purdue.edu Abstract We propose a novel rigorous approach for the analysis of Linsker's unsupervised Hebbian learning network. The behavior of this model is determined by the underlying nonlinear dynamics which are parameterized by a set of parameters originating from the Hebbian rule and the arbor density of the synapses. These parameters determine the presence or absence of a specific receptive field (also referred to as a 'connection pattern') as a saturated fixed point attractor of the model. In this paper, we perform a qualitative analysis of the underlying nonlinear dynamics over the parameter space, determine the effects of the system parameters on the emergence of various receptive fields, and predict precisely within which parameter regime the network will have the potential to develop a specially designated connection pattern. In particular, this approach exposes, for the first time, the crucial role played by the synaptic density functions, and provides a complete precise picture of the parameter space that defines the relationships among the different receptive fields. Our theoretical predictions are confirmed by numerical simulations. lian/eng Feng, H. Pan, V. P. Roychowdhury 320 1 Introduction For the purpose of understanding the self-organization mechanism of primary visual system, Linsker has proposed a multilayered unsupervised Hebbian learning network with random un correlated inputs and localized arborization of synapses between adjacent layers (Linsker, 1986 & 1988). His simulations have shown that for appropriate parameter regimes, several structured connection patterns (e.g., centre-surround and oriented afferent receptive fields (aRFs)) occur progressively as the Hebbian evolution of the weights is carried out layer by layer. The behavior of Linsker's model is determined by the underlying nonlinear dynamics which are parameterized by a set of parameters originating from the Hebbian rule and the arbor density of the synapses. For a nonlinear system, usually, there coexist several attractors for the same set of system parameters. That is, for a given set of the parameters, the state space comprises several attractive basins, each corresponding to a steady state respectively. The initial condition determines which attractor will be eventually reached. At the same time, a nonlinear system could have a different group of coexisting attractors for a different set of system parameters. That is, one could make the presence or absence of a specific state as a fixed point attractor by varying the set of the parameters. For a development model like Linsker's network, what is expected to be observed is that the different aRFs could emerge under different sets of parameters but should be relatively not sensitive to the initial conditions. In other words, the dynamics should avoid the coexistence of several attractors in an appropriate way. The purpose of this paper is to gain more insights into the dynamical mechanism of this self-organization model by performing a rigorous analysis on its parameter space without any approximation. That is, our goal is to reveal the effects of the system parameters on the stability of aRFs, and to predict precisely within which parameter regime the network will have the potential to develop a specially designated aRF. The novel rigorous approach presented here applies not only to the Linsker-type Hebbian learning but also to other related self-organization models about neural development. In Linsker's network, each cell in the present layer M receives synaptic inputs from a number of cells in the preceding layer C. The density of these synaptic connections decreases monotonically with distance rC, from the point underlying the M-cell's position. Since the synaptic weights change on a long time scale compared to the variation of random inputs, by averaging the Hebb rule over the ensemble of inputs in layer C, the dynamical equation for the development of the synaptic strength wT(i) between a M-cell and i-th C-cell at time Tis Nc. WT+1 (i) = f{wT(i) + k1 + l:)Qj + k2]r(j)wT(j)} (1) j=l where k1' k2 are system parameters which are particular combinations of the constants of the Hebb rule, r(?) is a non-negative normalized synaptic density function (SDF) 1, and L:iEC, rei) = 1, and 10 is a limiter function defined by I(x) = W max , if x > W max ; = x, if Ix I ~ W max ; and = -W max , if x < -w max ? The covariance IThe SDF is explicitly incorporated into the dynamics (1) which is equivalent to Linsker's formulation. A rigorous explanation for this equivalence is given in MacKay & Miller, 1990. A Rigorous Analysis of Linsker-type Hebbian Learning 321 matrix {Qij} of the layer C describes the correlation of activities of the i-th and the j-th C-cells. Actually, the covariance matrix of each layer is determined by SDFs r(?) of all layers preceding the layer under consideration. The idea of this paper is the following. It is well known that in general it is intractable to characterize the behavior of a nonlinear dynamics, since the nonlinearity is the cause of the coexistence of many attractors. And one has the difficulty in obtaining the complete characteristics of attractive basins in the state space. But usually for some cases, it is relatively easy to derive a necessary and sufficient condition to check whether a given state is a fixed point of the dynamics. In terms of this condition, the whole parameter regime for the emergence of a fixed point of the dynamics may be obtained in the parameter space. If we are further able to prove the stability of the fixed point, which implies that this fixed point is a steady state if the initial condition is in a non empty vicinity in the state space, we can assert the occurrence of this fixed point attractor in that parameter regime. For Linsker's network, fortunately, the above idea can be carried out because of the specific form of the nonlinear function 1(?). Due to space limitations, the rigorous proofs are in (Feng, Pan, & Roychowdhury, 1995). 2 The Set Of Saturated Fixed Point Attractors And The Criterion For The Division Of Parameter Regimes In fact, Linsker's model is a system of first-order nonlinear difference equations, taking the form = j[wr(i) + hi(Wr' kl' k2)], Wr = {wr(j),j = 1, .. . , Nc}, (2) where hi(W r , kl' k2) = kl + 2:f~dQ~ + k2]r(j)wr (j). And the aRFs observed in Linsker's simulation are the saturated fixed point attractors of this nonlinear system (2). Since the limiter function 1(?) is defined on a hypercube n = [-w max , wmax]N wr+l(i) C in weight state space within which the dynamics is dominated by the linear system wr+l(i) = wr(i) + hi(Wr, kl' k 2 ) , the short-time behaviors of evolution dynamics of connection patterns can be fully characterized in terms of the properties of eigenvectors and their eigenvalues. But this method of stability analysis will not be suitable for the long-time evolution of equation (1) or (2), provided the hypercube constraint is reached as the first largest component of W reaches saturation. However, it is well-known that a fixed point or an equilibrium state of dynamics (2) satisfies (3) Because of the special form of the nonlinear function 1(?), the fixed point equation (3) implies that 3T, such that for T > T, I wr(i) + hi(Wr, kl' k 2) I ~ Wmax , if hi (w, kl , k2) i= o. So a saturated fixed point Wr (i) must have the same sign as hi(wr , kl' k 2 ), i.e. wr(i)hi(Wr, kl' k 2 ) > o. By using the above idea, our Theorems 1 & 2 (proven in Feng, Pan, & Roychowdhury, 1995) state that the set of saturated fixed point attractors of the dynamics in 322 lian/eng Feng, H. Pan, V. P. Roychowdhury equation (1) is given by nFP = {w I w(i)hi(W r ,kl,k2) > 0,1::; i::; N.d, and w E nFP is stable, where the weight vector w belongs to the set of all extreme points of the hypercube n (we assume W max = 1 without loss of generality) . We next derive an explicit necessary and sufficient condition for the emergence of structured aRFs, i.e., we derive conditions to determine whether a given w belongs to nFP. Define J+(w) = {i I wei) = 1} as the index set of cells at the preceding layer C with excitatory weight for a connection pattern w, and J-(w) = {i I wei) = -1} as the index set of C-cells with inhibitory weight for w. Note from the property of fixed point attractors that a connection pattern w is an attractor of the dynamics (1) if and only if for i E J+(w), we have w(i){k 1 + l)Q~ + k2]r(j)w(j)} = j w(i){k 1 + EjEJ+(w)[Qt + k2]r(j)w(j) + EjEJ-(w)[Qt + k2]r(j)w(j)} > O. By the definition of J+(w) and J-(w), we deduce from the above inequality that kl 2: + [Q~ 2: + k2]r(j) - jEJ+(w) [Q~ + k2]r(j) > 0 jEJ-(w) namely kl + k2[ 2: 2: r(j) - jEJ+(w) r(j)] > jEJ-(w) 2: Q~r(j) - jEJ-(w) 2: Q~r(j). jEJ+(w) Inequality above is satisfied for all i in J+(w), and the left hand is independent of i. Hence, kl + k2[ 2: r(j) - jEJ+(w) 2: r(j)] > . max [ \EJ+(w) jEJ-(w) I: I: Q~r(j) - jEJ-(w) Q~r(j)]. jEJ+(w) On the other hand, for i E J-(w), we can similarly deduce that kl + k2[ 2: r(j) - jEJ+(w) 2: r(j)] < . min [ jEJ-(w) IEJ-(w) 2: Q~r(j) - jEJ-(w) 2: Q~r(j)] . jEJ+(w) We introduce the slope function: c(w) ~f 2: jEJ+(w) r(j) - 2: r(j) jEJ-(w) which is the difference of sums of the SDF r( ?) over J+(w) and J-(w), and two kl -intercept functions: d1(w) ~ { maxtEJ+(w)(EjEJ-(w) Qf;r(j) - EjEJ+(w) Qtr(j)), -00, if J+(w) # 0 if J+(w) = 0 A Rigorous Analysis of Linsker-type Hehbian Learning 323 A ~---k , (a) (b) o o E Figure 1: The parameter subspace of (kl, k 2 ). (a) Parameter regime of (kl' k 2 ) to ensure the emergence of all-excitatory (regime A) and all-inhibitory (regime B) connection patterns. The dark grey regime C is the coexistence regime for both all-excitatory and all-inhibitory connection patterns. And the regime D without texture are the regime that Linsker's simulation results are based on , in which both all-excitatory and all-inhibitory connection patterns are no longer an attractor. (b) The principal parameter regimes. Now from our Theorem 3 in Feng, Pan, & Roychowdhury, 1995, for every layer of Linsker's network, the new rigorous criterion for the division of stable parameter regimes to ensure the development of various structured connection patterns is d2 (w) > ki + c(w)k 2 > dl(w). That is, for a given SDF r(-), the parameter regime of (kl' k2) to ensure that w is a stable attractor of dynamics (1) is a band between two parallel lines ki + c(w)k 2 > dl(w) and ki +c(w)k 2 < d2 (w) (See regimes E and F in Fig.1(b)). It is noticed that as dl(w) > d2 (w), there is no regime of (kl' k2) for the occurrence of that aRF w as an attractor of equation (1). Therefore, the existence of such a structured aRF w as an attractor of equation (1) is determined by k1-intercept functions dIe) and d2 (?), and therefore by the covariance matrix Q.c or SDFs r(?) of all preceding layers. 3 Parameter Regimes For aRFs Between Layers BAnd C Based on our general theorems applicable to all layers, we mainly focus on describing the stabilization process of synaptic development from the 2nd (B) to the 3rd layer (C) by considering the effect of the system parameters on the weight development. For the sake of convenience, we assume that the input at 1st layer (A) is independent normal distribution with mean 0 and variance 1, and the connection strengths from layer A to B are all-excitatory same as in Linsker's simulations. The emergence of various aRFs between layer Band C have been previously studied in the literature, and in this paper we mention only the following new results made possible by our approach: (1) For the cell in layer C, the all-excitatory and the all-inhibitory connection patterns still have the largest stable regimes. Denote both SDFs from layer A to B and from B to C as r AB ( .,. ) and r BC (-) respectively. The parameter plane of (kl' k2) 324 lianfeng Feng, H. Pan, V. P. Roychowdhury Table 1: The Principal Parameter Regimes TYPE Regime A ATTRACTOR All-excitatory aRF Regime B All-inhibitory aRF Regime All-excitatory and all-inhibitory aRFs coexist The structured aRFs may have separate parameter regimes Any connection pattern in which the excitatory connections constitute the majority Any connection pattern in which the inhibitory connections constitute the majority A small coexistence regime of many connection patterns around the origin point of the parameter plane of ( kl , k2) =AnB Regime F d2{w) > kl where Regime G d2(W 1) > + C{W)k2 > d1(w) c(w) < 0 kl + c(w 1)k2 > d 1 (wI) =EnFnAnB is divided into four regimes by for all-excitatory pattern and for all-inhibitory pattern (See Fig.I(a?. (2) The parameter with large and negative k2 and approximately -1 < -kdk2 < 1 is favorable for the emergence of various structured connection patterns (e.g., ONcenter cells, OFF-center cells, bi-Iobed cells, and oriented cells) . This is because this regime (See regime D in Fig.I) is removed from the parameter regime where both all-excitatory and all-inhibitory aRFs are dominant, including the coexistence regime of many kind of at tractors around the origin point (See regime G in Fig.I(b The above results provide a precise picture about the principal parameter regimes summarized in Table 1. ?. (3) The relative size of the radiuses of two SDFs r AS (-,.) and r Sc (-) plays a key role in the evolution of various structured aRFs from B to C. A given SDF r.cM (i, j), i E M, j E e will be said to have a range r M if r.cM (i, j) is 'sufficient small' for lIi- jll ~ rM. For a Gaussian SDF r.cM(j,k) '" exp(-lIj-kll/r~), j E e,k EM, the range r M is its standard deviation. We give the analytic prediction about the influence of the SDF's ranges rs, rc on the dynamics by changing rs from the smallest extreme to the largest one with respect to rc. For the smallest extreme of rs (i.e. the A Rigorous Analysis of Linsker-type Hebbian Learning 325 synaptic connections from A to B are concentrated enough, and those from layer B to C are fully feedforward connected), we proved that any kind of connection pattern has a stable parameter regime and emerge under certain parameters, because each synaptic connection within an aRF is developed independently. As rB is changed from the smallest to the largest extreme, the development of synaptic connections between layer Band C will depend on each other stronger and stronger in the sense that most of connections have the same sign as their neighbors in an aRF. So for the largest extreme of rB (i.e. the weights from layer A to B are fully feedforward but there is no constraint on the SDF r BC (.)), any structured aRFs except for the all-excitatory and the all-inhibitory connection patterns will never arise at all, although there exist correlation in input activities (for a proof see Feng, Pan, & Roychowdhury, 1995). Th_erefore, without localized SDF, there would be no structured covariance matrix Q = {[Qij + k 2 ]r(j)} which embodies localized correlation in afferent activities. And without structured covariance matrix Q, no structured aRFs would emerge. (4) As another application of our analyses, we present several numerical results on the parameter regimes of (kl' k2' rB, rc) for the formation of various structured aRFs (Feng & Pan, 1993; Feng, Pan, & Roychowdhury, 1995) (where we assume that rAB(i,j) "" exp(-lli-jll/r~), i E B,j E A, and rBC(i) "" exp(-llill/r2), i E B as in (Linsker, 1986 & 1988)). For example, we show that various aRFs as at tractors have different relative stability. For a fixed rc, the SDF's range rB of the preceding layer as the third system parameter has various critical values for different attractors. That is, an attractor will no longer be stable if rB exceeds its corresponding critical value (See Fig. 2). For circularly symmetric ON-center cells, those aRFs with large ON-center core (which have positive or small negative slope value c(w) ~ -kt/k 2 ) always have a stable parameter regime. But for those ON-center cells with large negative slope value c(w), their stable parameter regimes decrease in size with c(w). Similarly, circularly symmetric OFF-center cells with large OFF-center core (which have negative or small positive slope value c(w)) will be more stable than those with large positive average of weights. But for non-circularly-symmetric patterns (e.g., bi-Iobed cells and oriented cells), only those at tractors with zero average synaptic strength might always have a stable parameter regime (See regime H in Fig.1(b)). If the third parameter rB is large enough to exceed its critical values for other aRFs and k2 is large and negative, then ON-center aRFs with positive c(w) and OFFcenter aRFs with negative c(w) will be almost only at tractors in regime DnE and regime DnF respectively. This conclusion makes it clear why we usually obtain ON-center aRFs in regime DnE and OFF-center aRFs in regime DnF much more easily than other patterns. 4 Concluding Remarks One advantage of our rigorous approach to this kind of unsupervised Hebbian learning network is that, without approximation, it unifies the treatment of many diverse problems about dynamical mechanisms. It is important to notice that there is no assumption on the second item hi(w T ) on the right hand side of equation (1), and there is no restriction on the matrix Q. Our Theorems 1 and 2 provide the general framework for the description of the fixed point attractors for any difference equation of the type stated in (2) that uses a limiter function. Depending on the 326 Jianfeng Feng, H. Pan. V. P. Roychowdhury ~ __-O-N-C-e-"-te-r-a--,R_FS~~____.(r c= 10) ~~ __-O-rie-"-ted---,a.R.-Fs------._,(r c= 10) . I~~~~~~~~~?~' . I~~~~~~~~~?~ 000@@@@ ??? 000~~~~ ? ? ? ~ __.O.... FF.-c ....8n ___ ter___a-R-Fs---__(r c= 10) ~ __-B-I-I-obed--a-R-F.. s ..--_ _ _(rc= 10) .I~~~~~~~~~?~' ooooo~a ??? Figure 2: The critical values of the SDF's range ra for different connection patterns. structure of the second item, hi(w T ) , it is not difficult to adapt our Theorem 3 to obtain the precise relationship among system parameters in other kind of models as long as 10 is a limiter function. Since the functions in the necessary and sufficient condition are computable (like our slope and k1-intercept functions), one is always able to check whether a designated fixed point is stable for a specific set of parameters. Acknowledgements The work ofV. P. Roychowdhury and H. Pan was supported in part by the General Motors Faculty Fellowship and by the NSF Grant No. ECS-9308814. J. Feng was partially supported by Chinese National Key Project of Fundamental Research "Climbing Program" and CNR of Italy. References R. Linsker. (1986) From basic network principle to neural architecture (series). Proc. Natl. Acad. Sci. USA 83: 7508-7512,8390-8394,8779-8783. R. Linsker. (1988) Self-organization in a perceptual network. Computer 21(3): 105-117. D. MacKay, & K. Miller. (1990) Analysis of Linsker's application of Hebbian rules to linear networks. Network 1: 257-297. J. Feng, & H. Pan. (1993) Analysis of Linsker-type Hebbian learning: Rigorous results. Proc. 1993 IEEE Int. Con! on Neural Networks - San Francisco Vol. III, 1516-1521. Piscataway, NJ: IEEE. J. Feng, H. Pan, & V. P. Roy chowdhury. (1995) Linsker-type Hebbian learning: A qualitative analysis on the parameter space. (submitted). 

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Template-Based Algorithms for Connectionist Rule Extraction Jay A. Alexander and Michael C. Mozer Department of Computer Science and Institute for Cognitive Science University of Colorado Boulder, CO 80309--0430 Abstract Casting neural network weights in symbolic terms is crucial for interpreting and explaining the behavior of a network. Additionally, in some domains, a symbolic description may lead to more robust generalization. We present a principled approach to symbolic rule extraction based on the notion of weight templates, parameterized regions of weight space corresponding to specific symbolic expressions. With an appropriate choice of representation, we show how template parameters may be efficiently identified and instantiated to yield the optimal match to a unit's actual weights. Depending on the requirements of the application domain, our method can accommodate arbitrary disjunctions and conjunctions with O(k) complexity, simple n-of-m expressions with O( k!) complexity, or a more general class of recursive n-of-m expressions with O(k!) complexity, where k is the number of inputs to a unit. Our method of rule extraction offers several benefits over alternative approaches in the literature, and simulation results on a variety of problems demonstrate its effectiveness. 1 INTRODUCTION The problem of understanding why a trained neural network makes a given decision has a long history in the field of connectionist modeling. One promising approach to this problem is to convert each unit's weights and/or activities from continuous numerical quantities into discrete, symbolic descriptions [2, 4, 8]. This type of reformulation, or rule extraction, can both explain network behavior and facilitate transfer of learning. Additionally, in intrinsically symbolic domains, there is evidence that a symbolic description can lead to more robust generalization [4]. 610 Jay A. Alexander, Michael C. Mozer We are interested in extracting symbolic rules on a unit-by-unit basis from connectionist nets that employ the conventional inner product activation and sigmoidal output functions. The basic language of description for our rules is that of n-of-m expressions. An n-of-m expression consists of a list of m subexpressions and a value n such that 1 ~ n ~ m. The overall expression is true when at least n of the m subexpressions are true. An example of an n-of-m expression stated using logical variables is the majority voter function X = 2 of (A, B, C). N-of-m expressions are interesting because they are able to model behaviors intermediate to standard Boolean OR (n = 1) and AND (n = m) functions. These intermediate behaviors reflect a limited form of two-level Boolean logic. (To see why this is true, note that the expression for X above is equivalent to AB + BC + AC.) In a later section we describe even more general behaviors that can be represented using recursive forms of these expressions. N-of-m expressions fit well with the activation behavior of sigmoidal units, and they are quite amenable to human comprehension. To extract an n-of-m rule from a unit's weights, we follow a three-step process. First we generate a minimal set of candidate templates, where each template is parameterized to represent a given n-of-m expression. Next we instantiate each template's parameters with optimal values. Finally we choose the symbolic expression whose instantiated template is nearest to the actual weights. Details on each of these steps are given below. 2 TEMPLATE-BASED RULE EXTRACTION 2.1 Background Following McMillan [4], we define a weight template as a parameterized region of weight space corresponding to a specific symbolic function. To see how weight templates can be used to represent symbolic functions, consider the weight vector for a sigmoidal unit with four inputs and a bias: w = WI w2 w3 w4 b Now consider the following two template vectors: t1 t2 = = -p P 0 -p P -p P P J.5p -D.5p These templates are parameterized by the variable p . Given a large positive value of p (say 5.0) and an input vector 1 (whose components are approximately 0 and 1), t1 describes the symbolic expression J of('lz, 12, 14), while t2 describes the symbolic expression 2 of(Ib 12? h. 14), A general description for n-of-m templates of this form is the following: 1. M of the weight values are set to ?p , P > 0; all others are set to O. (+p is used for normal subexpressions, -p for negated sUbexpressions) 2. The bias value is set to (0.5 + m neg - n)p, where m neg represents the number of negated subexpressions. When the inputs are Boolean with values -1 and +1, the form of the templates is the same, except the template bias takes the value (J + m - 2n)p. This seemingly trivial difference turns out to have a significant effect on the efficiency of the extraction process. Template-Based Algorithms for Connectionist Rule Extraction 611 2.2 Basic extraction algorithm Generating candidate templates Given a sigmoidal unit with k inputs plus a bias, the total number of n-of-m expressions that unit may compute is an exponential function of k: /c T /c m = ?.J ~ ~ ?.J m=ln=1 2m(k) m = ?.J ~ m=1 2 mk! (k-m)!(m-l)! = 2k3/C-1 For example, Tk=1O is 393,660, while Tk=20 is over 46 billion. Fortunately we can apply knowledge of the unit's actual weights to explore this search space without generating a template for each possible n-of-m expression. Alexander [1] proves that when the -11+1 input representation is used, we need consider at most one template for each possible choice of n and m. For a given choice of nand m, a template is indicated when sign(1 + m - 2n) =sign(b). A required template is formed by setting the template weights corresponding to the m highest absolute value actual weights to sp, where s represents the sign of the corresponding actual weight. The template bias is set to (1 + m - 2n)p. This reduces the number of templates required to a polynomial function of k: Values for Tk=1O and Tk=20 are now 30 and 110, respectively, making for a very efficient pruning of the search space. When 011 inputs are used, this simple procedure does not suffice and many more templates must be generated. For this reason, in the remainder of this paper we focus on the -11+ 1 case and assume the use of symmetric sigmoid functions. Instantiating template parameters Instantiating a weight template t requires finding a value for p such that the Euclidean distance d = lit - wl1 2 is minimized. Letting Uj = 1 if template weight tj is nonzero, Uj =0 otherwise, the value of p that minimizes this distance for any -11+ 1 template is given by: /c p* = L IWjIUj + (1 +m-2n)b ==-I!...-_ _ _ _ _~- I:....: ? m + (1 + m - 2n) 2 Finding the nearest template and checking extraction validity Once each template is instantiated with its value of p*, the distance between the template and the actual weight vector is calculated, and the minimal distance template is selected as the basis for rule extraction. Having found the nearest template t*, we can use its values as part of a rudimentary check on extraction validity. For example, we can define the extraction error as 100% x Ilt*-wI1 2/1IwI1 2 to measure how well the nearest symbolic rule fits the actual weights. We can also examine the value of p* used in t*. Small values of p* translate into activation levels in the linear regime of the sigmoid functions, compromising the assumption of Boolean outputs propagating to subsequent inputs. 612 Jay A. Alexander, Michael C. Mozer 2.3 Extending expressiveness While the n-of-m expressions treated thus far are fairly powerful, there is an interesting class of symbolic behaviors that cannot be captured by simple n-of-m expressions. The simplest example of this type of behavior may be seen in the single hidden unit version of xor described in [6]. In this 2-1-1 network the hidden unit H learns the expression AND(h 12), while the output unit (which connects to the two inputs as well as to the hidden unit) learns the expression AND[OR(h 12 ),Hj. This latter expression may be viewed as a nested or recursive form of n-of-m expression, one where some of the m subexpressions may themselves be n-of-m expressions. The following two forms of recursive n-of-m expressions are linearly separable and are thus computable by a single sigmoidal unit: OR [Cn-of-m ' COR j AND [C n-of-m , CAND j where is a nested n-of-m expression (1 S; n S; m) COR is a nested OR expression (n = 1) CAND is a nested AND expression (n =m) Cn-of-m These expressions may be seen to generalize simple n-of-m expressions in the same way that simple n-of-m expressions generalize basic disjunctions and conjunctions.1 We term the above forms augmented n-of-m expressions because they extend simple n-of-m expressions with additional disjuncts or conjuncts. Templates for these expressions (under the -1/+ 1 input representation) may be efficiently generated and instantiated using a procedure similar to that described for simple n-of-m expressions. When augmented expressions are included in the search, the total number of templates required becomes: This figure is O(k) worse than for simple n-of-m expressions, but it is still polynomial in k and is quite manageable for many problems. (Values for Tk=1O and Tk=20 are 150 and 1250, respectively.) A more detailed treatment of augmented n-of-m expressions is given in [1]. 3 RELATED WORK Here we briefly consider two alternative systems for connectionist rule extraction. Many other methods have been developed; a recent summary and categorization appears in [2]. 3.1 McMillan McMillan described the projection of actual weights to simple weight templates in [4]. McMillan's parameter selection and instantiation procedures are inefficient compared to those described here, though they yield equivalent results for the classes of templates he used. McMillan treated only expressions with m S; 2 and no negated sUbexpressions. 1 In fact the nesting may continue beyond one level. Thus sigmoidal units can compute expressions like OR[AND(Cn _of_m ' CAND), COR]. We have not yet experimented with extensions of this sort. Template-Based Algorithms for Connectionist Rule Extraction 613 3.2 Towell and Shavlik Towell and Shavlik [8] use a domain theory to initialize a connectionist network, train the network on a set of labeled examples, and then extract rules that describe the network's behavior. To perform rule extraction, Towell and Shavlik first group weights using an iterative clustering algorithm. After applying additional training, they typically check each training pattern against each weight group and eliminate groups that do not affect the classification of any pattern. Finally, they scan remaining groups and attempt to express a rule in purely symbolic n-of-m form. However, in many cases the extracted rules take the form of a linear inequality involving multiple numeric quantities. For example, the following rule was extracted from part of a network trained on the promoter recognition task [5] from molecular biology: Minus35 " -10 < + 5.0 * + 3.1 * + 1.9 * + 1.5 * - 1.5 * - 1.9 * - 3.1 * nt(@-37 nt(@-37 nt(@-37 nt(@-37 nt(@-37 nt(@-37 nt(@-37 '--T-G--A' ) '---GT---') '----C-CT' ) '---C--A-' ) '------GC' ) '--CAW---' ) '--A----C' ) where nt() returns the number of true subexpressions, @-37 locates the subexpressions on the DNA strand, and "_N indicates a don't-care subexpression. Towell and Shavlik's method can be expected to give more accurate results than our approach, but at a cost. Their method is very compute intensive and relies substantially on access to a fixed set of training patterns. Additionally, it is not clear that their rules are completely symbolic. While numeric expressions were convenient for the domains they studied, in applications where one is interested in more abstract descriptions, such expressions may be viewed as providing too much detail, and may be difficult for people to interpret and reason about. Sometimes one wants to determine the nearest symbolic interpretation of unit behavior rather than a precise mathematical description. Our method offers a simpler paradigm for doing this. Given these differences, we conclude that both methods have their place in rule extraction tool kits. 4 SIMULATIONS 4.1 Simple logic problems We used a group of simple logic problems to verify that our extraction algorithms could produce a correct set of rules for networks trained on the complete pattern space of each function. Table 1 summarizes the results. 2 The rule-plus-exception problem is defined as /= AB + 1\B CD; xor-l is the 2-1-1 version of xordescribed in Section 2.3; and xor-2 is a strictly layered (2-2-1) version of xor [6]. The negation problem is also described in [6]; in this problem one of the four inputs controls whether the other inputs appear normally or negated at the outputs. (As with xor-l, the network for negation makes use of direct inputJ output connections.) In addition to the perfect classification performance of the rules, the large values of p* and small values of extraction error (as defined in Section 2.2) provide evidence that the extraction process is very accurate. 614 Jay A. Alexander, MichaeL C. Mozer Averagep? Extraction Error Network Topology Hidden Unit Penalty Term Hidden Unit(s) Output Unit(s) Hidden Unit(s) Output Unit(s) Patterns Correctly Classified by Rules 4-2-1 - 2.72 6.15 0.8% 1.3% 100.0% xor-l 2-1-1 - 5.68 4.40 0.1 % 0.1 % 100.0 % xor-2 2-2-1 - 4.34 5.68 0.4% 1.0% 100.0 % negation 4-3-4 activation 5.40 5.17 0.2% 2.2% 100.0 % Problem rule-plus-exception Table 1: Simulation summary for simple logic problems Symbolic solutions for these problems often come in fonns different from the canonical fonn of the function. For example, the following rules for the rule-pLus-exception problem show a level of negation within the network: H1 H2 o OR (A, = B, c, D) AND (A, B) OR (H1' H2 ) Example results on xor- J show the expected use of an augmented n-of-m expression: 12) H OR (1 1 , o OR [AND(I 1 , 1 2 ), H! 4.2 The MONK's problems We tested generalization perfonnance using the MONK's problems [5,7], a set of three problems used to compare a variety of symbolic and connectionist learning algorithms. A summary of these tests appears in Table 2. Our perfonnance was equal to or better than all of the systems tested in [7] for the monks- J and monks-2 problems. Moreover, the rules extracted by our algorithm were very concise and easy to understand, in contrast to those produced by several of the symbolic systems. (The two connectionist systems reported in [7] were opaque, Le., no rules were extracted.) As an example, consider the following output for the monks-2 problem: HI 2 of (head_shape round, body_shape round, is_smiling yes, H2 3 of (head_shape round, body_shape round, is_smiling yes, holding sword, jacket_color red, has_tie not no) o AND holding sword, jacket_color red, has_tie yes) (HI' H2 ) The target concept for this problem is exactly 2 of the attributes have their first value. These rules demonstrate an elegant use of n-of-m expressions to describe the idea of "exactly 2" as "at least 2 but not 3". The monks-3 problem is difficult due to (intentional) training set noise, but our results are comparable to the other systems tested in [7]. 2 All results in this paper are for networks trained using batch-mode back propagation on the cross-entropy error function. Training was stopped when outputs were within 0.05 of their target values for each pattern or a fixed number of epochs (typically 10(0) was reached. Where indicated, a penalty term for nonBoolean hidden activations or hidden weight decay was added to the main error function. For the breast cancer problem shown in Table 4.3, hidden rules were extracted first and the output units were retrained briefly before extracting their rules. Results for the problems in Table 4.3 used leave-one-out testing or 100fold cross-validation (with 10 different initial orderings) as indicated. All results are averages over 10 replications with different initial weights. Template-Based Algorithms for Connectionist Rule Extraction 615 Problem Network Topology Hidden Unit Penalty Term monies-I 17-3--1 decay 124 100.0% 100.0% 432 100.0% 100.0% monles-2 17-2-1 decay 169 100.0% 100.0% 432 100.0% 100.0% monks-3 l7"'{)""l - 122 93.4% 93.4% 432 97.2% 97.2% Training Set Test Set #of Patterns Perf. of Network Perf. of Rules #of Patterns Perf. of Network Perf. of Rules Table 2: Simulation summary for the MONK's problems 4.3 VCI repository problems The final set of simulations addresses extraction performance on three real-world databases from the UCI repository [5]. Table 3 shows that good results were achieved. For the promoters task, we achieved generalization performance of nearly 88%, compared to 93-96% reported by Towell and Shavlik [8]. However, our results are impressive when viewed in light of the simplicity and comprehensibility of the extracted output. While Towell and Shavlik's results for this task included 5 rules like the one shown in Section 3.2, our single rule is quite simple: promoter = 5 of (@-45 'AA-------TTGA-A-----T------T-----AAA----C') Results for the house-votes-84 and breast-eaneer-wise problems are especially noteworthy since the generalization performance of the rules is virtually identical to that of the raw networks. This indicates that the rules are capturing a significant portion of the computation being performed by the networks. The following rule was the one most frequently extracted for the house-votes-84 problem, where the task is to predict party affiliation: V7 , V9 , V1O ' Vl l , V12 ) v4 Democrat OR [ 5 of (V 3 , where V3 voted for adoption-of-the-budget-resolution bill , 1 V4 voted for physician-fee-freeze bill V7 voted for anti-satellite-test-ban bill V9 = voted for rnx-missile bill V1 0 = voted for immigration bill Vll voted for synfuels-corporation-cutback bill V12 = voted for education-spending bill Shown below is a typical rule set extracted for the breast-eaneer-wise problem. Here the goal is to diagnose a tumor as benign or malignant based on nine clinical attributes. Malignant = AND (H 1 , H2 ) 4 of (thickness> 3, size> 1, adhesion> 1, epithelial> 5, nuclei> 3, chromatin> 1, normal> 2, mi toses > 1) = 30f(thickness>6,size>1,shape>1,epithelial>1, nuclei> 8, normal> 9) = not used Hl = H2 H3 As suggested by the rules, we used a thermometer (cumulative) coding of the nominally valued attributes so that less-than or greater-than subexpressions could be efficiently represented in the hidden weights. Such a representation is often useful in diagnosis tasks. We also limited the hidden weights to positive values due to the nature of the attributes. 616 Jay A. Alexander, Michael C. Mozer Training Set Test Set Problem Network Topology #I of Patterns Perf. of Network Perf. of Rules #I of Patterns Perf. of Network Perf. of Rules promoters 2284-1 105 100.0% 95.9% I 94.2% 87.6% house-votes-84 164-1 387 97.3 % 96.2% 43 95.7% 95.9% breast-cancer-wisc 81-3-1 630 98.5 % 96.3 % 70 95.8% 95.2% Table 3: Simulation summary for uel repository problems Taken as a whole our simulation results are encouraging, and we are conducting further research on rule extraction for more complex tasks. 5 CONCLUSION We have described a general approach for extracting various types of n-of-m symbolic rules from trained networks of sigmoidal units, assuming approximately Boolean activation behavior. While other methods for interpretation of this sort exist, ours represents a valuable price/performance point, offering easily-understood rules and good extraction performance with computational complexity that scales well with the expressiveness desired. The basic principles behind our approach may be flexibly applied to a wide variety of problems. References [1] Alexander, J. A. (1994). Template-based procedures for neural network interpretation. MS Thesis. Department of Computer Science, University of Colorado, Boulder, CO. [2] Andrews, R., Diederich, l, and Tickle, A. B. (1995). A survey and critique of techniques for extracting rules from trained artificial neural networks. To appear in Fu, L. M. (Ed.), Knowledge-Based Systems, Special Issue on Knowledge-Based Neural Networks. [3] Mangasarian, O. L. and Wolberg, W. H. (1990). Cancer diagnosis via linear programming. SIAM News 23:5, pages 1 & 18. [4] McMillan, C. (1992). Rule induction in a neural network through integrated symbolic and subsymbolic processing. PhD Thesis. Department of Computer Science, University of Colorado, Boulder, CO. [5] Murphy, P. M. and Aha, D. W. (1994). UCI repository of machine learning databases. [Machine-readable data repository]. Irvine, CA: University of California, Department of Information and Computer Science. Monks data courtesy of Sebastian Thrun, promoters data courtesy of M. Noordewier and J. Shavlik, congressional voting data courtesy of Jeff Schlimmer, breast cancer data courtesy of Dr. William H. Wolberg (see also [3] above). [6] Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986). Learning internal representations by error propagation. In Rumelhart, D. E., McClelland, l L., and the PDP Research Group, Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Volume 1: Foundations, pages 318-362. Cambridge, MA: MIT Press. [7] Thrun, S. B., and 23 other authors (1991). The MONK's problems - A performance comparison of different learning algorithms. Technical Report CS-CMU-91-197. Carnegie Mellon University, Pittsburgh, PA. [8] Towell, G. and Shavlik, J. W. (1992). Interpretation of artificial neural networks: Mapping knowledge-based neural networks into rules. In Moody, J. E., Hanson, S. J., and Lippmann, R. P. (Eds.), Advances in Neurallnfonnation Processing Systems, 4:977-984. San Mateo, CA: Morgan Kaufmann. 

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An Actor/Critic Algorithm that Equivalent to Q-Learning ? IS Robert H. Crites Computer Science Department University of Massachusetts Amherst, MA 01003 Andrew G. Barto Computer Science Department University of Massachusetts Amherst, MA 01003 crites~cs.umass.edu barto~cs.umass.edu Abstract We prove the convergence of an actor/critic algorithm that is equivalent to Q-Iearning by construction. Its equivalence is achieved by encoding Q-values within the policy and value function of the actor and critic. The resultant actor/critic algorithm is novel in two ways: it updates the critic only when the most probable action is executed from any given state, and it rewards the actor using criteria that depend on the relative probability of the action that was executed. 1 INTRODUCTION In actor/critic learning systems, the actor implements a stochastic policy that maps states to action probability vectors, and the critic attempts to estimate the value of each state in order to provide more useful reinforcement feedback to the actor. The result is two interacting adaptive processes: the actor adapts to the critic, while the critic adapts to the actor. The foundations of actor/critic learning systems date back at least to Samuel's checker program in the late 1950s (Samuel,1963). Examples of actor/critic systems include Barto, Sutton, & Anderson's (1983) ASE/ ACE architecture and Sutton's (1990) Dyna-PI architecture. Sutton (1988) notes that the critic in these systems performs temporal credit assignment using what he calls temporal difference (TD) methods. Barto, Sutton, & Watkins (1990) note a relationship between actor/critic 402 Robert Crites, Andrew G. Barto architectures and a dynamic programming (DP) algorithm known as policy iteration. Although DP is a collection of general methods for solving Markov decision processes (MDPs), these algorithms are computationally infeasible for problems with very large state sets. Indeed, classical DP algorithms require multiple complete sweeps of the entire state set. However, progress has been made recently in developing asynchronous, incremental versions of DP that can be run online concurrently with control (Watkins, 1989; Barto et ai, 1993). Most of the theoretical results for incremental DP have been for algorithms based on a DP algorithm known as value iteration. Examples include Watkins' (1989) Q-Iearning algorithm (motivated by a desire for on-line learning), and Bertsekas & Tsitsiklis' (1989) results on asynchronous DP (motivated by a desire for parallel implementations). Convergence proofs for incremental algorithms based on policy iteration (such as actor/critic algorithms) have been slower in coming. Williams & Baird (1993) provide a valuable analysis of the convergence of certain actor/critic learning systems that use deterministic policies. They assume that a model of the MDP (including all the transition probabilities and expected rewards) is available, allowing the use of operations that look ahead to all possible next states. When a model is not available for the evaluation of alternative actions, one must resort to other methods for exploration, such as the use of stochastic policies. We prove convergence for an actor/critic algorithm that uses stochastic policies and does not require a model of the MDP. The key idea behind our proof is to construct an actor/critic algorithm that is equivalent to Q-Iearning. It achieves this equivalence by encoding Q-values within the policy and value function of the actor and critic. By illustrating the way Qlearning appears as an actor/critic algorithm, the construction sheds light on two significant differences between Q-Iearning and traditional actor/critic algorithms. Traditionally, the critic attempts to provide feedback to the actor by estimating V1I', the value function corresponding to the current policy'll". In our construction, instead of estimating ~, the critic directly estimates the optimal value function V"'. In practice, this means that the value function estimate V is updated only when the most probable action is executed from any given state. In addition, our actor is provided with more discriminating feedback, based not only on the TD error, but also on the relative probability of the action that was executed. By adding these modifications, we can show that this algorithm behaves exactly like Q-Iearning constrained by a particular exploration strategy. Since a number of proofs of the convergence of Q-Iearning already exist (Tsitsiklis, 1994; Jaakkola et ai, 1993; Watkins & Dayan, 1992), the fact that this algorithm behaves exactly like Q-Iearning implies that it too converges to the optimal value function with probability one. 2 MARKOV DECISION PROCESSES Actor/critic and Q-Iearning algorithms are usually studied within the Markov decision process framework. In a finite MDP, at each discrete time step, an agent observes the state :z: from a finite set X, and selects an action a from a finite set Ax by using a stochastic policy'll" that assigns a probability to each action in Ax. The agent receives a reward with expected value R(:z:, a), and the state at the next An Actor/Critic Algorithm That Is Equivalent to Q-Learning 403 time step is y with probability pll(:z:,y). For any policy 7?' and :z: E X, let V""(:Z:) denote the ezpected infinite-horizon discounted return from :z: given that the agent uses policy 7?'. Letting rt denote the reward at time t, this is defined as: V""(:z:) = E7r [L:~o,trtl:z:o = :z:], (1) where :Z:o is the initial state, 0 :::; , < 1 is a factor used to discount future rewards, and E7r is the expectation assuming the agent always uses policy 7?'. It is usual to call V 7r (:z:) the value of:z: under 7?'. The function V"" is the value function corresponding to 7?'. The objective is to find an optimal policy, i.e., a policy,7?'*, that maximizes the value of each state :z: defined by (1). The unique optimal value function, V*, is the value function corresponding to any optimal policy. Additional details on this and other types of MOPs can be found in many references. 3 ACTOR/CRITIC ALGORITHMS A generic actor/critic algorithm is as follows: 1. Initialize the stochastic policy and the value function estimate. 2. From the current state :z:, execute action a randomly according to the current policy. Note the next state y, the reward r, and the TO error e = [r + ,V(y)] - V(:z:), where 0 :::; , < 1 is the discount factor. 3. Update the actor by adjusting the action probabilities for state :z: using the TO error. If e > 0, action a performed relatively well and its probability should be increased. If e < 0, action a performed relatively poorly and its probability should be decreased. 4. Update the critic by adjusting the estimated value of state :z: using the TO error: V(:z:) -- V(:z:) +a e where a is the learning rate. 5. :z: -- y. Go to step 2. There are a variety of implementations of this generic algorithm in the literature. They differ in the exact details of how the policy is stored and updated. Barto et al (1990) and Lin (1993) store the action probabilities indirectly using parameters w(:z:, a) that need not be positive, and need not sum to one. Increasing (or decreasing) the probability of action a in state :z: is accomplished by increasing (or decreasing) the value of the parameter w(:z:, a). Sutton (1990) modifies the generic algorithm so that these parameters can be interpreted as action value estimates. He redefines e in step 2 as follows: e = [r + ,V(y)] - w(:z:, a). For this reason, the Oyna-PI architecture (Sutton, 1990) and the modified actor/critic algorithm we present below both reward less probable actions more readily because of their lower estimated values. 404 Robert Crites, Andrew G. Barto Barto et al (1990) select actions by adding exponentially distributed random numbers to each parameter w(:z:, a) for the current state, and then executing the action with the maximum sum. Sutton (1990) and Lin (1993) convert the parameters w(:z:, a) into action probabilities using the Boltzmann distribution, where given a temperature T, the probability of selecting action i in state :z: is ew(x,i)/T '" L.JaEA .. ew(x,a)/T' In spite of the empirical success of these algorithms, their convergence has never been proven. 4 Q-LEARNING Rather than learning the values of states, the Q-Iearning algorithm learns the values of state/action pairs. Q(:z:, a) is the expected discounted return obtained by performing action a in state :z: and performing optimally thereafter. Once the Q function has been learned, an optimal action in state :z: is any action that maximizes Q(:z:, .). Whenever an action a is executed from state :z:, the Q-value estimate for that state/action pair is updated as follows: Q(:z:, a) +- Q(:z:, a) + O!xa(n) [r + "y maxbEAlI Q(y, b) - Q(:z:, a)), where O!xa (n) is the non-negative learning rate used the nth time action a is executed from state :z:. Q-Learning does not specify an exploration mechanism, but requires that all actions be tried infinitely often from all states. In actor/critic learning systems, exploration is fully determined by the action probabilities of the actor. 5 A MODIFIED ACTOR/CRITIC ALGORITHM For each value v E !R, the modified actor/critic algorithm presented below uses an invertible function, H.", that assigns a real number to each action probability ratio: H1J : (0,00) -+ !R. Each H." must be a continuous, strictly increasing function such that H.,,(l) and HH.. (Z2)(i;) = H",(Zl) for all Zl,Z2 = v, > o. = One example of such a class of functions is H.,,(z) T In(z) + v, v E !R, for some positive T. This class of functions corresponds to Boltzmann exploration in Qlearning. Thus, a kind of simulated annealing can be accomplished in the modified actor/critic algorithm (as is often done in Q-Iearning) by gradually lowering the "temperature" T and appropriately renormalizing the action probabilities. It is also possible to restrict the range of H." if bounds on the possible values for a given MDP are known a priori. For a state :z:, let Pa be the probability of action a, let Pmax be the probability of the most probable action, a max , and let Za = ~. An Actor/Critic Algorithm That Is Equivalent to Q-Leaming 405 The modified actor/critic algorithm is as follows: 1. Initialize the stochastic poliCj and the value function estimate. 2. From the current state :z:, execute an action randomly according to the current policy. Call it action i. Note the next state y and the immediate reward r, and let e = [r + -yV(y)] - Hy(X) (Zi). 3. Increase the probability of action i if e > 0, and decrease its probability if e < O. The precise probability update is as follows. First calculate zt = H~tX)[HY(x)(Zi) + aXi(n) e]. Then determine the new action probabilities by dividing by normalization factor N = zt + E#i Zj, as follows: a:~ +-:W, Pi and Pj +- a:' =jt, j =P i. 4. Update V(:z:) only if i = UomIlX' Since the action probabilities are updated after every action, the most probable action may be different before and after the update. If i = amllx both before and after step 3 above, then update the value function estimate as follows: V(:z:) Otherwise, if i = UomIlX +- V(:z:) + aXi(n) e before or after step 3: V(:z:) +- HY(x)(Npk), where action Ie is the most probable action after step 3. 5. :z: 6 +- y. Go to step 2. CONVERGENCE OF THE MODIFIED ALGORITHM The modified actor/critic algorithm given above converge6 to the optimal value function V? with probability one if: Theorem: 1. The 6tate and action 6et6 are finite. 2. E:=o axlI(n) =00 and E:=o a!lI(n) < 00. Space does not permit us to supply the complete proof, which follows this outline: 1. The modified actor/critic algorithm behaves exactly the same as a Q- learning algorithm constrained by a particular exploration strategy. 2. Q-Iearning converges to V? with probability one, given the conditions above (Tsitsiklis, 1993; Jaakkola et aI, 1993; Watkins & Dayan, 1992). 3. Therefore, the modified actor/critic algorithm also converges to V? with probability one. Robert Crites, Andrew G. Barto 406 The commutative diagram below illustrates how the modified actor/critic algorithm behaves exactly like Q-Iearning constrained by a particular exploration strategy. The function H recovers Q-values from the policy ?r and value function V. H- 1 recovers (?r, V) from the Q-values, thus determining an exploration strategy. Given the ability to move back and forth between (?r, V) and Q, we can determine how to change (?r, V) by converting to Q, determining updated Q-values, and then converting back to obtain an updated (?r, V). The modified actor/critic algorithm simply collapses this process into one step, bypassing the explicit use of Q-values. (7r , VA) t Modified Actor/Critic -------.. ( 7r , VA) t+1 H-l H A A (Jt ------------Q---L-ea-r-ru-?n-g----------~~ (Jt+1 Following the diagram above, (?r, V) can be converted to Q-values as follows: Going the other direction, Q-values can be converted to (?r, V) as follows: and The only Q-value that should change at time t is the one corresponding to the state/action pair that was visited at time tj call it Q(:z:, i). In order to prove the convergence theorem, we must verify that after an iteration ofthe modified actor/critic algorithm, its encoded Q-values match the values produced by Q-Iearning: Qt+1(:Z:, a) = Qt(:Z:, i) + Qx.(n) [r + "y max Qt(Y, b) - Qt(:Z:, i)], a = i. bEAli (2) (3) In verifying this, it is necessary to consider the four cases where Q(:z:, i) is, or is not, the maximum Q-value for state :z: at times t and t + 1. Only enough space exists to present a detailed verification of one case. Case 1: Qt(:Z:, i) = ma:z: Qt(:Z:,?) and Qt+l(:Z:, i) = ma:z: Qt+l(:Z:, .). In this case, Jli(t) = Pmax(t) and P.(t + 1) = Pmax(t + 1), since Hyt(x) and H yt +1 (x) are strictly increasing. Therefore Zi (t) = 1 and Zi (t + 1) = 1. Therefore, Vi (:z:) = HYt (x)[1] = HYt(x)[Zi(t)] = Qt(:Z:, i), and An Actor/Critic Algorithm That Is Equivalent to Q-Learning 407 H yt+1 (x) [Zi(t + 1)] Hyt+ 1 (x) [1] Vt+l(:C) Vt(:c) + O!xi(n) e Qt(:Z:, i) + O!xi(n) [r + "y max Qt(Y, b) - Qt(:Z:, i)]. bEAJI This establishes (2). To show that (3) holds, we have that Vt+l(:Z:) Vt(:c) + O!xi(n) e Qt(:z:, i) + O!xi(n) e HYt(x)[Zi(t)] + O!xi(n) e HYt(x)[H~t~x)[HYt(x)[Zi(t)] + O!xi(n) e]] (4) Hyt(x) [zt(t)] and H yt + 1 (x) [za(t + 1)] H [ Pa(t + 1) ] Yt+ 1 (x) Pmax(t + 1) H [Za(t)/lV] Yt+ 1 (x) zt(t)/lV H [Za(t)] Yt+ 1 (x) zt(t) i f a i= i za(t) HHfrt(.. )[zt(t)][zt(t) 1 by (4) Hyt(x) [za(t)] by a property of H Qt(:Z:, a). The other cases can be shown similarly. 7 CONCLUSIONS We have presented an actor/critic algorithm that is equivalent to Q-Iearning constrained by a particular exploration strategy. Like Q-Iearning, it estimates V" directly without a model of the underlying decision process. It uses exactly the same amount of storage as Q-Iearning: one location for every state/action pair. (For each state, IAI- 1 locations are needed to store the action probabilities, since they must sum to one. The remaining location can be used to store the value of that state.) One advantage of Q-Iearning is that its exploration is uncoupled from its value function estimates. In the modified actor/critic algorithm, the exploration strategy is more constrained. 408 Robert Crites, Andrew G. Barto It is still an open question whether other actor/critic algorithms are guaranteed to converge. One way to approach this question would be to investigate further the relationship between the modified actor/critic algorithm described here and the actor/critic algorithms that have been employed by others. Acknowledgements We thank Vijay GullapaUi and Rich Sutton for helpful discussions. This research was supported by Air Force Office of Scientific Research grant F49620-93-1-0269. References A. G. Barto, S. J. Bradtke &. S. P. Singh. (1993) Learning to act using real-time dynamic programming. Artificial Intelligence, Accepted. A. G. Barto, R. S. Sutton &. C. W. Anderson. (1983) Neuronlike adaptive elements that can solve difficult learning control problems. IEEE Transactions on Systems, Man, and Cybernetics 13:835-846. A. G. Barto, R. S. Sutton &. C. J. C. H. Watkins. (1990) Learning and sequential decision making. In M. Gabriel &. J. Moore, editors, Learning and Computational Neuroscience: Foundations of Adaptive Networks. MIT Press, Cambridge, MA. D. P. Bertsekas &. J. N. Tsitsiklis. (1989) Parallel and Distributed Computation: Numerical Metkods. Prentice-Hall, Englewood Cliffs, NJ. T. Jaakkola, M. 1. Jordan &. S. P. Singh. (1993) On the convergence of stochastic iterative dynamic programming algorithms. MIT Computational Cognitive Science Technical Report 9307. L. Lin. (1993) Reinforcement Learning for Robots Using Neural Networks. PhD Thesis, Carnegie Mellon University, Pittsburgh, PA. A. L. Samuel. (1963) Some studies in machine learning using the game of checkers. In E. Feigenbaum &. J. Feldman, editors, Computers and Tkougkt. McGraw-Hill, New York, NY. R. S. Sutton. (1988) Learning to predict by the methods of temporal differences. Mackine Learning 3:9-44. R. S. Sutton. (1990) Integrated architectures for learning, planning, and reacting based on approximating dynamic programming. In Proceedings of tke Seventk International Conference on Mackine Learning. J. N. Tsitsiklis. (1994) Asynchronous stochastic approximation and Q-Iearning. Mackine Learning 16:185-202. C. J. C. H. Watkins. (1989) Learning from Delayed Rewards. PhD thesis, Cambridge University. C. J. C. H. Watkins &. P. Dayan. (1992) Q-Iearning. Mackine Learning 8:279-292. R. J. Williams &. L. C. Baird. (1993) Analysis ofsome 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. PART V ALGORITHMS AND ARCIDTECTURES 

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Temporal Dynamics of Generalization Neural Networks Changfeng Wang Department of Systems Engineering University Of Pennsylvania Philadelphia, PA 19104 fwang~ender.ee.upenn.edu ? In Santosh S. Venkatesh Department of Electrical Engineering University Of Pennsylvania Philadelphia, PA 19104 venkateshGee.upenn.edu Abstract This paper presents a rigorous characterization of how a general nonlinear learning machine generalizes during the training process when it is trained on a random sample using a gradient descent algorithm based on reduction of training error . It is shown, in particular, that best generalization performance occurs, in general, before the global minimum of the training error is achieved. The different roles played by the complexity of the machine class and the complexity of the specific machine in the class during learning are also precisely demarcated . 1 INTRODUCTION In learning machines such as neural networks, two major factors that affect the 'goodness of fit' of the examples are network size (complexity) and training time. These are also the major factors that affect the generalization performance of the network. Many theoretical studies exploring the relation between generalization performance and machine complexity support the parsimony heuristics suggested by Occam's razor, to wit that amongst machines with similar training performance one should opt for the machine of least complexity. Multitudinous numerical experiments (cf. [5]) suggest, however , that machines of larger size than strictly necessary to explain the data can yield generalization performance similar to that of smaller machines (with 264 Changieng Wang. Santosh S. Venkatesh similar empirical error) if learning is optimally stopped at a critical point before the global minimum of the training error is achieved . These results seem to fly in contradiction with a learning theoretic interpretation of Occam's razor. In this paper, we ask the following question: How does the gradual reduction of training error affect the generalization error when a machine of given complexity is trained on a finite number of examples? Namely, we study the simultaneous effects of machine size and training time on the generalization error when a finite sample of examples is available. Our major result is a rigorous characterization of how a given learning machine generalizes during the training process when it is trained using a learning algorithm based on minimization of the empirical error (or a modification of the empirical error). In particular, we are enabled to analytically determine conditions for the existence of a finite optimal stopping time in learning for achieving optimal generalization. We interpret the results in terms of a time-dependent, effective machine size which forms the link between generalization error and machine complexity during learning viewed as an evolving process in time. Our major results are obtained by introducing new theoretical tools which allow us to obtain finer results than would otherwise be possible by direct applications of the uniform strong laws pioneered by Vapnik and Cervonenkis (henceforth refered to as VC-theory). The different roles played by the complexity of the machine class and the complexity of the specific machine in the class during learning are also precisely demarcated in our results. Since the generalization error is defined in terms of an abstract loss function, the results find wide applicability including but not limited to regression (square-error loss function) and density estimation (log-likelihood loss) problems. 2 THE LEARNING PROBLEM We consider the problem of learning from examples a relation between two vectors x and y determined by a fixed but unknown probability distribution P(x, y). This model includes, in particular, the input-output relation described by y = g(x, e) , (1) e, where g is some unknown function of x and which are random vectors on the same probability space. The vector x can be viewed as the input to an unknown system, a random noise term (possibly dependent on x) , and y the system's output. e The hypothesis class from which the learning procedure selects a candidate function (hypothesis) approximating g is a parametric family of functions 1{d = {!(x, fJ) : fJ E 8d ~ ?R d } indexed by a subset 8d of d-dimensional Euclidean space. For example, if x E ?R m and y is a scalar, 1{d can be the class of functions computed by a feedforward neural network with one hidden layer comprised of h neurons and activation function 1/;, viz ., Temporal Dynamics of Generalization in Neural Networks In the above, d 265 = (m + 2)h denotes the number of adjustable parameters. The goal of learning within the hypothesis class 1ld is to find the best approximation of the relation between x and y in 1ld from a finite set of n examples 'Dn = {(Xl , yd, ? .. , (xn , Yn)} drawn by independent sampling from the distribution P(x , y) . A learning algorithm is simply a map which, for every sample 'Dn (n ~ 1) , produces a hypothesis in 1ld. In practical learning situations one first selects a network of fixed structure (a fixed hypothesis class Jid), and then determines the "best" weight vector e* (or equivalently, the best function I(x, e*) in this class) using some training algorithm. The proximity of an approximation I( x, e) to the target function g( x, e) at each point X is measured by a loss function q : (J(x,e),g(x,e) I--t ~+. For a given hypothesis class, the function 1(?, e) is completely determined by the parameter vector e. With g fixed, the loss function may be written, with a slight abuse of notation, as a map q( x, y, e) from ~m x ~ x 8 into ~+ . Examples of the forms of loss (g(x,e) - f(x,e))2 functions are the familiar square-law loss function q(x,y,e) commonly used in regression and learning in neural networks, and the KulbackLeibler distance (or relative entropy) q( x, y, e) = In PP y f9 x, for density estimation, where p(Y I i(x , e?) denotes the conditional density function of y given i(x, e). = t t: ) The closeness of 1(-, e) to g(.) is measured by the expected (ensemble) loss or error ?(e, d) ~ J q(x, y, e)p(dx, dy) . Ie, = The optimal approximation e*) is such that ?(e*, d) min9EE>d ?(e, d). In similar fashion, we define the corresponding empirical loss (or training error) by 1 n ?n(e, d) = J q(x, y, e)Pn(dx, dy) = ~ ?= q(Xi, Yi , e) . ? =1 where Pn denotes the joint empirical distribution of input-output pairs (x, y). The global minimum of the empirical error over 8d is denoted by 0, namely, arg min9EE> ?n(e, d). An iterative algorithm for minimizing ?n(e, d) (or a modification of it) over 8 d generates at each epoch t a random vector et :'D n -+ 8 d . The quantity ?(e t , d) = E f q(x, et)p(dx, de) is referred to as the generalization error of et . We are interested in the properties of the process { et : t 1, 2, ... }, and the time-evolution of the sequence {?(et, d) : t 1,2, .. . }. e= = = Note that each et is a functional of Pn . When P = Pn , learning reduces to an optimization problem. Deviations from optimality arise intrinsically as a consequence of the discrepancy between Pn and P. The central idea of this work is to analyze the consequence of the deviation .6. n ~ Pn - P on the generalization error. To simplify notation , we henceforth suppress d and write simply 8 , ?(e) and ?n(e) instead of 8 d , ?(e, d), and ?n(e, d), respectively. 2.1 RegUlarity Conditions We will be interested in the local behavior of learning algorithms. Consequently, we assume that 8 is a compact set, and e* is the unique global minimum of ?(e) on 8. 266 Changfeng Wang, Santosh S. Venkatesh It can be argued that these assumptions are an idealization of one of the following situations: ? A global algorithm is used which is able to find the global minimum of En (9), and we are interested in the stage of training when 9t has entered a region 8 where 9* is the only global minimum of E(9); ? A local algorithm is used, and the algorithm has entered a region 8 which contains 9* as the unique global minimum of E(9) or as a unique local minimum with which we are content. In the sequel, we write () / {)9 to denote the gradient operator with respect to the vector 9, and likewise write ()2/{)02 to denote the matrix of operators L:l{/~;{/J:,j=l. In the rest of the development we assume the following regularity conditions: Al. The loss function q(x, y,.) is twice continuously differential for all 0 E 8 and for almost all (x, Y)j A2. P(x, y) has compact support; A3. The optimal network 9* is an interior point of 8; A4. The matrix ~(9*) = ~E(O*) is nonsingular. These assumptions are typically satisfied in neural network applications. We will also assume that the learning algorithm converges to the global minimum of En (0) over 8 (note that 8 may not be a true global minimum, so the assumption applies to gradient descent algorithms which converge locally). It is easy to demonstrate that for each such algorithm, there exists an algorithm which decreases the empirical error monotonically at each step of iteration. Thus, without loss of generality, we also assume that all the algorithms we consider have this monotonicity property. 3 3.1 GENERALIZATION DYNAMICS First Phase of Learning The quality of learning based on the minimization of the empirical error depends on the value of the quantity sUPe IEn(9) - E(9)1. Under the above assumptions, it is shown in [3] that E(O) = En((}) + Op ( Inn) ..;n and n) E(OA) = E(9*) + 0 (In -;;- . Therefore, for any iterative algorithm for minimizing En(O), in the initial phase of learning the reduction of training error is essentially equivalent to the reduction of generalization error. It can be further shown that this situation persists until the estimates ()t enter an n- 6,. neighborhood of 8, where bn - t 1/2. The basic tool we have used in arriving at this conclusion is the VC-method. The characterization of the precise generalization properties of the machine after Ot enters an n- 6 neighborhood of the limiting solution needs a more precise language than can be provided by the VC-method, and is the main content of the rest of this work. Temporal Dynamics of Generalization in Neural Networks 3.2 267 Learning by Gradient Descent In the following, we focus on generalization properties when the machine is trained using the gradient descent algorithm (Backpropagation is a Gauss-Seidel implementation of this algorithm); in particular, the adaptation is governed by the recurrence (t ~ 0), (2) where the positive quantity ( governs the rate of learning. Learning and generalization properties for other algorithms can be studied using similar techniques. Replace fn by f in (2) and let { (); , t ~ O} denote the generated sequence of vectors. We can show (though we will not do so here) that the weight vector ()t is asymptotically normally distributed with expectation (); and covariance matrix with all It is precisely the deviation of ()t from (); caused by the entries of order 0 ( perturbation of amount ~n = Pn - P to the true distribution P which results in interesting artifacts such as a finite optimal stopping time when the number of examples is finite. k). 3.3 The Main Equation of Generalization Dynamics Under the regularity conditions mentioned in the last section, we can find the generalization error at each epoch of learning as an explicit function of the number of iterations, machine parameters, and the initial error. Denote by Al 2: A2 2: ... ~ Ad the eigenvalues of the matrix <1>( ()*) and suppose T is the orthogonal diagonalizing matrix for <1>(()*), viz., T'?(()*)T = diag(Al, ... ,Ad). Set 8 = (8 l , ... ,8d)' ~ T( ()o - ()*) and for each i let Vi denote the ith diagonal element of the d x d matrix T'E { (:0 q( x, ()*?) (:0 q( x, ()*?) '} T. Also let S( (), p) denote the open ball of radius p at (). Under Assumptions Al-A4, the generalization error of the machine trained according to (2) is governed by the following equation for all starting points ()o E S(()*, n- r ) (0 < r ~ t), and uniformly for all t 2: 0: MAIN THEOREM d f(()t) = f(()*) + -.!.. ,,{ Vi [1 - (1- (Ai)t] 2 + 8; Ai(l- (Ai)2t} + O(n- 3r ) . 2n L...J A' i=l (3) ' If ()o (/:. S(()*, n-t), then the generalization dynamics is governed by the following equation valid for all r > 0: where tl is the smallest t such that E 1[1 - (<1>]t11 = An- r for some A > 0, and C(tl)' Ci(tl) ~ 0 are constants depending on network parameters and tlIn the special case when the data is generated by the following additive noise model y=g(x)+e, (5) 268 Changfeng Wang. Santosh S. Venkatesh with E [~Ix] = 0, and E [elx] = 0- 2 = constant, if g(x) = f(x,O*) and the loss function q(x, 0) is given by the square-error loss function, the above equation reduces to the following form: 2 ?(Ot) = ?(0*) + ;n d ?= {[1 - (1- f.~i)t] 2 + 61.Ai(1- f.Ai)2t} + O(n- 3r ). ,=1 In particular, if f(x , 0) is linear in 0, we obtain our previous result [4] for linear machines. The result (3) is hence a substantive extension of the earlier result to very general settings. It is noted that the extension goes beyond nonlinearity and the original additive noise data generating model-we no longer require that the 'true' model be contained in the hypothesis class. 3.4 Effective Complexity Write Ci ~ to be Vi /.Ai . The effective complexity of the nonlinear machine Ot at t is defined 2 d C(O*, d, t) ~ 2: ci (1 - (1- f.Ad) . i=1 Analysis shows that the term Ci indicates the level of sensitivity of output of the machine to the ith component of the normalized weight vector, 0; C(O*, d, t) denotes the degree to which the approximation power of the machine is invoked by the learning process at epoch t. Indeed, as t ---+ 00, C(O*, d, t) ---+ Cd = '2:.1=1 Ci, which is the complexity of the limiting machine {} which represents the maximal fitting of examples to the machine (i.e., minimized training error). For the additive noise data generating model (5) and square-error loss function, the effective complexity becomes, d C(O*, d, t) = 2:(1- (1- cAif)2 . i=1 The sum can be interpreted as the effective number of parameters used at epoch t. At the end of training, it becomes exactly the number of parameters of the machine . Now write 0; ~ 0* + (0 0 - 0* )(1- f.Ad. With these definitions, (3) can be rewritten to give the following approximation error and complexity error decomposition of generalization error in the learning process: (t ~ 0). (6) The first term on the right-hand-side, ?(0;), denotes the approximation error at epoch t and is the error incurred in using as an approximation of the 'truth.' Note that the approximation error depends on time t and the initial value 00 , but not the examples. Clearly, it is the error one would obtain at epoch t in minimizing the function ?(0) (as opposed to ?n(O? using the same learning algorithm and starting with the same step length f and initial value 00 . The second term on the righthand-side is the complexity error at epoch t. This is the part of the generalization error at t due to the substitution of ?n(O) for ?(0). 0; Temporal Dynamics of Generalization in Neural Networks 269 The overfitting phenomena in learning is often intuitively attributed to the 'fitting of noise.' We see that is only partly correct: it is in fact due to the increasing use of the capacity of the machine, that the complexity penalty becomes increasingly large, this being true even when the data is clean, i.e., when == O! Therefore, we see that (6) gives an exact trade-off of the approximation error and complexity error in the learning process. e For the case of large initial error, we see from the main theorem that the complexity error is essentially the same as that at the end of training , when the initial error is reduced to about the same order as before. The reduction of the training error leads to monotone decrease in generalization error in this case. 3.5 Optimal Stopping Time We can phrase the following succinct open problem in learning in neural networks: When should learning be ideally stopped? The question was answered for linear machines which is a special form of neural networks in [4] . This section extends the result to general nonlinear machines (including neural networks) in regular cases. For this purpose, we write the generalization error in the following form: where <p(t) ~ : 2 d ?= {li(l- dd 2t - dj (1- fAdt} , s=l and dj and Ii are machine parameters. The time-evolution of generalization error during the learning process is completely determined by the function <p(t). Define tmin E { T ~ 0 : ?(OT) ~ ?(Ot) for all t ~ O}, that is tmin denotes an epoch at which the generalization error is minimized. The smallest such number will be referred to as the optimal stopping time of learning. In general we have Cj > a for all i . In this case, it is possible to determine that there is a finite optimal stopping time. More specifically, there exists two constants t/ and tu which depend on the machine parameters such that t/ ~ tmin ~ tu. Furthermore, it can be shown that the function <p(t) decreases monotonically for t ~ t/ and increases monotonically for all t ~ tu. Finally, we can relate the generalization performance when learning is optimally stopped to the best achievable performance by means of the following inequality: ?(0 . ) < ?(0.) tmm - + (1 - IC)Cd 2n ' where IC = O(nO) is a constant depending on Ii and dj's, and is in the interval (O,~], and Cd denotes, as before, the limiting value of the effective machine complexity C(O?, d, t) as t -+ 00. = In the pathological case where there exists i such that Cj 0, there may not exist a finite optimal stopping time. However, even in such cases, it can be shown that if In(l - fAdl In(l - fAd) < 2, a finite optimal stopping time still exists. 270 4 Changfeng Wang, Santosh S. Venkatesh CONCLUDING REMARKS This paper describes some major results of our recent work on a rigorous characterization of the generalization process in neural network types of learning machines. In particular, we have shown that reduction of training error may not lead to improved generalization performance. Two major techniques involved are the uniform weak law (VC-theory) and differentiable statistical functionals, with the former delivering an initial estimate, and the latter giving finer results. The results shows that the complexity (e.g. VC-dimension) of a machine class does not suffice to describe the role of machine complexity in generalization during the learning process; the appropriate complexity notion required is a time-varying and algorithm-dependent concept of effective machine complexity. Since results in this work contain parameters which are typically unknown, they cannot be used directly in practical situations. However, it is possible to frame criteria overcoming such difficulties. More details of the work described here and its extensions and applications can be found in [3]. The methodology adopted here is also readily adapted to study the dynamical effect of regularization on the learning process [3]. Acknowledgements This research was supported in part by the Air Force Office of Scientific Research under grant F49620-93-1-0120. References [1] Kolmogorov, A. and V. Tihomirov (1961). f-entropy and f-capacity of sets in functional spaces. Amer. Math. Soc. Trans. (Ser. 2), 17:277-364. [2] Vapnik, V. (1982). Estimation of Dependences Based on Empirical Data. Springer-Verlag, New York. [3] Wang, C. (1994). A Theory of Generalization in Learning Machines. Ph. D. Thesis, University of Pennsylvania. [4] Wang, C., S. S. Venkatesh, and J. S. Judd (1993) . Optimal stopping and effective machine size in learning. Proceedings of NIPS'93. [5] Weigend, A. (1993). On overtraining and the effective number of hidden units. Proceedings of the 1993 Connectionist Models Summer School. 335-342. Ed. Mozer, M. C. et al. Hillsdale , NJ: Erlbaum Associates. 

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Limits on Learning Machine Accuracy Imposed by Data Quality Corinna Cortes, L. D. Jackel, and Wan-Ping Chiang AT&T Bell Laboratories Holmdel, NJ 07733 Abstract Random errors and insufficiencies in databases limit the performance of any classifier trained from and applied to the database. In this paper we propose a method to estimate the limiting performance of classifiers imposed by the database. We demonstrate this technique on the task of predicting failure in telecommunication paths. 1 Introduction Data collection for a classification or regression task is prone to random errors, e.g. inaccuracies in the measurements of the input or mis-labeling of the output. Missing or insufficient data are other sources that may complicate a learning task and hinder accurate performance of the trained machine. These insufficiencies of the data limit the performance of any learning machine or other statistical tool constructed from and applied to the data collection - no matter how complex the machine or how much data is used to train it. In this paper we propose a method for estimating the limiting performance of learning machines imposed by the quality of the database used for the task. The method involves a series of learning experiments. The extracted result is, however, independent of the choice of learning machine used for these experiments since the estimated limiting performance expresses a characteristic of the data. The only requirements on the learning machines are that their capacity (VC-dimension) can be varied and can be made large, and that the learning machines with increasing capacity become capable of implementing any function. 240 Corinna Cortes. L. D. Jackel. Wan-Ping Chiang We have applied the technique to data collected for the purpose of predicting failures in telecommunication channels of the AT&T network. We extracted information from one of AT&T's large databases that continuously logs performance parameters of the network. The character and amount of data comes to more material than humans can survey. The processing of the extracted information is therefore automated by learning machines. We conjecture that the quality of the data imposes a limiting error rate on any learning machine of,... 25%, so that even with an unlimited amount of data, and an arbitrarily complex learning machine, the performance for this task will not exceed ,... 75% correct. This conjecture is supported by experiments. The relatively high noise-level of the data, which carries over to a poor performance of the trained classifier, is typical for many applications: the data collection was not designed for the task at hand and proved inadequate for constructing high performance classifiers. 2 Basic Concepts of Machine Learning We can picture a learning machine as a device that takes an unknown input vector and produces an output value. More formally, it performs some mapping from an input space to an output space. The particular mapping it implements depends of the setting of the internal parameters of the learning machine. These parameters are adjusted during a learning phase so that the labels produced on the training set match, as well as possible, the labels provided. The number of patterns that the machine can match is loosely called the "capacity" of the machine. Generally, the capacity of a machine increases with the number of free parameters. After training is complete, the generalization ability of of the machine is estimated by its performance on a test set which the machine has never seen before. The test and training error depend on both the the number of training examples I, the capacity h of the machine, and, of course, how well suited the machine is to implement the task at hand. Let us first discuss the typical behavior of the test and training error for a noise corrupted task as we vary h but keep the amount I of training data fixed. This scenario can, e.g., be obtained by increasing the number of hidden units in a neural network or increasing the number of codebook vectors in a Learning Vector Quantization algorithm [6]. Figure la) shows typical training and test error as a function of the capacity of the learning machine. For h < I we have many fewer free parameters than training examples and the machine is over constrained. It does not have enough complexity to model the regularities of the training data, so both the training and test error are large (underfitting). As we increase h the machine can begin to fit the general trends in the data which carries over to the test set, so both error measures decline. Because the performance of the machine is optimized on only part of the full pattern space the test error will always be larger than the training error. As we continue to increase the capacity of the learning machine the error on the training set continues to decline, and eventually it reaches zero as we get enough free parameters to completely model the training set. The behavior of the error on the test set is different. Initially it decreases, but at some capacity, h*, it starts to rise. The rise occurs because the now ample resources of the training machine are applied to learning vagaries of the training Limits on Learnillg Machine Accuracy Imposed by Data Quality % error a) % error b) 241 c) % error ? . ?. ...,..- --- ~ Ie " capacity ( h) -- training .. training set size ( I) .. . intrinsic noise level capacity ( h) Figure 1: Errors as function of capacity and training set size. Figure la) shows characteristic plots of training and test error as a function of the learning machine capacity for fixed training set size. The test error reaches a minimum at h = h* while the training error decreases as h increases. Figure Ib) shows the training and test errors at fixed h for varying I. The dotted line marks the asymptotic error Eoo for infinite I. Figure lc) shows the asymptotic error as a function of h. This error is limited from below by the intrinsic noise in the data. set, which are not reproduced in the test set (overfitting). Notice how in Figure la) the optimal test error is achieved at a capacity h* that is smaller than the capacity for which zero error is achieved on the training set. The learning machine with capacity h* will typically commit errors on misclassified or outlying patterns of the training set. We can alternatively discuss the error on the test and training set as a function of the training set size I for fixed capacity h of the learning machine. Typical behavior is sketched in Figure Ib) . For small I we have enough free parameters to completely model the training set, so the training error is zero. Excess capacity is used by the learning machine to model details in the training set, leading to a large test error. As we increase the training set size I we train on more and more patterns so the test error declines. For some critical size of the training set, Ie, the machine can no longer model all the training patterns and the training error starts to rise. As we further increase I the irregularities of the individual training patterns smooth out and the parameters of the learning machine is more and more used to model the true underlying function . The test error declines, and asymptotically the training and test error reach the same error value Eoo . This error value is the limiting performance of the given learning machine to the task . In practice we never have the infinite amount of training data needed to achieve Eoo. However, recent theoretical calculations [8, 1, 2, 7, 5] and experimental results [3] have shown that we can estimate Eoo by averaging the training and test errors for I> Ie. This means we can predict the optimal performance of a given machine. For a given type of learning machine the value of the asymptotic error Eoo of the machine depends on the quality of the data and the set of functions it can implement. The set of available functions increases with the capacity of the machine: 242 Corinna Cortes, L. D. Jackel, Wall-Ping Chiang low capacity machines will typically exhibit a high asymptotic error due to a big difference between the true noise-free function of the patterns and the function implemented by the learning machine, but as we increase h this difference decreases. If the learning machine with increasing h becomes a universal machine capable of modeling any function the difference eventually reaches zero, so the asymptotic error Eoo only measures the intrinsic noise level of the data. Once a capacity of the machine has been reached that matches the complexity of the true function no further improvement in Eoo can be achieved. This is illustrated in Figure lc). The intrinsic noise level of the data or the limiting performance of any learning machine may hence be estimated as the asymptotic value of Eoo as obtained for asymptotically universal learning machines with increasing capacity applied to the task. This technique will be illustrated in the following section. 3 Experimental Results In this section we estimate the limiting performance imposed by the data of any learning machine applied to the particular prediction task. 3.1 Task Description To ensure the highest possible quality of service, the performance parameters of the AT&T network are constantly monitored. Due to the high complexity of the network this performance surveillance is mainly corrective: when certain measures exceed preset thresholds action is taken to maintain reliable, high quality service. These reorganizations can lead to short, minor impairments of the quality of the communication path. In contrast, the work reported here is preventive: our objective is to make use of the performance parameters to form predictions that are sufficiently accurate that preemptive repairs of the channels can be made during periods of low traffic. In our study we have examined the characteristics of long-distance, 45 Mbitsfs communication paths in the domestic AT&T network. The paths are specified from one city to another and may include different kinds of physical links to complete the paths. A path from New York City to Los Angeles might include both optical fiber and coaxial cable. To maintain high-quality service, particular links in a path may be switched out and replaced by other, redundant links. There are two primary ways in which performance degradation is manifested in the path. First is the simple bit-error rate, the fraction of transmitted bits that are not correctly received at the termination of the path. Barring catastrophic failure (like a cable being cut), this error rate can be measured by examining the error-checking bits that are transmitted along with the data. The second instance of degradation, ''framing error" , is the failure of synchronization between the transmitter and receiver in a path. A framing error implies a high count of errored bits. In order to better characterize the distribution of bit errors, several measures are historically used to quantify the path performance in a 15 minutes interval. These measures are: Low-Rate The number of seconds with exactly 1 error. Limits Oil 243 Learning Machine Accllracy Imposed by Data Quality "No-Trouble" patterns: Frame-Error Rate-High I ? ? .. Rate-Medlum IJ Rate-Low I , o "Trouble" pattems: In time (days) 21 II I ,,' , I l"e Figure 2: Errors as function of time. The 3 top patterns are members of the "No-Trouble" class. The 3 bottom ones are members of the "Trouble" class. Errors are here plotted as mean values over hours. Medium-Rate The number of seconds with more than one but less than 45 errors. High-Rate The number of seconds with 45 or more errors, corresponding to a bit error rate of at least 10- 6 ? Frame-Error The number of seconds with a framing error. A second with a frameerror is always accompanied by a second of High-Rate error. Although the number of seconds with the errors described above in principle could be as high as 900, any value greater than 255 is automatically clipped back to 255 so that each error measure value can be stored in 8 bits. Daily data that include these measures are continuously logged in an AT&T database that we call Perf(ormance)Mon(itor). Since a channel is error free most of the time, an entry in the database is only made if its error measures for a 15 minute period exceed fixed low thresholds, e.g. 4 Low-Rate seconds, 1 Medium- or HighRate second, or 1 Frame-Error. In our research we "mined" PerfMon to formulate a prediction strategy. We extracted examples of path histories 28 days long where the path at day 21 had at least 1 entry in the PerfMon database. We labeled the examples according to the error-measures over the next 7 days. If the channel exhibited a 15-minute period with at least 5 High-Rate seconds we labeled it as belonging to the class "Trouble". Otherwise we labeled it as member of "No-Trouble" . The length of the history- and future-windows are set somewhat arbitrarily. The history has to be long enough to capture the state of the path but short enough that our learning machine will run in a reasonable time. Also the longer the history the more likely the physical implementation of the path was modified so the error measures correspond to different media. Such error histories could in principle be eliminated from the extracted examples using the record of the repairs and changes 244 Corinna Cortes, L. D. lackel, Wan-Ping Chiang of the network. The complexity of this database, however, hinders this filtering of examples. The future-window of7 days was set as a design criterion by the network system engineers. Examples of histories drawn from PerfMon are shown in Figure 2. Each group of traces in the figure includes plots of the 4 error measures previously described. The 3 groups at the top are examples that resulted in No-Trouble while the examples at the bottom resulted in Trouble. Notice how bursty and irregular the errors are, and how the overall level of Frame- and High-Rate errors for the Trouble class seems only slightly higher than for the No-Trouble class, indicating the difficulty of the classification task as defined from the database PerfMon. PerfMon constitutes, however, the only stored information about the state of a given channel in its entirety and thus all the knowledge on which one can base channel end-to-end predictions: it is impossible to install extra monitoring equipment to provide other than the 4 mentioned end-to-end error measures. The above criteria for constructing examples and labels for 3 months of PerfMon data resulted in 16325 examples from about 900 different paths with 33.2% of the examples in the class Trouble. This means, that always guessing the label of the largest class, No-Trouble, would produce an error rate of about 33%. 3.2 Estimating Limiting Performance The 16325 path examples were randomly divided into a training set of 14512 examples and a test set of 1813 examples. Care was taken to ensure that a path only contributes to one of the sets so the two sets were independent, and that the two sets had similar statistical properties. Our input data has a time-resolution of 15 minutes. For the results reported here the 4 error measures of the patterns were subsampled to mean values over days yielding an input dimensionality of 4 x 21. We performed two sets of independent experiments. In one experiment we used fully connected neural networks with one layer of hidden units. In the other we used LVQ learning machines with an increasing number of codebook vectors. Both choices of machine have two advantages: the capacity of the machine can easily be increased by adding more hidden units, and by increasing the number of hidden units or number of codebook vectors we can eventually model any mapping [4]. We first discuss the results with neural networks. Baseline performance was obtained from a threshold classifier by averaging all the input signals and thresholding the result. The training data was used to adjust the single threshold parameter. With this classifier we obtained 32% error on the training set and 33% error on the test set. The small difference between the two error measures indicate statistically induced differences in the difficulty of the training and test sets. An analysis of the errors committed revealed that the performance of this classifier is almost identical to always guessing the label of the largest class "No-Trouble": close to 100% of the errors are false negative. A linear classifier with about 200 weights (the network has two output units) obtained 28% error on the training set and 32% error on the test set. Limits on Learning Machine Accuracy Imposed by Data Quality 40 classification error. % 40 245 classification error. % test 30 30 20 20 10~ ____- .__________ 3 weights (log 10) training ~~ 4 10 ~-------------~ 2 3 codebook vectors (log 1(} Figure 3: a) Measured classification errors for neural networks with increasing number of weights (capacity). The mean value between the test and training error estimates the performance of the given classifier trained with unlimited data. b) Measured classification errors for LVQ classifiers with increasing number of codebook vectors. Further experiments exploited neural nets with one layer of respectively 3, 5, 7, 10, 15, 20, 30, and 40 hidden units. All our results are summarized in Figure 3a). This figure illustrates several points mentioned in the text above. As the complexity of the network increases, the training error decreases because the networks get more free parameters to memorize the data. Compare to Figure 1a). The test error also decreases at first, going through a minimum of 29% at the network with 5 hidden units. This network apparently has a capacity that best matches the amount and character of the available training data. For higher capacity the networks overfit the data at the expense of increased error on the test set. Figure 3a) should also be compared to Figure 1c). In Figure 3a) we plotted approximate values of Eoo for the various networks - the minimal error of the network to the given task. The values of Eoo are estimated as the mean of the training and test errors. The value of Eoo appears to flatten out around the network with 30 units, asymptotically reaching a value of 24% error. An asymptotic Eoo-value of 25% was obtained from LVQ-experiments with increasing number of codebook vectors. These results are summarized in Figure 3b). We therefore conjecture that the intrinsic noise level of the task is about 25%, and this number is the limiting error rate imposed by the quality of the data on any learning machine applied to the task. 246 4 Corinna Cortes, L. D. Jackel, Wan-Ping Chiang Conclusion In this paper we have proposed a method for estimating the limits on performance imposed by the quality of the database on which a task is defined. The method involves a series of learning experiments. The extracted result is, however, independent of the choice of learning machine used for these experiments since the estimated limiting performance expresses a characteristic of the data. The only requirements on the learning machines are that their capacity can be varied and be made large, and that the machines with increasing capacity become capable of implementing any function. In this paper we have demonstrated the robustness of our method to the choice of classifiers: the result obtained with neural networks is in statistical agreement with the result obtained for LVQ classifiers. Using the proposed method we have investigated how well prediction of upcoming trouble in a telecommunication path can be performed based on information extracted from a given database. The analysis has revealed a very high intrinsic noise level of the extracted information and demonstrated the inadequacy of the data to construct high performance classifiers. This study is typical for many applications where the data collection was not necessarily designed for the problem at hand. Acknowledgments We gratefully acknowledge Vladimir Vapnik who brought this application to the attention of the Holmdel authors. One of the authors (CC) would also like to thank Walter Dziama, Charlene Paul, Susan Blackwood, Eric Noel, and Harris Drucker for lengthy explanations and helpful discussions of the AT&T transport system. References [1] [2] [3] [4] [5] [6] [7] [8] s. Bos, W. Kinzel, and M. Opper. The generalization ability of perceptrons with continuous output. Physical Review E, 47:1384-1391, 1993. Corinna Cortes. Prediction of Generalization Ability in Learning Machines. PhD thesis, University of Rochester, NY, 1993. Corinna Cortes, L. D. Jackel, Sara A. So1la, V. Vapnik, and John S. Denker. Learning curves: Asymptotic value and rate of convergence. In Advances in Neural Information Processing Systems, volume 6. Morgan Kaufman, 1994. G. Cybenko, K. Hornik, M. Stinchomb, and H. White. Multilayer feedforward neural networks are universal approximators. Neural Networks, 2:359-366, 1989. T. L. Fine. Statistical generalization and learning. Technical Report EE577, Cornell University, 1993. Teuvo Kohonen, Gyorgy Barna, and Ronald Chrisley. Statistical pattern recognition with neural networks: Benchmarking studies. In Proc. IEEE Int. Con! on Neural Networks, IJCNN-88, volume 1, pages 1-61-1-68, 1988. N. Murata, S. Yoshizawa, and S. Amari. Learning curves, model selection, and complexity of neural networks. In Advances in Neural Information Processing Systems, volume 5, pages 607-614. Morgan Kaufman, 1992. H. S. Seung, H. Sompolinsky, and N. Tishby. Statistical mechanics of learning from examples. Physical Review A, 45:6056-6091, 1992. 

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Diffusion of Credit in Markovian Models Yoshua Bengio? Dept. I.R.O., Universite de Montreal, Montreal, Qc, Canada H3C-3J7 bengioyCIRO.UMontreal.CA Paolo Frasconi Dipartimento di Sistemi e Informatica Universita di Firenze, Italy paoloCmcculloch.ing.unifi.it Abstract This paper studies the problem of diffusion in Markovian models, such as hidden Markov models (HMMs) and how it makes very difficult the task of learning of long-term dependencies in sequences. Using results from Markov chain theory, we show that the problem of diffusion is reduced if the transition probabilities approach 0 or 1. Under this condition, standard HMMs have very limited modeling capabilities, but input/output HMMs can still perform interesting computations. 1 Introduction This paper presents an important new element in our research on the problem of learning long-term dependencies in sequences. In our previous work [4J we found theoretical reasons for the difficulty in training recurrent networks (or more generally parametric non-linear dynamical systems) to learn long-term dependencies. The main result stated that either long-term storing or gradient propagation would be harmed, depending on whether the norm of the Jacobian of the state to state function was greater or less than 1. In this paper we consider a special case in which the norm of the Jacobian of the state to state function is constrained to be exactly 1 because this matrix is a stochastic matrix. We consider both homogeneous and non-homogeneous Markovian models. Let n be the number of states and At be the transition matrices (constant in the homogeneous case): Aij(ut} = P(qt = j I qt-l = i, Ut; e) where Ut is an external input (constant in the homogeneous case) and e is a vector of parameters. In the homogeneous case (e.g., standard HMMs), such models can learn the distribution of output sequences by associating an output distribution to each state. In ?also, AT&T Bell Labs, Holmdel, NJ 07733 554 Yoshua Bengio. Paolo Frasconi the non-homogeneous case, transition and output distributions are conditional on the input sequences, allowing to model relationships between input and output sequences (e.g. to do sequence 'regression or classification as with recurrent networks). We thus called Input/Output HMM (IOHMM) this kind of non-homogeneous HMM . In [3, 2] we proposed a connectionist implementation of IOHMMs. In both cases, training requires propagating forward probabilities and backward probabilities, taking products with the transition probability matrix or its transpose. This paper studies in which conditions these products of matrices might gradually converge to lower rank, thus harming storage and learning of long-term context. However, we find in this paper that IOHMMs can better deal with this problem than homogeneous HMMs . 2 2.1 Mathematical Preliminaries Definitions A matrix A is said to be non-negative, written A 2:: 0, if Aij 2:: 0 Vi, j . Positive matrices are defined similarly. A non-negative square matrix A E R nxn is called row stochastic (or simply stochastic in this paper) if 'L,'l=1 Aij = 1 Vi = 1 . . . n. A non-negative matrix is said to be row [column} allowable if every row [column] sum is positive. An allowable matrix is both row and column allowable. A nonnegative matrix can be associated to the directed transition graph 9 that constrains the Markov chain. An incidence matrix A corresponding to a given non-negative matrix A replaces all positive entries of A by 1. The incidence matrix of A is a connectivity matrix corresponding to the graph 9 (assumed to be connected here). Some algebraic properties of A are described in terms of the topology of g. Definition 1 (Irreducible Matrix) A non-negative n x n matrix A is said to be irreducible if for every pair i,j of indices, :3 m = m(i,j) positive integer s.t. (Amhj > O. A matrix A is irreducible if and only if the associated graph is strongly connected (i.e., there exists a path between any pair of states i,j) . If :3k s.t . (Ak)ii > 0, d(i) is called the period of index i ifit is the greatest common divisor (g .c.d.) of those k for which (Ak)ii > O. In an irreducible matrix all the indices have the same period d, which is called the period of the matrix. The period of a matrix is the g .c.d. of the lengths of all cycles in the associated transition graph. Definition 2 (Primitive matrix) A non-negative matrix A is said to be primitive if there exists a positive integer k S.t. Ak > O. An irreducible matrix is either periodic or primitive (i.e. of period 1). A primitive stochastic matrix is necessarily allowable. 2.2 The Perron-Frobenius Theorem Theorem 1 (See [6], Theorem 1.1.) Suppose A is an n x n non-negative primitive matrix. Then there exists an eigenvalue r such that: 1. r is real and positive; 2. with r can be associated strictly positive left and right eigenvectors; 3. r> 1>'1 for any eigenvalue>. 1= r; 4? the eigenvectors associated with r are unique to constant multiples. 5. If 0 S B A and f3 is an eigenvalue of B, then 1f31 r . Moreover, 1f31 = r implies B = A. s s 555 Diffusion of Credit in Markovian Models 6. r is simple root of the characteristic equation of A. A simple consequence of the theorem for stochastic matrices is the following: Corollary 1 Suppose A is a primitive stochastic matrix. Then its largest eigehvalue is 1 and there is only one corresponding right eigenvector, which is 1 = [1, 1 .. ?1]'. Furthermore, all other eigenvalues < 1. Proof. A1 = 1 by definition of stochastic matrices. This eigenvector is unique and all other eigenvalues < 1 by the Perron-Frobenius Theorem. If A is stochastic but periodic with period d, then A has d eigenvalues of module 1 which are the d complex roots of 1. 3 Learning Long-Term Dependencies with HMMs In this section we analyze the case of a primitive transition matrix as well as the general case with a canonical re-ordering of the matrix indices. We discuss how ergodicity coefficients can be used to measure the difficulty in learning long-term dependencies. Finally, we find that in order to avoid all diffusion, the transitions should be deterministic (0 or 1 probability). 3.1 Training Standard HMMs Theorem 2 (See [6], Theorem 4.2.) If A is a primitive stochastic matrix, then as t -+ 00, At -+ 1V' where v' is the unique stationary distribution of the Markov chain. The rate of approach is geometric. Thus if A is primitive, then liIDt-+oo At converges to a matrix whose eigenvalues are all 0 except for ,\ = 1 (with eigenvector 1), i.e. the rank of this product converges to 1, i.e. its rows are equal. A consequence oftheorem 2 is that it is very difficult to train ordinary hidden Markov models, with a primitive transition matrix, to model long-term dependencies in observed sequences. The reason is that the distribution over the states at time t > to becomes gradually independent of the distribution over the states at time to as t increases. It means that states at time to become equally responsible for increasing the likelihood of an output at time t. This corresponds in the backward phase of the EM algorithm for trainin~ HMMs to a diffusion of credit over all the states. In practice we train HMMs WIth finite sequences. However , training will become more and more numerically ill-conditioned as one considers longer term dependencies. Consider two events eo (occurring at to) and et (occurring at t), and suppose there are also "interesting" events occurring in between. Let us consider the overall influence of states at times 1" < t upon the likelihood of the outputs at time t. Because of the phenomenon of diffusion of credit, and because gradients are added together, the influence of intervening events (especially those occurring shortly before t) will be much stronger than the influence of eo . Furthermore, this problem gets geometrically worse as t increases. Clearly a positive matrix is primitive. Thus in order to learn long-term dependencies, we would like to have many zeros in the matrix of transition probabilities. Unfortunately, this generally supposes prior knowledge of an appropriate connectivity graph . 3.2 Coefficients of ergodicity To study products of non-negative matrices and the loss of information about initial state in Markov chains (particularly in the non-homogeneous case), we introduce the projective distance between vectors x and y: x?y? d(x',y') = ~~ In(--.:..l.). I ,) XjYi Clearly, some contraction takes place when d(x'A,y'A) ::; d(x',y'). 556 Yoshua Bengio, Paolo Frasconi Definition 3 BirkhofJ's contraction coefficient TB(A), for a non-negative columnallowable matrix A, is defined in terms of the projective distance: TB(A) = sup x ,y> Ojx;t>.y d(x' A, y' A) d(x', y') Dobrushin's coefficient Tl(A), for a stochastic matrix A, is defined as follows: Tl(A) = 1 2 s~p L I,) laik - ajkl? k = Both are proper ergodicity coefficients: 0 ~ T(A) ~ 1 and T(A) 0 if and only if A has identical rows. Furthermore, T(AIA2) ~ T(Al)T(A2)(see [6]). 3.3 Products of Stochastic Matrices Let A (1 ,t) = A 1 A 2 ??? A t - 1 A t denote a forward product of stochastic matrices AI, A 2, ... At. From the properties of TB and Tl, if T(At} < 1, t > 0 then limt-l-oo T(A(l,t?) = 0, i.e. A(l,t) has rank 1 and identical rows. Weak ergodicity is then defined in terms of a proper ergodic coefficient T such as TB and Tl: Definition 4 (Weak Ergodicity) The products of stochastic matrices weakly ergodic if and only if for all to ~ 0 as t -+ 00, T(A(to,t?) -+ O. A(p,r) are Theorem 3 (See [6], Lemma 3.3 and 3.4.) Let A(l,t) a forward product of non-negative and allowable matrices, then the products A(l,t) are weakly ergodic if and only if the following conditions both hold: 1. 3to S.t. A(to,t) > 0 Vt > to A(to,t) 2. A (;~,t) - -+ Wij (t) > 0 as t -+ 00, i. e. rows of A (to,t) tend to proportionality. ),k For stochastic matrices, row-proportionality is equivalent to row-equality since rows sum to 1. limt-l-oo ACto,t) does not need to exist in order to have weak ergodicity. 3.4 Canonical Decomposition and Periodic Graphs Any non-negative matrix A can be rewritten by relabeling its indices in the following canonical decomposition [6], with diagonal blocks B i , C i and Q: A= Bl 0 0 B2 ( ..... . ..... . 0 . .. . . . ... 0 0 Ll L2 C'+ 1 0 0 0 ... 0 . 0 . . .. . . . " . .... . .. Cr Lr 0 0 Q ) (1 ) where Bi and Ci are irreducible, Bi are primitive and Ci are periodic. Define the corresponding sets of states as SBi' Se" Sq. Q might be reducible, but the groups of states in Sq leak into the B or C blocks, i.e., Sq represents the transient part of the state space. This decomposition is illustrated in Figure 1a. For homogeneous and non-homogeneous Markov models (with constant incidence matrix At = Ao), because P(qt E Sqlqt-l E Sq) < 1, liIl1t-l-oo P(qt E Sqlqo E Sq) = O. Furthermore, because the Bi are primitive, we can apply Theorem 1, and starting from a state in SB" all information about an initial state at to is gradually lost. 557 Diffusion of Credit in Markovian Models (b) Figure 1: (a): Transition graph corresponding to the canonical decomposition. (b): Periodic graph 91 becomes primitive (period 1) 92 when adding loop with states 4,5. A more difficult case is the one of (A(to ,t))jk with initial state j ESc, . Let d i be the period of the ith periodic block Cj. It can be shown r6] that taking d products of periodic matrices with the same incidence matrix and period d yields a blockdiagonal matrix whose d blocks are primitive. Thus C(to ,t) retains information about the initial block in which qt was. However, for every such block of size > 1, information will be gradually lost about the exact identity of the state within that block. This is best demonstrated through a simple example. Consider the incidence matrix represented by the graph 91 of Figure lb. It has period 3 and the only non-deterministic transition is from state 1, which can yield into either one of two loops. When many stochastic matrices with this graph are multiplied together, information about the loop in which the initial state was is gradually lost (i.e. if the initial state was 2 or 3, this information is gradually lost). What is retained is the phase information, i.e. in which block ({O}, {I}, or {2,3}) of a cyclic chain was the initial state. This suggests that it will be easy to learn about the type of outputs associated to each block of a cyclic chain, but it will be hard to learn anything else. Suppose now that the sequences to be modeled are slightly more complicated, requiring an extra loop of period 4 instead of 3, as in Figure lb. In that case A is primitive: all information about the initial state will be gradually lost. 3.5 Learning Long-Term Dependencies: a Discrete Problem? We might wonder if, starting from a positive stochastic matrix, the learning algorithm could learn the topology, i.e. replace some transition probabilities by zeroes. Let us consider the update rule for transition probabilities in the EM algorithm: A oL ij 8A;j (2) A ij ~ " oL . wj Aij oA.j Starting from Aij > 0 we could obtain a new Aij = 0 only if O~~j = 0, i.e. on a local maximum of the likelihood L. Thus the EM training algorithm will not exactly obtain zero probabilities. Transition probabilities might however approach O. It is also interesting to ask in which conditions we are guaranteed that there will not be any diffusion (of influence in the forward phase, and credit in the backward phase of training). It requires that some of the eigenvalues other than Al = 1 have a norm that is also 1. This can be achieved with periodic matrices C (of period 558 Yoshua Bengio, PaoLo Frasconi 5 -: Periodic_ . .~:;:~. -"~::::-~:-~,~~:.~~:-~:-~~:~--.-.-.~~~~~"'-"" -10 :l:~; %?.';:"'?.:: 1: .??.?:.??. II ...: ill) Left-to-right- '-.'-, .. l?:t """ '".,''. 'I d ?.?..?.?.:.' ? .??..:?.. ?: .?: .. ? .? '/,.. .. -15 ~ --20 / Full connected -25 , , Left-to-right (triangular) -30 5 10 15 T 20 25 30 (a) II t=4 t=3 (b) Figure 2: (a) Convergence of Dobrushin's coefficient (see Definition 3. (b) Evolution of products A(l,t) for fully connected graph. Matrix elements are visualized with gray levels. d), which have d eigenvalues that are the d roots of 1 on the complex unit circle. To avoid any loss of information also requires that Cd = I be the identity, since any diagonal block of Cd with size more than 1 will yield to a loss of information (because of diffusion in primitive matrices) . This can be generalized to reducible = I. matrices whose canonical form is composed of periodic blocks Ci with ct The condition we are describing actually corresponds to a matrix with only 1 's and O's_ If At is fixed, it would mean that the Markov chain is also homogeneous. It appears that many interesting computations can not be achieved with such constraints (i.e. only allowing one or more cycles of the same period and a purely deterministic and homogeneous Markov chain). Furthermore, if the parameters of the system are the transition probabilities themselves (as in ordinary HMMs), such solutions correspond to a subset of the corners of the 0-1 hypercube in parameter space. Away from those solutions, learning is mostly influenced by short term dependencies, because of diffusion of credit. Furthermore, as seen in equation 2, algorithms like EM will tend to stay near a corner once it is approached. This suggests that discrete optimization algorithms, rather continuous local algorithms, may be more appropriate to explore the (legal) corners of this hypercube. 4 Experiments 4.1 Diffusion: Numerical Simulations Firstly, we wanted to measure how (and if) different kinds of products of stochastic matrices converged, for example to a matrix of equal rows. We ran 4 simulations, each with an 8 states non-homogeneous Markov chain but with different constraints on the transition graph: 1) 9 fully connected; 2) 9 is a left-to-right model (i.e. A is upper triangular); 3) 9 is left-to-right but only one-state skips are allowed (i.e. A is upper bidiagonal); 4) At are periodic with period 4. Results shown in Figure 2 confirm the convergence towards zero of the ergodicity coefficient 1 , at a rate that depends on the graph topology. In Figure 2, we represent visually the convergence of fully connected matrices, in only 4 time steps, towards equal columns. lexcept for the experiments with periodic matrices, as expected Diffusion of Credit in Markovian Models 559 100,----~-..-???-~/~???~~~\-.~-:-/~:--~-~-~--~~-~--yg-iV-en~ 80 / \? __ -----'\ \\ ,' ..,\ .........- ~ 40stales \ ~ Fully <XlOIIeCIed, .. 16 llitale.1II \ " \ Randomly co",,..;ted. \ \. 24" ... , \ ?./ .... \ ./ 'I Fully conrect.:ted. \ \. \ ~_. \ Fully connected, ',\ 20 .1 ? \\ \"--24 sl.ate." \ ,, \ - _ ?? -yoo,........ Cb.1 (a) 10 Span 1000 (b) Figure 3: (a): Generating HMM. Numbers out of state circles denote output symbols. (b): Percentage of convergence to a good solution (over 20 trials) for various series o( experiments as the span of dependencies is increased. 4.2 Training Experiments To evaluate how diffusion impairs training, a set of controlled experiments were performed, in which the training sequences were generated by a simple homogeneous HMM with long-term dependencies, depicted in Figure 3a. Two branches generate similar sequences except for the first and last symbol. The extent of the longterm context is controlled by the self transition probabilities of states 2 and 5, A = P(qt = 2lqt-l - 2) = P(qt = 5lqt-l = 5). Span or "half-life" is log(.5)/ log(A), i.e. Aspan = .5). Following [4], data was generated for various span of long-term dependencies (0.1 to 1000). For each series of experiments, varying the span, 20 different training trials were run per span value, with 100 training sequences 2 . Training was stopped either after a maximum number of epochs (200), of after the likelihood did not improve significantly, i.e., (L(t) - L(t - l))/IL(t)1 < 10- 5 , where L(t) is the logarithm of the likelihood of the training set at epoch t. If the HMM is fully connected (except for the final absorbing state) and has just the right number of states, trials almost never converge to a good solution (1 in 160 did). Increasing the number of states and randomly putting zeroes helps. The randomly connected HMMs had 3 times more states than the generating HMM and random connections were created with 20% probability. Figure 3b shows the average number of converged trials for these different types of HMM topology. A trial is considered successful when it yields a likelihood almost as good or better than the likelihood of the generating HMM on the same data. In all cases the number of successful trials rapidly drops to zero beyond some value of span. 5 Conclusion and Future Work In previous work on recurrent networks we had found that propagating credit over the long term was incompatible with storing information for the long term. For Markovian models, we found that when the transition probabilities are close to 1 and 0, information can be stored for the long term AND credit can be prop2it appeared sufficient since the likelihood of the generating HMM did not improve much when trained on this data 560 Yoshua Bengio, PaoLo Frasconi agated over the long term. However, like for recurrent networks, this makes the problem of learning long-term dependencies look more like a discrete optimization problem. Thus it appears difficult for local learning algorithm such as EM to learn optimal transition probabilities near 1 or 0, i.e. to learn the topology, while taking into account long-term dependencies. The arguments presented are essentially an application of established mathematical results on Markov chains to the problem of learning long term dependencies in homogeneous and non-homogeneous HMMs. These arguments were also supported by experiments on artificial data, studying the phenomenon of diffusion of credit and the corresponding difficulty in training HMMs to learn long-term dependencies . IOHMMs [1] introduce a reparameterization of the problem: instead of directly learning the transition probabilities, we learn parameters of a function of an input sequence. Even with a fully connected topology, transition probabilities computed at each time step might be very close to and 1. Because of the non-stationarity, more interestin~ computations can emerge than the simple cycles studied above. For example in l3] we found IOHMMs effective in grammar inference tasks. In [1] comparative experiments were performed with a preliminary version of IOHMMs and other algorithms such as recurrent networks, on artificial data on which the span of long-term dependencies was controlled. IOHMMs were found much better than the other algorithms at learning these tasks. ? Based on the analysis presented here, we are also exploring another approach to learning long-term dependencies that consists in building a hierarchical representation of the state. This can be achieved by introducing several sub-state variables whose Cartesian product corresponds to the system state. Each of these sub-state variables can operate at a different time scale, thus allowing credit to propagate over long temporal spans for some of these variables . Another interesting issue to be investigated is whether techniques of symbolic prior knowledge injection (such as in (5]) can be exploited to choose good topologies. One advantage, compared to traditIOnal neural network approaches, is that the model has an underlying finite state structure and is thus well suited to inject discrete transition rules . Acknowledgments We would like to thank Leon Bottou for his many useful comments and suggestions, and the NSERC and FCAR Canadian funding agencies for support. References [1] Y. Bengio and P. Frasconi. Credit assignment through time: Alternatives to backpropagation. In J. D. Cowan, et al., eds., Advances in Neural Information Processing Systems 6. Morgan Kaufmann, 1994. [2] Y. Bengio and P. Frasconi. An Input Output HMM Architecture. In this volume: J. D. Cowan, et al., eds., Advances in Neural Information Processing Systems 7. Morgan Kaufmann, 1994. [3] Y. Bengio and P. Frasconi. An EM approach to learning sequential behavior. Technical Report RT-DSI-ll/94 , University of Florence, 1994. [4] Y. Bengio, P. Simard, and P. Frasconi. Learning long-term dependencies with gradient descent is difficult. IEEE Trans. Neural Networks, 5(2):157- 166, 1994. [5] P. Frasconi, M. Gori, M. Maggini, and G . Soda. Unified integration of explicit rules and learning by example in recurrent networks. IEEE Trans. on Knowledge and Data Engineering, 7(1), 1995. [6] E. Seneta. Nonnegative Matrices and Markov Chains. Springer, New York, 1981. 

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544 MURPHY: A Robot that Learns by Doing Bartlett W. Mel Center for Complex Systems Research University of Illinois 508 South Sixth Street Champaign, IL 61820 January 2, 1988 Abstract MURPHY consists of a camera looking at a robot arm, with a connectionist network architecture situated in between. By moving its arm through a small, representative sample of the 1 billion possible joint configurations, MURPHY learns the relationships, backwards and forwards, between the positions of its joints and the state of its visual field. MURPHY can use its internal model in the forward direction to "envision" sequences of actions for planning purposes, such as in grabbing a visually presented object, or in the reverse direction to "imitate", with its arm, autonomous activity in its visual field. Furthermore, by taking explicit advantage of continuity in the mappings between visual space and joint space, MURPHY is able to learn non-linear mappings with only a single layer of modifiable weights. Background Current Focus Of Learning Research Most connectionist learning algorithms may be grouped into three general catagories, commonly referred to as supenJised, unsupenJised, and reinforcement learning. Supervised learning requires the explicit participation of an intelligent teacher, usually to provide the learning system with task-relevant input-output pairs (for two recent examples, see [1,2]). Unsupervised learning, exemplified by "clustering" algorithms, are generally concerned with detecting structure in a stream of input patterns [3,4,5,6,7]. In its final state, an unsupervised learning system will typically represent the discovered structure as a set of categories representing regions of the input space, or, more generally, as a mapping from the input space into a space of lower dimension that is somehow better suited to the task at hand. In reinforcement learning, a "critic" rewards or penalizes the learning system, until the system ultimately produces the correct output in response to a given input pattern [8]. It has seemed an inevitable tradeoff that systems needing to rapidly learn specific, behaviorally useful input-output mappings must necessarily do so under the auspices of an intelligent teacher with a ready supply of task-relevant training examples. This state of affairs has seemed somewhat paradoxical, since the processes of Rerceptual and cognitive development in human infants, for example, do not depend on the moment by moment intervention of a teacher of any sort. Learning by Doing The current work has been focused on a fourth type of learning algorithm, i.e. learning-bydoing, an approach that has been very little studied from either a connectionist perspective ? American Institute of Physics 1988 545 or in the context of more traditional work in machine learning. In its basic form, the learning agent ? begins with a repertoire of actions and some form of perceptual input, ? exercises its repertoire of actions, learning to predict i) the detailed sensory consequences of its actions, and, in the other direction, ii) its actions that are associated with incoming sensory patterns, and ? runs its internal model (in one or both directions) in a variety of behaviorally-relevant tasks, e.g. to "envision" sequences of actions for planning purposes, or to internally "imitate" , via its internal action representation, an autonomously generated pattern of perceptual activity. In comparison to standard supenJised learning algorithms, the crucial property of learning-by-doing is that no intelligent teacher is needed to provide input-output pairs for learning. Laws of physics simply translate actions into their resulting percepts, both of which are represented internally. The learning agent need only notice and record this relationship for later use. In contrast to traditional unsupervised learning approaches, learning-by-doing allows the acquisition of specific, task-relevant mappings, such as the relationship between a simultaneously represented visual and joint state. Learning-bydoing differs as well from reinforcement paradigms in that it can operate in the absence of a critic, i.e. in situations where reward or penalty for a particular training instance may be inappropriate. Learning by doing may therefore by described as an unsupeMJised associative algorithm, capable of acquiring rich, task-relevant associations, but without an intelligent teacher or critic. Abridged History of the Approach The general concept of leaning by doing may be attributed at least to Piaget from the 1940's (see [9] for review). Piaget, the founder of the "constructivist" school of cognitive development, argued that knowledge is not given to a child as a passive observer, but is rather discovered and constructed by the child, through active manipulation of the environment. A handful of workers in artificial intelligence have addressed the issue of learning-by-doing, though only in highly schematized, simulated domains, where actions and sensory states are represented as logical predicates [10,11,12,13]. Barto & Sutton [14] discuss learning-by-doing in the context of system identification and motor control. They demonstrated how a simple simulated automaton with two actions and three sensory states can build a model of its environment through exploration, and subsequently use it to choose among behavioral alternatives. In a similar vein, Rumelhart [15] has suggested this same approach could be used to learn the behavior of a robot arm or a set of speech articulators. Furthermore, the forward-going "mental model" , once learned, could be used internally to train an inverse model using back-propagation. In previous work, this author [16] described a connectionist model (VIPS) that learned to perform 3-D visual transformations on simulated wire-frame objects. Since in complex sensory-motor environments it is not possible, in general, to learn a direct relationship between an outgoing command state and an incoming sensory state, VIPS was designed to predict changes in the state of its visual field as a function of its outgoing motor command. VIPS could then use its generic knowledge of motor-driven visual transformations to "mentally rotate" objects through a series of steps. 546 Recinolopic Visual RepresemaDOII -- - Value-Coded loint RepresenlatiOll - -- I Figure 1: MURPHY's Connectionist Architecture. 4096 coarsely-tuned visual units are organized in a square, retinotopic grid. These units are bi-directionally interconnected with a population of 273 joint units. The joint population is subdivided into 3 sUbpopulations, each one a value-coded representation of joint angle for one of the three joints. During training, activity in the joint unit population determines the physical arm configuration. Inside MURPY The current work has sought to further explore the process of learning-by-doing in a complex sensory-motor domain, extending previous work in three ways. First, the learning of mappings between sensory and command (e.g. motor) representations should be allowed to proceed in both directions simultaneously during exploratory behavior, where each mapping may ultimately subserve a very different behavioral goal. Secondly, MURPHY has been implemented with a real camera and robot arm in order to insure representational realism to the greatest degree possible. Third, while the specifics of MURPHY's internal structures are not intended as a model of a specific neural system, a serious attempt has been made to adhere to architectural components and operations that have either been directly suggested by nervous system structures, or are at least compatible with what is currently known. Detailed biological justification on this point awaits further work. MURPHY's Body MURPHY consists of a 512 x 512 JVC color video camera pointed at a Rhino XR-3 robotic arm. Only the shoulder, elbow, and wrist joints are used, such that the arm can move only in the image plane of the camera. (A fourth, waist joint is not used). White spots are stuck to the arm in convenient places; when the image is thresholded, only the white spots appear in the image. This arrangement allows continuous control over the complexity of the visual image of the arm, which in turn affects time spent both in computing visual features and processing weights during learning. A Datacube image processing system is used for the thresholding operation and to "blur" the image in real time with a gaussian mask. The degree of blur is variable and can be used to control the degree of coarse-coding (i.e. receptive field overlap) in the camera-topic array of visual units. The arm is software controllable, with a stepper motor for each joint. Arm dynamics are not considered in this work. 547 MURPHY's Mind MURPHY is currently based on two interconnected populations of neuron-like units. The first is organized as a rectangular, visuotopically-mapped 64 x 64 grid of coarsely-tuned visual units that each responds when a visual feature (such as a white spot on the arm) falls into its receptive field (fig. 1). Coarse coding insures that a single visual feature will activate a small population of units whose receptive fields overlap the center of stimulation. The upper trace in figure 2 shows the unprocessed camera view, and the center trace depicts the resulting pattern of activation over the grid of visual units. The second population of 273 units consists of three subpopulations, representing the angles of each of the three joints. The angle of each joint is value-coded in a line of units dedicated to that joint (fig. 1). Each unit in the population is "centered" at a some joint angle, and is maximally activated when the joint is to be sent to that angle. Neighboring joint units within a joint sub population have overlapping "projective fields" and progressively increasing joint-angle centers. It may be noticed that both populations of units are coarsely tuned, that is, the units have overlapping receptive fields whose centers vary in an orderly fashion from unit to neighboring unit. This style of representation is extremely common in biological sensory systems [17,18,19]' and has been attributed a number ofrepresentational advantages (e.g. fewer units needed to encode range of stimuli, increased immunity to noise and unit malfunction, and finer stimulus discriminations). A number of additional advantages of this type of encoding scheme are discussed in section, in relation to ease of learning, speed of learning, and efficacy of generalization. MURPHY's Education By moving its arm through a small, representative sample (approximately 4000) of the 1 billion possible joint configurations, MURPHY learns the relationships, backwards and forwards, between the positions of its joints and the state of its visual field. During training, the physical environment to which MURPHY's visual and joint representations are wired enforces a particular mapping between the states of these two representations. The mapping comprises both the kinematics of the arm as well as the optical parameters and global geometry of the camera/imaging system. It is incrementally learned as each unit in population B comes to "recognize" , through a process of weight modification, the states of population A in which it has been strongly activated. After sufficient training experience therefore, the state of popUlation A is sufficient to generate a "mental image" on population B, that is, to predictively activate the units in B via the weighted interconnections developed during training. In its current configuration, MURPHY steps through its entire joint space in around 1 hour, developing a total of approximately 500,000 weights between the two popUlations . The Learning Rule Tradeoffs in Learning and Representation It is well known in the folklore of connectionist network design that a tradeoff exists between the choice of representation (i.e. the "semantics") at the single unit level and the consequent ease or difficulty of learning within that representational scheme. At one extreme, the single-unit representation might be completely decoded, calling for a separate unit for each possible input pattern. While this scheme requires a combinatorially explosive number of units, and the system must "see" every possible input pattern during training, the actual weight modification rule is rendered very simple. At another extreme, the single unit representation might be chosen in a highly encoded fashion with complex interactions among input units. In this case, the activation of an output unit 548 may be a highly non-linear or discontinuous function of the input pattern, and must be learned and represented in mUltiple layers of weights. Research in connectionism has often focused on Boolean functions [20,21,1,22,23], typified by the encoder problem [20], the shifter problem [21] and n-bit parity [22]. Since Boolean functions are in general discontinuous, such that two input patterns that are close in the sense of Hamming distance do not in general result in similar outputs, much effort has been directed toward the development of sophisticated, multilayer weight-modification rules (e.g. back-propagation) capable of learning arbitrary discontinuous functions. The complexity of such learning procedures has raised troubling questions of scaling behavior and biological plausibility. The assumption of continuity in the mappings to be learned, however, can act to significationly simplify the learning problem while still allowing for full generalization to novel input patterns. Thus, by relying on the continuity assumption, MURPHY's is able to learn continuous non-linear functions using a weight modification procedure that is simple, locally computable, and confined to a single layer of modifiable weights. How MURPHY learns For sake of concrete illustration, MURPHY's representation and learning scheme will be described in terms of the mapping learned from joint units to visual units during training. The output activity of a given visual unit may be described as a function over the 3dimensional joint space, whose shape is determined by the kinematics of the arm, the location of visual features (i.e. white spots) on the arm, the global properties of the camera/imaging system, and the location of the visual unit's receptive field. In order for the function to be learned, a visual unit must learn to "recognize" the regions of joint space in which it has been visually activated during training. In effect, each visual unit learns to recognize the global arm configurations that happen to put a white spot in its receptive field. It may be recalled that MURPHY's joint unit population is value-coded by dimension, such that each unit is centered on a range of joint angles (overlapping with neighboring units) for one of the 3 joints. In this representation, a global arm configuration can be represented as the conjunctive activity of the k (where k 3) most active joint units. MURPHY's visual units can therefore learn to recognize the regions of joint space in which they are strongly activated by simply ''memorizing'' the relevant global joint configurations as conjunctive clusters of input connections from the value-coded joint unit population. To realize this conjunctive learning scheme, MURPHY's uses sigma-pi units (see [24]), as described below. At training step S, the set of k most active joint units are first identified. Some subset of visual units is also strongly activated in state S, each one signalling the presence of a visual feature (such as a white spot) in its receptive fields. At the input to each active visual unit, connections from the k most highly active joint units are formed as a multiplicative k-tuple of synaptic weights. The weights Wi on these connections are initially chosen to be of unit strength. The output Cj of a given synaptic conjunction is computed by multiplying the k joint unit activation values Xi together with their weights: = The output y of the entire unit is computed as a weighted sum of the outputs of each conjunction and then applying a sigmoidal nonlinearity: y = u(L WjCj). i Sigma-pi units of this kind may be thought of as a generalization of a logical disjunction of conjunctions (OR of AND's). The multiplicative nature of the conjunctive clusters insures 549 that every input to the conjunct is active in order for the conjunct to have an effect on the unit as a whole. If only a single input to a conjunct is inactive, the effect of the conjunction is nullified. Specific-Instance Learning in Continuous Domains MURPHY's learning scheme is directly reminiscent of specific-instance learning as discussed by Hampson &: Vol per [23] in their excellent review of Boolean learning and representational schemes. Specific-instance learning requires that each unit simply ''memorize" all relevant input states, i.e. those states in which the unit is intended to fire. Unfortunately, simple specific-instance learning allows for no generalization to novel inputs, implying that each desired system responses will have to have been explicitly seen during training. Such a state of affairs is clearly impractical in natural learning contexts. Hampson &: Volper [23] have further shown that random Boolean functions will require an exponential number of weights in this scheme. For continous functions on the other hand, two kinds of generalization are possible within this type of specific-instance learning scheme. We consider each in turn, once again from the perspective of MURPHY's visual units learning to recognize the regions in joint space in which they are activated. Generalization by Coarse-Coding When a visual unit is activated in a given joint configuration, and acquires an appropriate conjunct of weights from the set of highly active units in the joint popUlation, by continuity the unit may assume that it should be at least partially activated in nearby joint configurations as well. Since MURPHY's joint units are coarse-coded in joint angle, this will happen automatically: as the joints are moved a small distance away from the specific training configuration, the output of the conjunct encoding that training configuration will decay smoothly from its maximum. Thus, a visual unit can "fire" predictively in joint configurations that it has never specifically seen during training, by interpolating among conjuncts that encode nearby joint configurations. This scheme suggests that training must be sufficiently dense in joint space to have seen configurations ''nearby'' to all points in the space by some criterion. In practice, the training step size is related to the degree of coarse-coding in the joint population, which is chosen in tUrn such that a joint pertubation equal to the radius of a joint unit's projective field (i.e. the range of joint angles over which the unit is active) should on average push a feature in the visual field a distance of about one visual receptive field radius. As a rule of thumb, the visual receptive field radius is chosen small enough so as to contain only a single feature on average. Generalization by Extrapolation The second type of generalization is based on a heuristic principle, again illustrated in terms of learning in the visual population. If a visual unit has during training been very often activated over a large, easy-to-specify region of joint space, such as a hyperrectangular region, then it may be assumed that the unit is activated over the entire region of joint space, i.e. even at points not yet seen. At the synaptic level, "large regions" can be represented as conjuncts with fewer terms. In its simplest form, this kind of generalization amounts to simply throwing out one or more joints as irrelevant to the activation of a given visual unit. What synaptic mechanism can achieve this effect? Competition among joint unit afferents can be used to drive irrelevant variables from the sigma-pi conjuncts. Thus, if a visual unit is activated repeatedly during training, and the elbow and shoulder angle units are constantly active while the most active wrist unit varies from step to step, then the weighted connections from the repeatedly activated elbow and shoulder units 550 -- . ? ? ? -- ? , ?? ? ?? ? ? ? ,.. til I .; , ?.. ? ? ,., I Figure 2: Three Visual Traces. The top trace shows the unprocessed camera view of MURPHY's arm. White spots have been stuck to the arm at various places, such that a thresholded image contains only the white spots. This allows continuous control over the visual complexity of the image. The center trace represents the resulting pattern of activation over the 64 x 64 grid of coarsely-tuned visual units. The bottom trace depicts an internally-produced "mental" image of the arm in the same configuration as driven by weighted connections from the joint population. Note that the "mental" trace is a sloppy, but highly recognizable approximation to the camera-driven trace. will become progressively and mutually reinforced at the expense of the set of wrist unit connections, each of which was only activated a single time. This form of generalization is similar in function to a number of "covering" algorithms designed to discover optimal hyper-rectangular decompositions (with possible overlap) of a set of points in a multi-dimensional space (e.g. [25,26]). The competitive feature has not yet been implemented explicitly at the synaptic level, rather, the full set of conjuncts acquired during training are currently collapsed en masse into a more compact set of conjuncts, according to the above heuristics. In a typical run, MURPHY is able to eliminate between 30% and 70% of its conjuncts in this way. What MURPHY Does Grabbing A Visually Presented Target Once MURPHY has learned to image its arm in an arbitrary joint configuration, it can use heuristic search to guide its arm "mentally" to a visually specified goal. Figure 3(a) depicts a hand trajectory from an initial position to the location of a visually presented target. Each step in the trajectory represents the position of the hand (large blob) at an intermediate joint configuration. MURPHY can visually evaluate the remaining distance to the goal at each position and use best-first search to reduce the distance. Once a complete trajectory has been found, MURPHY can move its arm to the goal in a single physical step, dispensing with all backtracking dead-ends, and other wasted operations (fig. 3(b? . It would also be possible to use the inverse model, i.e. the map from a desired visual into an internal joint image, to send the arm directly to its final position. Unfortunately, MURPHY has no means in its current early state of development to generate a full-blown 551 - _... - M1RPI!Y'. HInt&! Tra jeetory Figure 3: Grabbing a.n Object. (a) Irregular trajectory represents sequence of "mental" steps taken by MURPHY in attempting to "grab" a visually-presented target (shown in (b) as white cross). Mental image depicts MURPHY's arm in its final goal configuration, i.e. with hand on top of object. Coarse-coded joint activity is shown at right. (b) Having mentally searched a.nd found the target through a series of steps, MURPHY moves its arm phY6ically in a single step to the target, discarding the intermediate states of the trajectory that are not relevant in this simple problem. visual image of its arm in one of the final goal positions, of which there are many possible. Sending the tip of a robot arm to a given point in space is a classic task in robotics. The traditional approach involves first writing explicit kinematic equations for the arm based on the specific geometric details of the given arm. These equations take joint angles as inputs and produce manipUlator coordinates as outputs. In general, however, it is most often useful to specify the coordinates of the manipulator tip (i.e. its desired final position), and compute the joint angles necessary to achieve this goal. This involves the solution of the kinematic equations to generate an inverse kinematic model. Deriving such expressions has been called "the most difficult problem we will encounter" in vision-based robotics [27]. For this reason, it is highly desirable for a mobile agent to learn a model of its sensory-motor environment from scratch, in a way that depends little or not at all on the specific parameters of the motor apparatus, the sensory apparatus, or their mutual interactions. It is interesting to note that in this reaching task, MURPHY appears from the outside to be driven by an inverse kinematic model of its arm, since its first official act after training is to reach directly for a visually-presented object. While it is clear that best-first search is a weak method whose utility is limited in complex problem solving domains, it may be speculated that given the ability to rapidly image arm configurations, combined with a set of simple visual heuristics and various mechanism for escaping local minima (e.g. send the arm home), a number of more interesting visual planning problems may be within MURPHY's grasp, such as grabbing an object in the presence of obstacles. Indeed, for problems that are either difficult to invert, or for which the goal state is not fully known a priori, the technique of iterating a forward-going model has a long history (e.g. Newton's Method). 552 Imitating Autonomous Arm Activity A particularly interesting feature of "learning-by-doing" is that for every pair of unit populations present in the learning system, a mapping can be learned between them both backwards and forwards. Each such mapping may enable a unique and interesting kind of behavior. In MURPHY's case, we have seen that the mapping from a joint state to a visual image is useful for planning arm trajectories. The reverse mapping from a visual state to ajoint image has an altogether different use, i.e. that of "imitation". Thus, if MURPHY's arm is moved passively, the model can be used to ''follow'' the motion with an internal command (i.e. joint) trace. Or, if a substitute arm is positioned in MURPHY's visual field, MURPHY can "assume the position", i.e . imitate the model arm configuration by mapping the afferent visual state into a joint image, and using the joint image to move the arm. As of this writing, the implementation of this behavior is still somewhat unreliable. Discussion and Future Work In short, this work has been concerned with learning-by-doing in the domain of visionbased robotics. A number of features differentiate MURPHY from most other learning systems, and from other approaches to vision-based robotics: ? No intelligent teacher is needed to provide input-ouput pairs. MURPHY learns by exercising its repertoire of actions and learning the relationship between these actions and the sensory images that result. ? Mappings between populations of units, regardless of modality, can be learned in both directions simultaneously during exploratory behavior. Each mapping learned can support a distinct class of behaviorally useful functions. ? MURPHY uses its internal models to first solve problems "mentally". Plans can therefore be developed and refined before they are actually executed. ? By taking explicit advantage of continuity in the mappings between visual and joint spaces, and by using a variant of specific-instance learning in such a way as to allow generalization to novel inputs, MURHPY can learn "difficult" non-linear mappings with only a single layer of modifiable weights. Two future steps in this endeavor are as follows: ? Provide MURPHY with direction-selective visual and joint units both, so that it may learn to predict relationships between rates of change in the visual and joint domains. In this way, MURPHY can learn how to perturb its joints in order to send its hand in a particular direction, greatly reducing the current need to search for hand trajectories. ? Challenge MURPHY to grab actual objects, possibly in the presence of obstacles, where path of approach is crucial. The ability to readily envision intermediate arm configurations will become critical for such tasks. . Acknowledgements Particular thanks are due to Stephen Omohundro for his unrelenting scientific and moral support, and for suggesting vision and robotic kinematics as ideal domains for experimentation. 553 References [1] T.J. Sejnowski & C.R. Roaenberg, Complex Systems, 1, 145, (1969). [2] G.J. Tesauro & T.J. Sejnowski. A parallel network that leMIUI to play backgammon. Submitted for publication. [3] S. Grossberg, BioI. Cybern., f3, 187, (1976). [4] T. Kohonen, Self organization and auociati"e memory., (Springer-Verlag, Berlin 1984). [5] D.E. Rumelhart & D. Zipser. In Parallel diatri6uted proceuing: e:rplorationa in tA.e microatructure oj cognition, "01. 1, D.E. Rumelhart, J.L. McClelland, Eds., (Bradford: Cambridge, MA, 1986), p. 151. [6] R. Linsker, Proc. Natl. Acad. Sci., 83, 8779, (1986). [7] G.E. Hinton & J.L. McClelland. Learning representations by recirculation. Oral presentation, IEEE conference on Neural Information Processing Systems, Denver, 1987. [8] A.G. Barto, R.S. Sutton, & C.W. Anderson, IEEE Trans. on Sy ?. , Man, Cybern., amc-13, 834, (1983). [9] H. Ginsburg & S. Opper, Piaget'a tA.eor, oj intellectual de"elopment., (Prentice Hall, New Jersey, 1969). [10] J.D. Becker. In Computer modela Jor tA.ougA.t and language., R. Schank & K.M. Colby, Eds., (FreelllAIl, San Francisco, 1973). [11] R.L. Rivest & R.E. Schapire. In Proc. of the 4th into workshop on ma.ch.ine learning, 364-375, (1987). [12] J .G. Carbonell & Y. Gil. In Proc. of the 4th into workshop on machlne learning, 256-266, (1987). [13] K. Chen, Tech Report, Dept. of Computer Science, University of illinois, 1987. [14] A.G. Barto & R.S. Sutton, AFWAL-TR-81-1070, Avionics Laboratory, Air Force Wright Aeronautical Laboratories, Wright-Patterson AFB, Ohio 45433, 1981. [15] D.E. Rumelhart, "On learning to do what you want". Talk given at CMU Connectionist Summer School,1986a. [16] B.W. Melin Proc. of 8th Ann. Con!. of the Cognitive Science Soc., 562-571, (1986). [17] D.H. Ballard, G.E. Hinton, & T.J Sejnowski, Nature, 306, 21, (1983). [18] R.P. Erikson, American Scientist, May-June 1984, p. 233. [19] G.E. Hinton, J.L. McClelland, & D.E. Rumelhart. In Parallel diatri6uted proceuing: e:rplorationa in tA.e microatructure oj cognition, "01. 1, D.E. Rumelhart, J .L. McClelland, Eds., (Bradford, Cambridge, 1986), p. 77. [20] D.H. Ackley, G.E. Hinton, & T.J. Sejnowski, Cognitive Science, 9, 147, (1985). [21] G.E. Hinton & T .J. Sejnowski. In Parallel diatri6uted proceuing: e:rplorationa in tA.e microatructure oj cognition, "01. 1, D.E. Rumelhart, J.L. McClelland, Eds., (Bradford, Cambridge, 1986), p. 282. [22] G.J. Tes&llro, Complex Systems, 1,367, (1987). [23] S.E. Hampson & D.J. Volper, Biological Cybernetics, 56, 121, (1987). [24] D.E. Rumelhart, G.E. Hinton, & J.L. McClelland. In Parallel diatri6uted proceuing: e:rplorationa in tA.e microatructure oj cognition, "01. 1, D.E. Rumelhart, J.L. McClelland, Eds., (Bradford, Cambridge, 1986), p. 3. [25] R.S. Michalski, J.G. Carbonell, & T.M. Mitchell, Eds., MacA.ine learning: an artificial intelligence approacA., Vois. I and II, (Morgan KauflllAll, Los Altos, 1986). [26] S. Omohundro, Complex Systems, 1, 273, (1987). [27] Paul, R. R060t manipulatora: matA.ematica, programming, and control., (MIT Press, Cambridge, 1981). 

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The NilOOO: High Speed Parallel VLSI for Implementing Multilayer Perceptrons Michael P. Perrone Thomas J. Watson Research Center P.O. Box 704 Yorktown Heights, NY 10598 mppGwatson.ibm.com Leon N Cooper Institute for Brain and Neural Systems Brown University Providence, Ri 02912 IncGcns.brown.edu Abstract In this paper we present a new version of the standard multilayer perceptron (MLP) algorithm for the state-of-the-art in neural network VLSI implementations: the Intel Ni1000. This new version of the MLP uses a fundamental property of high dimensional spaces which allows the 12 -norm to be accurately approximated by the It -norm. This approach enables the standard MLP to utilize the parallel architecture of the Ni1000 to achieve on the order of 40000, 256-dimensional classifications per second. 1 The Intel NilOOO VLSI Chip The Nestor/Intel radial basis function neural chip (Ni1000) contains the equivalent of 1024 256-dimensional artificial digital neurons and can perform at least 40000 classifications per second [Sullivan, 1993]. To attain this great speed, the Ni1000 was designed to calculate "city block" distances (Le. the II-norm) and thus to avoid the large number of multiplication units that would be required to calculate Euclidean dot products in parallel. Each neuron calculates the city block distance between its stored weights and the current input: neuron activity = w, L IWi - :eil (1) where is the neuron's stored weight for the ith input and :ei is the ith input. Thus the Nil000 is ideally suited to perform both the RCE [Reillyet al., 1982] and 748 Michael P. Perrone. Leon N. Cooper PRCE [Scofield et al., 1987] algorithms or any of the other commonly used radial basis function (RBF) algorithms. However, dot products are central in the calculations performed by most neural network algorithms (e.g. MLP, Cascade Correlation, etc.). Furthermore, for high dimensional data, the dot product becomes the computation bottleneck (i.e. most ofthe network's time is spent calculating dot products). If the dot product can not be performed in parallel there will be little advantage using the NilOOO for such algorithms. In this paper, we address this problem by showing that we can extend the NilOOO to many of the standard neural network algorithms by representing the Euclidean dot product as a function of Euclidean norms and by then using a city block norm approximation to the Euclidean norm. Section 2, introduces the approximate dot productj Section 3 describes the City Block MLP which uses the approximate dot productj and Section 4 presents experiments which demonstrate that the City Block MLP performs well on the NIST OCR data and on human face recognition data. 2 Approximate Dot Product Consider the following approximation [Perrone, 1993]: 1 11Z11 ~ y'n1Z1 (2) z where is some n-dimensional vector, II? II is the Euclidean length (i.e. the 12 norm) and I? I is the City Block length (i.e. the 11-norm). This approximation is motivated by the fact that in high dimensional spaces it is accurate for a majority of the points in the space. In Figure 1, we suggest an intuitive interpretation of why this approximation is reasonable. It is clear from Figure 1 that the approximation is reasonable for about 20% of the points on the arc in 2 dimensions. 1 As the dimensionality of the data space increases, the tangent region in Figure 1 expands asymptotically to fill the entire space and thus the approximation becomes more valid. Below we examine how accurate this approximation is and how we can use it with the NilOOO, particularly in the MLP context. Given a set of vectors, V, all with equal city block length, we measure the accuracy of the approximation by the ratio of the variance of the Euclidean lengths in V to the squared mean Euclidean lengths in V. If the ratio is low, then the approximation is good and all we must do is scale the city block length to the mean Euclidean length to get a good fit. 2 In particular, it can be shown that assuming all the vectors of the space are equally likely, the following equation holds [Perrone, 1993]: O'~ < (a~(!n+ 1) -1)ILfower, (3) where n is the dimension of the space; ILn is the average Euclidean length of the set of vectors with fixed city block length Sj O'~ is the variance about the average Euclidean length; ILlower is the lower bound for ILn and is given by ILlower == an S / Vnj 1 In fact, approximately 20% of the points are within 1% of each other and 40% of the points are within 5% of each other. 2 Note that in Equation 2 we scale by 1/ fo. For high dimensional spaces this is a good approximation to the ratio of the mean Euclidean length to the City Block length. VLSI for Implementing Multilayer Perceptrons 749 Figure 1: Two dimensional interpretation of the city block approximation. The circle corresponds to all of the vectors with the same Euclidean length. The inner square corresponds to all of the vectors with city block length equal the Euclidean length of the vectors in the circle. The outer square (tangent to the circle) corresponds to the set of vectors over which we will be making our approximation. In order to scale the outer square to the inner square, we multiple by 11 Vn where n is the dimensionality of the space. The outer square approximates the circle in the regions near the tangent points. In high dimensional spaces, these tangent regions approximate a large portion of the total hypersphere and thus the city block distance is a good approximation along most of the hypersphere. and an is defined by an - nn = --1 +-1 J1+ 2?r( -2 ~ +-nen- 1) (n) n+2 1 . (4) From this equation we see that the ratio of O"~ to /-L~ower in the large n limit is bounded above by 0.4. This bound is not very tight due to the complexity of the calculations required; however Figure 3 suggests that a much tighter bound must exist. A better bound exists if we are willing to add a minor constraint to our high dimensional space [Perrone, 1993]. In the case in which each dimension of the vector is constrained such that the entire vector cannot lie along a single axis, 3 we can show that 0"2 :::::: 2(n -1) ( ~ _ 1)2 /-L~ower (5) n (n + 1) 2 S a~ , V where S is the city block length of the vector in question. Thus in this case, the ratio of O"~ to /-L~ower decreases at least as fast as lin since nl S will be some fixed constant independent of n. 4 This dependency on nand S is shown in Figure 2. This result suggests that the approximation will be very accurate for many real-world pattern 3For example, when the axes are constrained to be in the range [D, 1] and the city block length of the vector is greater than 1. Note that this is true for the majority of the points in a n dimensional unit hypercube. ~Thus the accuracy improves as S increases towards its maximum value. 750 Michael P. Perrone, Leon N. Cooper recognition tasks such as speech and high resolution image recognition which can typically have thousand or even tens of thousands of dimensions. 1 0.8 SIn - 0.025 SIn - 0.05 SIn - 0.1 SIn - 0 .2 SIn - 0 .3 0.6 0.4 0 .2 ,----.---~~:~?~~~::~~ --------o L\.--__ :? ?~~:2~:~ ??--u -- ~--?_-?_??_-?_ ? -~--_?-_ . ._-._-._ . ..~.-~.-~-._ ..----------------_.. ~ .. -_.._.._._--_.. ~.-._.-_ : :_~~:-~.:_~_~-_:;~~_ : :~~:~~~-:~~:~ --~7 o 100 200 300 400 500 Figure 2: Plot of unj I-'lower vs. n for constrained vectors with varying values of Sin. As S grows the ratio shrinks and consequently, accuracy improves. If we assume that all of the vectors are uniformly distributed in an n-dimensional unit hypercube, it is easy to show that the average city block length is nj2 and the variance of the city block length is n/12. Since Sjn will generally be within one standard deviation ofthe mean, we find that typically 0.2 < Sjn < 0.8. We can use the same analysis on binary valued vectors to derive similar results. We explore this phenomenon further by considering the following Monte Carlo simulation. We sampled 200000 points from a uniform distribution over an n-dimensional cube. The Euclidean distance of each of these points to a fixed corner of the cube was calculated and all the lengths were normalized by the largest possible length, ~. Histograms of the resulting lengths are shown in Figure 3 for four different values of n. Note that as the dimension increases the variance about the mean drops. From Figure 3 we see that for as few as 100 dimensions, the standard deviation is approximately 5% of the mean length. 3 The City Block MLP In this section we describe how the approximation explained in Section 2 can be used by the NilOOO to implement MLPs in parallel. Consider the following formula for the dot product (6) VLSI for Implementing Multilayer Perceptrons 751 0.45 0 . 4 0.35 g ~ ~ooo D~mQg~ons ~ 100 10 :2 D.1.mes.1.ons D;l.mas1.ons D.1.mas1.ons - ..... -- -c:.--- ...... - .- 0.3 0.25 {} e 0.2 t>. 0 . 15 0.1 0.05 0.5 0.45 0.55 0.6 Norma1izad 0.65 LenQth 0.7 0.75 O.B Figure 3: Probability distributions for randomly draw lengths. Note that as the dimension increases the variance about the mean length drops. where II? II is the Euclidean length (i.e. 12-norm).5 Using Equation 2, we can approximation Equation 6 by ...... Y ~ - 1(1"';;'12 :t! + YI 4n :t! ? I'":t! - ;;'12) YI (7) where n is the dimension of the vectors and I . I is the city block length. The advantage to the approximation in Equation 7 is that it can be implemented in parallel on the Ni1000 while still behaving like a dot product. Thus we can use this approximation to implement MLP's on an Ni1000. The standard functional form for MLP's is given by [Rumelhart et al., 1986] N h(:t!;a,f3) d = u(aok + Lajk u (!30j + Lf3ij:t!i?) j=1 (8) i=1 were u is a fixed ridge function chosen to be u(:t!) = (1 + e N is the number of hidden units; k is the class label; d is the dimensionality of the data space; and a and f3 are adjustable parameters. The alternative which we propose, the City Block MLP, is given by [Perrone, 1993] -:t) 1 N gk(:t!; a, f3) = u(aok + d 1 - \ d L ajk u (f30j + 4(L lf3ij + :t!i1)2 - 4(L lf3ij -:t!i 1)2?) n i=1 j=1 n i=1 riNote also that depending on the information available to us, we could use either i? y= }(IIi + yW -lIiW -11?7W) or i? y= }<lIiIl2 + IIYlI2 -Iii - Y112). (9) 752 Michael P. Perrone, Leon N. Cooper DATA SET Faces Numbers Lowercase Uppercase HIDDEN UNITS 12 10 20 20 STANDARD CITYBLOCK % CORRECT % CORRECT 94.6?1.4 98.4?0.17 88.9?0.31 90.5?0.39 92.2?1.9 97.3?0.26 84.0?0.48 85.6?O.89 ENSEMBLE CITYBLOCK 96.3 98.3 88.6 90.7 Table 1: Comparison of MLPs classification performance with and with out the city block approximation to the dot product. The final column shows the effect of function space averaging. where the two city block calculation would be performed by neurons on the Nil000 chip. 6 The City Block MLP learns in the standard way by minimizing the mean square error (MSE), (10) MSE = (glc (Xi; 0:, (3) - tlci) 2 2: ilc where tic; is the value of the data at Xi for a class k. The MSE is minimized using the backpropagation stochastic gradient descent learning rule [Werbos, 1974]: For a fixed stepsize f'/ and each k, randomly choose a data point Xi and change 'Y by the amount A _ o(MSEi) L.l.'Y - -f'/ o'Y , (11) where 'Y is either 0: or {3 and MSEi is the contribution to the MSE of the ith data point. Note that although we have motivated the City Block MLP above as an approximation to the standard MLP, the City Block MLP can also be thought of as special case of radial basis function network. 4 Experimental Results This section describes experiments using the City Block MLP on a 120-dimensional representation of the NIST Handwritten Optical Character Recognition database and on a 2294-dimensional grayscale human face image database. The results indicate that the performance of networks using the approximation is as good as networks using the exact dot product [Perrone, 1993]. In order to test the performance of the City Block MLP, we simulated the behavior of the NilOOO on a SPARC station in serial. We used the approximation only on the first layer of weights (i.e. those connecting the inputs to the hidden units) where the dimensionality is highest and the approximation is most accurate. The approximation was not used in the second layer of weights (i.e. those connecting the hidden units to the output units were calculated in serial) since the number of hidden units was low and therefore do not correspond to a major computational bottleneck. It should be noted that for a 2 layer MLP in which the number of hidden units and output units are much lower than the input dimensionality, the 6The dot product between the hidden and the output layers may also be approximated in the same way but it is not shown here. In fact, the NilOOO could be used to perform all of the functions required by the network. VLSI for Implementing Multilayer Perceptrons DATA SET Numbers Lowercase Uppercase HIDDEN UNITS 10 20 20 STANDARD FOM 92.1?0.57 59.7?1.7 60.0?1.8 753 CITYBLOCK FOM 87.4?0.83 44.4?2.0 44.6?4.5 ENSEMBLE CITYBLOCK 92.5 62.7 66.4 Table 2: Comparison of MLPs FOM. The FOM is defined as the 100 minus the number rejected minus 10 time the number incorrect . majority of the computation is in the calculation of the dot products in the first weight layer. So even using the approximation only in the first layer will significantly accelerate the calculation. Also, the Nil000 on-chip math coprocessor can perform a low-dimensional, second layer dot product while the high-dimensional, first layer dot product is being approximated in parallel by the city block units. In practice, if the number of hidden units is large, the approximation to the dot product may also be used in the second weight layer. In the simulations, the networks used the approximation when calculating the dot product only in the feedforward phase of the algorithm. For the feedback ward phase (i.e. the error backpropagation phase), the algorithm was identical to the original backward propagation algorithm. In other words the approximation was used to calculate the network activity but the stochastic gradient term was calculated as if the network activity was generated with the real dot product. This simplification does not slow the calculation because all the terms needed for the backpropagation phase are calculated in the forward propagation phase In addition, it allows us to avoid altering the backpropagation algorithm to incorporate the derivative of the city block approximation. We are currently working on simulations which use city block calculations in both the forward and backward passes. Since these simulations will use the correct derivative for the functional form of the City Block MLP, we expect that they will have better performance. In practice, the price we pay for making the approximation is reduced performance. We can avoid this problem by increasing the number of hidden units and thereby allow more flexibility in the network. This increase in size will not significantly slow the algorithm since the hidden unit activities are calculated in parallel. In Table 1 and Table 2, we compare the performance of a standard MLP without the city block approximation to a MLP using the city block approximation to calculate network activity. In all cases, a population of 10 neural networks were trained from random initial weight configurations and the means and standard deviations were listed. The number of hidden units was chosen to give a reasonable size network while at the same time reasonably quick training. Training was halted by cross-validating on an independent hold-out set. From these results, one can see that the relative performances with and with out the approximation are similar although the City Block is slightly lower. We also perform ensemble averaging [Perrone, 1993, Perrone and Cooper, 1993] to further improve the performance of the approximate networks. These results are given in the last column of the table. From these data we see that by combining the city block approximation with the averaging method, we can generate networks which have comparable and sometimes better performance than the standard MLPs. In addition, because the Nil000 is running in parallel, there is minimal additional computational overhead for using 754 Michael P. Perrone, Leon N. Cooper the averaging. 7 5 Discussion We have described a new radial basis function network architecture which can be used in high dimensional spaces to approximate the learning characteristics of a standard MLP without using dot products. The absence of dot products allows us to implement this new architecture efficiently in parallel on an NilOOO; thus enabling us to take advantage of the Ni1000's extremely fast classification rates. We have also presented experimental results on real-world data which indicate that these high classifications rates can be achieved while maintaining or improving classification accuracy. These results illustrate that it is possible to use the inherent high dimensionality of real-world problems to our advantage. References [Perrone, 1993] Perrone, M. P. (1993). Improving Regression Estimation: Averaging 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. [Perrone and Cooper, 1993] Perrone, M. P. and Cooper, L. N. (1993). When networks disagree: Ensemble method for neural networks. In Mammone, R. J., editor, Artificial Neural Networks for Speech and Vision. Chapman-Hall. Chapter 10. [Reillyet al., 1982] Reilly, D. L., Cooper, L. N., and Elbaum, C. (1982). A neural model for category learning. Biological Cybernetics, 45:35-41. [Rumelhart et al., 1986] Rumelhart, D. E., McClelland, J. L., and the PDP Research Group (1986). Parallel Distributed Processing, Volume 1: Foundations. MIT Press. [Scofield et al., 1987] Scofield, C. L., Reilly, D. L., Elbaum, C., and Cooper, L. N. (1987). Pattern class degeneracy in an unrestricted storage density memory. In Anderson, D. Z., editor, Neural Information Processing Systems. American Institute of Physics. [Sullivan, 1993] Sullivan, M. (1993). Intel and Nestor deliver second-generation neural network chip to DARPA: Companies launch beta site program. Intel Corporation News Release. Feb. 12. [Werbos, 1974] Werbos, P. (1974). Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences. PhD thesis, Harvard University. 7The averaging can also be applied to the standard MLPs with a corresponding improvement in performance. However, for serial machines averaging slows calculations by a factor equal to the number of averaging nets. 

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PCA-Pyramids for Image Compression* Horst Bischof Department for Pattern Recognition and Image Processing Technical University Vienna Treitlstraf3e 3/1832 A-1040 Vienna, Austria bis@prip.tuwien.ac.at Kurt Hornik Institut fur Statistik und Wahrscheinlichkeitstheorie Technische UniversiUit Wien Wiedner Hauptstraf3e 8-10/1071 A-1040 Vienna, Austria Kurt.Hornik@ci .tuwien.ac.at Abstract This paper presents a new method for image compression by neural networks. First, we show that we can use neural networks in a pyramidal framework, yielding the so-called PCA pyramids. Then we present an image compression method based on the PCA pyramid, which is similar to the Laplace pyramid and wavelet transform. Some experimental results with real images are reported. Finally, we present a method to combine the quantization step with the learning of the PCA pyramid. 1 Introduction In the past few years, a lot of work has been done on using neural networks for image compression, d . e.g. (Cottrell et al., 1987; Sanger, 1989; Mougeot et al., 1991; Schweizer et al., 1991)). Typically, networks which perform a Principal Component Analysis (PCA) were employed; for a recent overview of PCA networks, see (Baldi and Hornik, 1995). A well studied and thoroughly understood PCA network architecture is the linear autoassociative network, see (Baldi and Hornik, 1989; Bourlard and Kamp, 1988). This network consists of N input and output units and M < N hidden units, and is *This work was supported in part by a grant from the Austrian National Fonds zur Forderung der wissenschaftlichen Forschung (No. S7002MAT) to Horst Bischof. 942 Horst Bischof, Kurt Hornik trained (usually by back-propagation) to reproduce the input at the output units. All units are linear. Bourlard & Kamp (Bourlard and Kamp, 1988) have shown that at the minimum of the usual quadratic error function ?, the hidden units project the input on the space spanned by the first M principal components of the input distribution. In fact, as long as the output units are linear, nothing is gained by using non-linear hidden units. On average, all hidden units have equal variance. However, peA is not the only method for image compression. Among many others, the Laplace Pyramid (Burt and Adelson, 1983) and wavelets (Mallat, 1989) have successfully been used to compress images. Of particular interest is the fact that these techniques provide a hierarchical representation of the image which can be used for progressive image transmission. However, these hierarchical methods are not adaptive. In this paper, we present a combination of autoassociative networks with hierarchical methods. We propose the so-called peA pyramids, which can be seen as an extension of image pyramids with a learning algorithm as well as cascaded locally connected autoassociative networks. In other words, we combine the structure of image pyramids and neural network learning algorithms, resulting in learning pyramids. The structure of this paper is as follows . We first present image pyramids and, in particular, the peA pyramid. Then, we discuss how these pyramids can be used for image compression, and present some experimental results. Next, we discuss a method to combine the quantization step of compression with the transformation. Finally, we give some conclusions and an outline of further research. 2 The peA Pyramid Before we introduce the peA pyramid, let us describe regular image pyramids. For a discussion of irregular pyramids and their relation to neural networks, see (Bischof, 1993). In the simplest case, each successive level ofthe pyramid is obtained from the previous level by a filtering operation followed by a sampling operator. More general functions can be used to achieve the desired reduction. We therefore call them reduction functions. The structure of a pyramid is determined by the neighbor relations within the levels of the pyramid and by the "father-son" relations between adjacent levels. A cell (if it is not at the base level) has a set of children (sons) at the level directly below which provide input to the cell, a set of neighbors (brothers/sisters) at the same level, and (if it is not the apex of the pyramid) a set of parents (fathers) at the level directly above. We denote the structure of a (regular) pyramid by the expression n x nlr, where n x n (the number of sons) is the size of the reduction window and r the reduction factor which describes how the number of cells decreases from level to level. 2.1 peA Pyramids Since a pyramid reduces the information content of an image level by level, an objective for the reduction function would be to preserve as much information as possible, given the restrictions imposed by the structure of the pyramid, or equivalently, to minimize the information loss by the reduction function. This naturally PCA-Pyramids for Image Compression 943 leads to the idea of representing the pyramid by a suitable peA network. Among the many alternatives for such networks, we have chosen the autoassociative networks for two reasons. First, the analysis of Hornik & Kuan (Hornik and Kuan, 1992) shows that these networks are more stable than competing models. Second, autoassociative networks have the nice feature that they automatically provide us with the expansion function (weights from the hidden layer to output layer). Since the neural network should have the same connectivity as the pyramid (i.e., the same father-son relations), its topology is determined by the structure of the pyramid. In this paper, we confine ourselves to the 4 x 4/4 pyramid for two reasons. First, the 4 x 4/4 pyramid has the nice property that every cell has the same number of fathers, which results in homogeneous networks. Second, as experiments have shown (Bischof, 1993) the results achieved with this pyramid are similar to other structures, e.g. the 5 x 5/4 pyramid, using fewer weights. E R L-..J rn Ii. = E(I n+l ) = E(R(ln? (a) General Setting (b) 4/2 pyramid (c) Corresponding network Figure 1: From the structure of the pyramid to the topology of the network Figure 1 depicts the one-dimensional situation of a 4/2 pyramid (this is the onedimensional counterpart of the two-dimensional 4 x 4/2 pyramid). Figure 1a shows the general goal to be achieved and the notations employed; Figure 1b shows a 4/2 pyramid. When constructing the corresponding network, we start at the output layer (Le., I~). For an n/r pyramid we typically choose the size of the output layer as n. Next, we have to include all fathers of the cells in the output layer as hidden units. Finally, we have to include all sons of the hidden layer cells in the input layer. For the 4/2 pyramid, this results in an 8-3-4 network as shown in Figure 1c. A similar construction yields an 8 x 8-3 x 3-4 x 4 network for the 4 x 4/4 pyramid. The next thing to consider are the constraints on the network weights due to the overlaps in the pyramid. To completely cover the input image with output units, we can shift the network only by four cells in each direction. Therefore, the hidden units at the borders overlap. For the 4/2 pyramid, the left and right hidden units must have identical weights. In the case of the 4 x 4/4 pyramid, the network has four independent units. The thus constructed network can be trained by some suitable learning algorithm, typically of the back-propagation type, using batches of an image as input for trai- 944 Horst Bischof, Kurt Hornik ning the first pyramid level. After that, the second level of the pyramid can be trained in the same way using the first pyramid level as training data, and so on. 2.2 PeA-Laplace Pyramid and Image Compression Thus far, we have introduced a network which can learn the reduction function R and the expansion function E of a pyramid. Analogously to the Laplace pyramid and the wavelet transform we can now introduce the level Li of the PCA-Laplace pyramid, given by Li = Ii - I: = Ii - E(R(Ii)) It should be noted that during learning we exactly minimize the squared Laplace (a) First 2 levels of a Laplace pyramid (upper half) and peA-Laplace pyramid (lower half) (grey = 0) (9) Reconstruction error of house image with quantization of 3 bits, 4 bits, 7 bits, and reconstructed image Figure 2: Results of PCA-Laplace-Pyramid level. The original image 10 can be completely recovered from level In and the Laplace levels L o, ... ,Ln - 1 by 10 = E(??? E(E(In) + L n- 1 ) + L n- 2 )???) + Lo? Since the level In is rather small (e.g., 32 x 32 pixels) and the levels of the PCALaplace pyramid are typically sparse (i.e., many pixels are zero, see Figure 2a) and can therefore be compressed considerably by a conventional compression algorithm PCA-Pyramids for Image Compression 945 (e.g. Lempel-Ziv (Ziv and Lempel, 1977)), this image representation results in a lossless image compression algorithm . In order to achieve higher compression ratios we can quantize the levels of the PCALaplace pyramid. In this case, the compression is lossy, because the original image cannot be recovered exactly. The compression ratio and the amount of loss can be controlled by the number of bits used to quantize the levels of the PCA-Laplacian. To measure the difference between the compressed and the original image, we use the normalized mean squared error (NMSE) as in (Cottrell et al., 1987; Sanger, 1989) . The NMSE is given by the mean squared error divided by the average squared intensity of the image, i.e., NMSE = MSE = ((10 - C(10))2) (I~) (I~)' where 10 and C(lo) are the original and the compressed image, respectively. The compression ratio is measured by the amount of bits used to store 10 , divided by the amount of bits used to store C(1o). 2.3 Results For the results reported here we trained the networks by a conjugate gradient algorithm for 100 steps! and used a uniform quantization which is fixed for all levels of the pyramid. As was shown in (Burt and Adelson, 1983; Mayer and Kropatsch, 1989), the results could be improved by gradually increasing the quantization from bottom to top. Figure 2b shows the error images when the levels of the PCA-Laplacian pyramid are quantized with 3, 4, and 7 bits and the reconstructed image from the 7 bit Laplacian. Note that we used the same lookup-table for the error images. To compress the levels of the PCA-Laplacian pyramid, we employed the standard UNIX compress program which implements a Lempel-Ziv algorithm. iFrom these images one can see that the results with the 4 and 7 bit quantization are very good. Visually, no difference between the reconstructed and the original image can be perceived. Table 1 shows the compression ratios and the NMSEs on these images. We have performed experiments on 20 different images, the results on these images are comparable to the ones reported here. These results compare favorably with the results in the literature (see Table 1). We have also applied a 5 x 5/4 Laplace pyramid to the house image which gave a compression ratio of 3.42 with an NMSE of 0.000087 for quantization with four bits of the Laplace levels. We have also included results achieved with JPEG. One can see that our method gives considerably better results . We have also demonstrated experimentally what happens if we train a pyramid on one image and then apply this pyramid to another image without retraining. These experiments indicate that the errors are only a little bit larger for images not trained on. With five additional steps of training the errors are almost the same. iFrom lIn all our experiments the training algorithm converged (i.e. usually after 200 steps, however the improvements between steps 20 and convergence are negligible). 946 Horst Bischof, Kurt Hornik Quant. 3 Bit 4 Bit 7 Bit no Quant. Cottrell,~Cottrell et al., 1987) Sanger (Sanger, 1989) 5 x 5/4 Laplace JPEG JPEG Compression ratio 37.628 24.773 8.245 3.511 8.0 22.0 3.420 8.290 15.774 Bits/Pixel 0.212 0.323 0.970 2.279 1.000 0.360 2.339 0.965 0.507 NMSE 0.0172 0.0019 0.0000215 0.0 0.0059 0.043 0.000087 0.00139 0.00348 Table 1: Compression ratios and NMSE for various compression methods this results we can conclude that we do not need to retrain the pyramid for each new image. 3 Integration of Quantization For the results reported in the previous section we have used a fixed and uniform quantization scheme which can be improved by using adaptive quantizers like the Lloyd I algorithm, Kohonen's Feature Maps, learning vector quantization, or something similar. Such an approach as taken by Schweizer (Schweizer et al., 1991) who combined a Cottrell-type network with self-organizing feature maps. However, we can go further. With the PCA network we minimize the squared Laplace level which does not necessarily yield low compression errors. What we really want to minimize are the quantized Laplace levels. Usually, the Laplace levels have an unimodally shaped histogram centered at zero. However, for the result of the compression (i.e., compression ratio and NMSE), it is irrelevant if we shift the histogram to the left or the right as long as we shift the quantization intervals in the same way. The best results could be achieved if we have a multimodal histogram with peaks centered at the quantization points. Using neural networks for both PCA and quantization, this goal could e.g. be achieved by a modular network as in Figure 3 for the 4/2 pyramid. For quantization, we could either apply a vector quantizer to a whole patch of the Laplace level, or use a scalar quantizer (as depicted in Figure 3) for each pixel of the Laplace level. In the second case, we have to constrain the weights of the quantization network to be identical for every Laplace pixel. Since scalar quanti.zation is simpler to analyze and uses less free parameters, we only consider this case. As each quantization subnetwork can be treated separately (we only have to average the weight changes over all subnetworks), the following only considers the case of one output unit of the PCA network. PCA-Pyramids for Image Compression Quantization 947 peA Figure 3: PCA network and Quantization network The error to be minimized is the squared quantization error where p refers to the patterns in the training set, Ck is the kth weight of the quantization network, and 1 is the output of the PCA-Laplace unit. Changing the weights of the quantization network by gradient descent leads to the LVQl rule of Kohonen Ac - { 2a(lp k 0, Ck), if k = kp is the winning unit, otherwise. For the PCA network we can proceed similarly to back-propagation to obtain the rule AWij = -K 8Ep = _K 8Ep 8Wij alp = _K 8Ep alp 8Wij alp 8~p 8i~ 8z~ 8Wij = -2K(lp _ Ck) 8i~ . 8Wij Of course, this is only one out of many possible algorithms. More elaborate minimization techniques than gradient descent could be used; similarly, LVQl could be replaced by a different quantization algorithm. But the basic idea of letting the quantization step and the the compression step adapt to each other remains unchanged. 4 Conclusions In this paper, we presented a new image compression scheme based on neural networks. The PCA and PCA-Laplace pyramids were introduced, which can be seen as both an extension of image pyramids to learning pyramids and as cascaded, locally connected autoassociators. The results achieved are promising and compare favorably to work reported in the literature. A lot of work remains to be done to analyze these networks analytically. The convergence properties of the PCA pyramid are not known; we expect results similar 948 Horst Bischof, Kurt Hornik to the ones (Baldi and Hornik, 1989) for the autoassociative network. Also, for the PCA network it would be desirable to characterize the features which are extracted. Similarly, the integrated network needs to be analyzed. It is clear that for such networks, the usual error function has local minima, but maybe they can be avoided by a proper training regime (i.e. start training the PCA pyramid, then train the vector quantizer, and finally train them together). References Baldi, P. and Hornik, K. (1989). Neural Networks and principal component analysis: Learning from examples without local minima. Neural Networks, 2:53-58. Baldi, P. and Hornik, K. (1995). Learning in Linear Neural Networks: a Survey. IEEE Transactions on Neural Networks, to appear. Bischof, H. (1993). Pyramidal Neural Networks. PhD thesis, TU-Vienna, Inst. f. Automation, Dept. f. Pattern Recognition and Image Processing. Bourlard, H. and Kamp, Y. (1988). Auto-Association by Multilayer Perceptrons and Singular Value Decomposition. Biological Cybernetics, 59:291-294. Burt, P. J. and Adelson, E. H. (1983). The Laplacian pyramid as a compact image code. IEEE Transactions on Communications, Vol. COM-31(No.4):pp.532-540. Cottrell, G., Munro, P., and Zipser, D. (1987). Learning Internal Representations from Grey-Scale Images: An Example of Extensional Programming. In Ninth Annual Conference of the Cognitive Science Society, pages 462-473. Hillsdale Erlbaum. Hornik, K. and Kuan, C. (1992). Convergence analysis of local feature extraction algorithms. Neural Networks, 5(2):229-240. Mallat, S. G. (1989). A Theory for Multiresolution Signal Decomposition: The Wavelet Representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-ll(No. 7):pp. 674-693. Mayer, H. and Kropatsch, W. G. (1989). Progressive Bildubertragung mit der 3x3/2 Pyramide. In Burkhardt, H., H6hne, K., and Neumann, B., editors, Informatik Fachberichte 219: Mustererkennung 1989, pages 160-167, Hamburg. l1.DAGM - Symposium, Springer Verlag. Mougeot, M., Azencott, R., and Angeniol, B. (1991). Image Compression with Back Propagation: Improvement of the Visual Restoration using different Cost Functions. Neural Networks, 4:467-476. Sanger, T. (1989). Optimal Unsupervised learning in a Single-Layer Linear Feedforward Neural Network. Neural Networks, 2:433-459. Schweizer, L., Parladori, G., Sicranza, G., and Marsi, S. (1991). A fully neural approach to image compression. In Kohonen, T., Makissara, K., Simula, 0., and Kangas, J., editors, Artificial Neural Networks, volume I, pages 815-820. Ziv, J. and Lempel, A. (1977). A universal algorithm for sequential data compression. IEEE Trans. on Information Theory, 23(5):337 - 343. 

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Learning Saccadic Eye Movements Using Multiscale Spatial Filters Rajesh P.N. Rao and Dana H. Ballard Department of Computer Science University of Rochester Rochester, NY 14627 {rao)dana}~cs.rochester.edu Abstract We describe a framework for learning saccadic eye movements using a photometric representation of target points in natural scenes. The representation takes the form of a high-dimensional vector comprised of the responses of spatial filters at different orientations and scales. We first demonstrate the use of this response vector in the task of locating previously foveated points in a scene and subsequently use this property in a multisaccade strategy to derive an adaptive motor map for delivering accurate saccades. 1 Introduction There has been recent interest in the use of space-variant sensors in active vision systems for tasks such as visual search and object tracking [14]. Such sensors realize the simultaneous need for wide field-of-view and good visual acuity. One popular class of space-variant sensors is formed by log-polar sensors which have a small area near the optical axis of greatly increased resolution (the fovea) coupled with a peripheral region that witnesses a gradual logarithmic falloff in resolution as one moves radially outward. These sensors are inspired by similar structures found in the primate retina where one finds both a peripheral region of gradually decreasing acuity and a circularly symmetric area centmlis characterized by a greater density of receptors and a disproportionate representation in the optic nerve [3]. The peripheral region, though of low visual acuity, is more sensitive to light intensity and movement. The existence of a region optimized for discrimination and recognition surrounded by a region geared towards detection thus allows the image of an object of interest detected in the outer region to be placed on the more analytic center for closer scrutiny. Such a strategy however necessitates the existence of (a) methods to determine which location in the periphery to foveate next, and (b) fast gaze-shifting mechanisms to achieve this 894 Rajesh P. N. Rao, Dana H. Ballard foveation. In the case of humans, the "where-to-Iook-next" issue is addressed by both bottom-up strategies such as motion or salience clues from the periphery as well as topdown strategies such as search for a particular form or color. Gaze-shifting is accomplished via very rapid eye movements called saccades. Due to their high velocities, guidance through visual feedback is not possible and hence, saccadic movement is preprogrammed or ballistic: a pattern of muscle activation is calculated in advance that will direct the fovea almost exactly to the desired position [3]. In this paper, we describe an iconic representation of scene points that facilitates topdown foveal targeting. The representation takes the form of a high-dimensional vector comprised of the responses of different order Gaussian derivative filters, which are known to form the principal components of natural images [5], at variety of orientations and scales. Such a representation has been recently shown to be useful for visual tasks ranging from texture segmentation [7] to object indexing using a sparse distributed memory [11]. We describe how this photometric representation of scene points can be used in locating previously foveated points when a log-polar sensor is being used. This property is then used in a simple learning strategy that makes use of multiple corrective saccades to adaptively form a retinotopic motor map similar in spirit to the one known to exist in the deep layers of the primate superior colliculus [13]. Our approach differs from previous strategies for learning motor maps (for instance, [12]) in that we use the visual modality to actively supply the necessary reinforcement signal required during the motor learning step (Section 3.2) . 2 The Multiscale Spatial Filter Representation In the active vision framework, vision is seen as subserving a larger context of the encompassing behaviors that the agent is engaged in. For these behaviors, it is often possible to use temporary, iconic descriptions of the scene which are only relatively insensitive to variations in the view. Iconic scene descriptions can be obtained, for instance, by employing a bank of linear spatial filters at a variety of orientations and scales. In our approach, we use derivative of Gaussian filters since these are known to form the dominant eigenvectors of natural images [5] and can thus be expected to yield reliable results when used as basis functions for indexingl . The exact number of Gaussian derivative basis functions used is motivated by the need to make the representations invariant to rotations in the image plane (see [11] for more details). This invariance can be achieved by exploiting the property of steerability [4] which allows filter responses at arbitrary orientations to be synthesized from a finite set of basis filters. In particular, our implementation uses a minimal basis set of two firstorder directional derivatives at 0? and 90?, three second-order derivatives at 0?, 60? and 120?, and four third-order derivatives oriented at 0?, 45?, 90?, and 135?. The response of an image patch J centered at (xo, Yo) to a particular basis filter G~j can be obtained by convolving the image patch with the filter : ri,j(XO, Yo) = (G/g . * I)(xo, Yo) = If 9 (XO - x, Yo - y)J(x, y)dx dy G/ (1) lIn addition, these filters are endorsed by recent physiological studies [15] which show that derivative-of-Gaussians provide the best fit to primate cortical receptive field profiles among the different functions suggested in the literature. Learning Saccadic Eye Movements Using Multiscale Spatial Filters 895 The iconic representation for the local image patch centered at (xo, Yo) is formed by combining into a single high-dimensional vector the responses from the nine basis filters, each (in our current implementation) at five different scales: r(xo, Yo) =(ri,j,s) , i = 1,2, 3;j = 1, . .. , i + 1; S = Smin , . .. , Smax (2) where i denotes the order of the filter, j denotes the number of filters per order, and denotes the number of different scales. S The use of multiple scales increases the perspicuity of the representation and allows interpolation strategies for scale invariance (see [9] for more details). The entire representation can be computed using only nine convolutions done at frame-rate within a pipeline image processor with nine constant size 8 x 8 kernels on a five-level octave-separated low-passfiltered pyramid of the input image. The 45-dimensional vector representation described above shares some of the favorable matching properties that accrue to high-dimensional vectors (d. [6]). In particular, the distribution of distances between points in the 45-dimensional space of these vectors approximates a normal distribution; most of the points in the space lie at approximately the mean distance and are thus relatively uncorrelated to a given point [11]. As a result, the multiscale filter bank tends to generate almost unique location-indexed signatures of image regions which can tolerate considerable noise before they are confused with other image regions. 2.1 Localization Denote the response vector from an image point as fi and that from a previously foveated model point as Tm. Then one metric for describing the similarity between the two points is simply the square of the Euclidean distance (or the sum-of-squared-differences) between 112. The algorithm for locating model points in a their response vectors dim = llfi new scene can then be described as follows : r.n 1. For the response vector representing a model point m, create a distance image I m defined by Im(x,y) = min [Imax - t3dim , 0] (3) where t3 is a suitably chosen constant (this makes the best match the brightest point in Im). 2. Find the best match point (Xb~, Yb~) in the image using the relation (4) Figure 1 shows the use of the localization algorithm for targeting the optical axis of a uniform-resolution sensor in an example scene. 2.2 Extension to Space-Variant Sensing The localization algorithm as presented above will obviously fail for sensors exhibiting nonuniform resolution characteristics. However, the multiscale structure of the response vectors can be effectively exploited to obtain a modified localization algorithm. Since decreasing radial resolution results in an effective reduction in scale (in addition to some 896 Rajesh P. N. Rao, Dana H. Ballard (a) (b) (c) (d) Figure 1: Using response vectors to saccade to previously foveated positions. (a) Initial gaze point. (b) New gaze point; (c) To get back to the original point, the "distance image" is computed: the brightest spot represents the point whose response vector is closest to that of the original gaze point; (d) Location of best match is marked and an oculomotor command at that location can be executed to foveate that point. other minor distortions) of previously foveated regions as they move towards the periphery, the filter responses previously occuring at larger scales now occur at smaller scales. Responses usually vary smoothly between scales; it is thus possible to establish a correspondence between the two response vectors of the same point on an object imaged at different scales by using a simple interpolate-and-eompare scale matching strategy. That is, in addition to comparing an image response vector and a model response vector directly as outlined in the previous section, scale interpolated versions of the image vector are also compared with the original model response vector. In the simplest case, interpolation amounts to shifting image response vectors by one scale and thus, responses from a new image are compared with original model responses at second, third, .. , scales, then with model responses at third, fourth, ... scales, and so on upto some threshold scale. This is illustrated in Figure 2 for two discrete movements of a simulated log-polar sensor. 3 The M ultisaccade Learning Strategy Since the high speed of saccades precludes visual guidance, advance knowledge of the precise motor command to be sent to the extraocular muscles for fixation of a desired retinal location is required. Results from neurophysiological and psychophysical studies suggest that in humans, this knowledge is acquired via learning: infants show a gradual increase in saccadic accuracy during their first year [1, 2] and adults can adapt to changes (caused for example by weakening of eye-muscles) in the interrelation between visual input and the saccades needed for centering. An adaptive mechanism for automatically learning the transfer function from retinal image space into motor space is also desirable in the context of active vision systems since an autonomous calibration of the saccadic system would (a) avoid the need for manual calibration, which can sometimes be complicated, and (b) provide resilience amidst changing circumstances caused by, for instance, changes in the camera lens mechanisms or degradation of the motor apparatus. 3.1 Motor Maps In primates, the superior eollieulus (SC), a multilayered neuron complex located in the upper regions of the brain stem, is known to playa crucial role in the saccade generation [13]. The upper layers of the SC contain a retinotopie sensory map with inputs from Learning Saccadic Eye Movements Using MuLtiscaLe SpatiaL Filters (a) Scale I (a) (b) Scale 2 III' I " 111'1"" Scille I (b) Scale 3 II" Scale 4 (c) Scale 5 III, ' II " ,li'I"" 111'1 Scale 2 s1I3 seNe 4 Scale 5 111 ' 1 "'1 1" 111 " 11,, 1 " 1' ,11 ,,1 ' 11 1,1 Scale I (c) 897 Scale 2 Scaje 3 Scale 4 Scale 5 11"1 11 , II ' ,11'''1, ,111,, 111 11 1 ,1'11'1 I (d) Figure 2: Using response vectors with a log-polar sensor, (a) through (c) represent a sequence of images (in Cartesian coordinates) obtained by movement of a simulated log-polar sensor from an original point (marked by '+') in the foveal region (indicated by a circle) towards the right. (d) depicts the process of interpolating (in this case, shifting) and matching response vectors of the same point as it moves towards the periphery of the sensor (Positive responses are represented by proportional upward bars and negative ones by proportional downward bars with the nine smallest scale responses at the beginning and the nine largest ones at the end). the retina while the deeper layers contain a motor map approximately aligned with the sensory map. The motor map can be visualized as a topologically-organized network of neurons which reacts to a local activation caused by an input signal with a vectorial output quantity that can be transcoded into a saccadic motor command. The alignment of the sensory and motor maps suggests the following convenient strategy for foveation: an excitation in the sensory layer (signaling a foveal target) is transferred to the underlying neurons in the motor layer which deliver the required saccade. In our framework, the excitation in the sensory layer before a goal-directed saccade corresponds to the brightest spot (most likely match) in the distance image (Figure 1 (c) for example), The formation of sensory map can be achieved using Kohonen's well-known stochastic learning algorithm by using a Gaussian input density function as described in [12]. Our primary interest lies not in the formation of the sensory map but in the development of a learning algorithm that assigns appropriate motor vectors to each location in the corresponding retinotopically-organized motor map. In particular, our algorithm employs a visual reinforcement signal obtained using iconic scene representations to determine the error vector during the learning step. 3.2 Learning the Motor Map Our multisaccade learning strategy is inspired by the following observations in [2] : During the first few weeks after birth, infants appear to fixate randomly. At about 3 months of age, infants are able to fixate stimuli albeit with a number of corrective saccades of relatively large dispersion. There is however a gradual decrease in both the dispersion 898 Rajesh P. N. Rao, Dana H. Ballard and the number of saccades required for foveation in subsequent months (Figure 3 (a) depicts a sample set of fixations). After the first year, saccades are generally accurate, requiring at most one corrective saccade2 ? The learning method begins by assigning random values to the motor vectors at each location. The response vector for the current fixation point is first stored and a random saccade is executed to a different point. The goal then is to refixate the original point with the help of the localization algorithm and a limited number of multiple corrective saccades. The algorithm keeps track of the motor vector with minimum error during each run and updates the motor vectors for the neighborhood around the original unit whenever an improvement is observed. The current run ends when either the original point was successfully foveated or the limit MAX for the maximum number of allowable corrective saccades was exceeded. A more detailed outline of the algorithm is as follows: 1. Initialize the motor map by assigning random values (within an appropriate range) to the saccadic motor vectors at each location. Align the optical axis of the sensor so that a suitable salient point falls on the fovea. Initialize the run number to t := O. 2. Store in memory the filter response vector of the point p currently in the center of the foveal region. Let t := t + 1. 3. Execute a random saccade to move the fovea to a different location in the scene. 4. Use the localization algorithm described in Section 2.2 and the stored response vector to find the location [ of the previously foveated point in the current retinal image. Execute a saccade using the motor vector St stored in this location in the motor map. 5. If the currently foveated region contains the original point p, return to 2 (SI is accurate); otherwise, s:= St. (a) Initialize the number of corrective saccades N := 0 and let (b) Determine the new location /' of p in the new image as in (4) and let emin be the error vector, i.e. the vector from the foveal center to /', computed from the output of the localization algorithm. (c) Execute a saccade using the motor vector Stl stored at [' and let ebe the error vector (computed from the output of the localization algorithm) from the foveal center to the new location [II of point p found as in 4. Let N := N + 1 and let SI' . (d) If lie'll < lliminll, then let emin := e and update the motor vectors for the units k given by the neighborhood function N(l, t) according to the wellknown Kohonen rule: (5) where 'Y(t) is an appropriate gain function (0 < 'Y(t) < 1). (e) If the currently foveated region contains the original point p, return to 2; otherwise, if N < MAX, then determine the new location [' of p in the new image as in (4) and go to 5(c) (i.e. execute the next saccade); otherwise, return to 2. s:= s+ 2Large saccades in adults are usually hypometric i.e. they undershoot, necessitating a slightly slower corrective saccade. There is currently no universally accepted explanation for the need for such a two-step strategy. 899 Learning Saccadic Eye Movements Using Multiscale Spatial Filters !" 1 I I .. ? ......, + IMX-IO f.! ' ..... 1 N!.IBfIll _ _ (a) (b) ",~;:;!l-;;:;;;--l-;;;; ll1I)"'---;;:ll1Ol;;;--;;"=-lI)---;",=-,C:;;,.,:---;;:; ...;;-;:;!,,", N_la rillelllltN (c) Figure 3: (a) Successive saccades executed by a 3-month old (left) and a 5-month old (right) infant when presented with a single illuminated stimulus (Adapted from [2]) . (b) Graph showing % of saccades that end directly in the fovea plotted against the number of iterations of the learning algorithm for different values of MAX. (c) An enlarged portion of the same graph showing points when convergence was achieved. The algorithm continues typically until convergence or the completion of a maximum number of runs. The gain term -y(t) and the neighborhood N(l, t) for any location l are gradually decreased with increasing number of iterations t. 4 Results and Discussion The simulation results for learning a motor map comprising of 961 units are shown in Figures 3 (b) and (c) which depict the variation in saccadic accuracy with the number of iterations of the algorithm for values of MAX (maximum number of corrective saccades) of 1, 5 and 10. From the graphs, it can be seen that starting with an initially random assignment of vectors, the algorithm eventually assigns accurate saccadic vectors to all units. Fewer iterations seem to be required if more corrective saccades are allowed but then each iteration itself takes more time. The localization algorithm described in Section 2.1 has been implemented on a Datacube MaxVideo 200 pipeline image processing system and takes 1-2 seconds for location of points. Current work includes the integration of the multisaccade learning algorithm described above with the Datacube implementation and further evaluation of the learning algorithm. One possible drawback of the proposed algorithm is that for large retinal spaces, learning saccadic motor vectors for every retinal location can be time-consuming and in some cases, even infeasible [1]. In order to address this problem, we have recently proposed a variation of the current learning algorithm which uses a sparse motor map in conjunction with distributed coding of the saccadic motor vectors. This organization bears some striking similarities to Kanerva's sparse distributed memory model [6] and is in concurrence with recent neurophysiological evidence [8] supporting a distributed population encoding of saccadic movements in the superior colliculus. We refer the interested reader to [10] for more details. 900 Rajesh P. N. Rao, Dana H. Ballard Acknowledgments We thank the NIPS*94 referees for their helpful comments. This work was supported by NSF research grant no. CDA-8822724, NIH/PHS research grant no. 1 R24 RR06853, and a grant from the Human Science Frontiers Program. References [1] Richard N. Aslin. Perception of visual direction in human infants. In C. Granlund, editor, Visual Perception and Cognition in Infancy, pages 91-118. Hillsdale, NJ: Lawrence Erlbaum Associates, 1993. [2] Gordon W. Bronson. The Scanning Patterns of Human Infants: Implications for Visual Learning. Norwood, NJ : Ablex, 1982. [3] Roger H.S. Carpenter. Movements of the Eyes. London: Pion, 1988. [4] William T . Freeman and Edward H. Adelson. The design and use of steerable filters. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(9):891-906, September 1991. [5] Peter J.B. Hancock, Roland J. Baddeley, and Leslie S. Smith. The principal components of natural images. Network, 3:61-70, 1992. [6] Pentti Kanerva. Sparse Distributed Memory. Bradford Books, Cambridge, MA, 1988. [7] Jitendra Malik and Pietro Perona. A computational model of texture segmentation. In IEEE Conference on Computer Vision and Pattern Recognition, pages 326-332, June 1989. [8] James T. McIlwain. Distributed spatial coding in the superior colliculus: A review. Visual Neuroscience, 6:3-13, 1991. [9] Rajesh P.N . Rao and Dana H. Ballard. An active vision architecture based on iconic representations. Technical Report 548, Department of Computer Scienc~, University of Rochester, 1995. [10] Rajesh P.N. Rao and Dana H. Ballard. A computational model for visual learning of saccadic eye movements. Technical Report 558, Department of Computer Science, University of Rochester, January 1995. [11] Rajesh P.N. Rao and Dana H. Ballard. Object indexing using an iconic sparse distributed memory. Technical Report 559, Department of Computer Science, University of Rochester, January 1995. [12] Helge Ritter, Thomas Martinetz, and Klaus Schulten. Neural Computation and SelfOrganizing Maps: An Introduction. Reading, MA: Addison-Wesley, 1992. [13] David L. Sparks and Rosi Hartwich-Young. The deep layers of the superior collicuIus. In R.H. Wurtz and M.E. Goldberg, editors, The Neurobiology of Saccadic Eye Movements, pages 213-255. Amsterdam: Elsevier, 1989. [14] Massimo Tistarelli and Giulio Sandini. Dynamic aspects in active vision. Computer Vision, Graphics, and Image Processing: Image Understanding, 56(1):108-129, 1992. [15] R.A. Young. The Gaussian derivative theory of spatial vision: Analysis of cortical cell receptive field line-weighting profiles. General Motors Research Publication GMR4920, 1985. 

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Extracting Rules from Artificial Neural Networks with Distributed Representations Sebastian Thrun University of Bonn Department of Computer Science III Romerstr. 164, D-53117 Bonn, Germany E-mail: thrun@carbon.informatik.uni-bonn.de Abstract Although artificial neural networks have been applied in a variety of real-world scenarios with remarkable success, they have often been criticized for exhibiting a low degree of human comprehensibility. Techniques that compile compact sets of symbolic rules out of artificial neural networks offer a promising perspective to overcome this obvious deficiency of neural network representations. This paper presents an approach to the extraction of if-then rules from artificial neural networks. Its key mechanism is validity interval analysis, which is a generic tool for extracting symbolic knowledge by propagating rule-like knowledge through Backpropagation-style neural networks. Empirical studies in a robot arm domain illustrate the appropriateness of the proposed method for extracting rules from networks with real-valued and distributed representations. 1 Introduction In the last few years artificial neural networks have been applied successfully to a variety of real-world problems. For example, neural networks have been successfully applied in the area of speech generation [12] and recognition [18], vision and robotics [8], handwritten character recognition [5], medical diagnostics [11], and game playing [13]. While in these and other approaches neural networks have frequently found to outperform more traditional approaches, one of their major shortcomings is their low degree of human comprehensibility. In recent years, a variety of approaches for compiling rules out of networks have been proposed. Most approaches [1, 3,4,6, 7, 16, 17] compile networks into sets of rules with equivalent structure: Each processing unit is mapped into a separate rule-or a smal1 set of rules-, and the ingoing weights are interpreted as preconditions to this rule. Sparse connectivity facilitates this type rule extraction, and so do binary activation values. In order to enforce such properties, which is a necessary prerequisite for these techniques to work effectively, some approaches rely on specialized training procedures, network initializations 506 Sebastian Thrun and/or architectures. While such a methodology is intriguing, as it draws a clear one-to-one correspondence between neural inference and rule-based inference, it is not universally applicable to arbitrary Backpropagation-style neural networks. This is because artificial neural networks might not meet the strong representational and structural requirements necessary for these techniques to work successfully. When the internal representation of the network is distributed in nature, individual hidden units typically do not represent clear, logical entities. One might argue that networks, if one is interested in extracting rules, should be constructed appropriately. But this would outrule most existing network implementation~, as such considerations have barely played a role. In addition, such an argument would suppress the development of distributed, non-discrete internal representations, which have often be attributed for the generalization properties of neural networks. It is this more general class of networks that is at stake in this paper. This paper presents a rule extraction method which finds rules by analyzing networks as a whole. The rules are of the type "if X then y," where both x and y are described by a linear set of constraints. The engine for proving the correspondence of rule and network classification is VI-Analysis. Rules extracted by VI-Analysis can be proven to exactly describe the network. 2 Validity-Interval Analysis Validity Interval Analysis (in short: VI-Analysis) is a generic tool for analyzing the inputoutput behavior of Backpropagation-style neural networks. In short, they key idea of VIAnalysis is to attach intervals to the activation range of each unit (or a subset of all units, like input and output units only), such that the network's activations must lie within these intervals. These intervals are called validity intervals. VI-Analysis checks whether such a set of intervals is consistent, i.e., whether there exists a set of network activations inside the validity intervals. It does this by iteratively refining the validity intervals, excluding activations that are provably inconsistent with other intervals. In what follows we will present the general VI-Analysis algorithm, which can be found in more detail elsewhere [14], Let n denote the total number of units in the network, and let Xi denote the (output) activation of unit i (i 1, ... , n). If unit i is an input unit, its activation value will simply be the external input value. If not, i.e., if i refers to a hidden or an output unit, let P( i) denote the set of units that are connected to unit i through a link. The activation Xi is computed in two steps: with WikXk + Oi = L kEP(i) The auxiliary variable neti is the net-input of unit i, and Wik and Oi are the weights and biases, respectively. O'j denotes the transfer function (squashing function), which usually is given by 1 + e- net , Validity intervals for activation values Xi are denoted by [ai, bi ]. If necessary, validity intervals are projected into the net-input space of unit i, where they will be denoted by [a~, b~]. Let T be a set of validity intervals for (a subset of) all units. An activation vector (XI, .. " xn) is said to be admissible with respect to T, if all activations lie in T. A set of intervals T is consistent, if there exists an admissible activation vector. Otherwise T is inconsistent. Assume an initial set of intervals, denoted by T, is given (in the next section we will present a procedure for generating initial intervals). VI-Analysis refines T iteratively using linear Extracting Rules from Artificial Neural Networks with Distributed Representations 507 non-linear .<;quashing functioll' CJ linear equations Figure 1: VI-Analysis in a single weight layer. Units in layer P are connected to the units in layer S. A validity interval [aj, bj ] is assigned to each unit j E PuS. By projecting the validity intervals for all i E S, intervals [a~, b~] for the net-inputs netj are created. These, plus the validity intervals for all units k E P, form a set of linear constraints on the activations x k in layer P. Linear programming is now employed to refine all interval bounds one-by-one. programming [9], so that those activation values which are inconsistent with other intervals are excluded. In order to simplify the presentation, let us assume without loss of ?enerality (a) that the network is layered and fully connected between two adjacent layers, and (b) that there is an interval [aj, bj ] ~ [0,1] in I for every unit in P and S.2 Consider a single weight layer, connecting a layer of preceding units, denoted by p, to a layer of succeeding units, denoted by S (cf Fig. 1). In order to make linear programming techniques applicable, the non-linearity of the transfer function must be eliminated. This is achieved by projecting [ai, bi ] back to the corresponding net-input intervals 3 [ai, biJ = {T-I([ai' biD E ~2 for all i E S. The resulting validity intervals in P and S form the foIIowing set of linear constraints on the activation values in P: Vk E P: Xk > ak and Xk < bk Vi E S: WikXk + ()d WjkXk + ()j > a~, [by substituting neti = (1) kEP kEP WikXk + ()i] WikXk + ()j < b~, [by substituting netj = L L L L kEP kEP Notice that all these constraints are linear in the activation values Xk (k E P). Linear programming allows to maximize or minimize arbitrary linear combinations of the variables x j while not violating a set of linear constraints [9]. Hence, linear programming can be applied to refine lower and upper bounds for validity intervals one-by-one. In VI-Analysis, constraints are propagated in two phases: 1. Forward phase. To refine the bounds aj and bj for units i E S, new bounds iii and hi are 'This assumption simplifies the description of VI-Analysis, although VI-Analysis can also be applied to arbitrary non-layered, partially connected network architectures, as well as recurrent networks not examined here. 2The canonical interval [0, I] corresponds to the state of maximum ignorance about the activation of a unit, and hence is the default interval if no more specific interval is known. 3Here ~ denotes the set of real numbers extended by ?oo. Notice that this projection assumes that the transfer function is monotonic. 508 derived: Sebastian Thrun with aA'z? min neti L: max L: min WikXk + Oi WikXk + OJ kE1' with = max neti kE1' If o'i > ai, a tighter lower bound is found and ai is updated by o'i . Likewise, b i is set to hi if hi < bi . Notice that the minimax operator is computed within the bounds imposed by Eq. I, using the Simplex algorithm (linear programming) [9]. 2. Backward phase. In the backward phase the bounds ak and b k of all units k E Pare refined. li k minxk and hk = max Xk As in the forward phase, ak is updated by o'k if li k > ak, and h is updated by hk if hk < bk. If the network has multiple weight layers, this process is applied to all weight layers one-byone. Repetitive refinement results in the propagation of interval constraints through multiple layers in both directions. The convergence of VI-Analysis follows from the fact that the update rule that intervals are changed monotonically, since they can only shrink or stay the same. Recall that the "input" of VI-Analysis is a set of intervals I ~ [0, l]n that constrain the activations of the network. VI-Analysis generates a refined set of intervals, I' ~ I, so that all admissible activation values in the original intervals I are also in the refined intervals I'. In other words, the difference between the original set of intervals and the refined set of intervals I - I' is inconsistent. In summary, VI-Analysis analyzes intervals I in order to detect inconsistencies. If I is found to be inconsistent, there is provably no admissible activation vector in I . Detecting inconsistencies is the driving mechanism for the verification and extraction of rules presented in turn. 3 Rule Extraction The rules considered in this paper are propositional if-then rules. Although VI-Analysis is able to prove rules expressed by arbitrary linear constraints [14], for the sake of simplicity we will consider only rules where the precondition is given by a set of intervals for the individual input values, and the output is a single target category. Rules of this type can be written as: !linput E some hypercube I then class is C (or short: I - - C) for some target class C. The compliance of a rule with the network can be verified through VI-Analysis. Assume, without loss of generality, the network has a single output unit, and input patterns are classified as members of class C if and only if the output activation, Xout, is larger than a threshold e (see [14] for networks with multiple output units). A rule conjecture I - - C is then verified by showing that there is no input vector i E I that falls into the opposite class, ,C. This is done by including the (negated) condition Xout E [0, e] into the set of intervals: Ineg = 1+ {xout E [0, e]}. If the rule is correct, Xout will never be in [0, e]. Hence, if VI-Analysis finds an inconsistency in Ineg, the rule I - - ,C is proven to be incorrect, and thus the original rule I - - C holds true for the network at hand. This illustrates how rules are verified using VI-Analysis. It remains to be shown how such conjectures can be generated in a systematic way. Two major classes of approaches can be distinguished, specific-to-general and general-to-specific. Extracting Rules from Artificial Neural Networks with Distributed Representations 509 Figure 2: Robot Ann. (a) Front view of two arm configurations. (b) Two-dimensional side view. The grey area indicates the workspace, which partially intersects with the table. 1. Specific-to-general. A generic way to generate rules, which forms the basis for the experimental results reported in the next section, is to start with rather specific rules which are easy to verify, and gradually generalize those rules by enlarging the corresponding validity intervals. Imagine one has a training instance that, without loss of generality, falls into a class C. The input vector of the training instance already forms a (degenerate) set of validity intervals I. VI-Analysis will, applied to I, trivially confirm the membership in C, and hence the single-point rule I ~ C. Starting with I, a sequence of more general rule preconditions I C II C I2 C ... can be obtained by enlarging the precondition of the rule (i.e., the input intervals I) by small amounts, and using VI-Analysis to verify if the new rule is still a member of its class. In this way randomly generated instances can be used as "seeds" for rules, which are then generalized via VI-Analysis. 2. General-to-specific. An alternative way to extract rules, which has been studied in more detail elsewhere [14], works from general to specific. General-to-specific rule search maintains a list of non-proven conjectures, R. R is initialized with the most general rules (like "everything is in C" and "nothing is in C"). VI-Analysis is then applied to prove rules in R. If it successfully confirms a rule, the rule and its complement is removed from R. If not, the rule is removed, too, but instead new rules are added to R. These new rules form a specialized version of the old rule, so that their disjunct is exactly the old rule. For example, new rules can be generated by splitting the hypercube spanned by the old rule into disjoint regions, one for each new rule. Then, the new set R is checked with VI-Analysis. The whole procedure continues till R is empty and the whole input domain is described by rules. In discrete domains, such a strategy amounts to searching directed acyclic graphs in breadth-first manner. Obviously, there is a variety of alternative techniques to generate meaningful rule hypotheses. For example, one might employ a symbolic learning technique such as decision tree learning [10] to the same training data that was used for training the network. The rules, which are a result of the symbolic approach, constitute hypotheses that can be checked using VI-Analysis. 4 Empirical Results In this section we will be interested in extracting rules in a real-valued robot arm domain. We trained a neural network to model the forward kinematics function of a 5 degree-of-freedom robot arm. The arm, a Mitsubishi RV-Ml, is depicted in Fig. 2. Its kinematic function determines the position of the tip of the manipulator in (x, y, z) workspace coordinates and 510 Sebastian Thrun coverage first 10 rules first 100 rules first 1 000 rules first 10000 rules average (per rule) 9.79% 2.59% 1.20% 0.335% cumulative 30.2% 47.8% 61.6% 84.4% Table 1: Rule coverage in the robot arm domain. These numbers inc1ude rules for both concepts, SAFE and UNSAFE. the angle of the manipulator h to the table based on the angles of the five joints. As can be seen in Fig. 2, the workspace intersects with the table on which the arm is mounted. Hence, some configurations of the joints are safe, namely those for which z ~ 0, whiJe others can physically not be reached without a col1ision that would damage the robot (unsafe). When operating the robot arm one has to be able to tell safe from unsafe. Henceforth, we are interested in a set of rules that describes the subspace of safe and unsafe joint configurations. A total of 8192 training examples was used for training the network (four input, five hidden and four output units), resulting in a considerably accurate model of the kinematics of the robot arm. Notice that the network operates in a continuous space. Obviously, compiling the network into logical rules node-by-node, as frequently done in other approaches to rule extraction, is difficult due to the real-valued and distributed nature of the internal representation. Instead, we applied VI-Analysis using a specific-to-general mechanism as described above. More specifically, we incrementally constructed a collection of rules that gradually covered the workspace of the robot arm. Rules were generated whenever a (random) joint configuration was not covered by a previously generated rule. Table 1 shows average results that characterize the extraction of rules. Initially, each rule covers a rather large fraction of the 5-dimensional joint configuration space. As few as 11 rules, on average, suffice to cover more than 50% (by volume) of the whole input space. However, these 50% are the easy half. As the domain gets increasingly covered by rules, gradually more specific rules are generated in regions closer to the c1ass boundary. After extracting 10,000 rules, only 84.4% of the input space is covered. Since the decision boundary between the two c1asses is highly non-linear, finitely many rules will never cover the input space completely. How general are the rules extracted by VI-Analysis? Genera])y speaking, for joint configurations c10se to the c1ass boundary, i.e., where the tip of the manipulator is close to the table, we observed that the extracted rules were rather specific. If instead the initial configuration was closer to the center of a class, VI-Analysis was observed to produce more general rules that had a larger coverage in the workspace. Here VI-Analysis managed to extract surprisingly general rules. For example, the configuration a = (30 0 ,80 0 ,200 ,600 , -200 ), which is depicted in Fig. 3, yields the rule !!.a2 ~ 90.5 0 and a3 ~ 27.3 0 then SAFE. Notice that out of 10 initial constraints, 8 were successfully removed by VI-Analysis. The rule lacks both bounds on a), a4 and as and the lower bounds on a2 and a3. Fig. 3a shows the front view of the initial arm configuration and the generalized rule (grey area). Fig. 3b shows a side view of the arm, along with a slice of the rule (the base joint a) is kept fixed). Notice that this very rule covers 17.1 % of the configuration space (by volume). Such general rules were frequently found in the robot arm domain. This conc1 udes the brief description of the experimental results. Not mentioned here are results with different size networks, and results obtained for the MONK's benchmark problems. For example, in the MONK's problems [15], VI-Analysis successfully extracted compact target Extracting Rules from Artificial Neural Networks with Distributed Representations 511 Figure 3: A single rule, extracted from the network. (a) Front view. (b) Two-dimensional side view. The grey area indicates safe positions for the tip of the manipulator. concepts using the originally published weight sets. These results can be found in [14]. 5 Discussion In this paper we have presented a mechanism for the extraction of rules from Backpropagationstyle neural networks. There are several limitations of the current approach that warrant future research. (a) Speed. While the one-to-one compilation of networks into rules is fast, rule extraction via VI-Analysis requires mUltiple runs of linear programming, each of which can be computationally expensive [9]. Searching the rule space without domain-specific search heuristics can thus be a most time-consuming undertaking. In all our experiments, however, we observed reasonably fast convergence of the VI-Algorithm, and we successfully managed to extract rules from larger networks in reasonable amounts of time. Recently, Craven and Shavlik proposed a more efficient search method which can be applied in conjunction with VI-Analysis [2]. (b) Language. Currently VI-Analysis is limited to the extraction of if-then rules with linear preconditions. While in [14] it has been shown how to generalize VI-Analysis to rules expressed by arbitrary linear constraints, a more powerful rule language is clearly desirable. (c) Linear optimization. Linear programming analyzes multiple weight layers independently, resulting in an overly careful refinement of intervals. This effect can prevent from detecting correct rules. If linear programming is replaced by a non-linear optimization method that considers multiple weight layers simultaneously, more powerful rules can be generated. On the other hand, efficient non-linear optimization techniques might find rules which do not describe the network accurately. Moreover, it is generally questionable whether there will ever exist techniques for mapping arbitrary networks accurately into compact rule sets. Neural networks are their own best description, and symbolic rules might not be appropriate for describing the input-output behavior of a complex neural network. A key feature of of the approach presented in this paper is the particular way rules are extracted. Unlike other approaches to the extraction of rules, this mechanism does not compile networks into structurally equivalent set of rules. Instead it analyzes the input output relation of networks as a whole. As a consequence, rules can be extracted from unstructured networks with distributed and real-valued internal representations. In addition, the extracted rules describe the neural network accurately, regardless of the size of the network. This makes VI-Analysis a promising candidate for scaling rule extraction techniques to deep networks, in which approximate rule extraction methods can suffer from cumulative errors. We conjecture that such properties are important if meaningful rules are to be extracted in today's and tomorrow's successful Backpropagation applications. 512 Sebastian Thrun Acknowledgment The author wishes to express his gratitude to Marc Craven, Tom Dietterich, Clayton McMillan. Tom Mitchell and Jude Shavlik for their invaluable feedback that has influenced this research . References [I] M. W. Craven and J. W. Shavlik. Learning symbolic rules using artificial neural networks. In Paul E. 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From Data Distributions to Regularization in Invariant Learning Todd K. Leen Department of Computer Science and Engineering Oregon Graduate Institute of Science and Technology 20000 N.W. Walker Rd Beaverton, Oregon 97006 tieen@cse.ogi.edu Abstract Ideally pattern recognition machines provide constant output when the inputs are transformed under a group 9 of desired invariances. These invariances can be achieved by enhancing the training data to include examples of inputs transformed by elements of g, while leaving the corresponding targets unchanged. Alternatively the cost function for training can include a regularization term that penalizes changes in the output when the input is transformed under the group. This paper relates the two approaches, showing precisely the sense in which the regularized cost function approximates the result of adding transformed (or distorted) examples to the training data. The cost function for the enhanced training set is equivalent to the sum of the original cost function plus a regularizer. For unbiased models, the regularizer reduces to the intuitively obvious choice a term that penalizes changes in the output when the inputs are transformed under the group. For infinitesimal transformations, the coefficient of the regularization term reduces to the variance of the distortions introduced into the training data. This correspondence provides a simple bridge between the two approaches. Todd Leen 22 4 1 A pproaches to Invariant Learning In machine learning one sometimes wants to incorporate invariances into the function learned. Our knowledge of the problem dictates that the machine outputs ought to remain constant when its inputs are transformed under a set of operations gl. In character recognition, for example, we want the outputs to be invariant under shifts and small rotations of the input image. In neural networks, there are several ways to achieve this invariance 1. The invariance can be hard-wired by weight sharing in the case of summation nodes (LeCun et al. 1990) or by constraints similar to weight sharing in higher-order nodes (Giles et al. 1988). 2. One can enhance the training ensemble by adding examples of inputs transformed under the desired inval"iance group, while maintaining the same targets as for the raw data. 3. One can add to the cost function a regularizer that penalizes changes in the output when the input is transformed by elements of the group (Simard et al. 1992). Intuitively one expects the approaches in 3 and 4 to be intimately linked. This paper develops that correspondence in detail. 2 The Distortion-Enhanced Input Ensemble Let the input data x be distributed according to the density function p( x). The conditional distribution for the corresponding targets is denoted p(tlx). For simplicity of notation we take t E R. The extension to vector targets is trivial. Let f(x; w) denote the network function, parameterized by weights w. The training procedure is assumed to minimize the expected squared error ?(w) = JJdtdx p(tlx) p(x) [t - f(x; w)]2 . (1) vVe wish to consider the effects of adding new inputs that are related to the old by transformations that correspond to the desired invariances. These transformations, or distortions, of the inputs are carried out by group elements g E g. For Lie groups2, the transformations are analytic functions of parameters a E Rk x -t x' = g(x;a) , (2) with the identity transformation corresponding to parameter value zero g(x;O) = x . (3) In image processing, for example, we may want our machine to exhibit invariance with respect to rotation, scaling, shearing and translations of the plane. These 1 We assume that the set forms a group. 2See for example (Sattillger, 1986). From Data Distributions to Regularization in Invariant Learning 225 transformations form a six-parameter Lie group3. By adding distorted input examples we alter the original density p( x). To describe the new density, we introduce a probability density for the transformation parameters p(a). Using this density, the distribution for the distortion-enhanced input ensemble is p(x') = j j dadx p(x'lx,a) p(a) p(x) = j jdadxt5(x'-g(x;a?p(a)p(x) where t5(.) is the Dirac delta function 4 Finally we impose that the targets remain unchanged when the inputs are transformed according to (2) i.e., p(tlx') = p(tlx). Substituting p(x') into (1) and using the invariance of the targets yields the cost function t J = j j dtdxda p(tlx)p(x)p(a) [t - f(g(x;a);w)]2 (4) Equation (4) gives the cost function for the distortion-enhanced input ensemble. 3 Regularization and Hints The remainder of the paper makes precise the connection between adding transformed inputs, as embodied in (4), and various regularization procedures. It is straightforward to show that the cost function for the distortion-enhanced ensemble is equivalent to the cost function for the original data ensemble (1) plus a regularization term. Adding and subtracting f(x; w) to the term in square brackets in (4), and expanding the quadratic leaves t = E + ER , (5) where the regularizer is ER = EH J + Ec da p(a) j dx p(x) [f(x, w) - f(g(x; a); w)]2 - 2 JJ j dtdxda p(tlx) p(x) p(a) x[t - f{x;w)] [f(g(x;a);w) - f(x;w)] (6) 3The parameters for rotations, scaling and shearing completely specify elements of G L2, the four parameter group of 2 x 2 invertible matrices. The translations carry an additional two degrees of freedom. 4 In general the density on 0 might vary through the input space, suggesting the conditional density p(o I :1'). This introduces rather minor changes in the discussion that will not be considered here. 226 ToddLeen Training with the original data ensemble using the cost function (5) is equivalent to adding transformed inputs to the data ensemble. The first term of the regularizer ?H penalizes the average squared difference between I(x;w) and I(g(x;a);w). This is exactly the form one would intuitively apply in order to insure that the network output not change under the transformation x -4 g( x, a). Indeed this is the similar to the form of the invariance "hint" proposed by Abu-Mostafa (1993). The difference here is that there is no arbitrary parameter multiplying the term. Instead the strength of the regularizer is governed by the average over the density pea). The term ?H measures the error in satisfying the invariance hint. The second term ?a measures the correlation between the error in fitting to the data, and the errol' in satisfying the hint. Only when these correlations vanish is the cost function for the enhanced ensemble equal to the original cost function plus the invariance hint penalty. The correlation term vanishes trivially when either 1. The invariance I (g( x; a); w) = I (x; w) is satisfied, or 2. The network function equals the least squares regression on t I(x; w) = J dt p(tlx) t =E[tlx] . (7) The lowest possible ? occurs when I satisfies (7), at which ? becomes the variance in the targets averaged over p( x ). By substituting this into ?a and carrying out the integration over dt p( tlx), the correlation term is seen to vanish. If the minimum of t occurs at a weight for which the invariance is satisfied (condition 1 above). then minimizing t (w) is equivalent to minimizing ? (w). If the minimum of t occurs at a weight for which the network function is the regression (condition 2), then minimizing t is equivalent to minimizing the cost function with the intuitive regularizer ? H 5. 3.1 Infinitesimal Transformations Above we enumerated the conditions under which the correlation term in ?R vanishes exactly for unrestricted transformations. If the transformations are analytic in the paranleters 0', then by restricting ourselves to small transformations (those close to the identity) we can-show how the correlation term approximately vanishes for unbiased models. To implement this, we assume that p( a) is sharply peaked up about the origin so that large transformations are unlikely. 51? the data is to be fit optimally, with enough freedom left over to satisfy the invariance hint, then there must be several weight values (perhaps a continuum of such values) for which the network function satisfies (7). That is, the problem must be under-specified. If this is the case, then the interesting part weight space is just the subset on which (7) is satisfied. On this subset the correlation term in (6) vanishes and the regularizer assumes the intuitive form. From Data Distributions to Regularization in Invariant Learning 227 t We obtain an approximation to the cost function by expanding the integrands in (6) in power series about 0 = 0 and retaining terms to second order. This leaves t = c+ JJ JJJ dxdo p(x) p(o) (Oi :~ L=o :!" f dt dx do p(tlx)p(x)p(o) [t-f(x;w)] x -2 [ ( Og"l 0?- ? OOi 0=0 0 2 gIl 1 2' +-0?0? J OOi (0) I0=0 )(ax"of) - (8) where x P and gP denote the pth components of x and g, OJ denotes the ith component of the transformation parameter vector, repeated Greek and Roman indices are summed over, and all derivatives are evaluated at 0 o. Note that we have used the fact that Lie group transformations are analytic in the parameter vector o to derive the expansion. = Finally we introduce two assumptions on the distribution p(o). First 0 is assumed to be zero mean. This corresponds, in the linear approximation, to a distribution of distortions whose mean is the identity transformation. Second, we assume that the components of 0 are uncorrelated so that the covariance matrix is diagonal with elements ul, i = 1 ... k. 6 With these assumptions, the cost function for the distortion-enhanced ensemble simplifies to t = J p( - L (T~ JJ c+~ (Tr ~ .=1 dx x) ( ~g"l va. a=O : f vx" ) 2 k dx dt p(tlx) p(X) { (f(x; w) - t ) .=1 X [ ~:; 10=0 ( :~ ) + :~ L=o :~ L=o (ox~2?x" )]} This last expression provides a simple bridge between the the methods of adding transformed examples to the data, and the alternative of adding a regularizer to the cost function: The coefficient of the regularization term in the latter approach is equal to the variance of the transformation parameters in the former approach. 6Note that the transformed patterns may be correlated in parts of the pattern space. For example the results of applying the shearing and rotation operations to an infinite vertical line are indistinguishable. In general, there may be regions of the pattern space for which the action of several different group elements are indistinguishable; that is x' = g(x; a) = g(x; (3). However this does not imply that a and (3 are statistically correlated. 228 3.1.1 Todd Leen Unbiased Models For unbiased models the regularizer in E( w) assumes a particularly simple form. Suppose the network function is rich enough to form an unbiased estimate of the least squares regression on t for the un distorted data ensemble. That is, there exists a weight value Wo such that f(x;wo) =J (10) dt tp(tlx) == E[tlx] This is the global minimum for the original error ?( w). The arguments of section 3 apply here as well. However we can go further. Even if there is only a single, isolated weight value for which (10) is satisfied, then to O( 0- 2 ) the correlation term in the regularizer vanishes. To see this note that by the implicit function theorem the modified cost function (9) has its global minimum at the new weight 7 (11) At this weight, the network function is no longer the regression on t, but rather f(x;wo) = E[tlx] + 0(0- 2 ) (12) ? Substituting (12) into (9), we find that the minimum of (9) is, to 0(0- 2 ), at the same weight as the minimum of t = ? + L.k .=1 o-~ JdX p(x) [ oglJ 0 Q'j I Q=O of (x, w) ] 2 oxlJ To 0(0- 2 ), minimizing (13) is equivalent to minimizing (9). So we regard effective cost function. (13) t as the The regularization term in (13) is proportional to the average square of the gradient of the network function along the direction in the input space generated by the lineal' part of g. The quantity inside the square brackets is just the linear part of [f (g( X; Q')) - f (x)] from (6). The magnitude of the regularization term is just the variance of the distribution of distortion parameters. This is precisely the form of the regularizer given by Simard et al. in their tangent prop algorithm (Simard et aI, 1992). This derivation shows the equivalence (to 0(0"2)) between the tangent prop regularizer and the alternative of modifying the input distribution. Furthermore, we see that with this equivalence, the constant fixing the strengt.h of the regularization term is simply the variance of the distortions introduced into the original training set. We should stress that the equivalence between the regularizer, and the distortionenhanced ensemble in (13) only holds to 0(0- 2 ). If one allows the variance of the 7We assume that the Hessian of ? is nonsingular at woo From Data Distributions to Regularization in Invariant Learning 229 distortion parameters u 2 to become arbitrarily large in an effort to mock up an arbitrarily large regularization term, then the equivalence expressed in (13) breaks down since terms of order O( ( 4 ) can no longer be neglected. In addition, if the transformations are to be kept small so that the linearization holds (e.g. by restricting the density on a to have support on a small neighborhood of zero), then the variance will bounded above. 3.1.2 Smoothing Regularizers In the previous sections we showed the equivalence between modifying the input distribution and adding a regularizer to the cost function. We derived this equivalence to illuminate mechanisms for obtaining invariant pattern recognition. The technique for dealing with infinitesimal transformations in section ?3.1 was used by Bishop (1994) to show the equivalence between added input noise and smoothing regularizers. Bishop's results, though they preceded our own, are a special case of the results presented here. Suppose the group 9 is restricted to translations by random vectors g( X; a) X + a where a is spherically distributed with variance u!. Then the regularizer in (13) is = (14) This regularizer penalizes large magnitude gradients in the network function and is, as pointed out by Bishop, one of the class of generalized Tikhonov regularizers. 4 Summary We have shown that enhancing the input ensemble by adding examples transformed under a group x -? g(x;a), while maintaining the target values, is equivalent to adding a regularizer to the original cost function. For unbiased models the regulatizer reduces to the intuitive form that penalizes the mean squared difference between the network output for transformed and untransformed inputs - i.e. the error in satisfying the desired invariance. In general the regularizer includes a term that measures correlations between the error in fitting the data, and the error in satisfying the desired inva.riance. For infinitesimal transformations, the regularizer is equivalent (up to terms linear in the variance of the transformation parameters) to the tangent prop form given by Simard et a1. (1992), with regularization coefficient equal to the variance of the transformation parameters. In the special case that the group transformations are limited to random translations of the input, the regularizer reduces to a standard smoothing regularizer. \Ve gave conditions under which enhancing the input ensemble and adding the intuitive regularizer ?H are equivalent. However tins equivalence is only with regard to the optimal weight. We have not compared the training dynamics for the two approaches. In particular, it is quite possible that the full regularizer ?H + ?c exhibits different training dynamics from the intuitive form ?H. For the approach in which data are added to the input ensemble, one can easily construct datasets and distributions p( a) that either increase the condition number of the Hessian, or decrease it. Finally, it may be that the intuitive regularizer can have either detrimental or positive effects on the Hessian as well. 230 ToddLeen Acknowledgments I thank Lodewyk Wessels, Misha Pavel, Eric Wan, Steve Rehfuss, Genevieve Orr and Patrice Simard for stimulating and helpful discussions, and the reviewers for helpful comments. I am grateful to my father for what he gave to me in life, and for the presence of his spirit after his recent passing. This work was supported by EPRI under grant RP8015-2, AFOSR under grant FF4962-93-1-0253, and ONR under grant N00014-91-J-1482. References Yasar S. Abu-Mostafa. A method for learning from hints. In S. Hanson, J. Cowan, and C. Giles, editors, Advances in Neural Information Processing Systems, vol. 5, pages 73-80. Morgan Kaufmann, 1993. Chris M. Bishop. Training with noise is equivalent to Tikhonov regularization. To appear in Neural Computation, 1994. C.L. Giles, R.D. Griffin, and T. Maxwell. Encoding geometric invariances in higherorder neural networks. In D.Z.Anderson, editor, Neural Information Processing Systems, pages 301-309. American Institute of Physics, 1988. Y. Le Cun, B. Boser, J.S. Denker, D. Henderson, R.E. Howard, W. Hubbard, and L.D. Jackel. Handwritten digit recognition with a back-propagation network. In Advances in Neural Information Processing Systems, vol. 2, pages 396-404. Morgan Kaufmann Publishers, 1990. Patrice Simard, Bernard Victorri, Yann Le Cun, and John Denker. Tangent prop a formalism for specifying selected invariances in an adaptive network. In John E. Moody, Steven J. Hanson, and Richard P. Lippmann, editors, Advances in Neural Information Processing Systems 4, pages 895-903. Morgan Kaufmann, 1992. D.H. Sattinger and O.L. Weaver. 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On the Computational Complexity of Networks of Spiking Neurons (Extended Abstract) Wolfgang Maass Institute for Theoretical Computer Science Technische Universitaet Graz A-80lO Graz, Austria e-mail: maass@igi.tu-graz.ac.at Abstract We investigate the computational power of a formal model for networks of spiking neurons, both for the assumption of an unlimited timing precision, and for the case of a limited timing precision. We also prove upper and lower bounds for the number of examples that are needed to train such networks. 1 Introduction and Basic Definitions There exists substantial evidence that timing phenomena such as temporal differences between spikes and frequencies of oscillating subsystems are integral parts of various information processing mechanisms in biological neural systems (for a survey and references see e.g. Abeles, 1991; Churchland and Sejnowski, 1992; Aertsen, 1993). Furthermore simulations of a variety of specific mathematical models for networks of spiking neurons have shown that temporal coding offers interesting possibilities for solving classical benchmark-problems such as associative memory, binding, and pattern segmentation (for an overview see Gerstner et al., 1992). Some aspects of these models have also been studied analytically, but almost nothing is known about their computational complexity (see Judd and Aihara, 1993, for some first results in this direction). In this article we introduce a simple formal model SNN for networks of spiking neurons that allows us to model the most important timing phenomena of neural nets (including synaptic modulation), and we prove upper and lower bounds for its computational power and learning complexity. Further 184 Wolfgang Maass details to the results reported in this article may be found in Maass, 1994a,1994b, 1994c. An SNN N consists of - a finite directed graph {V, E} (we refer to the elements of V as "neurons" and to the elements of E as "synapses") - a subset Yin S; V of input neurons - a subset Vout S; V of output neurons Definition of a Spiking Neuron Network (SNN): - for each neuron v E V - Yin a threshold-function 9 v : R+ {where R+ := {x E R : x ~ O}) - for each synapse {u, v} E E a response-function weight- function Wu,v : R+ -+ R . ?u,v : -+ R+ R U {oo} -+ R and a We assume that the firing of the input neurons v E Yin is determined from outside of N, i.e. the sets Fv S; R+ of firing times ("spike trains") for the neurons v E Yin are given as the input of N. Furthermore we assume that a set T S; R+ of potential firing times 7iiiSfjeen fixed. For a neuron v E V - Yin one defines its set Fv of firing times recursively. The first element of Fv is inf{t E T : Pv(t) ~ 0 v (O)} , and for any s E Fv the next larger element of Fv is inf{t E T : t > sand Pv(t) ~ 0 v (t - s)} ,where the potential function Pv : R+ -+ R is defined by Pv(t) := 0 + L L wu,v(s) . ?u,v(t - s) u : {u, v} E EsE Fu : s < t The firing times ("spike trains") Fv of the output neurons v E Vout that result in this way are interpreted as the output of N. Regarding the set T of potential firing times we consider in this article the case T = R+ (SNN with continuous time) and the case T = {i? JJ : i E N} for some JJ with 1/ JJ E N (SNN with discrete time). We assume that for each SNN N there exists a bound TN E R with TN > 0 such that 0 v(x) = 00 for all x E (0, TN) and all v E V - Yin (TN may be interpreted as the minimum of all "refractory periods" Tref of neurons in N). Furthermore we assume that all "input spike trains" Fv with v E Yin satisfy IFv n [0, t]l < 00 for all t E R+. On the basis of these assumptions one can also in the continuous case easily show that the firing times are well-defined for all v E V - Yin (and occur in distances of at least TN)' Input- and Output-Conventions: For simulations between SNN's and Turing machines we assume that the SNN either gets an input (or produces an output) from {O, 1}* in the form of a spike-train (i.e. one bit per unit of time), or encoded into the phase-difference of just two spikes. Real-valued input or output for an SNN is always encoded into the phase-difference of two spikes. Remarks a) In models for biological neural systems one assumes that if x time-units have On the Computational Complexity of Networks of Spiking Neurons /85 passed since its last firing, the current threshold 0 11 (z) of a neuron v is "infinite" for z < TreJ (where TreJ = refractory period of neuron v), and then approaches quite rapidly from above some constant value. A neuron v "fires" (i.e. it sends an "action potential" or "spike" along its axon) when its current membrane potential PII (t) at the axon hillock exceeds its current threshold 0 11 . PII (t) is the sum of various postsynaptic potentials W U ,II(S). t: U ,II(t - s). Each of these terms describes an excitatory (EPSP) or inhibitory (IPSP) postsynaptic potential at the axon hillock of neuron v at time t, as a result of a spike that had been generated by a "presynaptic" neuron u at time s, and which has been transmitted through a synapse between both neurons. Recordings of an EPSP typically show a function that has a constant value c (c = resting membrane potential; e.g. c = -70m V) for some initial time-interval (reflecting the axonal and synaptic transmission time), then rises to a peak-value, and finally drops back to the same constant value c. An IPSP tends to have the negative shape of an EPSP. For the sake of mathematical simplicity we assume in the SNN-model that the constant initial and final value of all response-functions t: U ,1I is equal to 0 (in other words: t: U ,1I models the difference between a postsynaptic potential and the resting membrane potential c). Different presynaptic neurons u generate postsynaptic potentials of different sizes at the axon hillock of a neuron v, depending on the size, location and current state of the synapse (or synapses) between u an? v. This effect is modelled by the weight-factors W U ,II(S). The precise shapes of threshold-, response-, and weight-functions vary among different biological neural systems, and even within the same system. Fortunately one can prove significant upper bounds for the computational complexity of SNN's N without any assumptions about the specific shapes of these functions of N. Instead, we only assume that they are of a reasonably simple mathematical structure. b) In order to prove lower bounds for the computational complexity of an SNN N one is forced to make more specific assumptions about these functions . All lower bound results that are reported in this article require only some rather weak basic assumptions about the response- and threshold-functions. They mainly require that EPSP's have some (arbitrarily short) segment where they increase linearly, and some (arbitrarily short) segment where they decrease linearly (for details see Maass, 1994a, 1994b). c) Although the model SNN is apparently more "realistic" than all models for biological neural nets whose computational complexity has previously been analyzed, it deliberately sacrifices a large number of more intricate biological details for the sake of mathematical tractability. Our model is closely related to those of (Buhmann and Schulten, 1986), and (Gerstner, 1991, 1992). Similarly as in (Buhmann and Schulten, 1986) we consider here only the deterministic case. d) The model SNN is also suitable for investigating algorithms that involve synaptic modulation at various time-scales. Hence one can investigate within this framework not only the complexity of algorithms for supervised and unsupervised learning, but also the potential computational power of rapid weight-changes within the course of a computation. In the theorems of this paper we allow that the value of a weight W U,II(S) at a firing time s E Fu is defined by an algebraic computation tree (see van Leeuwen, 1990) in terms of its value at previous firing times s' E Fu with s' < s, some preceding firing times s < s of arbitrary other neurons, and arbitrary realvalued parameters. In this way WU,II(S) can be defined by different rational functions /86 Wolfgang Maass of the abovementioned arguments, depending on the numerical relationship between these arguments (which can be evaluated by comparing first the relative size of arbitrary rational functions of these arguments). As a simple special case one can for example increase wu ?tI (perhaps up to some specified saturation-value) as long as neurons u and v fire coherently, and decrease w u ?tI otherwise. For the sake of simplicity in the statements of our results we assume in this extended abstract that the algebraic computation tree for each weight w U ? tI involves only 0(1) tests and rational functions of degree 0(1) that depend only on 0(1) of the abovementioned arguments. Furthermore we assume in Theorems 3, 4 and 5 that either each weight is an arbitrary time-invariant real, or that each current weight is rounded off to bit-length poly(1ogpN') in binary representation, and does not depend on the times of firings that occured longer than time 0(1) ago. Furthermore we assume in Theorems 3 and 5 that the parameters in the algebraic computation tree are rationals of bit-length O(1ogpN'). e) It is well-known that the Vapnik-Chervonenkis dimension {"VC-dimension"} of a neural net N (and the pseudo-dimension for the case of a neural net N with realvalued output, with some suitable fixed norm for measuring the error) can be used to bound the number of examples that are needed to train N (see Haussler, 1992). Obviously these notions have to be defined differently for a network with timedependent weights. We propose to define the VC-dimension (pseudo-dimension)of an SNN N with time-dependent weights as the VC-dimension (pseudo-dimension) of the class of all functions that can be computed by N with different assignments of values to the real-valued (or rational-valued) parameters of N that are involved in the definitions of the piecewise rational response-, threshold-, and weight-functions of N. In a biological neural system N these parameters might for example reflect the concentrations of certain chemical substances that are known to modulate the behavior of N. f) The focus in the investigation of computations in biological neural systems differs in two essential aspects from that of classical computational complexity theory. First, one is not only interested in single computations of a neural net for unrelated inputs z, but also in its ability to process an interrelated sequence ?(z( i), y( i)} )ieN of inputs and outputs, which may for example include an initial training sequence for learning or associative memory. Secondly, exact timing of computations is allimportant in biological neural nets, and many tasks have to be solved within a specific number of steps. Therefore an analysis in terms of the notion of a real-time computation and real-time simulation appears to be more adequate for models of biological neural nets than the more traditional analysis via complexity classes. One says that a sequence ?(z(i),y(i)})ieN is processed in real-time by a machine M, if for every i E N the machine M outputs y( i) within a constant number c of computation steps after having received input z(i). One says that M' simulates M in real-time (with delay factor ~), if every sequence that is processed in real-time by M (with some constant c), can also be processed in real-time by M' (with a constant ~ . c). For SNN's M we count each spike in M as a computation step. These definitions imply that a real-time simulation of M by M' is a special case of a linear-time simulation, and hence that any problem that can be solved by M with a certain time complexity ten), can be solved by M' with time complexity O(t(n? On the Computational Complexity of Networks of Spiking Neurons 187 (see Maass, 1994a, 1994b, for details). 2 Networks of Spiking Neurons with Continuous Time Theorem 1: If the response- and threshold-functions of the neurons satisfy some rather weak basic assumptions (see Maass, 1994a, 1994b), then one can build from such neurons for any given dEN an SNN NTM(d) of finite size with rational delays that can simulate with a suitable assignment of rational values from [0, 1] to its weights any Turing machine with at most d tapes in real-time. Furthermore NTM(2) can compute any function F : {0,1}* -- {0,1}* with a suitable assignment of real values from [0,"1] to its weights. The fixed SNN NTM(d) of Theorem 1 can simulate Turing machines whose tape content is much larger than the size of NTM (d), by encoding such tape content into the phase-difference between two oscillators. The proof of Theorem 1 transforms arbitrary computations of Turing machines into operations on such phase-differences. The last part of Theorem 1 implies that the VC-dimension of some finite SNN's is infinite. In contrast to that the following result shows that one can give finite bounds for the VC-dimension of those SNN's that only use a bounded numbers of spikes in their computation. Furthermore the last part of the claim of Theorem 2 implies that their VC-dimension may in fact grow linearly with the number S of spikes that occur in a computation. Theorem 2: The VC-dimension and pseudo-dimension of any SNN N with piecewise linear response- and threshold-functions, arbitrary real-valued parameters and time-dependent weights (as specified in section 1) can be bounded (even for realvalued inputs and outputs) by D(IEI . WI . S(log IVI + log S? if N uses in each computation at most S spikes. Furthermore one can construct SNN's (with any response- and threshold-functions that satisfy our basic assumptions, with fixed rational parameters and rational timeinvariant weights) whose VC-dimension is for computations with up to S spikes as large as O(IEI . S). We refer to Maass, 1994a, 1994c, for upper bounds on the computational power of SNN's with continuous time. 3 Networks of Spiking Neurons with Discrete Time In this section we consider the case where all firing times of neurons in N are multiples of some J.l with 1/ J.l EN. We restrict our attention to the biologically plausible case where there exists some tN ~ 1 such that for all z > tN all response functions ?U,II(Z) have the value and all threshold functions ell(z) have some arbitrary constant value. If tN is chosen minimal with this property, we refer to PN := rtN/J.ll as the timing-precision ofN. Obviously for PN = 1 the SNN is equivalent to a "non-spiking" neural net that consists of linear threshold gates, whereas a SNN with continuous time may be viewed as the opposite extremal case for PN -- 00. ? 188 Wolfgang Maass The following result provides a significant upper bound for the computational power of an SNN with discrete time, even in the presence of arbitrary real-valued parameters and weights. Its proof is technically rather involved. Theorem 3: Assume that N is an SNN with timing-precision PJII, piecewise polynomial response- and piecewise rational threshold-functions with arbitrary real-valued parameters, and weight-functions as specified in section 1. Then one can simulate N for boolean valued inputs in real-time by a Turing machine with poly(lVl, logpJII,log l/TJII) states and poly(lVl, logpJII, tJII/TJII) tape-cells. On the other hand any Turing machine with q states that uses at most s tapecells can be simulated in real-time by an SNN N with any response- and thresholdfunctions that satisfy our basic assumptions, with rational parameters and timeinvariant rational weights, with O(q) neurons, logpJII = O(s), and tJII/TJII = 0(1). The next result shows that the VC-dimension of any SNN with discrete time is finite, and grows proportionally to logpJII. The proof of its lower bound combines a new explicit construction with that of Maass, 1993. Theorem 4: Assume that the SNN N has the same properties as in Theorem 3. Then the VC-dimension and the pseudo-dimension of N (for arbitrary real valued inputs) can be bounded by O(IEI?IVI?logpJII), independently of the number of spikes in its computations. Furthermore one can construct SNN's N of this type with any response- and threshold-functions that satisfy our basic assumptions, with rational parameters and time-invariant rational weights, so that N has (already for boolean inputs) a VCdimension of at least O(IEI(logpJII + log IE!?. 4 Relationships to other Computational Models We consider here the relationship between SNN's with discrete time and recurrent analog neural nets. In the latter no "spikes" or other non-trivial timing-phenomena occur, but the output of a gate consists of the "analog" value of some squashingor activation function that is applied to the weighted sum of its inputs. See e.g. (Siegelmann and Sontag, 1992) or (Maass, 1993) for recent results about the computational power of such models. We consider in this section a perhaps more "realistic" version of such modelsN, where the output of each gate is rounded off to an integer multiple of some ~ (with a EN). We refer to a as the number of activation levels of N. It is an interesting open problem whether such analog neural nets (with gate-outputs interpreted as firing rates) or networks of spiking neurons provide a more adequate computational model for biological neural systems. Theorem 5 shows that in spite of their quite different structure the computational power of these two models is in fact closely related. On the side the following theorem also exhibits a new subclass of deterministic finite automata (DFA's) which turns out to be of particular interest in the context of neural nets. We say that a DFA M is a sparse DFA of size s if M can be realized by a Turing machine with s states and space-bound s (such that each step of M corresponds to one step of the Turing machine). Note that a sparse DFA may have exponentially in s many states, but that only poly(s) bits are needed to describe its On the Computational Complexity of Networks of Spiking Neurons 189 transition function. Sparse DFA's are relatively easy to construct, and hence are very useful for demonstrating (via Theorem 5) that a specific task can be carried out on a "spiking" neural net with a realistic timing precision (respectively on an analog neural net with a realistic number of activation levels). Theorem 5: The following classes of machines have closely related computational power in the sense that there is a polynomial p such that each computational model from any of these classes can be simulated in real-time (with delay-factor ~ p(s?) by some computational model from any other class (with the size-parameter s replaced by p(s?): ? sparse DFA's of size s ? SNN's with 0(1) neurons and timing precision 2 3 ? recurrent analog neural nets that consist of O( 1) gates with piecewise rational activation functions with 23 activation levels, and parameters and weights of bit-length $ s ? neural nets that consist of s linear threshold gates (with recurrencies) with arbitrary real weights. The result of Theorem 5 is remarkably stable since it holds no matter whether one considers just SNN's N with 0(1) neurons that employ very simple fixed piecewise linear response- and threshold-functions with parameters of bit-length 0(1) (with tN/TN = 0(1) and time-invariant weights of bit-length $ s), or if one considers SNN's N with s neurons with arbitrary piecewise polynomial response- and piecewise rational threshold-functions with arbitrary real-valued parameters, tN/TN ~ s, and time-dependent weights (as specified in section 1). 5 Conclusion We have introduced a simple formal model SNN for networks of spiking neurons, and have shown that significant bounds for its computational power and sample complexity can be derived from rather weak assumptions about the mathematical structure of its response-, threshold-, and weight-functions. Furthermore we have established quantitative relationships between the computational power of a model for networks of spiking neurons with a limited timing precision (i.e. SNN's with discrete time) and a quite realistic version of recurrent analog neural nets (with a bounded number of activation levels). The simulations which provide the proof of this result create an interesting link between computations with spike-coding (in an SNN) and computations with frequency-coding (in analog neural nets). We also have established such relationships for the case of SNN's with continuous time (see Maass 1994a, 1994b, 1994c), but space does not permit to report these results in this article. The Theorems 1 and 5 of this article establish the existence of mechanisms for simulating arbitrary Turing machines (and hence any common computational model) on an SNN. As a consequence one can now demonstrate that a concrete task (such as binding, pattern-matching, associative memory) can be carried out on an SNN by simply showing that some arbitrary common computational model can carry out that task. Furthermore one can bound the required timing-precision of the SNN in terms of the space needed on a Turing machine. 190 Wolfgang Maass Since we have based our investigations on the rather refined notion of a real-time simulation, our results provide information not only about the possibility to implement computations, but also adaptive behavior on networks of spiking neurons. Acknowledgement I would like to thank Wulfram Gerstner for helpful discussions. References M. Abeles. (1991) Corticonics: Neural Circuits of the Cerebral Cortex. Cambridge University Press. A. Aertsen. ed. (1993) Brain Theory: Spatio-Temporal Aspects of Brain Function. Elsevier. J. Buhmann, K. Schulten. (1986) Associative recognition and storage in a model network of physiological neurons. Bioi. Cybern. 54: 319-335. P. S. Churchland, T. J. Sejnowski. (1992) The Computational Brain. MIT-Press. W. Gerstner. (1991) Associative memory in a network of "biological" neurons. Advances in Neural Information Processing Systems, vol. 3, Morgan Kaufmann: 84-90. W. Gerstner, R. Ritz, J. L. van Hemmen. (1992) A biologically motivated and analytically soluble model of collective oscillations in the cortex. Bioi. Cybern. 68: 363-374. D. Haussler. (1992) Decision theoretic generalizations of the PAC model for neural nets and other learning applications. Inf and Comput. 95: 129-161. K. T. Judd, K. Aihara. (1993) Pulse propagation networks: A neural network model that uses temporal coding by action potentials. Neural Networks 6: 203-215. J. van Leeuwen, ed. (1990) Handbook of Theoretical Computer Science, vol. A: Algorithms and Complexity. MIT-Press. W. Maass. (1993) Bounds for the computational power and learning complexity of analog neural nets. Proc. 25th Annual ACM Symposium on the Theory of Computing, 335-344. W. Maass. (1994a) On the computational complexity of networks of spiking neurons (extended abstract). TR 393 from May 1994 of the Institutes for Information Processing Graz (for a more detailed version see the file maass.spiking.ps.Z in the neuroprose archive). W. Maass. (1994b) Lower bounds for the computational power of networks of spiking neurons. Neural Computation, to appear. W. Maass. (1994c) Analog computations on networks of spiking neurons (extended abstract). Submitted for publication. H. T. Siegelmann, E. D. Sontag. (1992) On the computational power of neural nets. Proc. 5th ACM- Workshop on Computational Learning Theory, 440-449. 

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Optimal Movement Primitives Terence D. Sanger Jet Propulsion Laboratory MS 303-310 4800 Oak Grove Drive Pasadena, CA 91109 (818) 354-9127 tds@ai.mit .edu Abstract The theory of Optimal Unsupervised Motor Learning shows how a network can discover a reduced-order controller for an unknown nonlinear system by representing only the most significant modes. Here, I extend the theory to apply to command sequences, so that the most significant components discovered by the network correspond to motion "primitives". Combinations of these primitives can be used to produce a wide variety of different movements. I demonstrate applications to human handwriting decomposition and synthesis, as well as to the analysis of electrophysiological experiments on movements resulting from stimulation of the frog spinal cord. 1 INTRODUCTION There is much debate within the neuroscience community concerning the internal representation of movement, and current neurophysiological investigations are aimed at uncovering these representations. In this paper, I propose a different approach that attempts to define the optimal internal representation in terms of "movement primitives" , and I compare this representation with the observed behavior. In this way, we can make strong predictions about internal signal processing. Deviations from the predictions can indicate biological constraints or alternative goals that cause the biological system to be suboptimal. The concept of a motion primitive is not as well defined as that of a sensory primitive 1024 Terence Sanger u p y z Figure 1: Unsupervised Motor Learning: The plant P takes inputs u and produces outputs y. The sensory map C produces intermediate variables z, which are mapped onto the correct command inputs by the motor network N. within the visual system, for example. There is no direct equivalent to the "receptive field" concept that has allowed interpretation of sensory recordings. In this paper, I will propose an internal model that involves both motor receptive fields and a set of movement primitives which are combined using a weighted sum to produce a large class of movements. In this way, a small number of well-designed primitives can generate the full range of desired behaviors. I have previously developed the concept of "optimal unsupervised motor learning" to investigate optimal internal representations for instantaneous motor commands. The optimal representations adaptively discover a reduced-order linearizing controller for an unknown nonlinear plant. The theorems give the optimal solution in general, and can be applied to special cases for which both linear and nonlinear adaptive algorithms exist (Sanger 1994b). In order to apply the theory to complete movements it needs to be extended slightly, since in general movements exist within an infinite-dimensional task space rather than a finite-dimensional control space. The goal is to derive a small number of primitives that optimally encode the full set of observed movements. Generation of the internal movement primitives then becomes a data-compression problem, and I will choose primitives that minimize the resultant mean-squared error. 2 OPTIMAL UNSUPERVISED MOTOR LEARNING Optimal Unsupervised Motor Learning is based on three principles: 1. Dimensionality Reduction 2. Accurate Reduced-order Control 3. Minimum Sensory error Consider the system shown in figure 1. At time t, the plant P takes motor inputs u and produces sensory outputs y. A sensory mapping C transforms the raw sensory data y to an intermediate representation z. A motor mapping takes desired values of z and computes the appropriate command u such that CPu = z. Note that the Optimal Movement Primitives 1025 loop in the figure is not a feedback-control loop, but is intended to indicate the flow of information. With this diagram in mind, we can write the three principles as: 1. dim[z] < dim[y] 2. GPNz=z 3. IIPNGy - yll is minimized We can prove the following theorems (Sanger 1994b): Theorem 1: For all G there exists an N such that G P N z = z. If G is linear and p- 1 is linear, then N is linear. Theorem 2: For any G, define an invertible map C such that GC-l =_1 on range[G]. Then liP NGy - yll is minimized when G is chosen such that Ily - G-1GII is minimized . If G and P are linear and the singular value decomposition of P is given by LT SR, then the optimal maps are G = Land N = RT S-I. For the discussion of movement, the linear case will be the most important since in the nonlinear case we can use unsupervised motor learning to perform dimensionality reduction and linearization of the plant at each time t. The movement problem then becomes an infinite-dimensional linear problem. Previously, I have developed two iterative algorithms for computing the singular value decomposition from input/output samples (Sanger 1994a). The algorithms are called the "Double Generalized Hebbian Algorithm" (DGHA) and the "Orthogonal Asymmetric Encoder" (OAE). DGHA is given by 8G 8N T 'Y(zyT - LT[zzT]G) 'Y(zu T - LT[zzT]N T ) while OAE is described by: 'Y(iyT - LT[ZiT]G) 'Y(Gy - LT[GGT]z)uT where LT[ ] is an operator that sets the above diagonal elements of its matrix argument to zero, y = Pu, z = Gy, z = NT u, and 'Y is a learning rate constant. Both algorithms cause G to converge to the matrix of left singular vectors of P, and N to converge to the matrix of right singular vectors of P (multiplied by a diagonal matrix for DGHA) . DGHA is used in the examples below. 3 MOVEMENT In order to extend the above discussion to allow adaptive discovery of movement primitives, we now consider the plant P to be a mapping from command sequences u(t) to sensory sequences y(t). We will assume that the plant has been feedback linearized (perhaps by unsupervised motor learning). We also assume that the sensory network G is constrained to be linear. In this case, the optimal motor network N will also be linear. The intermediate variables z will be represented by a vector. The sensory mapping consists of a set of sensory "receptive fields" gi(t) 1026 Terence Sanger A .. Motor Map I Sensory Map n1(t)zl +- Zl +- ---lEt): \ : nn(~)~ ~ ~n +- Figure 2: Extension of unsupervised motor learning to the case of trajectories . Plant input and output are time-sequences u(t) and y(t). The sensory and motor maps now consist of sensory primitives gi(t) and motor primitives ni(t). such that Zi = J gj(t)y(t)dt =< gily > and the motor mapping consists of a set of "motor primitives" ni(t) such that u(t) = L ni(t)zi i as in figure 2. If the plant is equal to the identity (complete feedback linearization), then gi(t) = ni(t). In this case, the optimal sensory-motor primitives are given by the eigenfunctions of the autocorrelation function of y(t) . If the autocorrelation is stationary, then the infinite-window eigenfunctions will be sinusoids. Note that the optimal primitives depend both on the plant P as well as the statistical distribution of outputs y(t). In practice, both u(t) and y(t) are sampled at discrete time-points {tic} over a finite time-window, so that the plant input and output is in actuality a long vector. Since the plant is linear, the optimal solution is given by the singular value decomposition, and either the DGHA or OAE algorithms can be used directly. The resulting sensory primitives map the sensory information y(t) onto the finite-dimensional z, which is usually a significant data compression. The motor primitives map Z onto the sequence u(t), and the resulting y(t) = P[u(t)] will be a linear projection of y(t) onto the space spanned by the set {Pni(t)}. 4 EXAMPLE 1: HANDWRITING As a simple illustration, I examine the case of human handwriting. We can consider the plant to be the identity mapping from pen position to pen position, and the Optimal Movement Primitives 1027 1. 5. 2. 6. 3. 7. 4. 8. Figure 3: Movement primitives for sampled human handwriting. 1028 Terence Sanger human to be taking desired sensory values of pen position and converting them into motor commands to move the pen. The sensory statistics then reflect the set of trajectories used in producing handwritten letters. An optimal reduced-order control system can be designed based on the observed statistics, and its performance can be compared to human performance. For this example, I chose sampled data from 87 different examples of lower-case letters written by a single person, and represented as horizontal and vertical pen position at each point in time. Blocks of 128 sequential points were used for training, and 8 internal variables Zi were used for each of the two components of pen position. The training set consisted of 5000 randomly chosen samples. Since the plant is the identity, the sensory and motor primitives are the same, and these are shown as "strokes" in figure 3. Linear combinations of these strokes can be used to generate pen paths for drawing lowercase letters. This is shown in figure 4, where the word "hello" (not present in the training set) is written and projected using increasing numbers of intermediate variables Zi. The bottom of figure 4 shows the sequence of values of Zi that was used (horizontal component only). Good reproduction of the test word was achieved with 5 movement primitives. A total of 7 128-point segments was projected , and these were recombined using smooth 50% overlap . Each segment was encoded by 5 coefficients for each of the horizontal and vertical components, giving a total of 70 coefficients to represent 1792 data points (896 horizontal and vertical components) , for a compression ratio of 25 :1. 5 EXAMPLE 2: FROG SPINAL CORD The second example models some interesting and unexplained neurophysiological results from microstimulation of the frog spinal cord. (Bizzi et al. 1991) measured the pattern of forces produced by the frog hindlimb at various positions in the workspace during stimulation of spinal interneurons. The resulting force-fields often have a stable" equilibrium point" , and in some cases this equilibrium point follows a smooth closed trajectory during tonic stimulation of the interneuron. However, only a small number of different force field shapes have been found, and an even smaller number of different trajectory types . A hypothesis to explain this result is that larger classes of different trajectories can be formed by combining the patterns produced by these cells. This hypothesis can be modelled using the optimal movement primitives described above . Figure 5a shows a simulation of the frog leg. To train the network, random smooth planar movements were made for 5000 time points. The plant output was considered to be 32 successive cartesian endpoint positions , and the plant input was the timevarying force vector field. Two hidden units Z were used . In figure 5b we see an example of the two equilibrium point trajectories (movement primitives) that were learned by DG HA. Linear combinations of these trajectories account for over 96% of the variance of the training data, and they can approximate a large class of smooth movements . Note that many other pairs of orthogonal trajectories can accomplish this, and different trials often produced different orthogonal trajectory shapes. 1029 Optimal Movement Primitives Original 5.~ 1. 2 3. 8.~~ Coefficients Figure 4: Projection of test-word "hello" using increasing numbers of intermediate variables Zi. 1030 Terence Sanger WorkSpace o. b. Figure 5: a. Simulation of frog leg configuration. b. An example of learned optimal movement primitives. 6 CONCLUSION The examples are not meant to provide detailed models of internal processing for human or frog motor control. Rather, they are intended to illustrate the concept of optimal primitives and perhaps guide the search for neurophysiological and psychophysical correlates of these primitives. The first example shows that generation of the lower-case alphabet can be accomplished with approximately 10 coefficients per letter, and that this covers a considerable range of variability in character production . The second example demonstrates that an adaptive algorithm allows the possibility for the frog spinal cord to control movement using a very small number of internal variables. Optimal unsupervised motor learning thus provides a descriptive model for the generation of a large class of movements using a compressed internal description. A set of fixed movement primitives can be combined linearly to produce the necessary motor commands, and the optimal choice of these primitives assures that the error in the resulting movement will be minimized. References Bizzi E., Mussa-Ivaldi F. A., Giszter S., 1991, Computations underlying the execution of movement: A biological perspective, Science, 253:287-29l. Sanger T. D., 1994a, Two algorithms for iterative computation of the singular value decomposition from input/output samples, In Touretzky D., ed., Advances in Neural Information Processing 6, Morgan Kaufmann, San Mateo, CA, in press. Sanger T. D., 1994b, Optimal unsupervised motor learning, IEEE Trans. Neural Networks, in press. 

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Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping in the Barn Owl Alexandre Pougett Cedric Deffayett Terrence J. Sejnowskit cedric@salk.edu terry@salk.edu alex@salk .edu tHoward Hughes Medical Institute The Salk Institute La Jolla, CA 92037 Department of Biology University of California, San Diego and tEcole Normale Superieure 45 rue d'Ulm 75005 Paris, France Abstract The auditory system of the barn owl contains several spatial maps . In young barn owls raised with optical prisms over their eyes, these auditory maps are shifted to stay in register with the visual map, suggesting that the visual input imposes a frame of reference on the auditory maps. However, the optic tectum, the first site of convergence of visual with auditory information, is not the site of plasticity for the shift of the auditory maps; the plasticity occurs instead in the inferior colliculus, which contains an auditory map and projects into the optic tectum . We explored a model of the owl remapping in which a global reinforcement signal whose delivery is controlled by visual foveation. A hebb learning rule gated by reinforcement learned to appropriately adjust auditory maps. In addition, reinforcement learning preferentially adjusted the weights in the inferior colliculus, as in the owl brain, even though the weights were allowed to change throughout the auditory system. This observation raises the possibility that the site of learning does not have to be genetically specified, but could be determined by how the learning procedure interacts with the network architecture . 126 Alexandre Pouget, Cedric Deffayet, Te"ence J. Sejnowski c,an c:::======:::::? ? ~_m Visual System Optic Tectum Inferior Colllc-ulus External nucleua .- t Forebrain Field L U~ a~ t Ovold.H. Nucleull ?"bala:m.ic Relay Inferior Colltculu. Cenlnll Nucleus C1ec) t Cochlea Figure 1: Schematic view of the auditory pathways in the barn owl. 1 Introduction The barn owl relies primarily on sounds to localize prey [6] with an accuracy vastly superior to that of humans. Figure 1A illustrates some of the nuclei involved in processing auditory signals. The barn owl determines the location of sound sources by comparing the time and amplitude differences of the sound wave between the two ears. These two cues are combined together for the first time in the shell and core of the inferior colliculus (ICc) which is shown at the bottom of the diagram . Cells in the ICc are frequency tuned and subject to spatial aliasing. This prevents them from unambiguously encoding the position of objects. The first unambiguous auditory map is found at the next stage, in the external capsule of the inferior colliculus (ICx) which itself projects to the optic tectum (OT). The OT is the first subforebrain structure which contains a multimodal spatial map in which cells typically have spatially congruent visual and auditory receptive fields. In addition, these subforebrain auditory pathways send one major collateral toward the forebrain via a thalamic relay. These collaterals originate in the ICc and are thought to convey the spatial location of objects to the forebrain [3]. Within the forebrain, two major structures have been involved in auditory processing: the archistriatum and field L. The archistriatum sends a projection to both the inferior colliculus and the optic tectum . Knudsen and Knudsen (1985) have shown that these auditory maps can adapt to systematic changes in the sensory input. Furthermore, the adaptation appears to be under the control of visual input, which imposes a frame of reference on the incoming auditory signals. In owls raised with optical prisms, which introduce a systematic shift in part of the visual field, the visual map in the optic tectum was identical to that found in control animals, but the auditory map in the ICx was shifted by the amount of visual shift introduced by the prisms. This plasticity ensures that the visual and auditory maps stay in spatial register during growth Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping 127 and other perturbations to sensory mismatch. Since vision instructs audition, one might expect the auditory map to shift in the optic tectum, the first site of visual-auditory convergence. Surprisingly, Brainard and Knudsen (1993b) observed that the synaptic changes took place between the ICc and the ICx, one synapse before the site of convergence. These observations raise two important questions: First, how does the animal knows how to adapt the weights in the ICx in the absence of a visual teaching signal? Second, why does the change take place at this particular location and not in the aT where a teaching signal would be readily available? In a previous model [7], this shift was simulated using backpropagation to broadcast the error back through the layers and by constraining the weights changes to the projection from the ICc to ICx. There is, however, no evidence for a feedback projection between from the aT to the ICx that could transmit the error signal; nor is there evidence to exclude plasticity at other synapses in these pathways. In this paper, we suggest an alternative approach in which vision guides the remapping of auditory maps by controlling the delivery of a scalar reinforcement signal. This learning proceeds by generating random actions and increasing the probability of actions that are consistently reinforced [1, 5] . In addition, we show that reinforcement learning correctly predicts the site of learning in the barn owl, namely at the ICx-ICc synapse, whereas backpropagation [8] does not favor this location when plasticity is allowed at every synapse. This raises a general issue: the site of synaptic adjustment might be imposed by the combination of the architecture and learning rule, without having to restrict plasticity to a particular synapse. 2 2.1 Methods Network Architecture The network architecture of the model based on the barn owl auditory system, shown in figure 2A, contains two parallel pathways. The input layer was an 8x21 map corresponding to the ICc in which units responded to frequency and interaural phase differences. These responses were pooled together to create auditory spatial maps at subsequent stages in both pathways. The rest of the network contained a series of similar auditory maps, which were connected topographically by receptive fields 13 units wide. We did not distinguish between field L and the archistriatum in the forebrain pathways and simply used two auditory maps, both called FBr. We used multiplicative (sigma-pi) units in the aT whose activities were determined according to: Yi = L,. w~Br yfBr WfkBr yfc:c (1) j The multiplicative interaction between ICx and FBr activities was an important assumption of our model. It forced the ICx and FBr to agree on a particular position before the aT was activated. As a result, if the ICx-aT synapses were modified during learning, the ICx-FBr synapses had to be changed accordingly. 128 Alexandre Pouget, Cedric Deffayet, Terrence J. Sejnowski Figure 2: Schematic diagram of weights (white blocks) in the barn owl auditory system. A) Diagram of the initial weights in the network. B) Pattern of weights after training with reinforcement learning on a prism-induced shift offour units. The remapping took place within the ICx and FBr. C) Pattern of weights after training with backpropagation. This time the ICx-OT and FBr-OT weights changed. Weights were clipped between 5.0 and 0.01, except for the FBr-ICx connections whose values were allowed to vary between 8.0 and 0.01. The minimum values were set to 0.01 instead of zero to prevent getting trapped in unstable local minima which are often associated with weights values of zero. The strong coupling between FBr and ICx was another important assumption of the model whose consequence will be discussed in the last section. Examples were generated by simply activating one unit in the ICc while keeping the others to zero, thereby simulating the pattern of activity that would be triggered by a single localized auditory stimulus. In all simulations, we modeled a prism-induced shift of four units. 2.2 Reinforcement learning We used stochastic units and trained the network using reinforcement learning [1]. The weighted sum of the inputs, neti, passed through a sigmoid, f(x) , is interpreted as the probability, Pi, that the unit will be active: Pi = f(neti) * 0.99 + 0.01 (2) were the output of the unit Yi was: ._{a y, - with probability 1 - Pi 1 with probability Pi (3) Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping 129 Because of the form of the equation for Pi, all units in the network had a small probability (0.01) of being spontaneously active in the absence of any inputs. This is what allowed the network to perform a stochastic search in action space to find which actions were consistently associated with positive reinforcement. We ensured that at most one unit was active per trial by using a winner-take-all competition in each layer. Adjustable weights in the network were updated after each training examples with hebb-like rule gated by reinforcement: (4) A trial consisted in choosing a random target location for auditory input (ICc) and the output of the OT was used to generate a head movement . The reinforcement , r , was then set to 1 for head movements resulting in the foveation of the stimulus and to -0.05 otherwise. 2.3 Backpropagation For the backpropagation network , we used deterministic units with sigmoid activation functions in which the output of a unit was given by: (5) where neti is the weighted sum of the inputs as before. The chain rule was used to compute the partial derivatives of the squared error, E , with respect to each weights and the weights were updated after each training example according to: (6) The target vectors were similar to the input vectors, namely only one OT units was required to be activated for a given pattern, but at a position displaced by 4 units compared to the input. 3 3.1 Results Learning site with reinforcement In a first set of simulation we kept the ICc-ICx and ICc-FBr weights fixed. Plasticity was allowed at these site in later simulations. Figure 2A shows the initial set of weights before learning starts. The central diagonal lines in the weight diagrams illustrate the fact that each unit receives only one non-zero weight from the unit in the layer below at the same location. 130 Alexandre Pouget, Cedric Deffayet, Terrence J. Sejnowski There are two solutions to the remapping: either the weights change within the ICx and FBr, or from the ICx and the FBr to the ~T. As shown in figure 2B , reinforcement learning converged to the first solution. In contrast, the weights between the other layers were unaltered, even though they were allowed to change. To prove that the network could have actually learned the second solution, we trained a network in which the ICc-ICx weights were kept fixed . As we expected, the network shifted its maps simultaneously in both sets of weights converging onto the OT, and the resulting weights were similar to the ones illustrated in figure 2C. However, to reach this solution , three times as many training examples were needed. The reason why learning in the ICx and FBr were favored can be attributed to probabilistic nature of reinforcement learning. If the probability of finding one solution is p, the probability of finding it twice independently is p2. Learning in the ICx and FBR is not independent because of the strong connection from the FBr to the ICx. When the remapping is learned in the FBR this connection automatically remapped the activities in the ICx which in turn allows the ICx-ICx weights to remap appropriately. In the OT on the other hand, the multiplicative connection between the ICx and FBr weights prevent a cooperation between this two sets of weights. Consequently, they have to change independently, a process which took much more training. 3.2 Learning at the ICc-ICx and ICc-FBr synapses The aliasing and sharp frequency tuning in the response of ICc neurons greatly slows down learning at the ICc-ICx and ICc-FBr synapses. We found that when these synapses were free to change, the remapping still took place within the ICx or FBr (figure 3). 3.3 Learning site with backpropagation In contrast to reinforcement learning, backpropagation adjusted the weights in two locations: between the ICx and the OT and between the Fbr and OT (figure 2C). This is the consequence of the tendency of the backpropagation algorithm to first change the weights closest to where the error is injected. 3.4 Temporal evolution of weights Whether we used reinforcement or supervised learning, the map shifted in a very similar way. There was a simultaneous decrease of the original set of weights with a simultaneous increase of the new weights, such that both sets of weights coexisted half way through learning. This indicates that the map shifted directly from the original setting to the new configuration without going through intermediate shifts. This temporal evolution of the weights is consistent the findings of Brainard and Knudsen (1993a) who found that during the intermediate phase of the remapping, cells in the inferior colli cuI us typically have two receptive fields. More recent work however indicates that for some cells the remapping is more continuous(Brainard and Knudsen , personal communication) , a behavior that was not reproduced by either of the learning rule. Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping 131 Figure 3: Even when the ICc-ICx weights are free to change, the network update the weights in the ICx first. A separate weight matrix is shown for each isofrequency map from the ICc to ICx. The final weight matrices were predominantly diagonal; in contrast, the weight matrix in ICx was shifted. 4 Discussion Our simulations suggest a biologically plausible mechanism by which vision can guide the remapping of auditory spatial maps in the owl's brain. Unlike previous approaches, which relied on visual signals as an explicit teacher in the optic tectum [7], our model uses a global reinforcement signal whose delivery is controlled by the foveal representation of the visual system. Other global reinforcement signals would work as well. For example, a part of the forebrain might compare auditory and visual patterns and report spatial mismatch between the two. This signal could be easily incorporated in our network and would also remap the auditory map in the inferior colli cuI us. Our model demonstrates that the site of synaptic plasticity can be constrained by the interaction between reinforcement learning and the network architecture. Reinforcement learning converged to the most probably solution through stochastic search. In the network, the strong lateral coupling between ICx and FBr and the multiplicative interaction in the OT favored a solution in which the remapping took place simultaneously in the ICx and FBr. A similar mechanism may be at work in the barn owl's brain. Colaterals from FBr to ICx are known to exist, but the multiplicative interaction has not been reported in the barn owl optic tectum. Learning mechanisms may also limit synaptic plasticity. NMDA receptors have been reported in the ICx, but they might not be expressed at other synapses. There may, however, be other mechanisms for plasticity. The site of remapping in our model was somewhat different from the existing observations. We found that the change took place within the ICx whereas Brainard and Knudsen [3] report that it is between the ICc and the ICx. A close examination of their data (figure 11 in [3]) reveals that cells at the bottom of ICx were not 132 Alexandre Pouget, Cedric Deffayet, Terrence J. Sejnowski remapped, as predicted by our model, but at the same time, there is little anatomical or physiological evidence for a functional and hierarchical organization within the ICx. Additional recordings are need to resolve this issue. We conclude that for the barn owl's brain, as well as for our model, synaptic plasticity within ICx was favored over changes between ICc and ICx. This supports the hypothesis that reinforcement learning is used for remapping in the barn owl auditory system. Acknowledgments We thank Eric Knudsen and Michael Brainard for helpful discussions on plasticity in the barn owl auditory system and the results of unpublished experiments. Peter Dayan and P. Read Montague helped with useful insights on the biological basis of reinforcement learning in the early stages of this project. References [1] A.G. Barto and M.1. Jordan. Gradient following without backpropagation in layered networks. Proc. IEEE Int. Conf. Neural Networks, 2:629-636, 1987. [2] M.S. Brainard and E.1. Knudsen. Dynamics of the visual calibration of the map of interaural time difference in the barn owl's optic tectum. In Society For Neuroscience Abstracts, volume 19, page 369.8, 1993. [3] M.S. Brainard and E.!. Knudsen. Experience-dependent plasticity in the inferior colliculus: a site for visual calibration of the neural representation of auditory space in the barn owl. The journal of Neuroscience, 13:4589-4608, 1993. [4] E. Knudsen and P. Knudsen. Vision guides the adjustment of auditory localization in the young barn owls. Science, 230:545-548, 1985. [5] P.R. Montague, P. Dayan, S.J. Nowlan, A. Pouget, and T.J. Sejnowski. Using aperiodic reinforcement for directed self-organization during development. In S.J. Hanson, J.D. Cowan, and C.L. Giles, editors, Advances in Neural Information Processing Systems, volume 5. Morgan-Kaufmann, San Mateo, CA, 1993. [6] R.S. Payne. Acoustic location of prey by barn owls (tyto alba). Journal of Experimental Biology, 54:535-573, 1970. [7] D.J. Rosen, D.E. Rumelhart, and E.I. Knudsen. A connectionist model of the owl's sound localization system. In Advances in Neural Information Processing Systems, volume 6. Morgan-Kaufmann, San Mateo, CA, 1994. [8] D.E. Rumelhart, G.E. Hinton, and R.J . Williams. Learning internal representations by error propagation. In D. E. Rumelhart, J. L. McClelland, and the PDP Research Group, editors, Parallel Distributed Processing, volume 1, chapter 8, pages 318-362. MIT Press, Cambridge, MA, 1986. 

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Hierarchical Mixtures of Experts Methodology Applied to Continuous Speech Recognition Ying Zhao, Richard Schwartz, Jason Sroka*: John Makhoul BBN System and Technologies 70 Fawcett Street Cambridge MA 02138 Abstract In this paper, we incorporate the Hierarchical Mixtures of Experts (HME) method of probability estimation, developed by Jordan [1], into an HMMbased continuous speech recognition system. The resulting system can be thought of as a continuous-density HMM system, but instead of using gaussian mixtures, the HME system employs a large set of hierarchically organized but relatively small neural networks to perform the probability density estimation. The hierarchical structure is reminiscent of a decision tree except for two important differences: each "expert" or neural net performs a "soft" decision rather than a hard decision, and, unlike ordinary decision trees, the parameters of all the neural nets in the HME are automatically trainable using the EM algorithm. We report results on the ARPA 5,OOO-word and 4O,OOO-word Wall Street Journal corpus using HME models. 1 Introduction Recent research has shown that a continuous-density HMM (CD-HMM) system can outperform a more constrained tied-mixture HMM system for large-vocabulary continuous speech recognition (CSR) when a large amount of training data is available [2]. In other work, the utility of decision trees has been demonstrated in classification problems by using the "divide and conquer" paradigm effectively, where a problem is divided into a hierarchical set of simpler problems. We present here a new CD-HMM system which **MIT, Cambridge MA 02139 860 Ying Zhao, Richard Schwartz, Jason Sroka, John Makhoul has similar properties and possesses the same advantages as decision trees, but has the additional important advantage of having automatically trainable "soft" decision boundaries. 2 Hierarchical Mixtures of Experts The method of Hierarchical Mixtures of Experts (HME) developed recently by Jordan [1] breaks a large scale task into many small ones by partitioning the input space into a nested set of regions, then building a simple but specific model (local expert) in each region. The idea behind this method follows the principle of divide-and-conquer which has been utilized in certain approaches to classification problems, such as decision trees. In the decision tree approach, at each level of the tree, the data are divided explicitly into regions. In contrast, the HME model makes use of "soft" splits of the data, i.e., instead of the data being explicitly divided into regions, the data may lie simultaneously in multiple regions with certain probabilities. Therefore, the variance-increasing effect of lopping off distant data in the decision tree can be ameliorated. Furthermore, the "hard" boundaries in the decision tree are fixed once a decision is made, while the "soft" boundaries in the HME are parameterized with generalized sigmoidal functions, which can be adjusted automatically using the Expectation-Maximization (EM) algorithm during the splitting. Now we describe how to apply the HME methodology to the CSR problem. For each state of a phonetic HMM, a separate HME is used to estimate the likelihood. The actual HME first computes a posterior probability P(llz, s), the probability of phoneme class I, given the input feature vector z and state s. That probability is then divided by the a priori probability of the phone class I at state s. A one-level HME performs the following computation: c P(llz, s) =L P(llci, z, s)P(cilz, s) (1) i=l where I = 1, ,.. , L indicates phoneme class, Ci represents a local region in the input space, and C is the number of regions. P(cilz, s) can be viewed as a gating network, while P(lICi, z, s) can be viewed as a local expert classifier (expert network) in the region [1]. In a two-level HME, each region Ci is divided in turn into C subregions. The term P(IICi, z, s) is then computed in a similar manner to equation (1), and so on. If in some of these subregions there are no data available, we back off to the parent network. c, 3 TECHNICAL DETAILS As in Jordan's paper, we use a generalized sigmoidal function to parameterize P(cilz) as follows: (2) where z can be the direct input (in a one-layer neural net) or the hidden layer vector (in a two-layer neural net), and v,, i = 1, .. " C are weights which need to train. Similarly, the local phoneme classifier in region Ci, P(llc" z), can be parameterized with a generalized Mixtures of Experts Applied to Continuous Speech Recognition 861 sigmoidal function also: (3) where 8;i,j = 1, ... , L are weights. The whole system consists of two set of parameters: Vi, i 1, ... , C and 8;i' j 1, ... , L, = {8;i' Vi}. All parameters are estimated by using the EM algorithm. = e = The EM is an iterative approach to maximum likelihood estimation. Each iteration of an EM algorithm is composed of two steps: an Expectation (E) step and a Maximization (M) step. The M step involves the maximization of a likelihood function that is redefined in each iteration by the E step. Using the parameterizations in (2) and (3), we obtain the following iterative procedure for computing parameters = {Vi, 8;i}: . .. I'lze Vi(0) an d 8(0) 1. lDltIa ;i f or 1.. = I , ... , C ,}' = 1, ... , L . 2. E-step: In each iteration n, for each data pair (z(t), l(t?, t = 1, ... ,N, compute e zi(tin ) = P(cilz(t), l(t), e(n~ = P(Ci Iz(t), v~n?p(l(t)lci' z(t), 8~~~,i) (4) where i = 1, ... , C. Zi(t)<n) represents the probability of the data t lying in the region i, given the current parameter estimation e(n) . It will be used as a weight for this data in the region i in the M-step. The idea of "soft" splitting reflects that these weights are probabilities between 0 and 1, instead of a "hard"decision 0 or 1. 3. M-step: (5) (6) 4. Iterate until 8;i' Vi converge. The first maximization means fitting a generalized sigmoidal model (3) using the labeled data (z(t), l(t? and weighting Zi(t)<n). The second one means fitting a generalized sigmoidal model (2) using inputs z(t) and outputs Zi(t)<n). The criterion for fitting is the cross-entropy. Typically, the fitting can be solved by the Newton-Raphson method. However, it is quite expensive. Viewing this type of fitting as a multi-class classification task, we developed a technique to invert a generalized sigmoidal function more efficiently, which will be described in the following. A common method in a multi-class classification is to divide the problem into many 2-c1ass classifications. However, this method results in a positive and negative training unbalance usually. To avoid the positive and negative training unbalance, the following technique can be used to solve multi-class posterior probabilities simultaneously. Suppose we have a labeled data set, (z(t), l(t?, t = 1, ... , N, where l(t) E {I, ... , L} is the label for t-th data. We use a generalized sigmoidal function to model the posterior 862 Ying Zhao, Richard Schwartz, Jason Sroka, John Makhoul probability P(llz), where 1 = I, ... , L as follows: e9'f'z P,(z) = P(llz) = (7) 9T L:k e "z Obviously, since these probabilities sum up to one, we have L-I PL(Z) L P,(z). =1 - (8) '=1 Now, a training sample z(t) with a class label let) can be interpreted as: P,(z(t? = { 1 = let) 1 =/l(t) 0.9 1:':1 If we define (9) P,(z) T 9, z (10) = log PL(z) equation (10) implies that (11) for 1 = I, ... , L with 9Lz = O. This expression is the generalized sigmoidal function in (7). This means, we can train parameters in (7) to satisfy Equation (10) from the data. Using a least squares criterion, the objective is 9T z(t) - . ,,[ mm L..J t P'(Z(t?] 2 PL(Z(t? (12) log - - - for 1 = I, ... , L - 1. Denote a data matrix as z(l) z(2) x= zeN) A least squares solution to (12) is 9, = (loga)(XT X)-I [L L z(t) - '(t)=l Z(t)] (13) '(t)=L for 1 = I, ... , L, where a =9(L - 1). Substituting (13) into (11), we get a P,(z) = ZT(X T X)-l ~ L.JI(I~I L:k azT(XT X)-l L: z(t) z(t) (14) 1(1)=" Equation (13) and (14) are very easy to compute. Basically, we only have to accumulate the matrix XT X and sum z(t) into different classes 1 = I, ... , L. We can obtain probabilities P,(z) by a single inversion of matrix XT X after a pass through the training data. Mixtures of Experts Applied to Continuous Speech Recognition 863 4 Relation to Other Work The work reported here is very different from our previous work utilizing neural nets for CSR. There, a single segmental neural network (SNN) is used to model a complete phonetic segment [3]. Here, each HME estimates the probability density for each state of a phonetic HMM. The work here is more similar to that by Cohen et al. [4], the major difference being that in [4], a single very large neural net is used to perform the probability density modeling. The training of such a large network requires the use of a specialized parallel processing machine, so that the training can be done in a reasonable amount of time. By using the HME method and dividing the problem into many smaller problems, we are able to perform the needed training computation on regular workstations. Most of the previous work on CD-HMM work has utilized mixtures of gaussians to estimate the probability densities of an HMM. Since a ' multilayer feedforward neural network is a universal continuous function approximator, we decided to explore the use of neural nets as an alternative approach for continuous density estimation. 5 Experimental Results HMM SNN HMM+SNN HME HME+HMM Prior-modified HME + HMM Word Error Rate 7.8 8.5 7.1 7.6 6.8 6.2 Table 1: Error Rates for the ARPA WSJ 5K Development Test, Trigram Grammar HMM HME+ HMM Word Error Rate 9.5 8.7 Table 2: Error Rates for the ARPA WSJ 40K Test Set, Trigram Grammar In our initial application of the HME method to large-vocabulary CSR, we used phonetic context-independent HMEs to estimate the likelihoods at each state of 5-state HMMs. We implemented a two-level HME, with the input space divided into 46 regions, and each of those regions is further divided into 46 subregions. The initial divisions were accomplished by supervised training, with each division trained to one of the 46 phonemes in the system. All gating and local expert networks in the HME had identical structures - a two-layer generalized sigmoidal network. The whole HME system was implemented within an N-best paradigm [3], where the recognized sequence was obtained as a result of a rescoring of an N-best list obtained from our baseline BYBLOS system (tied-mixture HMM) with a statistical trigram grammar. 864 Ying Zhao, Richard Schwartz, Jason Sroka, John Makhoul We then built a context-dependent HME system based on the structure of the contextindependent HME models described above. For each state, the whole training data was divided into 46 parts according to its left or right context. Then for each context, a separate HME model was built for that context. To be computationally feasible, we used only one-level HMEs here. We first experimented using a left-context and right-context model. We tested the HME implementation on the ARPA 5,OOO-word Wall Street Journal corpus (WSJl, H2 dev set). We report the word error rates on the same test set for a number of different systems. Table 1 shows the word error rates for i) the baseline HMM system; ii) the segment-based neural net system (SNN) iii) the hybrid SNNIHMM system iv) a HME system alone. v) a HME system combined with HMM; vi) a HME +HMM system with modified priors. From Table 1, The performance of the baseline tied-mixture HMM is 7.8%. The performance of the SNN system (8.5%) is comparable to the HMM alone. We see that the performance of a HME (7.6%) is as good as the HMM system, which is better than the SNN system. When combined with the baseline HMM system, the HME and SNN both improve performance over the HMM alone about 10% from 7.8% to 6.8% and from 7.8% to 7.1% respectively. We found out that the improvement could be made larger for a hybrid HMElHMM by adjusting the context-dependent priors with the context-independent priors, and then smooth context-dependent models with a context-independent model. More specifically, in a context-dependent HME model, we usually estimate the posterior probability phoneme I, P(llc, z, s), given left or right context c and the acoustic input z in a particular state s. Because the samples may be sparse for many of context models, it is necessary to regularize (smooth) context-dependent models with a contextindependent model, where there is much more data available. However, since the two models have different priors: P(llc, s) in a context-dependent model and P(lls) in a context-independent model, a simple interpolation between the two models which is c,.)P(' c.) . d d od I d P(ll z, s ) = P(xP(x , .)P(' .) P(ll c, z, s ) = P(x ,P(x c,.) 10 a context- epen ent mean .) in a context-independent model is inconsistent. To scale the context-dependent priors P(llc, s) with a context-independent prior P(lls), we weighted each input data point z with the weight for a prior adjusting. After this modification, a context-dependent :c.i'c:;) HME actually estimates P(z ~~:~('I'). It combines better with a context-independent model. For the same experiment we showed in Table 1, the word error for the HME (with HMM) droped from 6.8% to 6.2% when priors were modified. For this 5,OOO-word development set, we got a total of about 20% word error reduction over the tied-mixture HMM system using a HME-based neural network system. We then switched our experiment domain from a 5,OOO-word to 40,OOO-word the test set. During this year, the BYBLOS system has been improVed from a tied-mixture system to a continuous density system. We also switched to using this new continuous density BYBLOS in our hybrid HMElHMM system. The language model used here was a 40,OOO-word trigram grammar. The result is shown in Table 2. From Table 2, we see that there is about a 10% word error rate reduction over the continuous density HMM system by combining a context-dependent HME system. Compared with the 20% improvement over the tied-mixture system we made for the 5,OOO-word development set, the improvement over the continuous density system in this 40,OOO-word Mixtures of Experts Applied to Continuous Speech Recognition 865 development is less. This may be due to the big improvement of the HMM system itself. 6 CONCLUSIONS The method of hierarchical mixtures of experts can be used as a continous density estimator to speech recognition. Experimental results showed that estimations from this approach are consistent with the estimations from the HMM system. The frame-based neural net system using hierarchical mixtures of experts improves the performance of both the state-of-the-art tied mixture HMM system and the continuous density HMM system. The HME system itself has the same performance as the state-of-the-art tied mixture HME system. 7 Acknowledgments This work was funded by the Advanced Research Projects Agency of the Department of Defense. References [1] Michael Jordan, "Hierarchical Mixtures of Experts and the EM Algorithm," Neural Computation, 1994, in press. [2] D. Pallett, J. Fiscus, W. Fisher, J. Garofolo, B. Lund, and M. Pryzbocki, "1993 Benchmark Tests for the ARPA Spoken Language Program," Proc. ARPA Human Language Technology Workshop, Plainsboro, NJ, Morgan Kaufman Publishers, 1994. [3] G. Zavaliagkos, Y. Zhao, R. Schwartz and J. Makhoul, "A Hybrid Neural Net System for State-of-the-Art Continuous Speech Recognition," in Advances in Neural Information Processing Systems 5, S. J. Hanson, J. D. Cowan and C. L. Giles, eds., Morgan Kaufmann Publishers, 1993. [4] M. Cohen, H. Franco, N. Morgan, D. Rumelhart and V. Abrash, "Context-Dependent Multiple Distribution Phonetic Modeling with MLPS," in Advances in Neural Information Processing Systems 5, S. J. Hanson, 1. D. Cowan and C. L. Giles, eds., Morgan Kaufmann Publishers, 1993. 

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477 A COMPUTATIONA.LLY ROBUST ANATOlVIICAL MODEL FOR RETIN.AL DIRECTIONAL SELECTI\l ITY Norberto M. Grzywacz Center BioI. Inf. Processing MIT, E25-201 Cambridge, MA 02139 Franklin R. Amthor Dept. Psychol. Univ. Alabama Birmingham Birmingham, AL 35294 ABSTRACT We analyze a mathematical model for retinal directionally selective cells based on recent electrophysiological data, and show that its computation of motion direction is robust against noise and speed. INTROduCTION Directionally selective retinal ganglion cells discriminate direction of visual motion relatively independently of speed (Amthor and Grzywacz, 1988a) and with high contrast sensitivity (Grzywacz, Amthor, and Mistler, 1989). These cells respond well to motion in the "preferred" direction, but respond poorly to motion in the opposite, null, direction. There is an increasing amount of experimental work devoted to these cells. Three findings are particularly relevant for this paper: 1- An inhibitory process asymmetric to every point of the receptive field underlies the directional selectivity of ON-OFF ganglion cells of the rabbit retina (Barlow and Levick, 1965). This distributed inhibition allows small motions anywhere in the receptive field center to elicit directionally selective responses. 2- The dendritic tree of directionally selective ganglion cells is highly branched and most of its dendrites are very fine (Amthor, Oyster and Takahashi, 1984; Amthor, Takahashi, and Oyster, 1988). 3The distributions of excitatory and inhibitory synapses along these cells' dendritic tree appear to overlap. (Famiglietti, 1985). Our own recent experiments elucidated some ofthe spatiotemporal properties of these cells' receptive field. In contrast to excitation , which is transient with stimulus, the inhibition is sustained and might arise from sustained amacrine cells (Amthor and Grzywacz, 1988a). Its spatial distribution is wide, extending to the borders of the receptive field center (Grzywacz and Amthor, 1988). Finally, the inhibition seems to be mediated by a high-gain shunting, not hyperpolarizing, synapse, that is, a synapse whose reversal potential is close to cell's resting potential (Amthor and Grzywacz, 1989). 478 Grzywacz and Amthor In spite of this large amount of experimental work, theoretical efforts to put these pieces of evidence into a single framework have been virtually inexistent. We propose a directional selectivity model based on our recent data on the inhibition's spatiotemporal and nonlinear properties. This model, which is an elaboration of the model of Torre and Poggio (1978), seems to account for several phenomena related to retinal directional <;eledivity. THE ]\IIODEL Figure 1 illustrates the new model for retinal directional selectivity. In this modd, a stimulus moving in the null direction progressively activates receptive field regions linked to synapses feeding progressively more distal dendrites Of the ganglion cells . Every point in the receptive field center activates adjacent excitatory a~d inhibitory synapses. The inhibitory synapses are assumed to cause shunting inhibition. ("'"e also formulated a pre-ganglionic version of this model, which however, is outside the scope of this paper) . .-- NULL FIGURE 1. The new model for retinal directional selectivity. This model is different than that of Poggio and Koch (1987), where the motion axis is represented as a sequence of activation of different dendrites. Furthermore, in their model, the inhibitory synapses must be closer to the soma than the excitatory ones . (However, our model is similar a model proposed, and argued against, elsewhere (Koc,h, Poggio, and Torre, 1982). An advantage of our model is that it accounts for the large inhibitory areas t.o most points of the receptive field (Grzywacz and Amthor, 1988). Also, in the new model, the distributions of the excitatory and inhibitory synapses overlap along the dendritic tree, as suggested (Famiglietti, 1985). Finally, the dendritic tree of ONOFF directionally selective ganglion cells (inset- Amthor, Oyster, and Takahashi, A Computationally Robust Anatomical Model 1984) is consistent with our model. The tree's fine dendrites favor the multiplicity of directional selectivity and help to deal with noise (see below). In this paper, we make calculations with an unidimensional version of the model dealing with motions in the preferred and null directions. Its receptive field maps into one dendrite of the cell. Set the origin of coordinates of the receptive field to be the point where a dot moving in the null direction enters the receptive field. Let S be the size of the receptive field. Next, set the origin of coordinates in the dendri te to be the soma and let L be the length of the dendrite. The model postulates that a point z in the receptive field activates excitatory and inhibitory synapses in pain t z = zL/ S of the dendrite. The voltages in the presynaptic sites are assumed to be linearly related to the stimulus, [(z,t), that is, there are functions fe{t) and li(t) such that the excitatory, {3e(t), and inhibitory, {3i(t), presynaptic voltages of the synapses to position ;r in the dendrite are . J = e, . l, where "*" stands for convolution. We assume that the integral of Ie is zero, (the excitation is transient), and that the integral of Ii is positive. (In practice, gamma distribution functions for Ii and the derivatives of such functions for Ie were used in this paper's simulations.) The model postulates that the excitatory, ge, and inhibitory, gi, postsynaptic conductances are rectified functions of the presynaptic voltages. In some examples, we use the hyperbolic tangent as the rectification function: where Ij and T; are constants. In other examples, we use the rectification functions described in Grzywacz and Koch (1987), and their model of ON-OFF rectifications. For the postsynaptic site, we assume, without loss of generality, zero reversal potential and neglect the effect of capacitors (justified by the slow time-courses of excitation and inhibition). Also, we assume that the inhibitory synapse leads to shunting inhibition, that is, its conductance is not in series with a battery. Let Ee be the voltage of the excitatory battery. In general, the voltage, V, in different positions of the dendrite is described by the cable equation: d2~~:, t) = Ra (-Ee!}e (z, t) + V (z, t) (ge (z, t) + 9j (z, t) + 9,.)), where Ra is the axoplasm resistance per unit length, g,. is the resting membrane conductance, and the tilde indicates that in this equation the conductances are given per unit length. 479 480 Grzywacz and Amthor For the calculations illustrated in this paper, the stimuli are always delivered to the receptive field through two narrow slits. Thus, these stimuli activate synapses in two discrete positions of a dendrite. In this paper, we only show results for square wave and sinusoidal modulations of light, however, we also performed calculations for more general motions. The synaptic sites are small so that their resting conductances are negligible, and we assume that outside these sites the excitatory and inhibitory conductances are zero. In this case, the equation for outside the synapses is: d2U dy2 = U, = where we defined A 1/(Ra y,,)1/2 (the length constant), U = V/Ee, and y The boundary conditions used are = = Z/A. = where L L/ A, and where if R, is the soma's input resistance, then p R, /(RaA) (the dendritic-to-soma cond uctance ratio). The first condition means that currents do not flow through the tips of the dendrites. Finally, label by 1 the synapse proximal to the soma, and by 2 the distal one; the boundary conditions at the synapses are 1I~lIj = lim II> 111 1I<lIj lim U 1I~lIj U, j = 1,2, j = 1,2, (I ) where 7'e = geRa).. and 7'i = 9iRa)... It can be shown that the relative inhibitory strength for motions in the preferred direction decreases with L and increases with p. Thus, to favor conditions for multiplicity of direction selectivity in the receptive field, we perform calculations with L -+ 00 and p = 1. The strengths of the excitatory syna.pses are set such that their contribution to somatic voltage in the absence of inhibition is in variant with position. Finally, we ensure that the excitatory synapses never sat urate. Under these conditions, one can show that the voltage in the soma is: U (0) = {27'e,2 + (7'e,2 + 7'i,2 + 2) 7'e,d e2~y - (7'?,2 + 7'j,2) 7'e,1 , ((7'i,l +2)7'i,2 + 27'j,1 +4)e 26Y - 7'i,17'i,2 (2) where 6y is the distance between the synapses. A fina.l quantity, which is used in this paper is the directional selectivity index It. Let Up and Un be the total responses to the second slit in the apparent motion in the preferred and null directions respectively. {Alternatively, for the sinusoidal A Computationally Robust Anatomical Model motion, these quantities are the respective average responses .) \Ve follow Grzywacz and Koch (198i) and define (3) RESULTS This section presents the results of calcuhtions ba."ed on the modd. \Ve address: the multiple computations of directional seir->divity in the {.?ells? receptive fields; the robustness of these computations again~t noise: I h" robustness of these computations against speed. Figure 2 plots the degree of directional selectivity for apparent lIlotions activating two synapses as function of the synapses' distan,'e in a dendrite (computed from Equations 2 and 3). 1. ~--------------------------- 32 .".. 16 ...c:: -.... -.., ... B >~ '" .. .5 u ~ 2 c: a '" u "- 0.5 c'" .0 .00 l!~IG URE 1 .5 1.0 . 50 Dendritic Distance (>.1 2. Locality of lnkraction betwt't"'n 2.0 synap"'t'~ a(I1\-'(1l,'" hv apparent mo- tions. It can be shown that the critical parameter controlling whether a certain synaptic distance produces a criterion directional selectivity is rj (Equation 1). As the parameter rj increases, the criterion distance decreases , Thus , "ince in retinal directionally selective cells the inhibition has high gain (Amthor and Grzywa<."z, 1989) and the dendrites are fine (Amthor, Oyster and Takahashi, 1984; Amthor, Takahashi, and Oyster, 1988), then rj is high, and motions anywhere in receptive field should elicit directionally selective responses (Barlow and Levick, 1965). In other words, the model's receptive field computes motion direction multiple times. Next, we show that the high inhibitory gain and the cells' fine dendrites help to deal with noise, and thus . may explain the high contrast sensitivity (0.5% contrast- 481 482 Grzywacz and Amthor Grzywacz, Amthor, and Mistler, 1989) of the cells' directional selectivity. This conclusion is illustrated in Figure 3's multiple plots. .. - -Looo .. ~ 1.0 - OUTPUT NOISE J ~I -"11 INIIIIT DAY 1IfI\/1 NOIS? VCCJ1ATDA'f 1IfI\/1 NOI. !. ~1 , ".., ......a -NlgII , .. .50 .. N ,, ... oo-===--e:;..........::.-....;::.,---==--..::.. 00 10 ? .0 Response ?. 00 .0 10 .00 Response 4.0 . 1.0 Aupan .. FIGURE 3. The model's computatifm of direction is robust against additive noise in the output, and in the excitatory and inhibitory inputs. To generate this figure, we used Equation 2 assuming that a Gaussian noise is added to the cell's output, excitatory input, or inhibitory input. {In the latter case, we used an approximation that assumes small standard deviation for the inhibitory input's noise.) Once again the critical parameter is the ri defined in Equation 1. The larger this parameter is, the better the model deals with noise. In the case of output noise, an increase of the parameter separatt's the preferred and null mean responses. For noise in the excitatory input, a parameter increase not only separates the means, but also reduces the standard deviation: Shunting inhibition SHUnTS down the noise. Finally, the most dramatic improvement occurs when the noise is in the inhibitory input. (In all these plots, the parameter increase is always by a factor of three.) Since for retinal direc tionally selective ganglion cells, ri is high (high inhibitory gain and fine dendrites), we conclude that the cells' mechanism are particularly well suited to deal with noise. For sinusoidal motions, the directional selectivity is robust for a large range of temporal frequencies provided that the frequencies are sufficiently low (Figure 4). (Nevertheless, the cell's preferred direction responses may be sharply tuned to either temporal frequency or speed- Amthor and Grzywacz, 1988). A Computationally Robust Anatomical Model I -HI", rl - 'llHl rl I ! , \ I I I I I I i 2.00 ~ , I iis :: " I -- 20.0 200. --------------- ,i i i is 2.00 20 .0 FrtQlltncy ~I IE-roo FIGURE 4. Directional selectivity is robust against speed modulation. To generate this curve, we subtracted the average respons~ to a isolated flickering slit from the preferred and null average responses (from Equation 2). This robustness is due to the invariance with speed for low speeds of the relative temporal phase shift between inhibition and excitation. Since the excitation has band-pass characteristics, it leads the stimulus by a constant phase. On the other hand, the inhibition is delayed and advanced in the preferred and null directions respectively, due to the asymmdric spatial integration. The phase shifts due to this integration are also speed invariant. CONCLUSIONS We propose a new model for retinal directional selectivity. The shunting inhibition of ganglion cells (Torre and Poggio, 1978), which is temporally sustained, is the main biophysical mechanism of the model. It postulates that for null direction motion, the stimulus activates regions of the receptive field that are linked to excitatory and inhibitory synapses, which are progressively farther away from the soma. This models accounts for: 1- the distribution of inhibition around points of the receptive field (Grzywacz and Amthor, 1988); 2- the apparent full overlap of the distribution of excitatory and inhibitory synapses along the dendritic trees of directionally selective ganglion cells (Famiglietti, 1985); 3- the multiplicity of directionally selective regions (Barlow and Levick, 1965); 4- the high contrast sensitivity of the cells' directional selectivity (Grzywacz, Amthor, and Mistler, 1989); 5- the relative in variance of directional selectivity on stimulus speed (Amthor and Grzywacz, 1988). Two lessons of our model to neural network modeling are: Threshold is not the only neural mechanism, and the basic computational unit may not be a neuron 483 484 Grzywacz and Amthor but a piece of membrane (Grzywacz and Poggio, 1989). In our model, nonlinear interactions are relatively confined to specific dendritic tree branches (Torre and Poggio, 1978). This allows local computations by which single cells might generate receptive fields with multiple directionally selective regions, as observed by Barlow and Levick (1965). Such local computations could not occur if the inhibition only worked through a reduction in spike rate by somatic hyperpolarization. Thus, most neural network models may be biologically irrelevant, since they are built upon a too simple model of the neuron. The properties of a network depend strongly on its basic elements. Therefore, to understand the computations of biological networks. it may be essential to first understand the basic biophysical mechanisms of information processing before developing complex networks. ACKNOWLEDGMENTS We thank Lyle Borg-Graham and Tomaso Poggio for helpful discussions. Also, we thank Consuelita Correa for help with the figures. N .M.G. was supported by grant BNS-8809528 from the National Science Foundation, by the Sloan Foundation, and by a grant to Tomaso Poggio and Ellen Hildreth from the Office of Naval Research, Cognitive and Neural Systems Division. F.R.A. was supported by grants from the National Institute of Health (EY05070) and the Sloan Foundation. REFERENCES Amthor & Grzywacz (1988) Invest. Ophthalmol. Vi". Sci. 29:225. Amthor & Grzywacz (1989) Retinal Directional Selectivity Is Accounted for by Shunting Inhibition. Submitted for Publication. Amthor, Oyster & Takahashi (1984) Brain Res. 298:187. Amthor, Takahashi & Oyster (1989) J. Compo Neurol. In Press. Barlow & Levick (1965) J. Physiol. 178:477. Famiglietti (1985) Neuro"ci. Abst. 11:337. Grzywacz & Amthor (1988) Neurosci. Ab"t. 14:603. Grzywacz, Amthor & Mistler (1989) Applicability of Quadratic and Threshold Models to Motion Discrimination in the Rabbit Retina. Submitted for Publication. Grzywacz & Koch (1987) Synapse 1:417. Grzywacz & Poggio (1989) In An Introduction to Neural and Electronic Networks. Zornetzer, Davis & Lau, Eds. Academic Press, Orlando, USA. In Press. Koch, Poggio & Torre (1982) Philos . Tran". R. Soc. B 298:227. Poggio & Koch (1987) Sci. Am. 256:46. Torre & Poggio (1978) Proc. R. Soc. B 202:409. 

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FINANCIAL APPLICATIONS OF LEARNING FROM HINTS Yaser s. Abu-Mostafa California Institute of Technology and NeuroDollars, Inc. e-mail: yaser@caltech.edu Abstract The basic paradigm for learning in neural networks is 'learning from examples' where a training set of input-output examples is used to teach the network the target function. Learning from hints is a generalization of learning from examples where additional information about the target function can be incorporated in the same learning process. Such information can come from common sense rules or special expertise. In financial market applications where the training data is very noisy, the use of such hints can have a decisive advantage. We demonstrate the use of hints in foreign-exchange trading of the U.S. Dollar versus the British Pound, the German Mark, the Japanese Yen, and the Swiss Franc, over a period of 32 months. We explain the general method of learning from hints and how it can be applied to other markets. The learning model for this method is not restricted to neural networks. 1 INTRODUCTION When a neural network learns its target function from examples (training data), it knows nothing about the function except what it sees in the data. In financial market applications, it is typical to have limited amount of relevant training data, with high noise levels in the data. The information content of such data is modest, and while the learning process can try to make the most of what it has, it cannot create new information on its own. This poses a fundamental limitation on the 412 Yaser S. Abu-Mostafa learning approach, not only for neural networks, but for all other models as well. It is not uncommon to see simple rules such as the moving average outperforming an elaborate learning-from-examples system. Learning from hints (Abu-Mostafa, 1990, 1993) is a value-added feature to learning from examples that boosts the information content in the data. The method allows us to use prior knowledge about the target function, that comes from common sense or expertise, along with the training data in the same learning process. Different types of hints that may be available in a given application can be used simultaneously. In this paper, we give experimental evidence of the impact of hints on learning performance, and explain the method in some detail to enable the readers to try their own hints in different markets. Even simple hints can result in significant improvement in the learning performance. Figure 1 shows the learning performance for foreign exchange (FX) trading with and without the symmetry hint (see section 3), using only the closing price history. The plots are the Annualized Percentage Returns (cumulative daily, unleveraged, transaction cost included), for a sliding one-year test window in the period from April 1988 to November 1990, averaged over the four major FX markets with more than 150 runs per currency. The error bar in the upper left corner is 3 standard deviations long (based on 253 trading days, assuming independence between different runs). The plots establish a statistically significant differential in performance due to the use of hints. This differential holds for all four currencies. 10 r---------~--------_r--------~--------~r_--------~ average: without hint -;-I" average: with hint;..':':.--, I 8 " ~ 30 /"_,,, " 6 / ,-' " " " ,-'" iI'?.' 4 J ,-" .. , ..I , .....r';' _",'''.-~''''''''.'-'' ,~_" .." 2 .. ,,-" _A' ,-,,-, ..'" ,.1 o '.~:<.~.~........................................... -2 ______ ________L -________ 100 150 200 250 Test Day Number ~--------~--------~-- o 50 ~ ~ Figure 1: Learning performance with and without hint Since the goal of hints is to add information to the training data, the differential in performance is likely to be less dramatic if we start out with more informative training data. Similarly, an additional hint may not have a pronounced effect if Financial Applications of Learning from Hints 413 we have already used a few hints in the same application. There is a saturation in performance in any market that reflects how well the future can be forecast from the past. (Believers in the Efficient Market Hypothesis consider this saturation to be at zero performance). Hints will not make us forecast a market better than whatever that saturation level may be. They will, however, enable us to approach that level through learning. This paper is organized as follows. Section 2 characterizes the notion of very noisy data by defining the '50% performance range'. We argue that the need for extra information in financial market applications is more pronounced than in other pattern recognition applications. In section 3, we discuss our method for learning from hints. We give examples of different types of hints, and explain how to represent hints to the learning process. Section 4 gives result details on the use of the symmetry hint in the four major FX markets. Section 5 provides experimental evidence that it is indeed the information content of the hint, rather than the incidental regularization effect, that results in the performance differential that we observe. 2 FINANCIAL DATA This section provides a characterization of very noisy data that applies to the financial markets. For a broad treatment of neural-network applications to the financial markets, the reader is referred to (Abu-Mostafa et al, 1994). Other Information Input X Input X -- -- Target Ouput MARKET NEURAL NETWORK -... -- y Forecast y Figure 2: Illustration of the nature of noise in financial markets Consider the market as a system that takes in a lot of information (fundamentals, news events, rumors, who bought what when, etc.) and produces an output y (say up/down price movement for simplicity) . A model, e.g., a neural network, attempts 414 Yaser S. Abu-Mostafa to simulate the market (figure 2), but it takes an input x which is only a small subset of the information. The 'other information' cannot be modeled and plays the role of noise as far as x is concerned. The network cannot determine the target output y based on x alone, so it approximates it with its output y. It is typical that this approximation will be correct only slightly more than half the time. What makes us consider x 'very noisy' is that y and y agree only! + f. of the time (50% performance range). This is in contrast to the typical pattern recognition application, such as optical character recognition, where y and y agree 1 - f. of the time (100% performance range). It is not the poor performance per se that poses a problem in the 50% range, but rather the additional difficulty of learning in this range. Here is why. In the 50% range, a performance of ! + f. is good, while a performance of ! f. is disastrous. During learning, we need to distinguish between good and ~ad hypotheses based on a limited set of N examples. The problem with the 50% range is that the number of bad hypotheses that look good on N points is huge. This is in contrast to the 100% range where a good performance is as high as 1 - f.. The number of bad hypotheses that look good here is limited. Therefore, one can have much more confidence in a hypothesis that was learned in the 100% range than one learned in the 50% range. It is not uncommon to see a random trading policy making good money for a few weeks, but it is very unlikely that a random character recognition system will read a paragraph correctly. Of course this problem would diminish if we used a very large set of examples, because the law of large numbers would make it less and less likely that y and y can agree! + f. of the time just by 'coincidence'. However, financial data has the other problem of non-stationarity. Because of the continuous evolution in the markets, old data may represent patterns of behavior that no longer hold. Thus, the relevant data for training purposes is limited to fairly recent times. Put together, noise and non-stationarity mean that the training data will not contain enough information for the network to learn the function. More information is needed, and hints can be the means of providing it. 3 HINTS In this section, we give examples of different types of hints and discuss how to represent them to the learning process. We describe a simple way to use hints that allows the reader to try the method with minimal effort. For a more detailed treatment, please see (Abu-Mostafa, 1993). As far as our method is concerned, a hint is any property that the target function is known to have. For instance, consider the symmetry hint in FX markets as it applies to the U.S. Dollar versus the German Mark (figure 3). This simple hint asserts that if a pattern in the price history implies a certain move in the market, then this implication holds whether you are looking at the market from the U.S. Dollar viewpoint or the German Mark viewpoint. Formally, in terms of normalized prices, the hint translates to invariance under inversion of these prices. Is the symmetry hint valid? The ultimate test for this is how the learning performance is affected by the introduction of the hint. The formulation of hints is an art. 415 Financial Applications of Learning from Hints We use our experience, common sense, and analysis of the market to come up with a list of what we believe to be valid properties of this market. We then represent these hints in a canonical form as we will see shortly, and proceed to incorporate them in the learning process. The improvement in performance will only be as good as the hints we put in. ? ? u.s. GERMAN DOLLAR MARK Figure 3: Illustration of the symmetry hint in FX markets The canonical representation of hints is a more systematic task. The first step in representing a hint is to choose a way of generating 'virtual examples' of the hint. For illustration, suppose that the hint asserts that the target function y is an odd function of the input. An example of this hint would have the form y( -x) = -y(x) for a particular input x. One can generate as many virtual examples as needed by picking different inputs. After a hint is represented by virtual examples, it is ready to be incorporated in the learning process along with the examples of the target function itself. Notice that an example of the function is learned by minimizing an error measure, say (y(x) - y(x))2, as a way of ultimately enforcing the condition y(x) = y(x). In the same way, a virtual example of the oddness hint can be learned by minimizing (y(x) + y(-x))2 as a way of ultimately enforcing the condition y(-x) = -y(x). This involves inputting both x and -x to the network and minimizing the difference between the two outputs. It is easy to show that this can be done using backpropagation (Rumelhart et al, 1986) twice. The generation of a virtual example of the hint does not require knowing the value of the target function; neither y(x) nor y( -x) is needed to compute the error for the oddness hint. In fact, x and -x can be artificial inputs. The fact that we do not need the value of the target function is crucial, since it was the limited resource of examples for which we know the value of the target function that got us interested in hints in the first place. On the other hand, for some hints, we can take the examples of the target function that we have, and employ the hint to duplicate these examples. For instance, an example y(x) = 1 can be used to infer a second example y( -x) = -1 using the oddness hint. Representing the hint by duplicate examples is an easy way to try simple hints using the same software that we use for learning from examples. Yaser S. Abu-Mostafa 416 Let us illustrate how to represent two common types of hints. Perhaps the most common type is the invariance hint. This hint asserts that i)(x) = i)(x / ) for certain pairs x, x'. For instance, "i) is shift-invariant" is formalized by the pairs x, x' that are shifted versions of each other. To represent the invariance hint, an invariant pair (x, x') is picked as a virtual example. The error associated with this example is (y(x) - y(x /?2. Another related type of hint is the monotonicity hint. The hint asserts for certain pairs x, x' that i)(x) :5 i)(x / ). For instance, "i) is monotonically non decreasing in x" is formalized by the pairs x, x' such that x < x'. One application where the monotonicity hint occurs is the extension of personal credit. If person A is identical to person B except that A makes less money than B, then the approved credit line for A cannot exceed that of B. To represent the monotonicity hint, a monotonic pair (X,X/) is picked as a virtual example. The error associated with this example is given by (y(x) - Y(X /?2 if y(x) > y(x / ) and zero if y(x) :5 y(x'). 4 FX TRADING We applied the symmetry hint in the four FX markets of the U.S. Dollar versus the British Pound, the German Mark, the Japanese Yen, and the Swiss Franc. In each case, only the closing prices for the preceding 21 days were used for inputs. The objective (fitness) function we chose was the total return on the training set, and we used simple filtering methods on the inputs and outputs of the networks. In each run, the training set consisted of 500 days, and the test was done on the following 253 days. All four currencies show an improved performance when the symmetry hint is used. Roughly speaking, we are in the market half the time, each trade takes 4 days, the hit rate is close to 50%, and the A.P.R. without hint is 5% and with hint is 10% (the returns are annualized, unleveraged, and include the transaction cost; spread and average slippage). Notice that having the return as the objective function resulted in a fairly good return with a modest hit rate. 5 CROSS CHECKS In this final section, we report more experimental results aimed at validating our claim that the information content of the hint is the reason behind the improved performance. Why is this debatable? A hint plays an incidental role as a constraint on the neural network during learning, since it restricts the solutions the network may settle in. Because overfitting is a common problem in learning from examples, any restriction may improve the out-of-sample performance by reducing overfitting (Akaike, 1969, Moody, 1992). This is the idea behind regularization. To isolate the informative role from the regularizing role of the symmetry hint, we ran two experiments. In the first experiment, we used an uninformative hint, or 'noise' hint, which provides a random target output for the same inputs used in the examples of the symmetry hint. Figure 4 contrasts the performance of the noise hint with that of the real symmetry hint, averaged over the four currencies. Notice that the performance with the noise hint is close to that without any hint (figure 1), which is consistent with the notion of uninformative hint. The regularization effect seems to be negligible. 417 Financial Applications of Learning from Hints 10 ~--------~~--------~----------~----------~-----------n 8 6 4 2 o -2 ~ __________ o ~ ________ 50 ~ __________ 100 ~ __________ ~ __________-u 200 1 50 250 Test Day Number Figure 4: Performance of the real hint versus a noise hint 10 .------------r------------.------------r----------~r_----------~ with fa 1 s e ):l+tlt--:::=':with JJi!trI' hint ---- . I 3a ,,-.,,-?? 5 '~ ~ I ~ " +> Q) IX .. "'"~~.~.~.~.~.~.~.~~~~~~~.~.~.~ ~ ~-.~ -.-.?.-.- ---.-'-.~-.-.~'.~.?~-------.-.-.-.-. c .?. 0 .. . . - J .. - .. .. ....... ... ........... ..................................... . Q) "" f\1 +> C Q) tl ~ .,. -5 Q) "0 Q) .....-< N f\1 -10 " C ~ -15 -20 ~-- o ________ ~ 50 __________ ~ __________- L__________ 100 150 ~ __________--u 200 Test Day Number Figure 5: Performance of the real hint versus a false hint 2 50 418 Yaser S. Abu-Mostafa In the second experiment, we used a harmful hint, or 'false' hint, in place of the symmetry hint. The hint takes the same examples used in the symmetry hint and asserts antisymmetry instead. Figure 5 contrasts the performance of the false hint with that of the real symmetry hint. As we see, the false hint had a detrimental effect on the performance. This is consistent with the hypothesis that the symmetry hint is valid, since its negation results in worse performance than no hint at all. Notice that the transaction cost is taken into consideration in all of these plots, which works as a negative bias and amplifies the losses of bad trading policies. CONCLUSION We have explained learning from hints, a systematic method for combining rules and data in the same learning process, and reported experimental results of a statistically significant improvement in performance in the four major FX markets that resulted from using a simple symmetry hint. We have described different types of hints and simple ways of using them in learning, to enable the readers to try their own hints in different markets. Acknowledgements I would like to acknowledge Dr. Amir Atiya for his valuable input. I am grateful to Dr. Ayman Abu-Mostafa for his expert remarks. References Abu-Mostafa, Y. S. (1990), Learning from hints in neural networks, Journal of Complexity 6, pp. 192-198. Abu-Mostafa, Y. S. (1993), A method for learning from hints, Advances in Neural Information Processing Systems 5, S. Hanson et al (eds), pp. 73-80, MorganKaufmann. Abu-Mostafa, Y. S. et al (eds) (1994), Proceedings of Neural Networks in the Capital Markets, Pasadena, California, November 1994. Akaike, H. (l969), Fitting autoregressive models for prediction, Ann. Inst. Stat. Math. 21, pp. 243-247. Moody, J. (1992), The effective number of parameters: An analysis of generalization and regularization in nonlinear learning systems, in Advances in Neural Information Processing Systems 4, J. Moody et al (eds), pp. 847-854, Morgan Kaufmann. Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986), Learning internal representations by error propagation, Parallel Distributed Processing 1, D. Rumelhart et al, pp. 318-362, MIT Press. Weigend, A., Rumelhart, D., and Huberman, B. (1991), Generalization by weight elimination with application to forecasting, in Advances in Neural Information Processing Systems 3, R. Lippmann et al (eds), pp. 875-882, Morgan Kaufmann. 

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Asymptotics of Gradient-based Neural Network 'fraining Algorithms Sayandev Mukherjee Terrence L. Fine saymukh~ee.comell.edu tlfine~ee.comell.edu School of Electrical Engineering Cornell University Ithaca, NY 14853 School of Electrical Engineering Cornell University Ithaca, NY 14853 Abstract We study the asymptotic properties of the sequence of iterates of weight-vector estimates obtained by training a multilayer feed forward neural network with a basic gradient-descent method using a fixed learning constant and no batch-processing. In the onedimensional case, an exact analysis establishes the existence of a limiting distribution that is not Gaussian in general. For the general case and small learning constant, a linearization approximation permits the application of results from the theory of random matrices to again establish the existence of a limiting distribution. We study the first few moments of this distribution to compare and contrast the results of our analysis with those of techniques of stochastic approximation. 1 INTRODUCTION The wide applicability of neural networks to problems in pattern classification and signal processing has been due to the development of efficient gradient-descent algorithms for the supervised training of multilayer feedforward neural networks with differentiable node functions. A basic version uses a fixed learning constant and updates all weights after each training input is presented (on-line mode) rather than after the entire training set has been presented (batch mode) . The properties of this algorithm as exhibited by the sequence of iterates are not yet well-understood. There are at present two major approaches. Sayandev Mukherjee, Terrence L. Fine 336 Stochastic approximation techniques (Bucklew,Kurtz,Sethares, 1993; Finnoff, 1993; Kuan,Hornik, 1991; White, 1989) study the limiting behavior of the stochastic process that is the piecewise-constant or piecewise-linear interpolation of the sequence of weight-vector iterates (assuming infinitely many U.d. training inputs) as the learning constant approaches zero. It can be shown (Bucklew,Kurtz,Sethares, 1993; Finnoff, 1993) that as the learning constant tends to zero, the fluctuation between the paths and their limit, suitably normalized, tends to a Gaussian diffusion process. Leen and Moody (1993) and Orr and Leen (1993) have considered the Markov process formed by the sequence of iterates (again, assuming infinitely many Li.d. training inputs) for a fixed nonzero learning constant. This approach has the merit of dealing with the nonzero learning constant case anq of linking the study of the training algorithm with the well-developed literature on Markov processes. In particular, it is possible to solve (Leen,Moody, 1993) for the asymptotic distribution of the sequence of weight-vector iterates from the Chapman-Kolmogorov equation after certain assumptions have been used to simplify it considerably. However, the assumptions are unrealistic: in particular, the assumption of detailed balance does not hold in more than one dimension. This approach also fails to establish the existence of a limiting distribution in the general case. This paper follows the method of considering the sequence of weight-vector iterates as a discrete-time continuous state-space Markov process, when the learning constant is fixed and nonzero. We shall first seek to establish the existence of an asymptotic distribution, and then examine this distribution through its first few moments. It can be proved (Mukherjee, 1994), using Foster's criteria (Tweedie, 1976) for the positive-recurrence of a Markov process, that when a single sigmoidal node with one parameter is trained using the iterative form of the basic gradient-descent training algorithm (without batch-processing), the sequence of iterates of the parameter has a limiting distribution which is in general non-Gaussian, thereby qualifying the oftstated claims in the literature (see, for example, (Bucklew,Kurtz,Sethares, 1993; Finnoff, 1993; White, 1989)). However, this method proves to be intractable in the multiple parameter case. 2 THE GENERAL CASE AND LINEARIZATION IN W n The general version of this problem for a neural network", with scalar output involves training ", with the i. i.d. training sequence {( X n' Yn )}, loss function ? (;f, y, w) = ~ [y - ",(~, w W (~ E lRd , Y E lR, w E lRm) and the gradient-descent updating equation for the estimates of the weight vector given by W n+1 = Wn -/LV'1??(;f,y,w)I(X n + ,Yn+l,W n ) = W n + /L[Yn+1 - ",(W n' X n+1)]V'J?."'(w, ;f)lw",x n +1 ? 1 As is customary in this kind of analysis, the training set is assumed infinite, so that {W n}~=O forms a homogeneous Markov process in discrete time. In our analysis, the training data is assumed to come from the model Y = ",(WO,X) + Z, Asymptomatics of Gradient-Based Neural Network Training Algorithms 337 where Z and X are independent, and Z has zero mean and variance (J'2. Hence, the unrestricted Bayes estimator of Y given X, E(YIX) = "l(wO,X), is in the class of neural network estimators, and WO is the goal of training. For convenience, we define W = w - woo Assuming that J.L is small and that after a while, successive iterates, with high probability, jitter about in a close neighborhood of the optimal value wO, we make the important assumption that (1) for some 0 < k < 1 (see Section 4) 1. Applying Taylor series expansions to "l and \1 w"l and neglecting all terms Op(J.L1+ 2 k) and higher, we obtain the following linearized form of the updating equation: (2) where Bn+1 = J.LZn+l \11?"l( W, ;?)11?0,x n+l ' An+1 = G n+1 1m - J.L(\11?"l(w,;?)I~o ,Xn+l)(\11?"l(w,;?)I~o ,Xn+lf + J.L Z n+1 \1-w\1-w"l(w,;?) Iw- o'-n+l X (3) 1m - J.L(Gn+1 - Zn+1 J n+1), (4) = (\11!!."l(w,~)I~o ,Xn+l)(\11!!."l(w,~)I~o,Xn+l f oX I n +1 = (\1w\1w"l(w,;?)lw ) - '-n+l do not depend on W n. The matrices {(A n+1, B n +1 )} form an Li.d. sequence, but An+! and Bn+1 are dependent for each n. Hence the linearized W n again forms a homogeneous Markov process in discrete time. In what follows we analyze this process in the hope that its asymptotics agree with those of the original Markov process. 3 EXISTENCE OF A LIMITING DISTRIBUTION Let A, B, G, J denote random matrices with the common distributions of the i.i.d. sequences {An}, {Bn}, {G n }, and {In} respectively, and let T : lRm ---+ lRm be the random affine transformation w~Aw+B . The following result establishes the existence of a limiting distribution of W n. Lemma 1 (Berger Thm. V, p.162) Suppose E[1og+ IIAII + log+ IIBlll ElogIIA n A n - 1 ? ? ?AtII < < where log+ X = log x V Then the following conclusions hold: , 00? o for some n o. ll.e., (VE > O)(3M.)(Vn)lP(J.L- k IlW nil $ M.) ~ 1 - E. (5) (6) 338 Sayandev Mukherjee, Terrence L. Fine 1. Unique stationary distribution: There exists a unique mndom variable WEIRm, upto distribution, that is stationary with respect to T (i. e., W is independent of T, and TW has the same distribution as W). 2. Asymptotic stationarity: We have convergence in distribution: - 'D - Wn~W. Our choice of norm is the operator norm for the matrix A, /lAII = max IA(A)I, where {A(A)} are the eigenvalues of A, and the Euclidean norm for the vector B, m IIBII 2: IB iI = 2. i=1 We first verify (5). From the inequality "Ix E IR, log+ X ::; x 2 , it is easily seen that if r] is a feedforward net where all activation functions are twice-continuously differentiable in the weights, all hidden-layer activation functions are bounded and have bounded derivatives up to order 2, and if the training sequence (Xn' Yn ) is Li.d. with finite fourth moments, then (5) holds for the Euclidean norm for Band the Frobenius norm for A, IIAII2 = 2::1 2:7=1IAijI2. Since m (max IA(A)1)2 :::; m 2: IA(AW :::; 2: 2: IAij12, (7) i=1 j=1 we see that (5) also holds for the operator norm of A. Assumption (6) forces the product An'" A1 to tend to Omxm almost surely (Berger, 1993, p.146) and therefore removes the dependence of the asymptotic distribution of {W n} on that of the initial value ll:=o. A sufficient condition for (6) is given by the following lemma. Lemma 2 Suppose lEG is positive definite (note that it is positive semidefinite by definition), and for all n, lEAn < 00. Then (6) holds for sufficiently small, positive 11? Proof: By assumption, minA(lEG) = 8 > 0 for some 8. Let Hn = ~ 2:~=1 (G i - ZiJi)' By the Strong Law of Large Numbers applied to the LLd. random matrices (Gi - ZiJi) , we have Hn --+ lEG a.s., so minA(Hn) --+ minA(lEG) a.s. (8) Applying (7) to min A(Hn), it is easily shown that the same conditions on r] and the training sequence that are sufficient for (5) also give sUPn lE[min A(Hn )J2 < 00, which in turn implies that {min A(Hn)} are uniformly integrable. Together with (8), this implies (Loeve, 1977, p.165) that min A(Hn) --+ min A(lEG) in L1. Hence there Asymptomatics of Gradient-Based Neural Network Training Algorithms 339 exists some (nonrandom) N, say, such that ElminA(HN) - minA(EG)1 ::; 8/2. Since IEminA(HN) - minA(EG)1 ::; EI minA(HN) - minA(EG)1 ::; 8/2, we therefore have E [min'\ (~ t,(Gi - ZiJi?) 1? rnin'\(EG) - 8/2 = 8 - 8/2 = 8/2> O. (9) We shall prove that (6) holds for this N (;:::: m) by showing that Elog IIANAN- I ??? Ad 2 = 2lEiog IIANAN-I'" Alii < O. For our choice of norm, we therefore want E[log(max IA(AN .. . Al)I)2] < O. From Jensen's inequality, it is sufficient to have log E[max IA(AN . . . Ad 1]2 < 0, or equivalently, (10) Now, since N is fixed, we can choose /-l small enough that N AN'" Al = 1m - /-l2:)G i - ZiJi) + Op(/-l2). i=l Hence, A(AN ... AI) =1- /-lA(l:~1 (G i I'\(AN ... AI)I' = 1 - - ZiJi? + Op (/-l2), and N is fixed, so 21''\ (t,(G, - Z,J ,?) + Op (1"), giving max I'\(AN' .. Adl' <; 1 - 21' min'\ (t,(Gi - Z,J ,?) + Op (1"), EmaxIA(AN,,?AIW::; I-N8/-l+o(/-l), (11) where we use (9) and the observation that the structure of the last Op(/-l2) term is such that its expectation (guaranteed finite by the hypothesis EAN < 00) is o (/-l2) , or o(/-l), and we also restrict /-l < 1/N 8 so that 1- 2p.Ernin'\ (t,(G, - Z,J,?) > O. From (11), it is clear that (10) holds for all sufficiently small, positive /-l (<< 1/N8) . Therefore (6) holds for n = N. We can combine these two lemmas into the following theorem. Theorem 1 Let 'fJ be a feedforward net where all activation functions are twicecontinuously differentiable in the weights, all hidden-layer activation junctions are bounded and have bounded derivatives up to order 2, and let the training sequence (Xn' Y n ) be i.i.d. with finite moments. Further, assume that lEG is positive definite. Then, for all sufficiently small, positive /-l the sequence of random vectors {W n}~l obtained from the updating equation (2) has a unique limiting distribution. Sayandev Mukherjee, Terrence L. Fine 340 We circumvent the generally intractable problem of finding the limiting distribution by calculating and investigating the behavior of its moments. 4 MOMENTS OF THE LIMITING DISTRIBUTION Let us assume that the mean and variance of the limiting distribution exist, and that Z '" N(O, (1'2). From (2) and the form of An+! and B n+!, it is easy to show that EW = 0, or EW = wo, so the optimal value Wo is the mean of the limiting distribution of the sequence of iterates {W n}' It can also be shown (Mukherjee, 1994) that EWWT = (f.L(1'2/2) 1m, yielding W = Op(yTi). This is consistent with our assumption (1) with k = 1/2. In the one-dimensional case (d = m = 1), we have EW = 0 and EW2 = ~f.L(1'2 if lE[Xn+!'1}'(WO Xn+!W i= O. Using these results, the fact that Z '" N(O, (1'2), EW = 0, the independence of Z and X, and assuming that EX B < 00, it is not difficult to compute the expressions - 3 _ lEW - f.L2(1'4E[X3'1}"'1}'(1 - f.LX2'1}'2)] , E[X2'1}'2 - f.LX4( '1}'4 + (1'2'1}"2) + f.L2 X6'1}'2( '1}'4 /3 + (1'2'1}"2)] and where E[X2'1}'2(1 - f.LX2'1}'2)2 + f.LX4'1}'4 + 3f.L2 X6'1}"2'1}'2] K(f.L) 18f.L2(1'2E[X3'1}"'1}'(1 - f.LX2'1}'2)2 K(f.L) E[X2'1}'2 - + f.L4 (1' 4x 7 '1}"3'1}'] ~f.LX4('1}14 + (1'2'1}"2(1 2 f.LX2'1}'2)2) + f.L2 X6'1}'6 - ~f.L3 XB('1}'B + 3(1'4'1}"4)], and '1}' and '1}" are evaluated at the argument wO X for'1}. From the above expressions, it is seen that if'1}O = 1/[1 + e- O ] and X has a symmetric distribution (say N(O, 1)), then EW 3 i= 0 and lEW 4 i= 3(EW2)2, implying that W is non-Gaussian in general. This result is consistent with that obtained by direct application of Foster's criterion (Mukherjee, 1994). 5 RECONCILING LINEARIZATION AND STOCHASTIC APPROXIMATION METHODS The results of stochastic approximation analysis give a Gaussian distribution for W in the limit as f.L ---. 0 (Bucklew,Kurtz,Sethares, 1993; Finnoff, 1993). However, our results establish that the Gaussian distribution result is not valid for small nonzero f.L in general. To reconcile these results, recall W = Op (yTi). Hence, if we consider Asymptomatics of Gradient-Based Neural Network Training Algorithms 341 only moments of the normalized quantity W/ fo (and neglect higher-order terms in Op(fo)), we obtain E(W / fo)3 = 0 and E(W / fo)4 = 3[E(W / fo)2j2, which suggests that the normalized quantity W/ fo is Gaussian in the limit of vanishing {L, a conclusion also reached from stochastic approximation analysis. In support of this theoretical indication that the conclusions of our analysis (based on linearization for small {L) might tally with those of stochastic approximation techniques for small values of {L, simulations were done on the simple onedimensional training case of the previous section for 8 cases: {L = 0.1,0.2,0.3,0.5, and (72 = 0.1,0.5 for each value of {L, with w O fixed at 3. For each of the 8 cases, either 5 or 10 runs were made, with lengths (for the given values of {L) of 810000, 500000, 300000, and 200000 respectively. Each run gave a pair of sequences {Wn } obtained by starting off at Wo = 0 and training the network independently twice. Each resulting sequence {Wn} was then downs amp led at a large enough rate that the true autocorrelation of the downsampled sequence was less than 0.05, followed by deleting the first 10% of the samples of this downsampled sequence, so as to remove any dependence on initial conditions that might persist. (Autocorrelation at lag unity for this Markov Chain was so high that when {L = 0.1, a decimation rate of 9000 was required.) This was done to ensure that the elements of the resulting downsampled sequences could be assumed independent for the various hypothesis tests that were to follow. (a) For each run of each case, the empirical distribution functions of the two downsampled sequences thus generated were compared by means of the Kolmogorov-Smirnov test (Bickel,Doksum, 1977) at level 0.95, with the null hypothesis being that both sequences had the same actual cumulative distribution function (assumed continuous). This test was passed with ease on all trials, thereby showing that a limiting distribution existed and was attained by such a training algorithm. (b) For each run of each case, a skewness test and a kurtosis test (Bickel,Doksum, 1977) for normality were done at level 0.95 to test for normality. The sequences generated failed both tests for the ({L, (7) pair (0.1,0.1) and passed them both for the pairs (0 .1,0.5), (0.3,0.1), (0 .5,0.1), and (0.5,0.5). For the pair (0.2,0.5), the skewness test was passed and the kurtosis test failed, and for the pairs (0.2,0.1) and (0.3,0.5), the skewness test was failed and the kurtosis test passed. (c) All trials cleared the Kolmogorov tests (Bickel,Doksum, 1977) for normality at level 0.95, both when the normal distribution was taken to have the sample mean and variance (c.)mputed on the downsampled sequence), and when the normal distribution function had the asymptotic values of mean (zero) and variance ({L(72 /2). Hence we may conclude: 1. The limiting distribution of {Wn } exists. 2. For small values of {L, the deviation from Gaussianness is so small that the Gaussian distribution may be taken as a good approximation to the limiting distribution. 342 Sayandev Mukherjee, Terrence L. Fine In other words, though stochastic approximation analysis states that W/ yfji is Gaussian only in the limit of vanishing f.J" our simulation shows that this is a good approximation for small values of f.J, as well. Acknowledgements The research reported here was partially supported by NSF Grant SBR-9413001. References Berger, Marc A. An Introduction to Probability and Stochastic Processes. SpringerVerlag, New York, 1993. Bickel, Peter, and Doksum, Kjell. Mathematical Statistics: Basic Ideas and Selected Topics. Holden-Day, San Francisco, 1977. Bucklew, J.A., Kurtz, T.G., and Sethares, W.A. "Weak Convergence and Local Stability Properties of Fixed Step Size Recursive Algorithms," IEEE Trans. Inform. Theory, vol. 39, pp. 966-978, 1993. Finnoff, W. "Diffusion Approximations for the Constant Learning Rate Backpropagation Algorithm and Resistence to Local Minima." In Giles, C.L., Hanson, S.J., and Cowan, J.D., editors, Advances in Neural Information Processing Systems 5. Morgan Kaufmann Publishers, San Mateo CA, 1993, p.459 ff. Kuan, C-M, and Hornik, K. "Convergence of Learning Algorithms with Constant Learning Rates," IEEE Trans. Neural Networks, vol. 2, pp. 484-488, 1991. Leen, T.K., and Moody, J .E. "Weight Space Probability Densities in Stochastic Learning: 1. Dynamics and Equilibria," Adv. in NIPS 5, Morgan Kaufmann Publishers, San Mateo CA, 1993, p.451 ff. Loeve, M. Probability Theory I, 4th ed. Springer-Verlag, New York, 1977. Mukherjee, Sayandev. Asymptotics of Gradient-based Neural Network Training Algorithms. M.S. thesis, Cornell University, Ithaca, NY, 1994. Orr, G.B., and Leen, T.K. "Probability densities in stochastic learning: II. Transients and Basin Hopping Times," Adv. in NIPS 5, Morgan Kaufmann Publishers, San Mateo CA, 1993, p.507 ff. Rumelhart, D.E., Hinton, G.E., and Williams, R.J. "Learning interval representations by error propagation." In D.E. Rumelhart and J.L. McClelland, editors, Parallel Distributed Processing, Ch. 8, MIT Press, Cambridge MA, 1985. Tweedie, R.L. "Criteria for Classifying General Markov Chains," Adv. Appl. Prob., vol. 8,737-771, 1976. White, H. "Some Asymptotic Results for Learning in Single Hidden-Layer Feedforward Network Models," J. Am. Stat. Assn., vol. 84, 1003-1013, 1989. PART IV REThWORCEMENTLEARMNG 

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Instance-Based State Identification for Reinforcement Learning R. Andrew McCallum Department of Computer Science University of Rochester Rochester, NY 14627-0226 mccallumCcs.rochester.edu Abstract This paper presents instance-based state identification, an approach to reinforcement learning and hidden state that builds disambiguating amounts of short-term memory on-line, and also learns with an order of magnitude fewer training steps than several previous approaches. Inspired by a key similarity between learning with hidden state and learning in continuous geometrical spaces, this approach uses instance-based (or "memory-based") learning, a method that has worked well in continuous spaces. 1 BACKGROUND AND RELATED WORK When a robot's next course of action depends on information that is hidden from the sensors because of problems such as occlusion, restricted range, bounded field of view and limited attention, the robot suffers from hidden state. More formally, we say a reinforcement learning agent suffers from the hidden state problem if the agent's state representation is non-Markovian with respect to actions and utility. The hidden state problem arises as a case of perceptual aliasing: the mapping between states of the world and sensations of the agent is not one-to-one [Whitehead, 1992]. If the agent's perceptual system produces the same outputs for two world states in which different actions are required, and if the agent's state representation consists only of its percepts, then the agent will fail to choose correct actions. Note that even if an agent's state representation includes some internal state beyond its 378 R. Andrew McCallum immediate percepts, the agent can still suffer from hidden state if it does not keep enough internal state to uncover the non-Markovian-ness of its environment. One solution to the hidden state problem is simply to avoid passing through the aliased states. This is the approach taken in Whitehead's Lion algorithm [Whitehead, 1992]. Whenever the agent finds a state that delivers inconsistent reward, it sets that state's utility so low that the policy will never visit it again. The success of this algorithm depends on a deterministic world and on the existence of a path to the goal that consists of only unaliased states. Other solutions do not avoid aliased states, but do as best they can given a nonMarkovian state representation [Littman, 1994; Singh et al., 1994; Jaakkola et al., 1995]. They involve either learning deterministic policies that execute incorrect actions in some aliased states, or learning stochastic policies with action choice probabilities matching the proportions of the different underlying aliased world states. These approaches do not depend on a path of unaliased states, but they have other limitations: when faced with many aliased states, a stochastic policy degenerates into random walk; when faced with potentially harmful results from incorrect actions, deterministically incorrect or probabilistically incorrect action choice may prove too dangerous; and when faced with performance-critical tasks, inefficiency that is proportional to the amount of aliasing may be unacceptable. The most robust solution to the hidden state problem is to augment the agent's state representation on-line so as to disambiguate the aliased states. State identification techniques uncover the hidden state information-that is, they make the agent's internal state space Markovian. This transformation from an imperfect state information model to a perfect state information model has been formalized in the decision and control literature, and involves adding previous percepts and actions to the definition of agent internal state [Bertsekas and Shreve, 1978]. By augmenting the agent's perception with history information-.short-term memory of past percepts, actions and rewards-the agent can distinguish perceptually aliased states, and can then reliably choose correct actions from them. Predefined, fixed memory representations such as order n Markov models (also known as constant-sized perception windows, linear traces or tapped-delay lines) are often undesirable. When the length of the window is more than needed, they exponentially increase the number of internal states for which a policy must be stored and learned; when the length of the memory is less than needed, the agent reverts to the disadvantages of undistinguished hidden state. Even if the agent designer understands the task well enough to know its maximal memory requirements, the agent is at a disadvantage with constant-sized windows because, for most tasks, different amounts of memory are needed at different steps of the task. The on-line memory creation approach has been adopted in several reinforcement learning algorithms. The Perceptual Distinctions Approach [Chrisman, 1992] and Utile Distinction Memory [McCallum, 1993] are both based on splitting states of a finite state machine by doing off-line analysis of statistics gathered over many steps. Recurrent-Q [Lin, 1993] is based on training recurrent neural networks. Indexed Memory [Teller, 1994] uses genetic programming to evolve agents that use load and store instructions on a register bank. A chief disadvantage of all these techniques is that they require a very large number of steps for training. Instance-Based State Identification for Reinforcement Learning 2 379 INSTANCE-BASED STATE IDENTIFICATION This paper advocates an alternate solution to the hidden state problem we term instance-based state identification. The approach was inspired by the successes of instance-based (also called "memory-based") methods for learning in continuous perception spaces, (i.e. [Atkeson, 1992; Moore, 1992]). The application of instance-based learning to short-term memory for hidden state is driven by the important insight that learning in continuous spaces and learning with hidden state have a crucial feature in common: they both begin learning without knowing the final granularity of the agent's state space. The former learns which regions of continuous input space can be represented uniformly and which areas must be finely divided among many states. The later learns which percepts can be represented uniformly because they uniquely identify a course of action without the need for memory, and which percepts must be divided among many states each with their own detailed history to distinguish them from other perceptually aliased world states. The first approach works with a continuous geometrical input space, the second works with a percept-act ion-reward "sequence" space, (or "history" space). Large continuous regions correspond to less-specified, small memories; small continuous regions correspond to more-specified, large memories. Furthermore, learning in continuous spaces and sequence spaces both have a lot to gain from instance-based methods. In situations where the state space granularity is unknown, it is especially useful to memorize the raw previous experiences. If the agent tries to fit experience to its current, flawed state space granularity, it is bound to lose information by attributing experience to the wrong states. Experience attributed to the wrong state turns to garbage and is wasted. When faced with an evolving state space, keeping raw previous experience is the path of least commitment, and thus the most cautious about losing information. 3 NEAREST SEQUENCE MEMORY There are many possible instance-based techniques to choose from, but we wanted to keep the first application simple. With that in mind, this initial algorithm is based on k-nearest neighbor. We call it Nearest Sequence Memory, (NSM). It bears emphasizing that this algorithm is the most straightforward, simple, almost naive combination of instance-based methods and history sequences that one could think of; there are still more sophisticated instance-based methods to try. The surprising result is that such a simple technique works as well as it does. Any application of k-nearest neighbor consists of three parts: 1) recording each experience, 2) using some distance metric to find neighbors of the current query point, and 3) extracting output values from those neighbors. We apply these three parts to action-percept-reward sequences and reinforcement learning by Q-Iearning lWatkins, 1989] as follows: 1. For each step the agent makes in the world, it records the action, percept and reward by adding a new state to a single, long chain of states. Thus, each state in the chain contains a snapshot of immediate experience; and all the experiences are laid out in a time-connected history chain. 380 R. Andrew McCallum Learning in a Geometric Space k-nearest neighbor, k = 3 ? ? ? ? ? Learning in a Sequence Space k-nearest neighbor, k =3 00014001301201 ? ? match length 3~ 0 1 " action. percept. rewanl Figure 1: A continuous space compared with a sequence space. In each case, the "query point" is indicated with a gray cross, and the three nearest neighbors are indicated with gray shadows. In a geometric space, the neighborhood metric is defined by Euclidean distance. In a sequence space, the neighborhood metric is determined by sequence match length-the number of preceding states that match the states preceding the query point. 2. When the agent is about to choose an action, it finds states considered to be similar by looking in its state chain for states with histories similar to the current situation. The longer a state's string of previous experiences matches the agent's most recent experiences, the more likely the state represents where the agent is now. 3. Using the states, the agent obtains Q-values by averaging together the expected future reward values associated with the k nearest states for each action. The agent then chooses the action with the highest Q-value. The regular Q-Iearning update rule is used to update the k states that voted for the chosen action. Choosing to represent short-term memory as a linear trace is a simple, wellestablished technique. Nearest Sequence Memory uses a linear trace to represent memory, but it differs from the fixed-sized window approaches because it provides a variable memory-length-like k-nearest neighbor, NSM can represent varying resolution in different regions of state space. 4 DETAILS OF THE ALGORITHM A more complete description of Nearest Sequence Memory, its performance and its possible improvements can be found in [McCallum, 1995]. The interaction between the agent and its environment is described by actions, percepts and rewards. There is a finite set of possible actions, A {al,a2, ... ,am }, = Instance-Based State Identification for Reinforcement Learning 381 = a finite set of possible percepts, () {Ol, 02, ... , On}, and scalar range of possible rewards, = [x, y], x, Y E~. At each time step, t, the agent executes an action, at E A, then as a result receives a new percept, Ot E (), and a reward, rt E The agent records its raw experience at time t in a "state" data point, St. Also associated with St is a slot to hold a single expected future discounted reward value, denoted q(st). This value is associated with at and no other action. n n. 1. Find the k nearest neighbor (most similar) states for each possible future action. The state currently at the end of the chain is the "query point" from which we measure all the distances. The neighborhood metric is defined by the number of preceding experience records that match the experience records preceding the "query point" state. (Here higher values of n(s;, sJ) indicate that S; and Sj are closer neighbors.) if (a;-1 = aj-I) A (0;-1 otherwise _ _)_ { 1+n(s;_1,Sj-I), n ( S" SJ 0, Considering each of the possible future actions neighbors and give them a vote, v(s;). v(S;) ={ ~: ill = OJ-I) A (r;-1 = rj_I) (1) turn, we find the k nearest if n(st, s;) is among the k maxv$jlaj=a; n(st, Sj)'s otherwise (2) 2. Determine the Q-value for each action by averaging individual the q-values from the k voting states for that action. L Qt(a;) = (v(s;)/k)q(sj) (3) V$jlaj=a; 3. Select an action by maximum Q-value, or by random exploration. According to an exploration probability, e, either let at+1 be randomly chosen from A, or (4) 4. Execute the action chosen in step 3, and record the resulting experience. Do this by creating a new "state" representing the current state of the environment, and storing the action-percept-reward triple associated with it: Increment the time counter: t ~ t + 1. Create St; record in it at, Ot, rt. The agent can limit its storage and computational load by limiting the number of instances it maintains to N (where N is some reasonably large number) . Once the agent accumulates N instances, it can discard the oldest instance each time it adds a new one. This also provides a way to handle a changing environment. 5. Update the q-values by vote. Perform the dynamic programming step using the standard Q-Iearning rule to update those states that voted for the chosen action. Note that this actually involves performing steps 1 and 2 to get the next Q-values needed for calculating the utility of the agent's current state, Ut . (Here (3 is the learning rate.) Ut = (Vsda; = at-I) q(s;) ~ max Qt(a) (5) a (1- (3v(s;))q(s;) + (3v(s;)(r; + "YUt) (6) 382 R. Andrew McCallum Performance during learning 20 Nearest Sequence Memory Perceptual Distinctions Approach - Nearest Sequence Memory . Recurrent-a - . 60 ~ 50 15 .,c. li ~ 5 Steps per Trial during learning 70 10 40 30 5 20 1000 2000 3000 lD ~: t fBI 74 4000 5000 Steps 6000 7000 10 8000 40 ~: ...._._' 60 100 80 StepslOLMm 2500 -, StejM 10 L.", 238 _ o f ...... ~:; 153 20 Trials StejM 10 LMm 1500 0 Ii?iii :: ....borof ...... I0_~. ,."I0 0 =L-.'-...:39==5=-_ _ _Steps __ .....borof ...... Figure 2: Comparing Nearest Sequence Memory with three other algorithms: Perceptual Distinction Approach, Recurrent-Q and Utile Distinction Memory. In each case, NSM learns with roughly an order of magnitude fewer steps. 5 EXPERIMENTAL RESULTS The performance of NSM is compared to three other algorithms using the tasks chosen by the other algorithms' designers. In each case, NSM learns the task with roughly an order of magnitude fewer steps. Although NSM learns good policies quickly, it does not always learn optimal policies. In section 6 we will discuss why the policies are not always optimal and how NSM could be improved. The Perceptual Distinctions Approach [Chrisman, 1992] was demonstrated in a space ship docking application with hidden state. The task was made difficult by noisy sensors and actions. Some of the sensors returned incorrect values 30% of the time. Various actions failed 70, 30 or 20% of the time, and when they failed, resulted in random states. NSM used f3 = 0.2, I = 0.9, k = 8, and N = 1000. PDA takes almost 8000 steps to learn the task. NSM learns a good policy in less than 1000 steps, although the policy is not quite optimal. Utile Distinction Memory [McCallum, 1993] was demonstrated on several local perception mazes. Unlike most reinforcement learning maze domains, the agent perceives only four bits indicating whether there is a barrier to the immediately adjacent north, east, south and west. NSM used f3 = 0.9, I = 0.9, k = 4, and N = 1000. In two of the mazes, NSM learns the task in only about 1/20th the time required by UDM; in the other two, NSM learns mazes that UDM did not solve at all . Instance-Based State Identification for Reinforcement Learning 383 Recurrent-Q [Lin, 1993] was demonstrated on a robot 2-cup retrieval task. The env,jronment is deterministic, but the task is made difficult by two nested levels of hidden state and by providing no reward until the task is completely finished. NSM used {3 = 0.9, I = 0.9, k = 4, and N = 1000. NSM learns good performance in about 15 trials , Recurrent-Q takes about 100 trials to reach equivalent performance. 6 DISCUSSION Nearest Sequence Memory offers much improved on-line performance and fewer training steps than its predecessors. Why is the improvement so dramatic? I believe the chief reason lies with the inherent advantage of instance-based methods, as described in section 2: the key idea behind Instance-Based State Identification is the recognition that recording raw experience is particularly advantageous when the agent is learning a policy over a changing state space granularity, as is the case when the agent is building short-term memory for disambiguating hidden state. If, instead of using an instance-based technique, the agent simply averages new experiences into its current, flawed state space model, the experiences will be applied to the wrong states, and cannot be reused when the agent reconfigures its state space. Furthermore, and perhaps even more detrimentally, incoming data is always interpreted in the context of the flawed state space, always biased in an inappropriate way-not simply recorded, kept uncommitted and open to easy reinterpretation in light of future data. The experimental results in this paper bode well for instance-based state identification. Nearest Sequence Memory is simple-if such a simplistic implementation works as well as it does, more sophisticated approaches may work even better. Here are some ideas for improvement: The agent should use a more sophisticated neighborhood distance metric than exact string match length. A new metric could account for distances between different percepts instead of considering only exact matches. A new metric could also handle continuous-valued inputs. Nearest Sequence Memory demonstrably solves tasks that involve noisy sensation and action, but it could perhaps handle noise even better if it used some technique for explicitly separating noise from structure. K-nearest neighbor does not explicitly discriminate between structure and noise. If the current query point has neighbors with wildly varying output values, there is no way to know if the variations are due to noise , (in which case they should all be averaged), or due to fine-grained structure of. the underlying function (in which case only the few closest should be averaged). Because NSM is built on k-nearest neighbor, it suffers from the same inability to methodically separate history differences that are significant for predicting reward and history differences that are not. I believe this is the single most important reason that NSM sometimes did not find optimal policies. Work in progress addresses the structure/noise issue by combining instance-based state identification with the structure/noise separation method from Utile Distinction Memory [McCallum, 1993]. The algorithm, called Utile Suffix Memory, uses a tree-structured representation, and is related to work with Ron, Singer and Tishby's Prediction Suffix Trees, Moore's Parti-game, Chapman and Kaelbling's 384 R. Andrew McCallum G-algorithm, and Moore's Variable Resolution Dynamic Programming. See [McCallum, 1994] for more details as well as references to this related work. Aclmowledgments This work has benefited from discussions with many colleagues, including: Dana Ballard, Andrew Moore, Jeff Schneider, and Jonas Karlsson. This material is based upon work supported by NSF under Grant no. IRI-8903582 and by NIH/PHS under Grant no. 1 R24 RR06853-02. References [Atkeson, 1992] Christopher G. Atkeson. Memory-based approaches to approximating continuous functions. In M. Casdagli and S. Eubank, editors, Nonlinear Modeling and Forecasting, pages 503-521. Addison Wesley, 1992. [Bertsekas and Shreve, 1978] Dimitri. P. Bertsekas and Steven E. Shreve. Stochastic Optimal Control. Academic Press, 1978. [Chrisman, 1992] Lonnie Chrisman. Reinforcement learning with perceptual aliasing: The perceptual distinctions approach. In Tenth Nat'l Conf. on AI, 1992. [Jaakkola et al., 1995] Tommi Jaakkola, Satinder Pal Singh, and Michael 1. Jordan. Reinforcement learning algorithm for partially observable markov decision problems. In Advances of Neural Information Processing Systems 7. Morgan Kaufmann, 1995. [Lin, 1993] Long-Ji Lin. Reinforcement Learning for Robots Using Neural Networks. PhD thesis, Carnegie Mellon, School of Computer Science, January 1993. [Littman, 1994] Michael Littman. Memoryless policies: Theoretical limitations and practical results. In Proceedings of the Third International Conference on Simulation of Adaptive Behavior: From Animals to Animats, 1994. [McCallum, 1993] R. Andrew McCallum. Overcoming incomplete perception with utile distinction memory. In The Proceedings of the Tenth International Machine Learning Conference. Morgan Kaufmann Publishers, Inc., 1993. [McCallum, 1994] R. Andrew McCallum. Utile suffix memory for reinforcement learning with hidden state. TR 549, U. of Rochester, Computer Science, 1994. [McCallum, 1995] R. Andrew McCallum. Hidden state and reinforcement learning with instance-based state identification. IEEE Trans. on Systems, Man, and Cybernetics, 1995. (In press) [Earlier version available as U. of Rochester TR 502]. [Moore, 1992] Andrew Moore. Efficient Memory-based Learning for Robot Control. PhD thesis, University of Cambridge, November 1992. [Singh et al., 1994] Satinder Pal Singh, Tommi Jaakkola, and Michael 1. Jordan. Modelfree reinforcement learning for non-markovian decision problems. In The Proceedings of the Eleventh International Machine Learning Conference, 1994. [Teller, 1994] Astro Teller. The evolution of mental models. In Kim Kinnear, editor, Advances in Genetic Programming, chapter 9. MIT Press, 1994. [Watkins, 1989] Chris Watkins. Learning from delayed rewards. PhD thesis, Cambridge University, 1989. [Whitehead, 1992] Steven Whitehead. Reinforcement Learning for the Adaptive Control of Perception and Action. PhD thesis, Department of Computer Science, University of Rochester, 1992. 

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Learning with Preknowledge: Clustering with Point and Graph Matching Distance Measures Steven Gold!, Anand Rangarajan 1 and Eric Mjolsness 2 Department of Computer Science Yale University New Haven, CT 06520-8285 Abstract Prior constraints are imposed upon a learning problem in the form of distance measures. Prototypical 2-D point sets and graphs are learned by clustering with point matching and graph matching distance measures. The point matching distance measure is approx. invariant under affine transformations - translation, rotation, scale and shear - and permutations. It operates between noisy images with missing and spurious points. The graph matching distance measure operates on weighted graphs and is invariant under permutations. Learning is formulated as an optimization problem . Large objectives so formulated ('" million variables) are efficiently minimized using a combination of optimization techniques - algebraic transformations, iterative projective scaling, clocked objectives, and deterministic annealing. 1 Introduction While few biologists today would subscribe to Locke's description of the nascent mind as a tabula rasa, the nature of the inherent constraints - Kant's preknowl1 E-mail address of authors: lastname-firstname@cs.yale.edu 2Department of Computer Science and Engineering, University of California at San Diego (UCSD), La Jolla, CA 92093-0114. E-mail: emj@cs.ucsd.edu 714 Steven Gold, Anand Rangarajan, Eric Mjolsness edge - that helps organize our perceptions remains much in doubt. Recently, the importance of such preknowledge for learning has been convincingly argued from a statistical framework [Geman et al., 1992]. Researchers have proposed that our brains may incorporate preknowledge in the form of distance measures [Shepard, 1989]. The neural network community has begun to explore this idea via tangent distance [Simard et al., 1993], model learning [Williams et al., 1993] and point matching distances [Gold et al., 1994]. However, only the point matching distances have been invariant under permutations. Here we extend that work by enhancing both the scope and function of those distance measures, significantly expanding the problem domains where learning may take place. We learn objects consisting of noisy 2-D point-sets or noisy weighted graphs by clustering with point matching and graph matching distance measures. The point matching measure is approx. invariant under permutations and affine transformations (separately decomposed into translation, rotation, scale and shear) and operates on point-sets with missing or spurious points. The graph matching measure is invariant under permutations. These distance measures and others like them may be constructed using Bayesian inference on a probabilistic model of the visual domain. Such models introduce a carefully designed bias into our learning, which reduces its generality outside the problem domain but increases its ability to generalize within the problem domain. (From a statistical viewpoint, outside the problem domain it increases bias while within the problem domain it decreases variance). The resulting distance measures are similar to some of those hypothesized for cognition. The distance measures and learning problem (clustering) are formulated as objective functions. Fast minimization of these objectives is achieved by a combination of optimization techniques - algebraic transformations, iterative projective scaling, clocked objectives, and deterministic annealing. Combining these techniques significantly increases the size of problems which may be solved with recurrent network architectures [Rangarajan et al., 1994]. Even on single-cpu workstations non-linear objectives with a million variables can routinely be minimized. With these methods we learn prototypical examples of 2-D points set and graphs from randomly generated experimental data. 2 2.1 Distance Measures in Unsupervised Learning An Affine Invariant Point Matching Distance Measure The first distance measure quantifies the degree of dissimilarity between two unlabeled 2-D point images, irrespective of bounded affine transformations, i.e. differences in position, orientation, scale and shear. The two images may have different numbers of points. The measure is calculated with an objective that can be used to find correspondence and pose for unlabeled feature matching in vision. Given two sets of points {Xj} and {Yk}, one can minimize the following objective to find the affine transformation and permutation which best maps Y onto X : J Epm(m, t,A) K = L: L: mjkllXj j=lk=l with constraints: Yj Ef=l mjk ~ 1 , J t - A? Ykll 2 + g(A) - a K L: L: mjk j=lk=l Yk Ef=l mjk ~ 1 , Yjk mjk 2:: O. Learning with Preknowledge 715 A is decomposed into scale, rotation, vertical shear and oblique shear components. g(A) regularizes our affine transformation - bounding the scale and shear compo- nents. m is a fuzzy correspondence matrix which matches points in one image with corresponding points in the other image. The inequality constraint on m allows for null matches - that is a given point in one image may match to no corresponding point in the other image. The a term biases the objective towards matches. Then given two sets of points {Xj} and {Yk} the distance between them is defined as: D({Xj}, {Yk}) = min (Epm(m,t, A) I constraints on m) m,t,A This measure is an example of a more general image distance measure derived in [Mjolsness, 1992]: d(z, y) = mind(z, T(y? E [0, (0) T where T is a set of transformation parameters introduced by a visual grammar. Using slack variables, and following the treatment in [Peterson and Soderberg, 1989; Yuille and Kosowsky, 1994] we employ Lagrange multipliers and an z logz barrier function to enforce the constraints with the following objective: J Epm(m, t, A) J K = L: L: mjkllXj - t - A? Ykll 2 + g(A) - a j=lk=l L: L: mjk j=lk=1 1 J+1 K+1 +-p L: L: mjk(logmjk - K 1) + j=1 k=l J K+1 j=l k=l L: J.'j (L: mjk - 1) + K J+1 k=1 j=l L: lIk(L: mjk - 1) (1) In this objective we are looking for a saddle point. (1) is minimized with respect to m, t, and A which are the correspondence matrix, translation, and affine transform, and is maximized with respect to l' and 11, the Lagrange multipliers that enforce the row and column constraints for m. The above can be used to define many different distance measures, since given the decomposition of A it is trivial to construct measures which are invariant only under some subset of the transformations (such as rotation and translation) . The regularization and a terms may also be individually adjusted in an appropriate fashion for a specific problem domain. 2.2 Weighted Graph Matching Distance Measures The following distance measure quantifies the degree of dissimilarity between two unlabeled weighted graphs. Given two graphs, represented by adjacency matrices Gab and gij, one can minimize the objective below to find the permutation which best maps G onto g: ' Egm(m) A I B a=l i=l b=1 = L: L:(L: Gabmbi - J L: m jgji)2 a ;=1 Steven Gold, Anand Rangarajan, Eric Mjolsness 716 with constraints: 'Va 2::=1 mai = 1 , 'Vi 2::=1 mai = 1 , 'Vai mai ;::: O. These constraints are enforced in the same fashion as in (1). An algebraic fixed-point transformation and self-amplification term further transform the objective to: A I z: 1 J B Egm(m) = L L(J.'ai(L Gabmbi majDji) a=1 i=1 b=1 j=1 z: lAI +(j L a=1 i=1 A 2J.'~i - ,lTaimai z: I I + ~lT~i) A mai(logmai - 1) + L lI:a(Z: mai - 1) + Ai(L mai - 1) a=1 i=1 i=1 a=1 (2) In this objective we are also looking for a saddle point. A second, functionally equivalent, graph matching objective is also used clustering problem: A B I III J Egm/(m) = LZ:LZ:mai m bj(Gab-Dji)2 a=lb=li=lj=l with constraints: 'Va 2::=1 mai 2.3 =1 , the (3) 'Vi 2::=1 mai = 1 , 'Vai mai ;::: O. The Clustering Objective The learning problem is formulated as follows: Given a set of I objects, {Xi} find a set of A cluster centers {Ya } and match variables {Mia} defined as M. _ {I 0 la - if Xi is in Ya's cluster otherwise, such that each object is in only one cluster, and the total distance of all the objects from their respective cluster centers is minimized. To find {Ya} and {Mia} minimize the cost function, I A Eeltuter(Y,M) = LLMiaD(Xi' Ya) i=l a=l with the constraint that 'Vi 2:a Mia = 1 , 'Vai Mai ;::: O. D(Xi, Y a), the distance function, is a measure of dissimilarity between two objects. The constraints on M are enforced in a manner similar to that described for the distance measure, except that now only the rows of the matrix M need to add to one, instead of both the rows and the columns. I Eeituter(Y, M) 1 I A Z:Z:MiaD(Xi , Ya) i=l a=1 I + (j Z:Z: Mia(log Mia - 1) i=l a=1 A + Z:Ai(LMia i=l A a=1 1) (4) Learning with Pre knowledge 717 Here, the objects are point-sets or weighted graphs. If point-sets the distance measure D(Xi, Ya) is replaced by (1), if graphs it is replaced by (2) or (3). Therefore, given a set of objects, X, we construct Ecltuter and upon finding the appropriate saddle point of that objective, we will have Y, their cluster centers, and M, their cluster memberships. 3 The Algorithm The algorithm to minimize the clustering objectives consists of two loops - an inner loop to minimize the distance measure objective [either (1) or (2)] and an outer loop to minimize the clustering objective (4). Using coordinate descent in the outer loop results in dynamics similar to the EM algorithm [Jordan and Jacobs, 1994] for clustering. All variables occurring in the distance measure objective are held fixed during this phase. The inner loop uses coordinate ascent/descent which results in repeated row and column projections for m. The minimization of m, and the distance measure variables [either t, A of (1) or 1', (f of (2)], occurs in an incremental fashion, that is their values are saved after each inner loop call from within the outer loop and are then used as initial values for the next call to the inner loop. This tracking of the values of the distance measure variables in the inner loop is essential to the efficiency of the algorithm since it greatly speeds up each inner loop optimization. Most coordinate ascent/descent phases are computed analytically, further speeding up the algorithm. Some local minima are avoided, by deterministic annealing in both the outer and inner loops. The mUlti-phase dynamics maybe described as a clocked objective. Let {D} be the set of distance measure variables excluding m. The algorithm is as follows: Initialize {D} to the equivalent of an identity transform, Y to random values Begin Outer Loop Begin Inner Loop Initialize {D} with previous values Find m, {D} for each ia pair: Find m by softmax, projecting across j, then k, iteratively Find {D} by coordinate descent End Inner Loop Find M ,Y using fixed values of m, {D}, determined in inner loop: Find M by softmax, across i Find Y by coordinate descent Increase f3M, f3m End Outer Loop When analytic solutions are computed for Y the outer loop takes a form similar to fuzzy ISODATA clustering, with annealing on the fuzziness parameter. 4 Methods and Experimental Results Four series of experiments were ran with randomly generated data to evaluate the learning algorithms. Point sets were clustered in the first three experiments and weighted graphs were clustered in the fourth. In each experiment a set of object 718 Steven Gold, Anand Rangarajan, Eric Mjolsness models were randomly generated. Then from each object model a set of object instances were created by transforming the object model according to the problem domain assumed for that experiment. For example, an object represented by points in two dimensional space was translated, rotated, scaled, sheared, and permuted to form a new point set. A object represented by a weighted graph was permuted. Noise was added to further distort the object. Parts of the object were deleted and spurious features (points) were added. In this manner, from a set of object models, a larger number of object instances were created. Then with no knowledge of the original objects models or cluster memberships, we clustered the object instances using the algorithms described above. The results were evaluated by comparing the object prototypes (cluster centers) formed by each experimental run to the object models used to generate the object instances for that experiment. The distance measures used in the clustering were used for this comparison, i.e. to calculate the distance between the learned prototype and the original object. Note that this distance measure also incorporates the transformations used to create the object instances. The mean and standard deviations of these distances were plotted (Figure 1) over hundreds of experiments, varying the object instance generation noise. The straight line appearing on each graph displays the effect of the noise only. It is the expected object model-object prototype distance if no transformations were applied, no features were deleted or added, and the cluster memberships of the object instances were known. It serves as an absolute lower bound on our learning algorithm. The noise was increased in each series of experiments until the curve flattened - that is the object instances became so distorted by noise that no information about the original objects could be recovered by the algorithm. In the first series of experiments (Figure 1a), point set objects were translated, rotated, scaled, and permuted. Initial object models were created by selecting points with a uniform distribution within a unit square. The transformations to create the object instance were selected with a uniform distribution within the following bounds; translation: ?.5, rotation: ?27?, log(scale): ? log(.5). 100 object instances were generated from 10 object models. All objects contained 20 points.The standard deviation of the Gaussian noise was varied by .02 from .02 to .16. 15 experiments were run at each noise level. The data point at each error bar represents 150 distances (15 experiments times 10 model-prototype distances for each experiment). In the second and third series of experiments (Figures 1b and 1c), point set objects were translated, rotated, scaled, sheared (obliquely and vertically), and permuted. Each object point had a 10% probability of being deleted and a 5% probability of generating a spurious point. The point sets and transformations were randomly generated as in the first experiment, except for these bounds; log(scale): ? log(.7), log(verticalshear): ?log(.7), and log(obliqueshear): ?log(.7). In experiment 2, 64 object instances and 4 object models of 15 points each were used. In experiment 3, 256 object instances and 8 object models of 20 points each were used. Noise levels like experiment 1 were used, with 20 experiments run at each noise level in experiment 2 and 10 experiments run at each noise level in experiment 3. In experiment 4 (Figure 1d), object models were represented by fully connected weighted graphs. The link weights in the initial object models were selected with a uniform distribution between 0 and 1. The objects were then randomly permuted Learning with Preknowledge 719 (a) (b) 0.05 0.1 0.15 standard deviation (c) 0.05 0.1 0.15 standard deviation (d) 3r---~----~----~, Ir-'~(" CD2 ~ .- "01 0.05 0.1 0.15 standard deviation or o V ~,/ ~ ("It-' V - 0.05 0.1 0.15 standard deviation Figure 1: (a): 10 clusters, 100 point sets, 20 points each , scale ,rotation, translation, 120 experiments (b): 4 clusters, 64 point sets, 15 points each, affine, 10 % deleted , 5 % spurious, 140 experiments (c): 8 clusters, 256 point sets , 20 points each, affine , 10 % deleted, 5 % spurious, 70 experiments (d): 4 clusters, 64 graphs, 10 nodes each , 360 experiments to form the object instance and uniform noise was added to the link weights. 64 object instances were generated from 4 object models consisting of 10 node graphs with 100 links. The standard deviation of the noise was varied by .01 from .01 to .12. 30 experiments where run at each noise level. In most experiments at low noise levels (~ .06 for point sets, ~ .03 for graphs), the object prototypes learned were very similar to the object models. Even at higher noise levels object prototypes similar to the object models are formed, though less consistently. Results from about 700 experiments are plotted. The objective for experiment 3 contained close to one million variables and converged in about 4 hours on an SGI Indigo workstation. The convergence times of the objectives of experiments 1, 2 and 4 were 120, 10 and 10 minutes respectively. 5 Conclusions It has long been argued by many, that learning in complex domains typically associated with human intelligence requires some type of prior structure or knowledge. We have begun to develop a set of tools that will allow the incorporation of prior 720 Steven Gold, Anand Rangarajan, Eric Mjolsness structure within learning. Our models incorporate many features needed in complex domains like vision - noise, missing and spurious features, non-rigid transformations. They can learn objects with inherent structure, like graphs. Many experiments have been run on experimentally generated data sets. Several directions for future research hold promise. One might be the learning of OCR data [Gold et al., 1995]. Second a supervised learning stage could be added to our algorithms. Finally the power of the distance measures can be enhanced to operate on attributed relational graphs with deleted nodes and links [Gold and Rangarajan, 1995]. Acknowledgements ONR/DARPA: NOOOI4-92-J-4048 , AFOSR: F49620-92-J-0465 and Yale CTAN. References S. Geman , E. Bienenstock, and R. Doursat. (1992) Neural networks and the bias/variance dilemma. Neural Computation 4:1-58. S. Gold, E . Mjolsness and A. Rangarajan. (1994) Clustering with a domain-specific distance measure. In J . Cowan et al., (eds.), NIPS 6. Morgan Kaufmann. S. Gold, C. P. Lu, A. Rangarajan, S. Pappu and E. Mjolsness. (1995) New algorithms for 2D and 3D point matching: pose estimation and correspondence. In G. Tesauro et al., (eds.) , NIPS 7. San Francisco, CA: Morgan Kaufmann. S. Gold and A. Rangarajan (1995) A graduated assignment algorithm for graph matching. YALEU/DCS/TR-I062, Yale Univ., CS Dept. M. I. Jordan and R. A. Jacobs. (1994) Hierarchical mixtures of experts and the EM algorithm . Neural Computation, 6:181-214. E. Mjolsness. Visual grammars and their neural networks. (1992) SPIE Conference on the Science of Artificial Neural Networks, 1110:63-85 . C. Peterson and B. Soderberg. A new method for mapping optimization problems onto neural networks. (1989) International Journal of Neural Systems,I(1):3-22. A. Rangarajan , S. Gold and E. Mjolsness. (1994) A novel optimizing network architecture with applications. YALEU /DCS/TR- I036, Yale Univ. , CS Dept . R. Shepard. (1989). Internal representation of universal regularities: A challenge for connectionism. In L. Nadel et al. , (eds.), Neural Connections, Mental Computation. Cambridge, MA , London, England: Bradford/MIT Press. P. Simard, Y. Le Cun, and J. Denker. (1993) Efficient pattern recognition using a transformation distance. In S. Hanson et ai., (eds.), NIPS 5. San Mateo, CA: Morgan Kaufmann. C. Williams , R. Zemel, and M. Mozer. (1993) Unsupervised learning of object models. AAAI Tech. Rep . FSS-93-04, Univ. of Toronto, CS Dept. A. L. Yuille and J .J. Kosowsky. (1994) . Statistical physics algorithms that converge. Neural Computation, 6:341-356 . 

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Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping in the Barn Owl Alexandre Pougett Cedric Deffayett Terrence J. Sejnowskit cedric@salk.edu terry@salk.edu alex@salk .edu tHoward Hughes Medical Institute The Salk Institute La Jolla, CA 92037 Department of Biology University of California, San Diego and tEcole Normale Superieure 45 rue d'Ulm 75005 Paris, France Abstract The auditory system of the barn owl contains several spatial maps . In young barn owls raised with optical prisms over their eyes, these auditory maps are shifted to stay in register with the visual map, suggesting that the visual input imposes a frame of reference on the auditory maps. However, the optic tectum, the first site of convergence of visual with auditory information, is not the site of plasticity for the shift of the auditory maps; the plasticity occurs instead in the inferior colliculus, which contains an auditory map and projects into the optic tectum . We explored a model of the owl remapping in which a global reinforcement signal whose delivery is controlled by visual foveation. A hebb learning rule gated by reinforcement learned to appropriately adjust auditory maps. In addition, reinforcement learning preferentially adjusted the weights in the inferior colliculus, as in the owl brain, even though the weights were allowed to change throughout the auditory system. This observation raises the possibility that the site of learning does not have to be genetically specified, but could be determined by how the learning procedure interacts with the network architecture . 126 Alexandre Pouget, Cedric Deffayet, Te"ence J. Sejnowski c,an c:::======:::::? ? ~_m Visual System Optic Tectum Inferior Colllc-ulus External nucleua .- t Forebrain Field L U~ a~ t Ovold.H. Nucleull ?"bala:m.ic Relay Inferior Colltculu. Cenlnll Nucleus C1ec) t Cochlea Figure 1: Schematic view of the auditory pathways in the barn owl. 1 Introduction The barn owl relies primarily on sounds to localize prey [6] with an accuracy vastly superior to that of humans. Figure 1A illustrates some of the nuclei involved in processing auditory signals. The barn owl determines the location of sound sources by comparing the time and amplitude differences of the sound wave between the two ears. These two cues are combined together for the first time in the shell and core of the inferior colliculus (ICc) which is shown at the bottom of the diagram . Cells in the ICc are frequency tuned and subject to spatial aliasing. This prevents them from unambiguously encoding the position of objects. The first unambiguous auditory map is found at the next stage, in the external capsule of the inferior colliculus (ICx) which itself projects to the optic tectum (OT). The OT is the first subforebrain structure which contains a multimodal spatial map in which cells typically have spatially congruent visual and auditory receptive fields. In addition, these subforebrain auditory pathways send one major collateral toward the forebrain via a thalamic relay. These collaterals originate in the ICc and are thought to convey the spatial location of objects to the forebrain [3]. Within the forebrain, two major structures have been involved in auditory processing: the archistriatum and field L. The archistriatum sends a projection to both the inferior colliculus and the optic tectum . Knudsen and Knudsen (1985) have shown that these auditory maps can adapt to systematic changes in the sensory input. Furthermore, the adaptation appears to be under the control of visual input, which imposes a frame of reference on the incoming auditory signals. In owls raised with optical prisms, which introduce a systematic shift in part of the visual field, the visual map in the optic tectum was identical to that found in control animals, but the auditory map in the ICx was shifted by the amount of visual shift introduced by the prisms. This plasticity ensures that the visual and auditory maps stay in spatial register during growth Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping 127 and other perturbations to sensory mismatch. Since vision instructs audition, one might expect the auditory map to shift in the optic tectum, the first site of visual-auditory convergence. Surprisingly, Brainard and Knudsen (1993b) observed that the synaptic changes took place between the ICc and the ICx, one synapse before the site of convergence. These observations raise two important questions: First, how does the animal knows how to adapt the weights in the ICx in the absence of a visual teaching signal? Second, why does the change take place at this particular location and not in the aT where a teaching signal would be readily available? In a previous model [7], this shift was simulated using backpropagation to broadcast the error back through the layers and by constraining the weights changes to the projection from the ICc to ICx. There is, however, no evidence for a feedback projection between from the aT to the ICx that could transmit the error signal; nor is there evidence to exclude plasticity at other synapses in these pathways. In this paper, we suggest an alternative approach in which vision guides the remapping of auditory maps by controlling the delivery of a scalar reinforcement signal. This learning proceeds by generating random actions and increasing the probability of actions that are consistently reinforced [1, 5] . In addition, we show that reinforcement learning correctly predicts the site of learning in the barn owl, namely at the ICx-ICc synapse, whereas backpropagation [8] does not favor this location when plasticity is allowed at every synapse. This raises a general issue: the site of synaptic adjustment might be imposed by the combination of the architecture and learning rule, without having to restrict plasticity to a particular synapse. 2 2.1 Methods Network Architecture The network architecture of the model based on the barn owl auditory system, shown in figure 2A, contains two parallel pathways. The input layer was an 8x21 map corresponding to the ICc in which units responded to frequency and interaural phase differences. These responses were pooled together to create auditory spatial maps at subsequent stages in both pathways. The rest of the network contained a series of similar auditory maps, which were connected topographically by receptive fields 13 units wide. We did not distinguish between field L and the archistriatum in the forebrain pathways and simply used two auditory maps, both called FBr. We used multiplicative (sigma-pi) units in the aT whose activities were determined according to: Yi = L,. w~Br yfBr WfkBr yfc:c (1) j The multiplicative interaction between ICx and FBr activities was an important assumption of our model. It forced the ICx and FBr to agree on a particular position before the aT was activated. As a result, if the ICx-aT synapses were modified during learning, the ICx-FBr synapses had to be changed accordingly. 128 Alexandre Pouget, Cedric Deffayet, Terrence J. Sejnowski Figure 2: Schematic diagram of weights (white blocks) in the barn owl auditory system. A) Diagram of the initial weights in the network. B) Pattern of weights after training with reinforcement learning on a prism-induced shift offour units. The remapping took place within the ICx and FBr. C) Pattern of weights after training with backpropagation. This time the ICx-OT and FBr-OT weights changed. Weights were clipped between 5.0 and 0.01, except for the FBr-ICx connections whose values were allowed to vary between 8.0 and 0.01. The minimum values were set to 0.01 instead of zero to prevent getting trapped in unstable local minima which are often associated with weights values of zero. The strong coupling between FBr and ICx was another important assumption of the model whose consequence will be discussed in the last section. Examples were generated by simply activating one unit in the ICc while keeping the others to zero, thereby simulating the pattern of activity that would be triggered by a single localized auditory stimulus. In all simulations, we modeled a prism-induced shift of four units. 2.2 Reinforcement learning We used stochastic units and trained the network using reinforcement learning [1]. The weighted sum of the inputs, neti, passed through a sigmoid, f(x) , is interpreted as the probability, Pi, that the unit will be active: Pi = f(neti) * 0.99 + 0.01 (2) were the output of the unit Yi was: ._{a y, - with probability 1 - Pi 1 with probability Pi (3) Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping 129 Because of the form of the equation for Pi, all units in the network had a small probability (0.01) of being spontaneously active in the absence of any inputs. This is what allowed the network to perform a stochastic search in action space to find which actions were consistently associated with positive reinforcement. We ensured that at most one unit was active per trial by using a winner-take-all competition in each layer. Adjustable weights in the network were updated after each training examples with hebb-like rule gated by reinforcement: (4) A trial consisted in choosing a random target location for auditory input (ICc) and the output of the OT was used to generate a head movement . The reinforcement , r , was then set to 1 for head movements resulting in the foveation of the stimulus and to -0.05 otherwise. 2.3 Backpropagation For the backpropagation network , we used deterministic units with sigmoid activation functions in which the output of a unit was given by: (5) where neti is the weighted sum of the inputs as before. The chain rule was used to compute the partial derivatives of the squared error, E , with respect to each weights and the weights were updated after each training example according to: (6) The target vectors were similar to the input vectors, namely only one OT units was required to be activated for a given pattern, but at a position displaced by 4 units compared to the input. 3 3.1 Results Learning site with reinforcement In a first set of simulation we kept the ICc-ICx and ICc-FBr weights fixed. Plasticity was allowed at these site in later simulations. Figure 2A shows the initial set of weights before learning starts. The central diagonal lines in the weight diagrams illustrate the fact that each unit receives only one non-zero weight from the unit in the layer below at the same location. 130 Alexandre Pouget, Cedric Deffayet, Terrence J. Sejnowski There are two solutions to the remapping: either the weights change within the ICx and FBr, or from the ICx and the FBr to the ~T. As shown in figure 2B , reinforcement learning converged to the first solution. In contrast, the weights between the other layers were unaltered, even though they were allowed to change. To prove that the network could have actually learned the second solution, we trained a network in which the ICc-ICx weights were kept fixed . As we expected, the network shifted its maps simultaneously in both sets of weights converging onto the OT, and the resulting weights were similar to the ones illustrated in figure 2C. However, to reach this solution , three times as many training examples were needed. The reason why learning in the ICx and FBr were favored can be attributed to probabilistic nature of reinforcement learning. If the probability of finding one solution is p, the probability of finding it twice independently is p2. Learning in the ICx and FBR is not independent because of the strong connection from the FBr to the ICx. When the remapping is learned in the FBR this connection automatically remapped the activities in the ICx which in turn allows the ICx-ICx weights to remap appropriately. In the OT on the other hand, the multiplicative connection between the ICx and FBr weights prevent a cooperation between this two sets of weights. Consequently, they have to change independently, a process which took much more training. 3.2 Learning at the ICc-ICx and ICc-FBr synapses The aliasing and sharp frequency tuning in the response of ICc neurons greatly slows down learning at the ICc-ICx and ICc-FBr synapses. We found that when these synapses were free to change, the remapping still took place within the ICx or FBr (figure 3). 3.3 Learning site with backpropagation In contrast to reinforcement learning, backpropagation adjusted the weights in two locations: between the ICx and the OT and between the Fbr and OT (figure 2C). This is the consequence of the tendency of the backpropagation algorithm to first change the weights closest to where the error is injected. 3.4 Temporal evolution of weights Whether we used reinforcement or supervised learning, the map shifted in a very similar way. There was a simultaneous decrease of the original set of weights with a simultaneous increase of the new weights, such that both sets of weights coexisted half way through learning. This indicates that the map shifted directly from the original setting to the new configuration without going through intermediate shifts. This temporal evolution of the weights is consistent the findings of Brainard and Knudsen (1993a) who found that during the intermediate phase of the remapping, cells in the inferior colli cuI us typically have two receptive fields. More recent work however indicates that for some cells the remapping is more continuous(Brainard and Knudsen , personal communication) , a behavior that was not reproduced by either of the learning rule. Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping 131 Figure 3: Even when the ICc-ICx weights are free to change, the network update the weights in the ICx first. A separate weight matrix is shown for each isofrequency map from the ICc to ICx. The final weight matrices were predominantly diagonal; in contrast, the weight matrix in ICx was shifted. 4 Discussion Our simulations suggest a biologically plausible mechanism by which vision can guide the remapping of auditory spatial maps in the owl's brain. Unlike previous approaches, which relied on visual signals as an explicit teacher in the optic tectum [7], our model uses a global reinforcement signal whose delivery is controlled by the foveal representation of the visual system. Other global reinforcement signals would work as well. For example, a part of the forebrain might compare auditory and visual patterns and report spatial mismatch between the two. This signal could be easily incorporated in our network and would also remap the auditory map in the inferior colli cuI us. Our model demonstrates that the site of synaptic plasticity can be constrained by the interaction between reinforcement learning and the network architecture. Reinforcement learning converged to the most probably solution through stochastic search. In the network, the strong lateral coupling between ICx and FBr and the multiplicative interaction in the OT favored a solution in which the remapping took place simultaneously in the ICx and FBr. A similar mechanism may be at work in the barn owl's brain. Colaterals from FBr to ICx are known to exist, but the multiplicative interaction has not been reported in the barn owl optic tectum. Learning mechanisms may also limit synaptic plasticity. NMDA receptors have been reported in the ICx, but they might not be expressed at other synapses. There may, however, be other mechanisms for plasticity. The site of remapping in our model was somewhat different from the existing observations. We found that the change took place within the ICx whereas Brainard and Knudsen [3] report that it is between the ICc and the ICx. A close examination of their data (figure 11 in [3]) reveals that cells at the bottom of ICx were not 132 Alexandre Pouget, Cedric Deffayet, Terrence J. Sejnowski remapped, as predicted by our model, but at the same time, there is little anatomical or physiological evidence for a functional and hierarchical organization within the ICx. Additional recordings are need to resolve this issue. We conclude that for the barn owl's brain, as well as for our model, synaptic plasticity within ICx was favored over changes between ICc and ICx. This supports the hypothesis that reinforcement learning is used for remapping in the barn owl auditory system. Acknowledgments We thank Eric Knudsen and Michael Brainard for helpful discussions on plasticity in the barn owl auditory system and the results of unpublished experiments. Peter Dayan and P. Read Montague helped with useful insights on the biological basis of reinforcement learning in the early stages of this project. References [1] A.G. Barto and M.1. Jordan. Gradient following without backpropagation in layered networks. Proc. IEEE Int. Conf. Neural Networks, 2:629-636, 1987. [2] M.S. Brainard and E.1. Knudsen. Dynamics of the visual calibration of the map of interaural time difference in the barn owl's optic tectum. In Society For Neuroscience Abstracts, volume 19, page 369.8, 1993. [3] M.S. Brainard and E.!. Knudsen. Experience-dependent plasticity in the inferior colliculus: a site for visual calibration of the neural representation of auditory space in the barn owl. The journal of Neuroscience, 13:4589-4608, 1993. [4] E. Knudsen and P. Knudsen. Vision guides the adjustment of auditory localization in the young barn owls. Science, 230:545-548, 1985. [5] P.R. Montague, P. Dayan, S.J. Nowlan, A. Pouget, and T.J. Sejnowski. Using aperiodic reinforcement for directed self-organization during development. In S.J. Hanson, J.D. Cowan, and C.L. Giles, editors, Advances in Neural Information Processing Systems, volume 5. Morgan-Kaufmann, San Mateo, CA, 1993. [6] R.S. Payne. Acoustic location of prey by barn owls (tyto alba). Journal of Experimental Biology, 54:535-573, 1970. [7] D.J. Rosen, D.E. Rumelhart, and E.I. Knudsen. A connectionist model of the owl's sound localization system. In Advances in Neural Information Processing Systems, volume 6. Morgan-Kaufmann, San Mateo, CA, 1994. [8] D.E. Rumelhart, G.E. Hinton, and R.J . Williams. Learning internal representations by error propagation. In D. E. Rumelhart, J. L. McClelland, and the PDP Research Group, editors, Parallel Distributed Processing, volume 1, chapter 8, pages 318-362. MIT Press, Cambridge, MA, 1986. Dynamic Modelling of Chaotic Time Series with Neural Networks Jose C. Principe, Jyh-Ming Kuo Computational NeuroEngineering Laboratory University of Florida, Gainesville, FL32611 principe@synapse.ee.ufi.edu Abstract This paper discusses the use of artificial neural networks for dynamic modelling of time series. We argue that multistep prediction is more appropriate to capture the dynamics of the underlying dynamical system, because it constrains the iterated model. We show how this method can be implemented by a recurrent ANN trained with trajectory learning. We also show how to select the trajectory length to train the iterated predictor for the case of chaotic time series. Experimental results corroborate the proposed method. 1.0 Introduction The search for a model of an experimental time series has been an important problem in science. For a long time the linear model was almost exclusively used to describe the system that produced the time series [1], but recently nonlinear models have also been proposed to replace the linear ones [2]. Lapedes and Farber [3] showed how artificial neural networks (ANNs) can be used to identify the dynamics of the unknown system that produced the time series. He simply used a multilayer perceptron to predict the next point in state space, and trained this topology with backpropagation. This paper explores more complex neural topologies and training methods with the goal of improving the quality of the identification of the dynamical system, and to understand better the issues of dynamic modelling with neural networks which are far from being totally understood. According to Takens' embedding theorem, a map F: Jilm + 1 ~ Jilm + 1 exists that transforms the current reconstructed state y(t) to the next state y(t + 1) , i.e. y(t+ 1) = F(y(t? (1) 312 Jose Principe, Jyh-Ming Kuo or where m is the estimated dimension of the unknown dynamical system cI>. Note that the map contains several trivial (nonlinear) filters and a predictor. The predictive mapping r: Jilm +1~ R can be expressed as x(t+ 1) = r(x(t? (2) where x (t) = [x (t - 2m) ... x (t - 1) x (t)] T. This is actually the estimated nonlinear autoregressive model of the input time series. The existence of this predictive model lays a theoretical basis for dynamic modelling in the sense that we can build from a vector time series a model to approximate the mapping r . If the conditions of Takens embedding theorem are met, this mapping captures some of the properties of the unknown dynamical system cI> that produced the time series [7]. Presently one still does not have a capable theory to guarantee if the predictor has successfully identified the original model cI>. The simple point by point comparison between the original and predicted time series used as goodness of fit for non-chaotic time series breaks down for chaotic ones [5]. Two chaotic time series can be very different pointwise but be produced by the same dynamical system (two trajectories around the same attractor). The dynamic invariants (correlation dimension, Lyapunov exponents) measure global properties of the attractor, so they should be used as the rule to decide about the success of dynamic modelling. Hence, a pragmatic approach in dynamic modelling is to seed the predictor with a point in state space, feed the output to its input as an autonomous system, and create a new time series. If the dynamic invariants computed from this time series match the ones from the original time series, then we say that dynamic modelling was successful [5]. The long term behavior of the autonomous predictive model seems to be the key factor to find out if the predictor identified the original model. This is the distinguishing factor between prediction of chaotic time series and dynamic modelling. The former only addresses the instantaneous prediction error, while the latter is interested in long term behavior. In order to use this theory, one needs to address the choices of predictor implementation. Due to the universal mapping characteristics of multilayer perceptrons (MLPs) and the existence of well established learning rules to adapt the MLP coefficients, this type of network appears as an appropriate choice [3]. However, one must realize that the MLP is a static mapper, and in dynamic modelling we are dealing with time varying signals, where the past of the signal contains vital information to describe the mapping. The design considerations to select the neural network topology are presented elsewhere [4]. Wejust would like to say that the MLP has to be enhanced with short term memory mechanisms, and that the estimation of the correlation dimension should be used to set the size of the memory layer. The main goal of the paper is to establish the methodology to efficiently train neural networks for dynamic modelling. Dynamic Modelling of Chaotic Time Series with Neural Networks 313 2. Iterated versus Single Step Prediction. From eqn. 2 it seems that the resulting dynamic model F can be obtained through single step prediction. This has been the conventional way to handle dynamic modelling [2],[3]. The predictor is adapted by minimizing the error E _.1 L L = I dist (x (i + 1) -F (1 (i? ) (3) =2m + 1 _.1 where L is the length of the time series, x(i) is the itb data sample, F is the map developed by the predictor and dist() is a distance measure (normally the L2 norm). Notice that the training to obtain the mapping is done independently from sample to sample, i.e. _.1 x(i+ 1) = F (xU? +5 1 _.1 x (i + j) =F (1 (i + j -1? + 5j where 5j are the instantaneous prediction errors, which are minimized during training. Notice that the predictor is being optimized under the assumption that the previous point in state space is known without error. The problem with this approach can be observed when we iterate the predictor as an autonomous system to generate the time series samples. If one wants to produce two samples in the future from sample i the predicted sample i+ 1 needs to be utilized to generate sample i+2. The predictor was not optimized to do this job, because during training the true i+ 1 sample was assumed known. As long as 51 is nonzero (as will be always the case for nontrivial problems), errors will accumulate rapidly. Single step prediction is more associated with extrapolation than with dynamic modelling, which requires the identification of the unique mapping that produces the time series. When the autonomous system generates samples, past values are used as inputs to generate the following samples, which means that the training should constrain also the iterates of the predictive mapping. Putting it in a simple way, we should train the predictor in the same way we are going to use it for testing (Le. as an autonomous system). We propose multistep prediction (or trajectory learning) as the way to constrain the iterates of the mapping developed by the predictor. Let us define L" E = I (4) dist(x(i + I)-xU + 1? =2m + 1 x where k is the number of prediction steps (length ofthe trajectory) and (i + 1) is an estimate of the predictive map _.1 x(i+l) = F (i(i-2m), ... ,i(i? (5) 314 Jose Principe, Jyh-Ming Kuo with j (i) =[ X 0 SiS 2m (i) _.1. F (x(i-2m-l), ... ,x(i-l? Equation (5) states that (for i>2m), i.e. i (i) _ .1. -.1. i>2m is the i-2m iterate of the predictive part of the map - .1. i(i+l) = (F (F ( ... F (x(2m??) t- _.1. = (F 2m (x(2m?) (6) Hence, minimizing the criterion expressed by equation (4) an optimal multistep predictor is obtained. The number of constraints that are imposed during learning is associated with k, the number of prediction steps, which corresponds to the number of iterations of the map. The more iterations, the less likely a sub-optimal solution is found, but note that the training time is being proportionally increased. In a chaotic time series there is a more important consideration that must be brought into the picture, the divergence of nearby trajectories, as we are going to see in a following section. 3. Multistep prediction with neural networks Figure 1 shows the topology proposed in [4] to identify the nonlinear mapping. Notice that the proposed topology is a recurrent neural network. with a global feedback loop. This topology was selected to allow the training of the predictor in the same way as it will be used in testing, i.e. using the previous network outputs to predict the next point. This recurrent architecture should be trained with a mechanism that will constrain the iterates of the map as was discussed above. Single step prediction does not fit this requirement. With multistep prediction, the model system can be trained in the same way as it is used in testing. We seed the dynamic net with a set of input samples, disconnect the input and feed back the predicted sample to the input for k steps. The mean square error between the predicted and true sample at each step is used as the cost function (equation (4?. If the network topology was feed forward , batch learning could be used to train the network, and static backpropagation applied to train the net. However, as a recurrent topology is utilized, a learning paradigm such as backpropagation through time (BPTT) or real time recurrent learning (RTRL) must be utilized [6]. The use of these training methods should not come as a surprise since we are in fact fitting a trajectory over time, so the gradients are time varying. This learning method is sometimes called "trajectory learning" in the recurrent learning literature [6]. A criterion to select the length of the trajectory k will be presented below. The procedure described above must be repeated for several different segments of the time series. For each new training segment, 2m+ 1 samples of the original time series are used to seed the predictor. To ease the training we suggest that successive training sequences of length k overlap by q samples (q<k). For chaotic time series we also suggest that the error be weighted according to the largest Lyapunov exponent. Hence Dynamic Modelling of Chaotic Time Series with Neural Networks 315 the cost function becomes E = Lr L" (7) h(i) ?dist(x(i+jq+l)-i(i+jq+l? J = 01 = 2m+ I where r is the number of training sequences, and h (i) = (e ~ (8) A I -(i-2m-I) III/IX ) In this equation A.max is the largest Lyapunov exponent and L\t the sampling interval. With this weighting the errors for later iteration are given less credit, as they should since due to the divergence of trajectories a small error is magnified proportionally to the largest Lyapunov exponent [7]. 4. Finding the length of the trajectory From the point of view of dynamic modelling, each training sequences should preferably contain enough information to model the attractor. This means that each sequence should be no shorter than the orbital length around the attractor. We proposed to estimate the orbital length as the reciprocal of the median frequency of the spectrum of the time series [8]. Basically this quantity is the average time required for a point to return to the same neighborhood in the attractor. The length of the trajectory is also equivalent to the number of constraints we impose on the iterative map describing the dynamical model. However, in a chaotic time series there is another fundamental limitation imposed on the trajectory length - the natural divergence of trajectories which is controlled by A.max ' the largest Lyapunov exponent. If the trajectory length is too long, then instabilities in the training can be expected. A full discussion of this topic is beyond the scope of this paper, and is presented elsewhere [8]. We just want to say that when A.max is positive there is an uncertainty region around each predicted point that is a function of the number of prediction steps (due to cummulative error). If the trajectory length is too long the uncertainty regions from two neighboring trajectories will overlap, creating conflicting requirements for training (the model is requried to develop a map to follow both segments A and B- Figure 2). It turns out that one can approximately find the number of iterations is that will guarantee no overlap of uncertainty regions [8]. The length of the principal axis of the uncertainty region around a signal trajectory at iteration i can be estimated as ?; = toe ~III/lXiAI (9) where ?0 is the initial separation. Now assuming that the two principal axis of nearby trajectories are colinear, we should choose the number of iterations is such that the distance d j between trajectories is larger than the uncertainty region, i.e. d; ~ 2?; . I I The estimate of is should be averaged over a number of neighboring training sequences (-50 depending on the signal dynamics). Hence, to apply this method three quantities must be estimated: the largest Lyapunov 316 Jose Principe, Jyh-Ming Kuo exponent, using one of the accepted algorithms. The initial separation can be estimated from the one-step predictor. And is by averaging local divergence. The computation time required to estimate these quantities is usually much less than setting by trial and error the length of the trajectory until a reasonable learning curve is achieved. We also developed a method to train predictors for chaotic signals with large A.max ' but it will not be covered in this paper [8]. 5. Results We used this methodology to model the Mackey-Glass system (d=30, sampled at 116 Hz). A signal of 500 samples was obtained by 4th order Runge-Kutta integration and normalized between -1,1. The largest Lyapunov exponent for this signal is 0.0071 nats/sec. We selected a time delay neural network (TDNN) with topology 8-14-1. The output unit is linear, and the hidden layer has sigmoid nonlinearities. The number of taps in the delay line is 8. We trained a one-step predictor and the multistep predictor with the methodology developed in this paper to compare results. The single step predictor was trained with static backpropagation with no momentum and step size of 0.001. Trained was stopped after 500 iterations. The final MSE was 0.000288. After training, the predictor was seeded with the first 8 points of the time series and iterated for 3,000 times. Figure 3a shows the corresponding output. Notice that the waveform produced by the model is much more regular that the Mackey-Glass signal, showing that some fine detail of the attractor has not been captured. Next we trained the same TDNN with a global feedback loop (TDNNGF). The estimate of the is over the neighboring orbits provided an estimate of 14, and it is taken as the length of the trajectory. We displaced each training sequence by 3 samples (q=3 in eqn 7). BPTT was used to train the TDNNGF for 500 iterations over the same signal. The final MSE was 0.000648, higher than for the TDNN case. We could think that the resulting predictor was worse. The TDNNGF predictor was initialized with the same 8 samples of the time series and iterated for 3,000 times. Figure 3b shows the resulting waveform. It "looks" much closer to the original Mackey-Glass time series. We computed the average prediction error as a function of iteration for both predictors and also the theoretical rate of divergence of trajectories assuming an initial error EO (Casdagli conjecture, which is the square of eqn 9) [7]. As can be seen in Figure 4 the TDNNGF is much closer to the theoretical limit, which means a much better model. We also computed the correlation dimension and the Lyapunov exponent estimated from the generated time series, and the figures obtained from TDNNGF are closer to the original time series. Figure 5 shows the instability present in the training when the trajectory length is above the estimated value of 14. For this case the trajectory length is 20. As can be seen the MSE decreases but then fluctuates showing instability in the training. 6. Conclusions This paper addresses dynamic modelling with artificial neural networks. We showed 317 Dynamic Modelling of Chaotic Time Series with Neural Networks that the network topology should be recurrent such that the iterative map is constrained during learning. This is a necessity since dynamic modelling seeks to capture the long term behavior of the dynamical system. These models can also be used as a sample by sample predictors. Since the network topology is recurrent, backpropagation through time or real time recurrent learning should be used in training. In this paper we showed how to select the length of the trajectory to avoid instability during training. A lot more work needs to be done to reliably capture dynamical properties of time series and encapsulate them in artificial models. But we believe that the careful analysis of the dynamic characteristics and the study of its impact on the predictive model performance is much more promising than guess work. According to this (and others) studies, modelling of chaotic time series of low Amax seems a reality. We have extended some of this work for time series with larger Amax , and successfully captured the dynamics of the Lorenz system [8]. But there, the parameters for learning have to be much more carefully selected, and some of the choices are still arbitrary. The main issue is that the trajectories diverge so rapidly that predictors have a hard time to capture information regarding the global system dynamics. It is interesting to study the limit of predictability of this type of approach for high dimensional and high Amax chaos. Predictor Corr. Dim. Lyapunov MG30 2.70+/-0.05 0.0073+/-0.000 1 TDNNGF 2.65+/-0.03 0.0074+/-0.0001 TDNN 1.60+1-0.10 0.0063+/-0.0001 segment B Figure 1. Prop_osed recurrent architecture Figure 2. State space representation in (IDNNGF) training a model 7. Acknowledgments This work was partially supported by NSF grant #ECS-9208789, and ONR #1494-941-0858. 318 Jose Principe, Jyh-Ming Kuo -OJ! -OJ! m ~ ~ _ ~ ~ w ~ _ ~ _ Figure 3a. Generated sequence with the TDNN ! m TDNNerror 002 015 TDNNGF j /~ .005 o o asdagli conjecture ? ?? 5 , 0.2 / e~ ~ /1,/ 0.01 C -I~~~~!:--=--=--:!~~~--::!=----= o ~ '110. 110 ICIO ~ W JIO ~ . . II1II Figure 3b. uenerated sequence with the TDNNGF ? ????? ?? ?? to .-' ~ / _._.:::~ IS ~ ~ ? Figure 4. Comparison of predictors Figure 5. Instability in training 8. References [1] Box, G. E., and G. M. Jenkins, Time Series Analysis, Forecasting and Control, Holden Day, San Francisco, 1970. [2] Weigend, A. S., B. A. Huberman, and D. E. Rumelhart, "Predicting the future: a connectionist approach," International Journal of Neural Systems, vol. 1, pp. 193209, 1990. [3] Lapedes, R., and R. Farber, "Nonlinear signal processing using neural networks: prediction and system modelling," Technical Report LA-UR87-2662, Los Alamos National Laboratory, Los Alamos, New Mexico, 1987. [4] Kuo J-M., Principe J.C., "A systematic approach to chaotic time series modeling with neural networks", in IEEE Workshop on Neural Nets for Signal Processing, Ermioni, Greece, 1994. [5] Principe, J. C., A. Rathie, and J. M. Kuo, "Prediction of chaotic time series with neural networks and the issue of dynamic modeling," International Journal of Biburcation and Chaos, vol. 2, no. 4, pp. 989-996, 1992. [6] Hertz, J, A. Krogh, and R. G. Palmer, Introduction to the Theory of Neural Computation, Addison-Wesley, Redwood City, CA, 1991. [7] Casdagli, M., "Nonlinear prediction of chaotic time series," Physica D 35, pp.335-356, 1989. [8] Kuo, J.M., "Nonlinear Dynamic Modelling with Artificial neural networks", Ph.D. dissertation, University of FLorida, 1993. 

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A Model for Chemosensory Reception Rainer Malaka J Thomas Ragg Institut fUr Logik, Komplexitat und Oeduktionssysteme Universitat Karlsruhe, PO Box 0-76128 Karlsruhe, Germany e-mail: malaka@ira.uka.de.ragg@ira.uka.de Martin Hammer Institut fur Neurobiologie Freie Universitat Berlin 0-14195 Berlin, Germany e-mail: mhammer@castor.zedat.fu-berlin.de Abstract A new model for chemosensory reception is presented. It models reactions between odor molecules and receptor proteins and the activation of second messenger by receptor proteins. The mathematical formulation of the reaction kinetics is transformed into an artificial neural network (ANN). The resulting feed-forward network provides a powerful means for parameter fitting by applying learning algorithms. The weights of the network corresponding to chemical parameters can be trained by presenting experimental data. We demonstrate the simulation capabilities of the model with experimental data from honey bee chemosensory neurons. It can be shown that our model is sufficient to rebuild the observed data and that simpler models are not able to do this task. 1 INTRODUCTION Terrestrial animals, vertebrates and invertebrates, have developed very similar solutions for the problem of recognizing volatile substances [Vogt et ai., 1989]. Odor molecules bind to receptor proteins (receptor sites) at the cell membrane of the sensory cell. This interaction of odor molecules and receptor proteins activates a G-protein mediated second 62 Rainer Malaka, Thomas Ragg, Martin Hammer _____ odor molecules 'f! ~ /2 " 4! ___ Ins" "----- second messengers ----- . action potentials Ionic Influx Figure 1: Reaction cascade in chemosensory neurons. Volatile odor molecules reach receptor proteins at the surface of the chemosensory neuron. The odor bound binding proteins activate second messengers (e.g. G-proteins). The activated second messengers cause a change in the conductivity of ion channels. Through ionic influx a depolarization can build an action potential. messenger process. The concentrations of cAMP or IP3 rise rapidly and activate cyclicnucleotide-gated ion channels or IP3-gated ion channels [Breer et al., 1989, Shepherd, 1991]. As a result of this second messenger reaction cascade the conductivity of ion channels is changed and the cell can be hyperpolarized or depolarized, which can cause the generation of action potentials. It has been shown that one odor is able to activate different second messenger processes and that there is some interaction between the different second messenger processes [Breer & Boekhoff, 1992]. Figure 1 shows schematically the cascade of reactions from odor molecules over receptor proteins and second messengers up to the changing of ion channel conductance and the generation of action potentials. Responses of sensory neurons can be very complex. The response as a function of the odor concentration is highly non-linear. The response to mixtures can be synergistic or inhibitory relative to the response to the components of the compound. A synergistic effect occurs, if the response of one sensory cell to a binary mixture of two odors AI and A2 with concentrations [AI]' [A2l is higher than the sum of the responses to the odors AI, A2 at concentrations [A tl, [A2] alone. An inhibitory effect occurs, if the response to the mixture is smaller than either response to the single odors. In bee subplacode and placode recordings both effects can be observed [Akers & Getz, 1993]. Models of chemosensory reception should be complex enough to simulate the inhibitory and synergistic effects observed in sensory neurons, and they must provide a means for parameter fitting. We want to introduce a computational model which is constructed analogously to the chemical reaction cascade in the sensory neuron. The model can be expressed as an ANN and all unknown parameters can be trained with a learning algorithm. A Model for Chemosensory Reception 2 63 THE RECEPTOR TRANSDUCER MODEL The first step of odor reception is done by receptor proteins located at the cell membrane. There may be many receptor protein types in sensory cells at different concentrations and with different sensitivity to various odors. There is the possibility for different odors ligands Ai to react with a receptor protein Rj, but it is also possible for a single odor to react with different receptor proteins. The second step is the activation of second messengers. Ennis proposed a modelling of these complex reactions by a reaction step of activated odor-receptor complexes with transducer mechanisms [Ennis, 1991]. These transducers are a simplification of the second messengers processes. In Ennis' model transducers and receptor proteins are odor specific. We generalize Ennis' model by introducing transducer mechanisms T" that can be activated by odor-receptor complexes, and as with odors and receptor proteins we allow receptor proteins and transducers to react in any combination. Receptor proteins and transducer proteins are not required to be odor specific. The kinetics of the two reactions are given by + Rj AiRj + T" Ai ~ AiRj ~ (1) AiRjT". In a first reaction odor ligands Ai bind to receptor proteins Rj and build odor-receptor complexes AiRj, which can activate transducer mechanisms T" in a second reaction. Affinities lcij and Ij" describe the possibility of reactions between odor ligands Ai and receptor proteins Rj or between odor-receptor complexes Ai Rj and transducers T", respectively. The mass action equations are [AiRj] = lcij[Ad[Rj] (2) [Ai RjT,,] = Ij,,[AiRj][T,,]. The binding of odor-receptor complexes with transducer mechanisms is not dependent on the specific odor which is bound to the receptor protein, i.e. Ij" does not depend on i. It is only necessary that the receptor protein is bound. A sensory neuron can now be defined by the total concentration (or amount) of receptor proteins [H] and transducers [1']. The total concentration of either type corresponds to the sum of the free sites and the bound sites: [Hj] [Rj] + I)AiRj] (3) [T,,] + I)AiRjT,,] .1 (4) i,j Activated transducer mechanisms may elicit an excitatory or inhibitory effect depending on the kind of ion channel they open. Thus we divide the transducers T" into two types: inhibitory and excitatory transducers. With 8" = 1 {+ -1 , if transducer T" is excitatory , if transducer T" is inhibitory (5) IWe use the simplification [flj] = [Rj] + L:i[AiRJ' ] instead of [flj] = [Rj] + L:.[AiRj] L:i,k [AiRjTk], which is sufficient for [flj] > [tk], see also [Malaka & Ragg, 1993]. + 64 Rainer Malaka, Thomas Ragg, Martin Hammer Figure 2: ANN equivalent to the full receptor-transducer model. The input layer corresponds to the concentration of odor ligands [Ad, the first hidden layer to activated receptor protein types, the second to activated transducer mechanisms. The output neuron computes the effect E of the sensory cell. the effect can be set to the sum of all activated excitatory transducers minus the sum of all inhibitory transducers relative to the total amount of transducers. An additive constant (J is used to model spontaneous reactions. With this the effect of an odor can be set to (6) With Eqs.(2,4) and the hyperbolic function hyp(x) Eq.(6) can be reformulated to E= 1 A 2::L[Tk] '" = x/(l + x) the effect E defined in 2: hyp (2: Ijk[AiRj ]) 15 k ['h] L ?. '" t ,) + (J . (7) Analogously, we eliminate [AiRj] and [Rj): E= 1 A Lk[n] 2:hyp (2:Ijk[Rj]hYP(2:kij[Ad)) 8k[Tk]+(J k j . (8) i Equation (8) can now be regarded as an ANN with 4 feed-forward layers. The concentrations of the odor ligands [Ad represent the input layer, the two hidden layers correspond to activated receptor proteins and activated transducers, and one output element in layer 4 represents the effect caused by the input odor. The weight between the i-th element of the input layer to the j-th element of the first hidden layer is kij and from there to the k-th neuron of the second hidden layer Ij k [Hj]. The weight from element k of hidden layer 2 to the output element is 15k [1'k]/ 2::k[1'k]. The adaptive elements ofthe hidden layers have the hyperbolic activation functions hypo The network structure is shown in Figure 2. A Model for Chemosensory Reception 65 6 5 4 3 6 10 recept or protein types 20 50 mecanisms Figure 3: Mean error in spikes per output neuron for the model with different network sizes. Network sizes are varied in the number ofreceptor protein types and the number of transducing mechanisms. 3 SIMULATION RESULTS Applying learning algorithms like backpropagation or RProp to the model network, it is possible to find parameter settings for optimal (or local optimal) simulations of chemosensory cell responses with given response characteristics. In our simulations the best training results were achieved by using the fast learning algorithm RProp, which is an imprOVed version of backpropagation [Riedmiller & Braun, 1993]. For our simulations we used extracellular recordings made by Akers and Getz from single sensilla placodes of honey bee workers applying different stimuli and their binary mixtures to the antenna (see [Akers & Getz, 1992] for material and methods). The data set for training the ANNs consists of responses of 54 subplacodes to the four odors, geraniol, citral, limonene, linalool, their binary mixtures, and a mixture of all of four odors each at two concentration levels and to a blank stimulus, i.e. 23 responses to different odor stimulations for each subplacode. In a series of training runs with varying numbers of receptor protein types and transducer types the full model was trained to fit the data set. The networks were able to simulate the responses of the subplacodes, dependent on the network size. The size of the first hidden layer corresponds to the number of receptor protein types (R) in the model, the size of the second hidden layer corresponds to the number of transducing mechanisms (T). Figure 3 shows the mean error per output neuron in spikes for all combinations of one to six receptor types and one to six transducer mechanisms and for combinations with ten, twenty and fifty receptor protein types. The mean response over all subplacode responses is 18.15 spikes. The best results with errors less than two spikes per response were achieved with models with at least three receptor protein types and at least three transducer mechanisms. A model with only two transducer types is not sufficient to simulate the data. For generalization tests we generated a larger pattern set with our model. This training set 66 Rainer Malaka, Thomas Ragg, Martin Hammer spi kes spi kes Figure 4: Simulation results of our model (a.b) and the Ennis model (c.d). The responses of simulated sensory cells is given in spikes. The left column (a,c) represents receptor neuron responses to mixtures of geraniol and citral, the right column (b,d) represents sensory cell responses to mixtures of limonene and linalool. The concentrations of the odorants are depicted on a logarithmic scale from 2- 5 to 26 micrograms (0.03 to 64 micrograms). Measurement points and deviations from simulated data are given by crosses in the diagrams. was divided in a set of 23 training patterns and 88 test patterns. The training set had the same structure as the experimental data. Training of new randomly initialized networks provided a mean error on the test set that was approximately 1.6 times higher than on the training set. An overfitting effect was not observable during the training sequence of 10000 A Model for Chemosensory Reception 67 learning epochs. It is also possible to transform many other models for chemosensory perception into ANN s. We fitted the stimulus summation model and the stimulus substitution model [Carr & Derby, 1986] as well as the models proposed by Ennis [Ennis, 1991]. All of the other models were not able to reproduce the complex response functions observed in honey bee sensory neurons. Some of them are able to simulate synergistic responses to binary mixtures, but none were able to produce inhibitory effects. Figure 4 shows the simulation of a sensory neuron that shows very similar spike rates for the single odors geraniol and citral and to their binary mixture at the same concentration, while the mixture interaction of limonene and linalool shows a strong synergistic effect, i.e. the response to mixture of both odors is much higher than the responses to the single odors. As shown in Figure 4a) and b) our model is able to simulate this behavior, while the Ennis model is not sufficient to show the two different types of interaction for the binary mixtures geraniol-citral and limonene-linalool, as shown in Figure 4c) and d). The error for the Ennis model is greater than four spikes per output neuron and the error for our model with six receptor types and four transducer mechanisms is smaller than one spike per output neuron. The stimulus summation and stimulus substitution model have very similar results as the Ennis model, Figure 4 e) and t) show the simulation of the stimulus summation. 4 CONCLUSIONS Artificial neural networks are a powerful tool for the simulation of the responses of chemosensory cells. The use of ANNs is consistent with theoretical modelings. Many previously proposed models are expressible as ANNs. The new receptor transducer model described in this paper is also expressible as an ANN. The use of learning algorithms is a means to fit parameters for the simulation with given experimental response data. With this method it is possible to create simulation models of chemosensory cells, that can be used in further modelings of olfactory and chemosensory systems. Applying data from honey bee placode recordings we could also investigate the necessary complexity of chemosensory models. It could be shown that only the full receptor transducer model is able to simulate the complex response characteristics observed in honey bee chemosensory cells. Most other models can show only low synergistic mixture interactions and none of the other models is able to simulate inhibitory effects in odor perception. The found parameters of the ANN do not have to correspond to physiological entities, such as affinities between molecules. The learning or parameter fitting optimizes the parameters in a way that the difference between experimental data and simulation results is minimized. If there are several solutions to this task, one solution will be found, which might differ from the actual values. But it can be said, that a model is not sufficient if the learning algorithm is not able to fit the experimental data This implies that the smallest model, which is able to simulate the given data covers the minimum of complexity necessary. For honey bees this means that a competitive receptor transducer model is necessary with at least two transducer mechanisms and three receptor protein types. Any other model, such as the stimulus summation model, the stimulus substitution model and the Ennis model, is not sufficient. The model is not restricted to insect olfactory receptor neurons and can also be applied to many types of olfactory or gustatory receptor neurons in invertebrates and vertebrates. 68 Rainer Malaka, Thomas Ragg, Martin Hammer Acknowledgments We want to thank Pat Akers and Wayne Getz for giving us subplacode response data to train the ANNs used in our model, Heinrich Braun and Wayne Getz for fruitful discussions on our work. This work was supported by grants of the Deutsche Forschungsgemeinschaft (DFG), SPP Physiologie und Theorie neuronaler Netze, and the State of Baden-WUrttemberg. References [Akers & Getz, 1992] R.P. Akers & W.M. Getz. A test of identified response classes among olfactory receptor neurons in the honeybee worker. Chemical Senses, 17(2):191-209, 1992. [Akers & Getz, 1993] RP. Akers & W.M. Getz. Response of olfactory receptor neurons in honey bees to odorants and their binary mixtures. 1. Compo Physiol. A, 173:169-185, 1993. [Breer & Boekhoff, 1992] H. Breer & I. Boekhoff. Second messenger signalling in olfaction. Current Opinion in Neurobiology, 2:439-443,1992. [Breer et al., 1989] H. Breer, I. Boekhoff, J. Strotmann, K. Rarning, & E. Tareilus. Molecular elements of olfactory signal transduction in insect antennae. In D. Schild, editor, Chemosensory Information Processing, pages 75-86. Springer, 1989. [Carr & Derby, 1986] W.E.S . Carr & C.D. Derby. Chemically stimulated feeding behavior in marine animals: the importance of chemical mixtures and the involvement of mixture interactions. 1.Chem.Ecol., 12:987-1009,1986. [Ennis,1991] D.M. Ennis. Molecular mixture models based on competitive and noncompetitive agonism. Chemical Senses, 16(1):1-17,1991. [Malaka & Ragg, 1993] R. Malaka & T. Ragg. Models for chemosensory receptors: An approach using artificial neural networks. Interner Bericht 18/93, Institut fUr Logik, Komplexitlit und Deduktionssysteme, Universitlit Karlsruhe, 1993. [Riedmiller & Braun, 1993] M. Riedmiller & H. Braun. A direct adaptive method for faster backpropagation learning: The rprop algorithm. In Proceedings of the ICNN, 1993. [Shepherd, 1991] G.M. Shepherd. Computational structure of the olfactory system. In J.L. Davis & H. Eichenbaum, editors, Olfaction - A Model System for Computational Neuroscience, chapter 1, pages 3-41. MIT Press, 1991. [Yogt et al., 1989] RG. Yogt, R Rybczynski, & M.R. Lerner. The biochemistry of odorant reception and transduction. In D. Schild, editor, Chemosensory Information Processing, pages 33-76. Springer, 1989. 

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SARDNET: A Self-Organizing Feature Map for Sequences Daniel L. James and Risto Miikkulainen Department of Computer Sciences The University of Texas at Austin Austin, TX 78712 dljames,risto~cs.utexas.edu Abstract A self-organizing neural network for sequence classification called SARDNET is described and analyzed experimentally. SARDNET extends the Kohonen Feature Map architecture with activation retention and decay in order to create unique distributed response patterns for different sequences. SARDNET yields extremely dense yet descriptive representations of sequential input in very few training iterations. The network has proven successful on mapping arbitrary sequences of binary and real numbers, as well as phonemic representations of English words. Potential applications include isolated spoken word recognition and cognitive science models of sequence processing. 1 INTRODUCTION While neural networks have proved a good tool for processing static patterns, classifying sequential information has remained a challenging task. The problem involves recognizing patterns in a time series of vectors, which requires forming a good internal representation for the sequences. Several researchers have proposed extending the self-organizing feature map (Kohonen 1989, 1990), a highly successful static pattern classification method, to sequential information (Kangas 1991; Samarabandu and Jakubowicz 1990; Scholtes 1991). Below, three of the most recent of these networks are briefly described. The remainder of the paper focuses on a new architecture designed to overcome the shortcomings of these approaches. 578 Daniel L. James, Risto Miikkulainen Recently, Chappel and Taylor (1993) proposed the Temporal Kohonen Map (TKM) architecture for classifying sequences. The TKM keeps track of the activation history of each node by updating a value called leaky integrator potential, inspired by the membrane potential in biological neural systems. The activity of a node depends both on the current input vector and the previous input vectors, represented by the node's potential. A given sequence is processed by mapping one vector at a time, and the last winning node serves to represent the entire sequence. This way, there needs to be a separate node for every possible sequence, which is a disadvantage when the number of sequences to be classified is large. The TKM also suffers from loss of context. Which node wins depends almost entirely upon the most recent input vectors. For example, the string baaaa would most likely map to the same node as aaaaa, making the approach applicable only to short sequences. The SOFM-S network proposed by van Harmelen (1993) extends TKM such that the activity of each map node depends on the current input vector and the past activation of all map nodes. The SOFM-S is an improvement of TKM in that contextual information is not lost as quickly, but it still uses a single node to represent a sequence. The TRACE feature map (Zandhuis 1992) has two feature map layers. The first layer is a topological map of the individual input vectors , and is used to generate a trace (i.e. path) of the input sequence on the map. The second layer then maps the trace pattern to a single node . In TRACE, the sequences are represented by distributed patterns on the first layer, potentially allowing for larger capacity, but it is difficult to encode sequences where the same vectors repeat, such as baaaa. All a-vectors would be mapped on the same unit in the first layer, and any number of a-vectors would be indistinguishable. The architecture described in this paper, SARDNET (Sequential Activation Retention and Decay NETwork), also uses a subset of map nodes to represent the sequence of vectors. Such a distributed approach allows a large number of representations be "packed" into a small map-like sardines. In the following sections, we will examine how SARDNET differs from conventional self-organizing maps and how it can be used to represent and classify a large number of complex sequences. 2 THE SARDNET ARCHITECTURE Input to SARDNET consists of a sequence of n-dimensional vectors S = V I, V 2 , V 3 , ... , VI (figure 1) . The components of each vector are real values in the interval [0,1]. For example, each vector might represent a sample of a speech signal in n different frequencies, and the entire sequence might constitute a spoken word. The SARDNET input layer consists of n nodes, one for each component in the input vector, and their values are denoted as A = (aI, a2, a3, ... , an). The map consists of m x m nodes with activation Ojk , 1 ~ j, k ~ m. Each node has an n-dimensional input weight vector Wjk, which determines the node's response to the input activation. In a conventional feature map network as well as in SARDNET, each input vector is mapped on a particular unit on the map, called the winner or the maximally responding unit . In SARDNET, however, once a node wins an input, it is made SARDNET: A Self-Organizing Feature Map for Sequences 579 Sequence of Input vectors S V1 V2 V3 V4 V, ---II Previous winners Input weight vector wJk.l Winning unit jlc Figure 1: The SARDNET architecture. A sequence of input vectors activates units on the map one at a time. The past winners are excluded from further competition, and their activation is decayed gradually to indicate position in the sequence. INITIALIZATION: Clear all map nodes to zero. MAIN LOOP: While not end of seihence 1. Find unactivated weight vector t at best matches the input. 2. Assign 1.0 activation to that unit. 3. Adjust weight vectors of the nodes in the neighborhood. 4. Exclude the winning unit from subse~ent competition. S. Decrement activation values for all ot er active nodes. RESULT: Sequence representation = activated nodes ordered by activation values Table 1: The SARDNET training algorithm. uneligible to respond to the subsequent inputs in the sequence. This way a different map node is allocated for every vector in the sequence. As more vectors come in, the activation of the previous winners decays. In other words, each sequence of length 1 is represented by 1 active nodes on the map, with their activity indicating the order in which they were activated. The algorithm is summarized in table 1. Assume the maximum length ofthe sequences we wish to classify is I, and each input vector component can take on p possible values. Since there are pn possible input vectors, Ipn map nodes are needed to represent all possible vectors in all possible positions in the sequence, and a distributed pattern over the Ipn nodes can be used to represent all pnl different sequences. This approach offers a significant advantage over methods in which pnl nodes would be required for pnl sequences. The specific computations of the SARDNET algorithm are as follows: The winning node (j, k) in each iteration is determined by the Euclidean distance Djk of the 580 Daniel L. James, Risto Miikkulainen input vector A and the node 's weight vector W j k: n Djk = 1)Wjk ,i - a;)2. (1) i=O The unit with the smallest distance is selected as the winner and activated with 1.0. The weights of this node and all nodes in its neighborhood are changed according to the standard feature map adaptation rule: (2) where a denotes the learning rate. As usual, the neighborhood starts out large and is gradually decreased as the map becomes more ordered. As the last step in processing an input vector , the activation 7]jk of all active units in the map are decayed proportional to the decay parameter d: O<d<1. (3) As in the standard feature map, as the weight vectors adapt, input vectors gradually become encoded in the weight vectors of the winning units. Because weights are changed in local neighborhoods , neighboring weight vectors are forced to become as similar as possible , and eventually the network forms a topological layout of the input vector space. In SARDNET, however , if an input vector occurs multiple times in the same input sequence, it will be represented multiple times on the map as well. In other words, the map representation expands those areas of the input space that are visited most often during an input sequence. 3 EXPERIMENTS SARDNET has proven successful in learning and recognizing arbitrary sequences of binary and real numbers , as well as sequences of phonemic representations for English words. This section presents experiments on mapping three-syllable words. This dat a was selected because it shows how SARDNET can be applied to complex input derived from a real-world task . 3.1 INPUT DATA The phonemic word representations were obtained from the CELEX database of the Max Planck Institute for Psycholinguistics and converted into International Phonetic Alphabet (IPA)-compliant representation, which better describes similarities among the phonemes. The words vary from five to twelve phonemes in length. Each phoneme is represented by five values: place, manner, sound, chromacity and sonority. For example, the consonant p is represented by a single vector (bilabial, stop, unvoiced, nil , nil) , or in terms of real numbers, (.125, .167, .750,0 , 0). The diphthong sound ai as in "buy" , is represented by the two vectors (nil , vowel, voiced, front, low) and (nil, vowel , voiced , front-center, hi-mid), or in real numbers , (0, 1, .25, .2, 1) and (0, 1, .25, .4, .286). There are a total of 43 phonemes in this data set, including 23 consonants and 20 vowels. To represent all phonemic sequences of length 12, TKM and SOFM-S would SARDNET: A Self-Organizing Feature Map for Sequences 581 o.ee 0.118 0.87 0.118 0.85 .......- - Figure 2: Accuracy of SARDNET for different map and data set sizes. The accuracy is measured as a percentage of unique representations out of all word sequences. need to have 45 12 ~ 6.919 map nodes, whereas SARDNET would need only 45 x 12 = 540 nodes . Of course, only a very small subset of the possible sequences actually occur in the data. Three data sets consisting of 713 , 988, and 1628 words were used in the experiments. If the maximum number of occurrences of phoneme i in any single sequence is Cj I then the number of nodes SARDNET needs is C = L:~o Cj I where N is the number of phonemes. This number of nodes will allow SARDNET to map each phoneme in each sequence to a unit with an exact representation of that phoneme in its weights . Calculated this way, SARDNET should scale up very well with the number of words: it would need 81 nodes for representing the 713 word set , 84 for the 988 set and 88 for the 1628 set. 3.2 DENSENESS AND ACCURACY A series of experiments with the above three data sets and maps of 16 to 81 nodes were run to see how accurately SARDNET can represent the sequences. Self-organization was quite fast: each simulation took only about 10 epochs, with a = 0.45 and the neighborhood radius decreasing gradually from 5-1 to zero. Figure 2 shows the percentage of unique representations for each data set and map SIze. SARDNET shows remarkable representational power: accuracy for all sets is better than 97.7%, and SARDNET manages to pack 1592 unique representations even on the smallest 16-node map. Even when there are not enough units to represent each phoneme in each sequence exactly, the map is sometimes able to "reuse" units to represent multiple similar phonemes. For example, assume units with exact representations for the phonemes a and b exist somewhere on the map, and the input data does not contain pairs of sequences such as aba-abb, in which it is crucial to distinguished the second a from the second b. In this case, the second occurrence of both phonemes could be represented by the same unit with a weight vector that is the average of a and b. This is exactly what the map is doing: it is finding the most descriptive representation of the data, given the available resources. 582 Daniel L. James, Risto Miikkulainen Note that it would be possible to determine the needed C = L:f:o Cj phoneme representation vectors directly from the input data set, and without any learning or a map structure at all, establish distributed representations on these vectors with the SARDNET algorithm. However, feature map learning is necessary ifthe number of available representation vectors is less than C. The topological organization of the map allows finding a good set of reusable vectors that can stand for different phonemes in different sequences, making the representation more efficient. 3.3 REPRESENTING SIMILARITY Not only are the representations densely packed on the map, they are also descriptive in the sense that similar sequences have similar representations. Figure 3 shows the final activation patterns on the 36-unit, 713-word map for six example words. The first two words, "misplacement" and "displacement," sound very similar, and are represented by very similar patterns on the map. Because there is only one m in "displacement" , it is mapped on the same unit as the initial m of "misplacement." Note that the two IDS are mapped next to each other, indicating that the map is indeed topological, and small changes in the input cause only small changes in the map representation. Note also how the units in this small map are reused to represent several different phonemes in different contexts. The other examples in figure 3 display different types of similarities with "misplacement". The third word, "miscarried", also begins with "mis", and shares that subpart of the representation exactly. Similarly, "repayment" shares a similar tail and "pessimist" the subsequence "mis" in a different part or the word. Because they appear in a different context, these subsequences are mapped on slightly different units, but still very close to their positions with "misplacement." The last word, "burundi" sounds very different, as its representation on the map indicates. Such descriptive representations are important when the map has to represent information that is incomplete or corrupted with noise. Small changes in the input sequence cause small changes in the pattern, and the sequence can still be recognized. This property should turn out extremely important in real-world applications of SARDNET, as well as in cognitive science models where confusing similar patterns with each other is often plausible behavior. 4 DISCUSSION AND FUTURE RESEARCH Because the sequence representations on the map are distributed , the number of possible sequences that can be represented in m units is exponential in m, instead of linear as in most previous sequential feature map architectures. This denseness together with the tendency to map similar sequences to similar representations should turn out useful in real-world applications, which often require scale-up to large and noisy data sets. For example, SARDNET could form the core of an isolated word recognition system. The word input would be encoded in durationnormalized sequences of sound samples such as a string of phonemes, or perhaps representations of salient transitions in the speech signal. It might also be possible to modify SARDNET to form a more continuous trajectory on the map so that SARDNET itself would take care of variability in word duration. For example, a SARDNEf: A Self-Organizing Feature Map for Sequences (a) ~~t 583 (b) (c) (e) (f) ? ript.ItDt (d) Figure 3: Example map representations. sequence of redundant inputs could be reduced to a single node if all these inputs fall within the same neighborhood. Even though the sequence representations are dense, they are also descriptive. Category memberships are measured not by labels of the maximally responding units, but by the differences in the response patterns themselves. This sort of distributed representation should be useful in cognitive systems where sequential input must be mapped to an internal static representation for later retrieval and manipulation. Similarity-based reasoning on sequences should be easy to implement, and the sequence can be easily recreated from the activity pattern on the map. Given part of a sequence, SARDNET may also be modified to predict the rest of the sequence. This can be done by adding lateral connections between the nodes in the map layer. The lateral connections between successive winners would be strengthened during training. Thus, given part of a sequence, one could follow the strongest lateral connections to complete the sequence. 584 5 Daniel L. James, Risto Miikkulainen CONCLUSION SARDNET is a novel feature map architecture for classifying sequences of input vectors. Each sequence is mapped on a distributed representation on the map, making it possible to pack a remarkable large number of category representations on a small feature map . The representations are not only dense, they also represent the similarities of the sequences, which should turn out useful in cognitive science as well as real-world applications of the architecture. Acknowledgments Thanks to Jon Hilbert for converting CELEX data into the International Phonetic Alphabet format used in the experiments. This research was supported in part by the National Science Foundation under grant #IRI-9309273. References Chappel, G. J., and Taylor, J. G. (1993). The temporal Kohonen map. Neural Networks, 6:441-445. Kangas, J. (1991). Time-dependent self-organizing maps for speech recognition. In Proceedings of the International Conference on Artificial Neural Networks (Espoo, Finland), 1591-1594. Amsterdam; New York: North-Holland. Kohonen, T. (1989). Self-Organization and Associative Memory. Berlin; Heidelberg; New York: Springer. Third edition. Kohonen, T . (1990) . The self-organizing map. Proceedings of the IEEE, 78:14641480. Samarabandu, J. K., and Jakubowicz, O. G. (1990). Principles of sequential feature maps in multi-level problems. In Proceedings of the International Joint Conference on Neural Networks (Washington, DC), vol. II, 683-686. Hillsdale, NJ: Erlbaum. Scholtes, J. C. (1991). Recurrent Kohonen self-organization in natural language processing. In Proceedings of the International Conference on Artificial Neural Networks (Espoo, Finland), 1751-1754. Amsterdam; New York: NorthHolland. van Harmelen, H. (1993). Time dependent self-organizing feature map for speech recognition. Master's thesis, University of Twente, Enschede, the Netherlands. Zandhuis, J. A. (1992). Storing sequential data in self-organizing feature maps. Internal Report MPI-NL-TG-4/92, Max-Planck-Institute fur Psycholinguistik, Nijmegen, the Netherlands. 

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Estimating Conditional Probability Densities for Periodic Variables Chris M Bishop and Claire Legleye Neural Computing Research Group Department of Computer Science and Applied Mathematics Aston University Birmingham, B4 7ET, U.K. c.m.bishop@aston.ac.uk Abstract Most of the common techniques for estimating conditional probability densities are inappropriate for applications involving periodic variables. In this paper we introduce three novel techniques for tackling such problems, and investigate their performance using synthetic data. We then apply these techniques to the problem of extracting the distribution of wind vector directions from radar scatterometer data gathered by a remote-sensing satellite. 1 INTRODUCTION Many applications of neural networks can be formulated in terms of a multi-variate non-linear mapping from an input vector x to a target vector t. A conventional neural network approach , based on least squares for example, leads to a network mapping which approximates the regression of t on x. A more complete description of the data can be obtained by estimating the conditional probability density of t, conditioned on x , which we write as p(tlx). Various techniques exist for modelling such densities when the target variables live in a Euclidean space. However, a number of potential applications involve angle-like output variables which are periodic on some finite interval (usually chosen to be (0,271")). For example , in Section 3 642 Chris M. Bishop, Claire Legleye we consider the problem of determining the wind direction (a periodic quantity) from radar scatterometer data obtained from remote sensing measurements. Most of the existing techniques for conditional density estimation cannot be applied in such cases. A common technique for unconditional density estimation is based on mixture models of the form m pet) = L Cki?i(t) (1) i=l where Cki are called mixing coefficients, and the kernel functions ?i(t) are frequently chosen to be Gaussians. Such models can be used as the basis of techniques for conditional density estimation by allowing the mixing coefficients, and any parameters governing the kernel functions, to be general functions of the input vector x. This can be achieved by relating these quantities to the outputs of a neural network which takes x as input, as shown in Figure 1. Such an approach forms the basis of conditional probability density n p(tlx) 1\ 1\ 1\ parameter (> vector U mixture model z neural network Figure 1: A general framework for conditional density estimation is obtained by using a feed-forward neural network whose outputs determine the parameters in a mixture density model. The mixture model then represents the conditional probability density of the target variables, conditioned on the input vector to the network. the 'mixture of experts' model (Jacobs et al., 1991) and has also been considered by a number of other authors (White, 1992; Bishop, 1994; Lui, 1994). In this paper we introduce three techniques for estimating conditional densities of periodic variables, based on extensions of the above formalism for Euclidean variables. Estimating Conditional Probability Densities for Periodic Variables 2 643 DENSITY ESTIMATION FOR PERIODIC VARIABLES In this section we consider three alternative approaches to estimating the conditional density p(Blx) of a periodic variable B, conditioned on an input vector x. They are based respectively on a transformation to an extended domain representation, the use of adaptive circular normal kernel functions, and the use of fixed circular normal kernels. 2.1 TRANSFORMATION TO AN EXTENDED VARIABLE DOMAIN The first technique which we consider involves finding a transformation from the periodic variable B E (0,27r) to a Euclidean variable X E (-00,00), such that standard techniques for conditional density estimation can be applied in X-space. In particular, we seek a conditional density function p(xlx) which is to be modelled using a conventional Gaussian mixture approach as described in Section 1. Consider the transformation 00 L p(Blx) = p(B + L27rlx) (2) L=-oo Then it is clear by construction that the density model on the left hand side satisfies the periodicity requirement p(B + 27rlx) = p(Blx). Furthermore, if the density function p(xlx) is normalized, then we have {21f Jo p(Blx) dB = (3) and so the corresponding periodic density p(Olx) will also be normalized. We now model the density function p(xlx) using a mixture of Gaussians of the form m p(xlx) = L fri(X)4>i(xlx) (4) i=l where the kernel functions are given by 1 ( {X - Xi(X)P) 4>i(xlx) = (27r)1/2Uj(x) exp 2u;(x) (5) and the parameters fri(X), Uj(x) and Xi(X) are determined by the outputs of a feed-forward network. In particular, the mixing coefficients frj(x) are governed by a 'softmax' activation function to ensure that they lie in the range (0,1) and sum to Chris M. Bishop, Claire Legleye 644 unity; the width parameters O'i(X) are given by the exponentials ofthe corresponding network outputs to ensure their positivity; and the basis function centres Xi(X) are given directly by network output variables. The network is trained by maximizing the likelihood function, evaluated for set of training data, with respect to the weights and biases in the network. For a training set consisting of N input vectors xn and corresponding targets (r, the likelihood is given by N .c = II peon Ixn)p(xn) (6) n=l where p(x) is the unconditional density of the input data. Rather than work with .c directly, it is convenient instead to minimize an error function given by the negative log of the likelihood. Making use of (2) we can write this in the form E = -In.c ~ - 2: In 2:P'(on + L271'1xn) n (7) L where we have dropped the term arising from p(x) since it is independent of the network weights. This expression is very similar to the one which arises if we perform density estimation on the real axis, except for the extra summation over L, which means that the data point on recurs at intervals of 271' along the x-axis. This is not equivalent simply to replicating the data, however, since the summation over L occurs inside the logarithm, rather than outside as with the summation over data points n. In a practical implementation, it is necessary to restrict the summation over L. For the results presented in the next section, this summation was taken over 7 complete periods of 271' spanning the range (-771', 771'). Since the Gaussians have exponentially decaying tails, this represents an extremely good approximation in almost all cases, provided we take care in initializing the network weights so that the Gaussian kernels lie in the central few periods. Derivatives of E with respect to the network weights can be computed using the rules of calculus, to give a modified form of back-propagation. These derivatives can then be used with standard optimization techniques to find a minimum of the error function. (The results presented in the next section were obtained using the BFGS quasi-Newton algorithm). 2.2 MIXTURES OF CIRCULAR NORMAL DENSITIES The second approach which we introduce is also based on a mixture of kernel functions of the form (1), but in this case the kernel functions themselves are periodic, thereby ensuring that the overall density function will be periodic. To motivate this approach, consider the problem of modelling the distribution of a velocity vector v in two dimensions (this arises, for example, in the application considered in Section 3). Since v lives in a Euclidean plane, we can model the density function p(v) using a mixture of conventional spherical Gaussian kernels, where each kernel has 645 Estimating Conditional Probability Densities for Periodic Variables the form ? (V x , Vy ) - flx F = 211"cr1 2 exp ( { Vx 2cr 2 {V y - fly 2cr 2 F) (8) where (v x , v y ) are the Cartesian components of v, and (flx, fly) are the components of the center I-' of the kernel. From this we can extract the conditional distribution of the polar angle () of the vector v, given a value for v = I/vll. This is easily done with the transformation Vx = v cos (), Vy = v sin (), and defining ()o to be the polar angle of 1-', so that flx = fl cos ()o and fly = fl sin ()o, where fl = 111-'11? This leads to a distribution which can be written in the form ?(()) = 1 (A) exp {A cos(() - ()on 211"10 (9) where the normalization coefficient has been expressed in terms of the zeroth order modified Bessel function of the first kind, Io(A) . The distribution (9) is known as a circular normal or von Mises distribution (Mardia, 1972) . The parameter A (which depends on v in our derivation) is analogous to the (inverse) variance parameter in a conventional normal distribution. Since (9) is periodic, we can construct a general representation for the conditional density of a periodic variable by considering a mixture of circular normal kernels, with parameters given by the outputs of a neural network. The weights of the network can again be determined by maximizing the likelihood function defined over a set of training data. 2.3 FIXED KERNELS The third approach introduced here is again based on a mixture model in which the kernel functions are periodic, but where the kernel parameters (specifying their width and location) are fixed. The only adaptive parameters are the mixing coefficients, which are again determined by the outputs of a feed-forward network having a softmax final-layer activation function. Here we consider a set of equally-spaced circular normal kernels in which the width parameters are chosen to give a moderate degree of overlap between the kernels so that the resulting representation for the density function will be reasonably smooth. Again, a maximum likelihood formalism is employed to train the network. Clearly a major drawback of fixed-kernel methods is that the number of kernels must grow exponentially with the dimensionality of the output space. For a single output variable, however , they can be regarded as practical techniques. 3 RESULTS In order to test and compare the methods introduced above, we first consider a simple problem involving synthetic data, for which the true underlying distribution function is known . This data set is intended to mimic the central properties of the real data to be discussed in the next section. It has a single input variable x and an output variable () which lies in the range (0,211"). The distribution of () is governed 646 Chris M. Bishop, Claire Leg/eye by a mixture of two triangular functions whose parameters (locations and widths) are functions of x. Here we present preliminary results from the application of the method introduced in section 2.1 (involving the transformation to Euclidean space) to this data. Figure 2 shows a plot of the reconstructed conditional density in both the extended X variable, and in the reconstructed polar variable (), for a particular value of the input variable x . 0.6 r-----r-------,~-_, x=0.5 0.5,..-----"'T""-----., -network -- - - true p(X) sinO 0.0 0.3 0.0 L..-_---L_"----'-----';.......oooooL..-.........- - I -21? 21? o x ~-~---+-----~ , -0.5 '--_ _ _ _...L..-_ _ _ _.... -0.5 0.0 cosO 0.5 Figure 2: The left hand plot shows the predicted density (solid curve) together with the true density (dashed curve) in the extended X space. The right hand plot shows the corresponding densities in the periodic () space. In both cases the input variable is fixed at x = 0.5. One of the original motivations for developing the techniques described in this paper was to provide an effective, principled approach to the analysis of radar scatterometer data from satellites such as the European Remote Sensing Satellite ERS-I. This satellite is equipped with three C-band radar antennae which measure the total backscattered power (called 0"0) along three directions relative to the satellite track, as shown in Figure 3. When the satellite passes over the ocean, the strengths of the backscattered signals are related to the surface ripples of the water (on length-scales of a few cm.) which in turn are determined by the low level winds. Extraction of the wind speed and direction from the radar signals represents an inverse problem which is typically multi-valued. For example, a wind direction of (}1 will give rise to similar radar signals to a wind direction of (}1 + 1r. Often, there are additional such 'aliases' at other angles . A conventional neural network approach to this problem , based on least-squares, would predict wind directions which were given by conditional averages of the target data. Since the average of several valid wind directions is typically not itself a valid direction, such an approach would clearly fail. Here we aim to extract the complete distribution of wind directions (as a function of the three 0"0 values and on the angle of incidence of the radar beam) and hence avoid Estimating Conditional Probability Densities for Periodic Variables 647 satellite fore beam 785km mid beam aft beam 5COkm Figure 3: Schematic illustration of the ERS-l satellite showing the footprints of the three radar scatterometers. such difficulties. This approach also provides the most complete information for the next stage of processing (not considered here) which is to 'de-alias' the wind directions to extract the most probable overall wind field. A large data set of ERS-l measurements, spanning a wide range of meteorological conditions, has been assembled by the European Space Agency in collaboration with the UK Meteorological Office. Labelling of the data set was performed using wind vectors from the Meteorological Office Numerical Weather Prediction code. An example of the results from the fixed-kernel method of Section 2.3 are presented in Figure 4. This clearly shows the existence of a primary alias at an angle of 1r relative to the principal direction, as well as secondary aliases at ?1r/2. Acknowledgements We are grateful to the European Space Agency and the UK Meteorological Office for making available the ERS-l data. We would also like to thank lain Strachan and Ian Kirk of AEA Technology for a number of useful discussions relating to the interpretation of this data. References Bishop C M (1994). Mixture density networks. Neural Computing Research Group Report, NCRG/4288, Department of Computer Science, Aston University, Birmingham, U.K. Jacobs R A, Jordan M I, Nowlan S J and Hinton G E (1991). Adaptive mixtures 648 Chris M. Bishop, Claire Leg/eye 0.5 ~--~---r---~--""'" 0.0 1--------.....;;::~~-__1 -0.5 -1.0 -1.5 ~ -1.5 _ _~_----II.....-_---I_ _- - J -1.0 -0.5 0.0 0.5 Figure 4: An example of the results obtained with the fixed-kernel method applied to data from the ERS-1 satellite. As well as the primary wind direction, there are aliases at 1r and ?1r/2. of local experts. Neural Computation, 3 79-87. Lui Y (1994) Robust parameter estimation and model selection for neural network regression. Advances in Neural Information Processing Systems 6 Morgan Kaufmann, 192-199 .. Mardia K V (1972) . Statistics of Directional Data. Academic Press, London. White H (1992). Parametric statistical estimation with artificial neural networks. University of California, San Diego, Technical Report. 

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Learning Prototype Models for Tangent Distance Trevor Hastie? Statistics Department Sequoia Hall Stanford University Stanford, CA 94305 email: trevor@playfair .stanford .edu Patrice Simard AT&T Bell Laboratories Crawfords Corner Road Holmdel, NJ 07733 email: patrice@neural.att.com Eduard Siickinger AT &T Bell Laboratories Crawfords Corner Road Holmdel, NJ 07733 email: edi@neural.att.com Abstract Simard, LeCun & Denker (1993) showed that the performance of nearest-neighbor classification schemes for handwritten character recognition can be improved by incorporating invariance to specific transformations in the underlying distance metric - the so called tangent distance. The resulting classifier, however, can be prohibitively slow and memory intensive due to the large amount of prototypes that need to be stored and used in the distance comparisons. In this paper we develop rich models for representing large subsets of the prototypes. These models are either used singly per class, or as basic building blocks in conjunction with the K-means clustering algorithm. *This work was performed while Trevor Hastie was a member of the Statistics and Data Analysis Research Group, AT&T Bell Laboratories, Murray Hill, NJ 07974 . J 000 1 Trevor Hastie, Patrice Simard, Eduard Siickinger INTRODUCTION Local algorithms such as K-nearest neighbor (NN) perform well in pattern recognition, even though they often assume the simplest distance on the pattern space. It has recently been shown (Simard et al. 1993) that the performance can be further improved by incorporating invariance to specific transformations in the underlying distance metric - the so called tangent distance. The resulting classifier, however, can be prohibitively slow and memory intensive due to the large amount of prototypes that need to be stored and used in the distance comparisons. In this paper we address this problem for the tangent distance algorithm, by developing rich models for representing large subsets of the prototypes. Our leading example of prototype model is a low-dimensional (12) hyperplane defined by a point and a set of basis or tangent vectors. The components of these models are learned from the training set, chosen to minimize the average tangent distance from a subset of the training images - as such they are similar in flavor to the Singular Value Decomposition (SVD), which finds closest hyperplanes in Euclidean distance. These models are either used singly per class, or as basic building blocks in conjunction with K-means and LVQ. Our results show that not only are the models effective, but they also have meaningful interpretations. In handwritten character recognition, for instance, the main tangent vector learned for the the digit "2" corresponds to addition/removal of the loop at the bottom left corner of the digit; for the 9 the fatness of the circle. We can therefore think of some of these learned tangent vectors as representing additional invariances derived from the training digits themselves. Each learned prototype model therefore represents very compactly a large number of prototypes of the training set. 2 OVERVIEW OF TANGENT DISTANCE When we look at handwritten characters, we are easily able to allow for simple transformations such as rotations, small scalings, location shifts, and character thickness when identifying the character. Any reasonable automatic scheme should similarly be insensitive to such changes. Simard et al. (1993) finessed this problem by generating a parametrized 7dimensional manifold for each image, where each parameter accounts for one such invariance. Consider a single invariance dimension: rotation. If we were to rotate the image by an angle B prior to digitization, we would see roughly the same picture, just slightly rotated. Our images are 16 x 16 grey-scale pixelmaps, which can be thought of as points in a 256-dimensional Euclidean space. The rotation operation traces out a smooth one-dimensional curve Xi(B) with Xi(O) = Xi, the image itself. Instead of measuring the distance between two images as D(Xi,Xj) = IIX i - Xjll (for any norm 11?11), the idea is to use instead the rotation-invariant DI (Xi, Xj) = minoi,oj IIX i(B;) - Xj(Bj )11. Simard et al. (1993) used 7 dimensions of invariance, accounting fo:: horizontal and vertical location and scale, rotation and shear and character thickness. Computing the manifold exactly is impossible, given a digitized image, and would be impractical anyway. They approximated the manifold instead by its tangent Learning Prototype Models for Tangent Distance 1001 plane at the image itself, leading to the tangent model Xi(B) = Xi + TiB, and the tangent distance DT(Xi,Xj) = minoi,oj IIXi(Bd -Xj(Bj)ll. Here we use B for the 7-dimensional parameter, and for convenience drop the tilde. The approximation is valid locally, and thus permits local transformations. Non-local transformations are not interesting anyway (we don't want to flip 6s into 9s; shrink all digits down to nothing.) See Sackinger (1992) for further details. If 11?11 is the Euclidean norm, computing the tangent distance is a simple least-squares problem, with solution the square-root of the residual sum-of-squares of the residuals in the regression with response Xi - Xj and predictors (-Ti : Tj ). Simard et al. (1993) used DT to drive a 1-NN classification rule, and achieved the best rates so far-2.6%-on the official test set (2007 examples) of the USPS data base. Unfortunately, 1-NN is expensive, especially when the distance function is non-trivial to compute; for each new image classified, one has to compute the tangent distance to each of the training images, and then classify as the class of the closest. Our goal in this paper is to reduce the training set dramatically to a small set of prototype models; classification is then performed by finding the closest prototype. 3 PROTOTYPE MODELS In this section we explore some ideas for generalizing the concept of a mean or centroid for a set of images, taking into account the tangent families. Such a centroid model can be used on its own, or else as a building block in a K-means or LVQ algorithm at a higher level. We will interchangeably refer to the images as points (in 256 space). The centroid of a set of N points in d dimensions minimizes the average squared norm from the points: (1) 3.1 TANGENT CENTROID One could generalize this definition and ask for the point M that minimizes the average squared tangent distance: N MT = argm,Jn LDT(Xi,M)2 (2) i=l This appears to be a difficult optimization problem, since computation of tangent distance requires not only the image M but also its tangent basis TM. Thus the criterion to be minimized is 1002 Trevor Hastie, Patrice Simard, Eduard Sackinger where T(M) produces the tangent basis from M. All but the location tangent vectors are nonlinear functionals of M, and even without this nonlinearity, the problem to be solved is a difficult inverse functional. Fortunately a simple iterative procedure is available where we iteratively average the closest points (in tangent distance) to the current guess. Tangent Centroid Algorithm Initialize: Set M = ~ 2:~1 Xi, let TM = T(M) be the derived set of tangent vectors, and D = 2:i DT(Xi' M). Denote the current tangent centroid (tangent family) by M(-y) = M +TM"I. Iterate: each i find 11M + TM"I - Xi(8)11 1. For N 1 a 1'. and = min'Y.9 8i that solves 2. Set M +- N 2:'=1 (Xi(8i ) - TMi'i) and compute the new tangent subspace TM = T(M). 3. Compute D = 2:iDT(Xi,M) Until: D converges. A Note that the first step in Iterate is available from the computations in the third step. The algorithm divides the parameters into two sets: M in the one, and then TM, "Ii and 8, for each i in the other. It alternates between the two sets, although the computation of TM given M is not the solution of an optimization problem. It seems very hard to say anything precise about the convergence or behavior of this algorithm, since the tangent vectors depend on each iterate in a nonlinear way. Our experience has always been that it converges fairly rapidly ? 6 iterations). A potential drawback of this algorithm is that the TM are not learned, but are implicit in M. 3.2 TANGENT SUBSPACE Rather than define the model as a point and have it generate its own tangent subspace, we can include the subspace as part of the parametrization: M(-y) = M + V"I. Then we define this tangent subspace model as the minimizer of N MS(M, V) = L min 11M + V"Ii - Xi(8d1l 2 (3) . 1 'Yi. 9 i t= over M and V. Note that V can have an arbitrary number 0 ::; r ::; 256 of columns, although it does not make sense for r to be too large. An iterative algorithm similar to the tangent centroid algorithm is available, which hinges on the SVD decomposition for fitting affine subspaces to a set of points. We briefly review the SVD in this context. Let X be the N x 256 matrix with rows the vectors Xi - X where X = ~ 2:~1 Xi. Then SVD(X) = UDV T is a unique decomposition with UNxR and V25 6xR the 1003 Learning Prototype Models for Tangent Distance orthonormal left and right matrices of singular vectors, and R = rank( X). D Rx R is a diagonal matrix of decreasing positive singular values. A pertinent property of the SVD is: Consider finding the closest affine, rank-r subspace to a set of points, or 2 N min M,v(r),{9i} 2: i=1 IIXi - M - v(r)'hll where v(r) is 256 x r orthonormal. The solution is given by the SVD above, with M = X and v(r) the first r columns of V, and the total squared distance E;=1 D;j. The V( r) are also the largest r principal components or eigenvectors of the covariance matrix of the Xi. They give in sequence directions of maximum spread, and for a given digit class can be thought of as class specific invariances. We now present our Tangent subspace algorithm for solving (3); for convenience we assume V is rank r for some chosen r, and drop the superscript. Tangent subspace algorithm Initialize: Set M = ~ Ef:l Xi and let V correspond to the first r right singular vectors of X. Set D = E;=1 D;j, and let the current tangent subspace model be M(-y) = M + V-y. Iterate: 1. For each i find that (ji which solves IIM(-y) - Xi (8)11 = min N 2. Set M +- ~ Ei=1 (Xi (8 i )) and replace the rows of X by Xi({jd - M. Compute the SVD of X, and replace V by the first r right singular vectors. 3. Compute D = E;=l D;j A Until: D converges. The algorithm alternates between i) finding the closest point in the tangent subspace for each image to the current tangent subspace model, and ii) computing the SVD for these closest points. Each step of the alternation decreases the criterion, which is positive and hence converges to a stationary point of the criterion. In all our examples we found that 12 complete iterations were sufficient to achieve a relative convergence ratio of 0.001. One advantage of this approach is that we need not restrict ourselves to a sevendimensional V - indeed, we have found 12 dimensions has produced the best results . The basis vectors found for each class are interesting to view as images. Figure 1 shows some examples of the basis vectors found, and what kinds of invariances in the images they account for. These are digit specific features; for example, a prominent basis vector for the family of 2s accounts for big versus small loops. Trevor Hastie, Patrice Simard, Eduard Siickinger 1004 Each of the examples shown accounts for a similar digit specific invariance. None of these changes are accounted for by the 7-dimensional tangent models, which were chosen to be digit nonspecific. Figure 1: Each column corresponds to a particular tangent subspace basis vector for the given digit . The top image is the basis vector itself, and the remaining 3 images correspond to the 0.1 , 0.5 and 0.9 quantiles for the projection indices for the training data for that basis vector, showing a range of image models for that basis, keeping all the others at o. 4 SUBSPACE MODELS AND K-MEANS CLUSTERING A natural and obvious extension of these single prototype-per-class models, is to use them as centroid modules in a K-means algorithm. The extension is obvious, and space permits only a rough description. Given an initial partition of the images in a class into K sets: 1. Fit a separate prototype model to each of the subsets; 2. Redefine the partition based on closest tangent distance to the prototypes found in step 1. In a similar way the tangent centroid or subspace models can be used to seed LVQ algorithms (Kohonen 1989), but so far we have not much experience with them. 5 RESULTS Table 1 summarizes the results for some of these models. The first two lines correspond to a SVD model for the images fit by ordinary least squares rather than least tangent squares. The first line classifies using Euclidean distance to this model, the second using tangent distance. Line 3 fits a single 12-dimensional tangent subspace model per class, while lines 4 and 5 use 12-dimensional tangent subspaces as cluster Learning Prototype Models for Tangent Distance 1005 Table 1: Test errors for a variety of situations. In all cases the training data were 7291 USPS handwritten digits, and the test data the "official" 2007 USPS test digits. Each entry describes the model used in each class, so for example in row 5 there are 5 models per class, hence 50 in all. 0 1 2 3 4 5 6 7 8 Prototype 1-NN 12 dim SVD subspace 12 dim SVD subspace 12 dim Tangent subspace 12 dim Tangent subspace 12 dim Tangent subspace Tangent centroid (4) U (6) 1-NN Metric Euclidean Euclidean Tangent Tangent Tangent Tangent Tangent Tangent Tangent # Prototypes7Class Error Rate ~ ~ 700 1 1 1 3 5 20 23 700 0.053 0.055 0.045 0.041 0.038 0.038 0.038 0.034 0.026 centers within each class. We tried other dimensions in a variety of settings, but 12 seemed to be generally the best. Line 6 corresponds to the tangent centroid model used as the centroid in a 20-means cluster model per class; the performance compares with with K=3 for the subspace model. Line 7 combines 4 and 6, and reduces the error even further. These limited experiments suggest that the tangent subspace model is preferable, since it is more compact and the algorithm for fitting it is on firmer theoretical grounds. Figure 4 shows some of the misclassified examples in the test set. Despite all the matching, it seems that Euclidean distance still fails us in the end in some of these cases. 6 DISCUSSION Gold, Mjolsness & Rangarajan (1994) independently had the idea of using "domain specific" distance measures to seed K-means clustering algorithms. Their setting was slightly different from ours, and they did not use subspace models. The idea of classifying points to the closest subspace is found in the work of Oja (1989), but of course not in the context of tangent distance. We are using Euclidean distance in conjunction with tangent distance. Since neighboring pixels are correlated, one might expect that a metric that accounted for the correlation might do better. We tried several variants using Mahalanobis metrics in different ways, but with no success. We also tried to incorporate information about where the images project in the tangent subspace models into the classification rule. We thus computed two distances: 1) tangent distance to the subspace, and 2) Mahalanobis distance within the subspace to the centroid for the subspace. Again the best performance was attained by ignoring the latter distance. In conclusion, learning tangent centroid and subspace models is an effective way 1006 Trevor Hastie, Patrice Simard, Eduard Siickinger true: 6 true: 2 true: 5 true: 2 true: 9 true: 4 pred. pro). ( 0 ) pradoproj. ( 0 ) pred. pro). ( 8 ) pred. proj. ( 0 ) prado proj. ( 4 ) prado pro). ( 7 ) Figure 2: Some of the errorS for the test set corresponding to line (3) of table 4. Each case is displayed as a column of three images. The top is the true image, the middle the tangent projection of the true image onto the subspace model of its class, the bottom image the tangent projection of the image onto the winning class . The models are sufficiently rich to allow distortions that can fool Euclidean distance. to reduce the number of prototypes (and thus the cost in speed and memory) at a slight expense in the performance. In the extreme case, as little as one 12 dimensional tangent subspace per class and the tangent distance is enough to outperform classification using ~ 700 prototypes per class and the Euclidean distance (4.1 % versus 5.3% on the test data). References Gold, S., Mjolsness, E. & Rangarajan, A. (1994), Clustering with a domain specific distance measure, in 'Advances in Neural Information Processing Systems', Morgan Kaufman, San Mateo, CA. Kohonen, T. (1989), Self-Organization and Associative Memory (3rd edition), Springer-Verlag, Berlin. Oja, E. (1989), 'Neural networks, principal components, and subspaces', International Journal Of Neural Systems 1(1), 61-68. Sackinger, E. (1992), Recurrent networks for elastic matching in pattern recognition, Technical report, AT&T Bell Laboratories. Simard, P. Y, LeCun, Y. & Denker, J. (1993), Efficient pattern recognition using a new transformation distance, in 'Advances in Neural Information Processing Systems', Morgan Kaufman, San Mateo, CA, pp. 50-58. 

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195 LEARNING WITH TEMPORAL DERIVATIVES IN PULSE-CODED NEURONAL SYSTEMS Mark Gluck David B. Parker Eric S. Reifsnider Department of Psychology Stanford University Stanford, CA 94305 Abstract A number of learning models have recently been proposed which involve calculations of temporal differences (or derivatives in continuous-time models). These models. like most adaptive network models. are formulated in tenns of frequency (or activation), a useful abstraction of neuronal firing rates. To more precisely evaluate the implications of a neuronal model. it may be preferable to develop a model which transmits discrete pulse-coded information. We point out that many functions and properties of neuronal processing and learning may depend. in subtle ways. on the pulse-coded nature of the information coding and transmission properties of neuron systems. When compared to formulations in terms of activation. computing with temporal derivatives (or differences) as proposed by Kosko (1986). Klopf (1988). and Sutton (1988). is both more stable and easier when reformulated for a more neuronally realistic pulse-coded system. In reformulating these models in terms of pulse-coding. our motivation has been to enable us to draw further parallels and connections between real-time behavioral models of learning and biological circuit models of the substrates underlying learning and memory. INTRODUCTION Learning algorithms are generally defined in terms of continuously-valued levels of input and output activity. This is true of most training methods for adaptive networks. (e.g .? Parker. 1987; Rumelhart. Hinton. & Williams, 1986; Werbos. 1974; Widrow & Hoff, 1960). and also for behavioral models of animal and hwnan learning. (e.g. Gluck & Bower. 1988a, 1988b; Rescorla & Wagner. 1972). as well as more biologically oriented models of neuronal function (e.g .? Bear & Cooper, in press; Hebb, 1949; Granger. Abros-Ingerson, Staubli, & Lynch, in press; Gluck & Thompson, 1987; Gluck. Reifsnider. & Thompson. in press; McNaughton & Nadel. in press; Gluck & Rumelhart. in press). In spite of the attractive simplicity and utility of the "activation" construct 196 Parker, Gluck and Reifsnider neurons use discrete trains of pulses for the transmission of information from cell to cell. Frequency (or activation) is a useful abstraction of pulse trains. especially for bridging the gap between whole-animal and single neuron behavior. To more precisely evaluate the implications of a neuronal model. it may be preferable to develop a model which transmits discrete pulse-coded information; it is possible that many functions and properties of neuronal processing and learning may depend. in subtle ways. on the pulse-coded nature of the information coding and transmission properties of neuron systems. In the last few years, a number of learning models have been proposed which involve computations of temporal differences (or derivatives in continuous-time models). Klopf (1988) presented a formal real-time model of classical conditioning that predicts the magnitude of conditioned responses (CRs). given the temporal relationships between conditioned stimuli (eSs) and an unconditional stimulus (US). Klopf's model incorporates a "differential-Hebbian" learning algorithm in which changes in presynaptic levels of activity are correlated with changes in postsynaptic levels of activity. Motivated by the constraints and motives of engineering. rather than animal learning. Kosko (1986) proposed the same basic rule and provided extensive analytic insights into its properties. Sutton (1988) introduced a class of incremental learning procedures. called "temporal difference" methods. which update associative (predictive) weights according to the difference between temporally successive predictions. In addition to the applied potential of this class of algorithms. Sutton & Barto (1987) show how their model. like Klopf's (1988) model. provides a good fit to a wide range of behavioral data on classical conditioning. These models. all of which depend on computations involving changes over time in activation levels. have been successful both for predicting a wide range of behavioral animal learning data (Klopf. 1988; Sutton & Barto. 1987) and for solving useful engineering problems in adaptive prediction (Kosko. 1986; Sutton. 1988). The possibility that these models might represent the computational properties of individual neurons. seems, at first glance. highly unlikely. However. we show by reformulating these models for pulse-coded communication (as in neuronal systems) rather than in terms of abstract activation levels. the computational soundness as well as the biological relevance of the models is improved. By avoiding the use of unstable differencing methods in computing the time-derivative of activation levels. and by increasing the error-tolerance of the computations, pulse coding will be shown to improve the accuracy and reliability of these models. The pulse coded models will also be shown to lend themselves to a closer comparison to the function of real neurons than do models that operate with activation levels. As the ability of researchers to directly measure neuronal behavior grows. the value of such close comparisons will increase. As an example. we describe here a pulse-coded version of Klopf's differential-Hebbian model of classification learning. Further details are contained in Gluck. Parker. & Reifsnider. 1988. Learning with Temporal Derivatives Pulse-Coding in Neuronal Systems We begin by outlining the general theory and engineering advantages of pulse-coding and then describe a pulse-coded refonnulation of differential-Hebbian learning. The key idea is quite simple and can be summarized as follows: Frequency can be seen, loosely speaking, as an integral of pulses; conversely, therefore, pulses can be thought of as carrying infonnation about the derivatives of frequency. Thus, computing with the "derivatives of frequency" is analogous to computing with pulses. As described below, our basic conclusion is that differential-Hebbian learning (Klopf, 1988; Kosko, 1986) when refonnulated for a pulse-coded system is both more stable and easier to compute than is apparent when the rule is fonnulated in tenns of frequencies. These results have important implications for any learning model which is based on computing with timederivatives, such as Sutton's Temporal Difference model (Sutton, 1988; Sutton & Barto, 1987) There are many ways to electrically transmit analog information from point to point. Perhaps the most obvious way is to transmit the infonnation as a signal level. In electronic systems, for example, data that varies between 0 and 1 can be transmitted as a voltage level that varies between 0 volts and 1 volt This method can be unreliable, however, because the receiver of the information can't tell if a constant DC voltage offset has been added to the information, or if crosstalk has occurred with a nearby signal path. To the exact degree that the signal is interfered with, the data as read by the receiver will be erroneously altered. The consequences of faults appearing in the signal are particularly serious for systems that are based on derivatives of the signal. In such systems, even a small, but sudden, unintended change in signal level can drastically alter its derivative, creating large errors. A more reliable way to transmit analog information is to encode it as the frequency of a series of pulses. A receiver can reliably detennine if it has received a pulse, even in the face of DC voltage offsets or moderate crosstalk. Most errors will not be large enough to constitute a pulse, and thus will have no effect on the transmitted infonnation. The receiver can count the number of pulses received in a given time window to detennine the frequency of the pulses. Further infonnation on encoding analog infonnation as the frequency of a series of pulses can be found in many electrical engineeri.ng textbooks (e.g., Horowitz & Hill, 1980). As noted by Parker (1987), another advantage of coding an analog signal as the frequency of a series of pulses is that the time derivative of the signal can be easily and stably calculated: If x (t) represents a series of pulses (x equals 1 if a pulse is occuring at time t; otherwise it equals 0) then we can estimate the frequency, f (t), of the series of pulses using an exponentially weighted time average: f (t) = Jllx ('t)e-Jl{t-'t) d't 197 198 Parker, Gluck and Reifsnider where Jl is the decay constant. The well known formula for the derivative of 1 (t) is AtJP- = Jl~(t)-/(t)) Thus. the time derivative of pulse-coded information can be calculated without using any unstable differencing methods. it is simply a function of presence or absence of a pulse relative to the current expectation (frequency) of pulses. As described earlier. calculation of time derivatives is a critical component of the learning algorithms proposed by Klopf (1988). Kosko (1986) and Sutton (Sutton. 1988; Sutton & Barto 1987). They are also an important aspect of 2nd order (pseudo-newtonian) extensions of the backpropogation learning rule for multi-layer adaptive "connectionist" networks (parker. 1987). Summary 01 Klopf s Model Klopf (1988) proposed a model of classical conditioning which incorporates the same learning rule proposed by Kosko (1986) and which extends some of the ideas presented in Sutton and Barto's (1981) real-time generalization of Rescorla and Wagner's (1972) model of classical conditioning. The mathematical specification of Klopf s model consists of two equations: one which calculates output signals based on a weighted sum of input signals (drives) and one which determines changes in synapse efficacy due to changes in signal levels. The specification of signal output level is defined as where: y (t ) is the measure of postsynaptic frequency of firing at time t; Wi (t) is the efficacy (positive or negative) of the i th synapse; Xi (t) is the frequency of action potentials at the i th synapse; 9 is the threshold of firing; and n is the number of synapses on the "neuron". This equation expresses the idea that the postsynaptic firing frequency depends on the summation of the weighted presynaptic firing frequencies. Wi (t )Xi (t ). relative to some threshold. 9. The learning mechanism is defined as where: ~Wi (t) is the change in efficacy of the i th synapse at time t; ~y (t) is the change in postsynaptic firing at time t; 't' is the longest interstimulus interval over which delayed conditioning is effective. The Cj are empirically established learning rate constants -each corresponding to a different inter-stimulus interval. In order to accurately simulate various behavioral phenomena observed in classical conditioning. Klopf adds three ancillary assumptions to his model. First. he places a lower bound of 0 on the activation of the node. Second. he proposes that changes in synaptic 199 Learning with Temporal Derivatives weight, ~w; (t), be calculated only when the change in presynaptic signal level is positive -- that is, when Ax; (t-j) > O. Third, he proposes separate excitatory and inhibitory weights in contrast to the single real-valued associative weights in other conditioning models (e.g., Rescorla & Wagner, 1972; Sutton & Barto, 1981). It is intriguing to note that all of these assumptions are not only sufficiently justified by constraints from behavioral data but are also motivated by neuronal constraints. For a further examination of the biological and behavioral factors supporting these assumptions see Gluck, Parker, and Reifsnider (1988). The strength of Klopf's model as a simple formal behavioral model of classical conditioning is evident. Although the model has not yielded any new behavioral predictions, it has demonstrated an impressive ability to reproduce a wide, though not necessarily complete, range of Pavlovian behavioral phenomena with a minimum of assumptions. Klopf (1988) specifies his learning algorithm in terms of activation or frequency levels. Because neuronal systems communicate through the transmission of discrete pulses, it is difficult to evaluate the biological plausibility of an algorithm when so formulated. For this reason, we present and evaluate a pulse-coded reformulation of Klopf's model. A Pulse-Coded Reformulation of Klopf s Model We illustrate here a pulse-coded reformulation of Klopf's (1988) model of classical conditioning. The equations that make up the model are fairly simple. A neuron is said to have fired an output pulse at time t if vet) > e, where e is a threshold value and vet) is defined as follows: vet) = (l-d)v(t-l) + !:Wi(t)Xi(t) (1) where v (t) an auxiliary variable, d is a small positive constant representing the leakage or decay rate, Wi (t) is the efficacy of synapse i at time t, and Xi (t) is the frequency of presynaptic pulses at time t at synapse i. The input to the decision of whether the neuron will fire consists of the weights and efficacies of the synapses as well as information about previous activation levels at the neuronal output Note that the leakage rate, d, causes older information about activation levels to have less impact on current values of v (t) than does recent information of the same type. The output of the neuron, p (t), is: v (t) > e then p (t) = 1 (pulse generated) v (t ) ~ e then p (t) = 0 (no pulse generated) It is important that once p (t) has been determined, v (t) will need to be adjusted if 200 Parker, Gluck and Reifsnider To reflect the fact that the neuron has fired, (i.e., p (t) = 1) then v (t) = v (t) - 1. This decrement occurs after p (t) has been determined for the current t. Frequencies of pulses at the output node and at the synapses are calculated using the following equations: p (t) = 1. / (t) =/ (t-l) + 11/(t) where 11/ (t) = m(p (t) - / (t-l)) where / (t) is the frequency of outgoing pulses at time t; p (t) is the ouput (1 or 0) of the neuron at time t ; and m is a small positive constant representing a leakage rate for the frequency calculation. Following Klopf (1988), changes in synapse efficacy occur according to (2) where I1Wi(t) = Wi (t+l) - Wi(t) and l1y (t) and ru:i (t) are calculated analogously to 11/ (t); 't is the longest interstimulus interval (lSI) over which delay conditioning is effective; and Cj is an empirically established set of learning rates which govern the efficacy of conditioning at an lSI of j . Changes in Wi (t) are governed by the learning rule in Equation 2 which alters v (t) via Equation 1. Figure 1 shows the results of a computer simulation of a pulse-coded version of Klopf's conditioning model. The first graph shows the excitatory weight (dotted line) and inhibitory weight (dashed line) of the CS "synapse". Also on the same graph is the net synaptic weight (solid line), the sum of the excitatory and inhibitory weights. The subsequent graphs show CS input pulses, US input pulses, and the output (CR) pulses. The simulation consists of three acquisition trials followed by three extinction trials. 201 Learning with Temporal Derivatives en l! 0.4 .~ 02 . ; ~ (...............................J 0.6 0.0 ?0.2 .................................. (................................., ......... ~ ?V1???????????U .............................. ------------------------------------------~-----~----------- o 50 100 150 200 250 cycle Figure 1. Simulation of pulse.coded version of Klopf's conditioning model. Top panel shows excitatory and inhibitory weights as dashed lines and the net synaptic weight of the CS as a solid line. Lower panels show the CS and US inputs and the CR output. As expected, excitatory weight increases in magnitude over the three ~quisition trials, while inhibitory weight is stable. During the first two extinction trials, the excitatory and the net synaptic weights decrease in magnitude, while the inhibitory weight increases. Thus, the CS produces a decreasing amount of output pulses (the CR). During the third extinction trial the net synaptic weight is so low that the CS cannot produce output pulses, and so the CR is extinct. However, as net weight and excitatory weight remain positive, there are residual effects of the acquisition which will accelerate reacquisition. Because a threshold must be reached before a neuronal output pulse can be emitted, and because output must occur for weight changes to occur, pulse coding adds to the accelerated reacquisition effect that is evident in the original Klopf model; extinction is halted before net weight is zero, when pulses can no longer be produced. 300 202 Parker, Gluck and Reifsnider Discussion To facilitate comparison between learning algorithms involving temporal derivative computations and actual neuronal capabilities. we formulated a pulse-coded variation of Klopfs classical conditioning model. Our basic conclusion is that computing with temporal derivatives (or differences) as proposed by Kosko (1986). Klopf (1988). and Sutton (1988). is more stable and easier when reformulated for a more neuronally realistic. pulse-coded system. than when the rules are fonnulated in terms of frequencies or activation. It is our hope that further examination of the characteristics of pulse-coded systems may reveal facts that bear on the characteristics of neuronal function. In refonnulating these algorithms in terms of pulse-coding. our motivation has been to enable us to draw further parallels and connections between real-time behavioral models of learning and biological circuit models of the substrates underlying classical conditioning. (e.g .? Thompson. 1986; Gluck & Thompson. 1987; Donegan. Gluck. & Thompson. in press). More generally. noting the similarities and differences between algorithmic/behavioral theories and biological capabilities is one way of laying the groundwork for developing more complete integrated theories of the biological bases of associative learning (Donegan. Gluck. & Thompson. in press). Acknowledgments Correspondence should be addressed to: Mark A. Gluck. Dept of Psychology. Jordan Hall; Bldg. 420. Stanford. CA 94305. For their commentary and critique on earlier drafts of this and related papers. we are indebted to Harry Klopf. Bart Kosko. Richard Sutton. and Richard Thompson. This research was supported by an Office of Naval Research Grant to R. F. Thompson and M. A. Gluck. References Bear. M. F., & Cooper, L. N. (in press). Molecular mechanisms for synaptic modification in the visual cortex: Interaction between theory and experiment. In M. A. Gluck, & D. E. Rumelhart (Eds.), Neuroscience and Connectionist Theory. Hillsdale, N.J.: Lawrence Erlbaum Associates .. Donegan, N. H., Gluck, M. A., & Thompson, R. F. (1989). Integrating behavioral and biological models of classical conditioning. In R. D. Hawkins, & G. H. Bower (Eds.), Computational models of learning in simple neural systems (Volume 22 of the Psychology of Learning and Motivation). New York: Academic Press. Gluck, M. A., & Bower. G. H. (1988a). Evaluating an adaptive network model of human learning. Journal of Memory and Language, 27, 166-195. Gluck, M. A., & Bower, G. H. (1988b). From conditioning to category learning: An adaptive network model. Journal of Experimental Psychology: General, 117(3), 225-244. Learning with Temporal Derivatives Gluck, M. A., Parker, D. B., & Reifsnider, E. (1988). Some biological implications of a differential-Hebbian learning rule. Psychobiology, 16(3), 298-302. Gluck, M. A, Reifsnider, E. S., & Thompson, R. F. (in press). Adaptive signal processing and temporal coarse coding: Cerebellar models of classical conditioning and VOR Adaptation. In M. A. Gluck, & D. E. Rumelhart (Eds.), Neuroscience and Connectionist Theory. Hillsdale, N.1.: Lawrence Erlbaum Associates .. Gluck, M. A, & Rumelhart, D. E. (in press). Neuroscience and Connectionist Theory. Hillsdale, N.J.: Lawrence Erlbaum Associates .. Gluck, M. A., & Thompson, R. F. (1987). Modeling the neural substrates of associative learning and memory: A computational approach. Psychological Review, 94, 176-191. Granger, R., Ambros-Ingerson, 1., Staubli, U., & Lynch, G. (in press). Memorial operation of multipIe, interacting simulated brain structures. In M. A. Gluck, & D. E. Rumelhart (Eds.), Neuroscience and Connectionist Theory. Hillsdale, N.J.: Lawrence Erlbaum Associates .. Hebb, D. (1949). Organization of Behavior. New York: Wiley & Sons. Horowitz, P., & Hill, W. (1980). The Art of Electronics. Cambridge, England: Cambridge University Press. Klopf, A. H. (1988). A neuronal model of classical conditioning. Psychobiology, 16(2), 85-125. Kosko, B. (1986). Differential hebbian learning. In 1. S. Denker (Ed.), Neural Networksfor Computing, AlP Conference Proceedings 151 (pp. 265-270). New York: American Institute of Physics. McNaughton, B. L., & Nadel, L. (in press). Hebb-Marr networks and the neurobiological representation of action in space. In M. A. Gluck, & D. E. Rumelhart (Eds.), Neuroscience and Connectionist Theory. Hillsdale, N.J.: Lawrence Erlbaum Associates .. Parker, D. B. (1987). Optimal Algorithms for Adaptive Networks: Second Order Back Propagation, Second Order Direct Propagation, and Second Order Hebbian Learning. Proceedings of the IEEE First Annual Conference on Neural Networks. San Diego, California:, . Rescorla. R. A, & Wagner, A. R. (1972). A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and non-reinforcement. In A. H. Black, & W. F. Prokasy (Eds.), Classical conditioning II: Current research and theory. New York: AppletonCentury-Crofts. RumeIhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning internal representations by error propogation. In D. Rumelhart, & 1. McClelland (Eds.), Parallel distributed processing: Explorations in the microstructure of cognition (Vol. 1: Foundations). Cambridge, M.A.: MIT Press. Sutton, R. S. (1988). Learning to predict by the methods of temporal differences. Machine Learning, 3, 9-44. Sutton, R. S., & Barto, A. G. (1981). Toward a modem theory of adaptive networks: Expectation and prediction. Psychological Review, 88, 135-170. Sutton, R. S., & Barto, A. G. (1987). A temporal-difference model of classical conditioning. In Proceedings of the 9th Annual Conference of the Cognitive Science Society. Seattle, WA. Thompson, R. F. (1986). The neurobiology ofleaming and memory. Science, 233, 941-947. Werbos, P. (1974). Beyond regression: New tools for prediction and analysis in the behavioral sciences. Doctoral dissertation (Economics), Harvard University, Cambridge, Mass .. Widrow, B., & Hoff, M. E. (1960). Adaptive switching circuits. Institute of Radio Engineers, Western Electronic Show and Convention, Convention Record, 4,96-194. 203 Part II Application 

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Deterministic Annealing Variant of the EM Algorithm N aonori U eda Ryohei N alcano ueda@cslab.kecl.ntt.jp nakano@cslab.kecl.ntt.jp NTT Communication Science Laboratories Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02 Japan Abstract We present a deterministic annealing variant of the EM algorithm for maximum likelihood parameter estimation problems. In our approach, the EM process is reformulated as the problem of minimizing the thermodynamic free energy by using the principle of maximum entropy and statistical mechanics analogy. Unlike simulated annealing approaches, this minimization is deterministically performed. Moreover, the derived algorithm, unlike the conventional EM algorithm, can obtain better estimates free of the initial parameter values. 1 INTRODUCTION The Expectation-Maximization (EM) algorithm (Dempster, Laird & Rubin, 1977) is an iterative statistical technique for computing maximum likelihood parameter estimates from incomplete data. It has generally been employed to a wide variety of parameter estimation problems. Recently, the EM algorithm has also been successfully employed as the learning algorithm of the hierarchical mixture of experts (Jordan & Jacobs, 1993). In addition, it has been found to have some relationship to the learning of the Boltzmann machines (Byrne, 1992). This algorithm has attractive features such as reliable global convergence, low cost per iteration, economy of storage, and ease of programming, but it is not free from problems in practice. The serious practical problem associated with the algorithm 546 Naonori Veda, Ryohei Nakano is the local maxima problem. This problem makes the performance dependent on the initial parameter value. Indeed, the EM algorithm should be performed from as wide a choice of starting values as possible according to some ad hoc criterion. To overcome this problem, we adopt the principle of statistical mechanics. Namely, by using the principle of maximum entropy, the thermodynamic free energy is defined as an effective cost function that depends on the temperature. The maximization of log-likelihood is done by minimizing the cost function. Unlike simulated annealing (Geman & Geman, 1984) where stochastic search is performed on the given energy surface, this cost function is deterministically optimized at each temperature. Such deterministic annealing (DA) approach has been successfully adopted for vector quantization or clustering problems (Rose et al., 1992; Buhmann et al., 1993; Wong, 1993). Recently, Yuile et al.(Yuile, Stolorz, & Utans, 1994) have shown that the EM algorithm can be used in conjunction with the DA. In our previous paper, independent ofYuile's work, we presented a new EM algorithm with DA for mixture density estimation problems (Ueda & Nakano, 1994). The aim of this paper is to generalize our earlier work and to derive a DA variant of the general EM algorithm. Since the EM algorithm can be used not only for the mixture estimation problems but also for other parameter estimation problems, this generalization is expected to be of value in practice. 2 GENERAL THEORY OF THE EM ALGORITHM Suppose that a measure space Y of "unobservable data" exists corresponding to a measure space X of "observable data". An observable data sample ~(E X) with density p(~; @) is called incomplete and (~, y) with joint density p(~, y;@) is called complete, where y is an unobservable data sample 1 corresponding to~. Note that ~ E'R,n and y E 'R,m. @ is parameter of the density distribution to be estimated. Given incomplete data samples X = {~klk = 1"", N}, the goal of the EM algorithm is to compute the maximum-likelihood estimate of @ that maximizes the following log-likelihood function : N L(@;X) = Llogp(~k;@)' (1) 1:=1 by using the following complete data log-likelihood function: N Le(@;X) = I: logp(zk' Yk;@) ' (2) k=l In the EM algorithm, the parameter @ is iteratively estimated. Suppose that @(t) denotes the current estimate of @ after the tth iteration of the algorithm. Then @(t+1) at the next iteration is determined by the following two steps: 1 In such unsupervised learning as mixture problems, 1/ reduces to an integer value (1/ E {I, 2, ... , C}, where C is the number of components), indicating the component from which a data sample x originates. Deterministic Annealing Variant of the EM Algorithm 547 E-step: Compute the Q-function defined by the conditional expectation of the complete data log-likelihood given X and @(t): Q(@; @(t? ~ E{Lc(@;X)IX, @(t)}. M-step: Set @(t+l) equal to @ (3) which maximizes Q(@, @(t?. It has theoretically been shown that an iterative procedure for maximizing Q over will cause the likelihood L to monotonically increase, e.g., L(@(t+l? 2: L(@(t?. Eventually, L(@(t? converges to a local maximum. The EM algorithm is especially useful when the maximization of the Q-function can be more easily performed than that of L. @ By substituting Eq. 2 into Eq. 3, we have N Q(@;@{t? ~J = ... J{logp(~j:'Yj:;@)} k=l N IIp(Yjl~j;@(t))dYl . .. dYN j=l N ~ J {logp(~k' Yk; @)}p(Ykl~k; @(t?dYk' (4) k=l @ that maximizes Q(@; @(t? should satisfy aQla@ = 0, or equivalently, N ~ k=l Ja {a@ logp( ~k, Yk; @) }P(Yk I~k; @(t?dYk = o. (5) Here, p(Ykl~k' @(t? denotes the posterior and can be computed by the following Bayes rule: (6) It can be interpreted that the missing information is estimated by using the poste- rior. However, because the reliability of the posterior highly depends on the parameter @(t), the performance of the EM algorithm is sensitive to an initial parameter value @(O). This has often caused the algorithm to become trapped by some local maxima. In the next section, we will derive a new variant of the EM algorithm as an attempt at global maximization of the Q-function in the EM process. 3 3.1 DETERMINISTIC ANNEALING APPROACH DERIVATION OF PARAMETERIZED POSTERIOR Instead of the posterior given in Eq. 6, we introduce another posterior f(Ykl~k)' The function form of f will be specified later. Using f(Ykl~k)' we consider a new function instead of Q, say E, defined as: N E ~r ~ J {-logp(~k' Yk; @)}f(Yk l~k)dYk' k=l (7) 548 Naonori Ueda. Ryohei Nakano (N ote: E is always nonnegative.) One can easily see that (-E) is also the conditional expectation of the complete data log-likelihood but it differs from Q in that the expectation is taken with respect to !(lIk I:l:k) instead of the posterior given by -Q. Eq. 6. In other words, if !(Ykl:l:k) = P(lIkl:l:k; B(t?, then E = Since we do not have a priori knowledge about !(lIk I:l:k), we apply the principle of maximum entropy to specify it. That is, by maximizing the entropy given by: L J{log !(1I k I:l:k)} !(Yk I:l:k )dllb N H=- (8) k=1 J with respect to !, under the constraints of Eq. 7 and !dYk = 1, we can obtain the following Gibbs distribution: 1 !(lIk I:l:k) = exp{ -{J( -logp(:l:k.lIk; B?)}, (9) Z:l:" J where Z:I:" = exp{ -{J( -logp(:l:k, 11k; B?)}dllb and is called the partition function. The parameter {J is the Lagrange multiplier determined by the value E. From an analogy of the annealing, 1j {J corresponds to the "temperature". By simplifying Eq. 9, we obtain a new posterior parameterized by {J, !( I) lIk:l:k = p(:l:k, 11k; B)fJ Jp(:l:k, 11k; B)fJdllk . (10) Clearly, when {J = 1, !(lIkl:l:k) reduces to the original posterior given in Eq. 6. The effect of {J will be explained later. Since :1:1, ..? ,:l:N are identically and independently distributed observations, the partition function Zp(B) for X becomes Ok Z:l:". Therefore, II JP(lIbllk; B)Pdllk ? N Zp(B) = (11) k=l Once the partition function is obtained explicitly, using statistical mechanics analogy, we can define the free energy as an effective cost function that depends on the temperature: (12) At equilibrium, it is well known that a thermodynamic system settles into a configuration that minimizes its free energy. Hence, B should satisfy oFp(B)joB = O. It follows that (13) Interestingly, We have arrived at the same equation as the result ofthe maximization of the Q-function, except that the posterior P(lIk I:l:k; B(t? in Eq. 5 is replaced by !(lIk I:l:k). Deterministic Annealing Variant of the EM Algorithm 3.2 549 ANNEALING VARIANT OF THE EM ALGORITHM Let Qf3(@; @(I? be the expectation of the complete data log-likelihood by the parameterized posterior f(y" I~,,). Then, the following deterministic annealing variant of the EM algorithm can be naturally derived to maximize -Ff3(@). [Annealing EM (AEM) algorithm] 1. Set {3 +- {3min(O < {3min <t:: 1). 2. Arbitrarily choose an initial estimate @(O). Set t +- O. 3. Iterate the following two steps until convergence2 : E-step: Compute QfJ(@;@(t? N =~ "=1 J {logp(~", y,,; @)} I p(~" y . @(t?f3 , '" dy". p(~", y";@(t? f3 dy,, (14) M-step: Set @(t+l) equal to @ which maximizes Qf3(@; @(t?. 4. Increase {3. 5. If {3 < {3max, set t +- t + 1, and repeat from step 3; otherwise stop. One can see that in the proposed algorithm, an outer loop is added to the original EM algorithm for the annealing process. An important distinction to keep in mind is that unlike simulated annealing, the optimization in step 3 is deterministically performed at each {3. Now let's consider the effect of the posterior parameterization of Eq. 10. The annealing process begins at small {3 (high temperature). Clearly, at this time, since f(y"I~,,) becomes uniform, -Ff3(@) has only one global maximum. Hence, the maximum can be easily found. Then by gradually increasing {3, the influence of each ~" is gradually localized. At {3 > 0, function Qf3 will have several local maxima. However, at each step, it can be assumed that the new global maximum is close to the previous one. Hence, by this assumption, the algorithm can track the global maximum at each {3 while increasing {3. Clearly, when {3 1 the parameterized posterior coincides with the original one. Moreover, noting that -Fl(@) L(@), {3max ought be one. = = 4 Demonstration To visualize how the proposed algorithm works, we consider a simple onedimensional, two-component normal mixture problem. The mixture is given by p(x;m},m2) = 0.3kexp{-~(x - ml?} + 0.7*exp{-~(x - m2)2}. In this case, @ = (mb m2), y" E {I, 2}, and therefore, the joint density p(x, 1; ml, m2) = 0.3$ exp{ -~(x - ml)2}, while p(x, 2; ml, m2) 0.7 $ exp{ -~(x - m2)2}. = One hundred samples in total were generated from this mixture with ml = -2 and m2 2. Figure 1 shows contour plots of the -FfJ(ml,m2)/N surface. It is interesting to see how FfJ(m}, m2) varies with {3. One can see that a finer and truer structure emerges by increasing {3. Note that as explained before, when {3 1, = = 2When the sequence converges to a saddle point (e.g., when the Hessian matrix of -Fp(9) has at least one positive eigen value), a local line search in the direction of the eigen vector corresponding to the largest eigen value should be performed to escape the solution from the saddle point. 550 Naonori Ueda, Ryohei Nakano -4 -2 0 -2 2 (a) ~ .. 0.1 -4 -2 0 2 0 -4 2 -2 (b) ~ - 0.25 -4 4 ml -2 0 2 2 (c) ~ '" 0.3 -4 4 ml -2 0 (f) ~ (e) ~ .. 0.7 (d) ~ = 0.5 0 2 =1 Figure 1: Contour plots of -Fp(mhm2)fN surface. ("+" denotes a global maximum at each JJ.) -4 -2 0 2 (a) (~),m~~ = (4, -1) 4 ml -4 -2 0 (b) (m(~),m~~ 2 =(-2,-4) Figure 2: Trajectories for EM and AEM procedures. 4 ml Deterministic Annealing Variant of the EM Algorithm 551 -Fl(ml, m2) == L(mb m2). The maximum of -Fl (or L) occurs at m1 = -2.0 and m2 1.9. As initial points, (m~O),m~O? (4,-1),(-2,-4) were tested. Note that these points are close to a local maximum. Figure 2 shows how both algorithms converge from these starting points3. = = The original EM algorithm converges to (mp2), m~12? = (2.002, -1.687) (Figure (1.999, -1.696) (Figure 2(b?. In contrast, the pro2( a? and to (m~19), m~19? posed AEM algorithm converges to (m~43), m~43? = (-2.022,1.879) (Figure 2(a? and (m~43),m~43? = (-2.022,1.879) (Figure 2(b?. Since these starting points are close to the local maximum point, the original EM algorithm becomes trapped by the local maximum point, while the proposed algorithm successfully converges to the global maximum in both cases. = 5 Discussion A new view of the EM algorithm has recently been presented (Hathaway, 1986; Neal & Hinton, 1993). This view states that the EM steps can be regarded as a grouped version of the method of coordinate ascent of the following objective function: J ~f N N I: E (lIlo) {logp(:l:l:, 111:; @)} - I: E (lIlo) {logp(lIl:)} p p I: (15) I: over P(lIl:) and @. That is, the E-step corresponds to the maximization of J with respect to P(lIl:) with fixed @, while the M-step corresponds to that of J with respect to @ with fixed P(lIl:)' Neal & Hinton mentioned that apart from the sign change, J is analogous to the "free energy" well known in statistical physics. It is worth mentioning another interpretation of the derived parameterized posterior; i.e., it plays a central role in the AEM algorithm. Taking the logarithm on both sides of Eq. 10, we have (16) Moreover, taking the conditional expectation4 given :1:1:, summing over k, and using Eq. 12, we have N FfJ(@) = 1 N I: EJ(lIlol:l:lo){ -logp(:l:I:, 111:; @)}+ pI: EJ(lIlo 1:1:10) {log 1(111: 1:l:1:)}? (17) 1:=1 1:=1 = From Eqs. 7 and 8, Eq. 17 can be rewritten as FfJ(@) E - !H. Noting that l/f3 corresponds to the "temperature", one can see that this expression exactly agrees with the free energy, while Eq. 15 is completely without the "temperature". In other words, FfJ(@) can be interpreted as an annealing variant of -J. Clearly, when f3 = 1, FfJ(@) == -J. 3 Although the AEM procedure was actually performed for successive fJ (fJnew - fJ x 1.4), the results a.re superimposed on -Fl(ml, m2)/N surfa.ce for convenience. 4Since the LHS of Eq. 16 is independent of 111" it does not cha.nge a.fter expecta.tion. 552 Naonori Veda. Ryohei Nakano The proposed algorithm is also applicable to the learning of the (Generalized) Radial Basis Function (RBF, GRBF) networks. Indeed, Nowlan (Nowlan, 1990), for instance, proposes a maximum likelihood competitive learning algorithm for the RBF networks. In his study, "soft competition" and "hard competition" are eXperimentally compared and it is shown that the soft competition can give better performance. In our algorithm, on the other hand, the soft model exactly corresponds to the case f3 = 1, while the hard model corresponds to the case f3 -+ 00. Consequently, both models can be regarded as special cases in our algorithm. Acknowledgements We would like to thank Dr. Tsukasa Kawaoka, NTT CS labs, for his encouragement. References J. Buhmann & H. Kuhnel. (1993) Complexity optimized data clustering by competitive neural networks. Neural Computation, 5:75-88. W. Byrne. (1992) Alternating minimization and Boltzmann machine learning. IEEE Trans. Neural Networks, 3:612-620. A. P. Dempster, N. M. Laird & D. B. Rubin. (1977) Maximum-likelihood from incomplete data via the EM algorithm. J. Royal Statist. Soc. Ser. B (methodological),39:1-38. S. Geman & D. Geman. (1984) Stochastic relaxation, Gibbs distribution and the Baysian restortion in images. IEEE Trans. Pattern Anal. Machine Intel/" 6,6:721741. R. J. Hathaway. (1986) Another interpretation of the EM algorithm for mixture distributions. Statistics fj Probability Leiters, 4: 53-56. M. 1. Jordan & R. A. Jacob. (1993) Hierarchical mixtures of experts and the EM algorithm. MIT Dept. of Brain and Cognitive Science preprint. R. M. Neal & G. E. Hinton. (1993) A new view of the EM algorithm that justifies incremental and other variants. submitted to Biometrika. S. J. Nowlan. (1990) Maximum likelihood competitve learning. in D. S. Touretzky et al. eds., Advances in Neural Information Systems 2, Morgan Kaufmann. 574-582. K. Rose, E. Gurewitz & G. C. Fox. (1992) Vector quantization by deterministic annealing. IEEE Trans. Information Theory, 38,4:1249-1257. N. Ueda & R. Nakano. (1994) Mixture density estimation via EM algorithm with deterministic annealing. in proc. IEEE Neural Networks for Signal Processing, 69-77. Y. Wong. (1993) Clustering data by melting. Neural Computation, 5:89-104. A. 1. Yuille, P. Stolorz & J. Utans. (1994) Statistical physics, mixtures of distributions, and the EM algorithm. Neural Computation, 6:334-340. 

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Patterns of damage in neural networks: The effects of lesion area, shape and number Eytan Ruppin and James A. Reggia ? Department of Computer Science A.V. Williams Bldg. University of Maryland College Park, MD 20742 ruppin@cs.umd.edu reggia@cs.umd.edu Abstract Current understanding of the effects of damage on neural networks is rudimentary, even though such understanding could lead to important insights concerning neurological and psychiatric disorders. Motivated by this consideration, we present a simple analytical framework for estimating the functional damage resulting from focal structural lesions to a neural network. The effects of focal lesions of varying area, shape and number on the retrieval capacities of a spatially-organized associative memory. Although our analytical results are based on some approximations, they correspond well with simulation results. This study sheds light on some important features characterizing the clinical manifestations of multi-infarct dementia, including the strong association between the number of infarcts and the prevalence of dementia after stroke, and the 'multiplicative' interaction that has been postulated to occur between Alzheimer's disease and multi-infarct dementia. *Dr. Reggia is also with the Department of Neurology and the Institute of Advanced Computer Studies at the University of Maryland. 36 1 Eytan Ruppin, James A. Reggia Introduction Understanding the response of neural nets to structural/functional damage is important for a variety of reasons, e.g., in assessing the performance of neural network hardware, and in gaining understanding of the mechanisms underlying neurological and psychiatric disorders. Recently, there has been a growing interest in constructing neural models to study how specific pathological neuroanatomical and neurophysiological changes can result in various clinical manifestations, and to investigate the functional organization of the symptoms that result from specific brain pathologies (reviewed in [1, 2]). In the area of associative memory models specifically, early studies found an increase in memory impairment with increasing lesion severity (in accordance with Lashley's classical 'mass action' principle), and showed that slowly developing lesions have less pronounced effects than equivalent acute lesions [3]. Recently, it was shown that the gradual pattern of clinical deterioration manifested in the majority of Alzheimer's patients can be accounted for, and that different synaptic compensation rates can account for the observed variation in the severity and progression rate of this disease [4]. However, this past work is limited in that model elements have no spatial relationships to one another (all elements are conceptually equidistant). Thus, as there is no way to represent focal (localized) damage in such networks, it has not been possible to study the functional effect of focal lesions and to compare them with that caused by diffuse lesions. The limitations of past work led us to use spatially-organized neural network for studying the effects of different types of lesions (we use the term lesion to mean any type of structural and functional damage inflicted on an initially intact neural network). The elements in our model, which can be thought of as representing neurons, or micro-columnar units, form a 2-dimensional array (whose edges are connected, forming a torus to eliminate edge effects), and each unit is connected primarily to its nearby neighbors, as is the case in the cortex [5]. It has recently been shown that such spatially-organized attractor networks can function reasonably well as associative memory devices [6]. This paper presents the first detailed analysis of the effects of lesions of various size, form and number on the memory performance of spatially-organized attractor neural networks. Assuming that these networks are a plausible model of some frontal and associative cortical areas (see, e.g., [7]), our results shed light on the clinical progress of disorders such as stroke and dementia. In the next section, we derive a theoretical framework that characterizes the effects of focal lesions on an associative network's performance. This framework, which is formulated in very general terms, is then examined via simulations in Section 3, which show a remarkable quantitative fit with the theoretical predictions, and are compared with simulations examining performance with diffuse damage. Finally, the clinical significance of our results is discussed in Section 4. 2 Analyzing the effects of focal lesions Our analysis pertains to the case where in the pre-damaged network, all units have an approximately similar average level of activity 1. A focal structural lesion IThis is true in general for associative memory networks, when the activity of each unit is averaged over a sufficiently long time span. Patterns of Damage ill NeuraL Networks 37 (anatomical lesion), denoting the area of damage and neuronal death, is modeled by clamping the activity of the lesioned units to zero. As a result of this primary lesion, the activity of surrounding units may be decreased, resulting in a secondary area of functional lesion, as illustrated in Figure 1. We are primarily interested in large focal lesions , where the area s of the lesion is significantly greater than the local neighborhood region from which each unit receives its inputs. Throughout our analysis we shall hold the working assumption that, traversing from the border of the lesion outwards, the activity of units gradually rises from zero until it reaches its normal, predamaged levels, at some distance d from the lesion's border (see Figure 1). As s is large and d is determined by local interactions on the borders of the structural lesion, we may reasonably assume that the value of d is independent of the lesion size, and depends primarily on the specific network characteristics , such as it architecture, dynamics, and memory load. Figure 1: A sketch of a structural (dark shading) and surrounding functional (light shading) rectangular lesion. Let the intact baseline performance level of the network be denoted as P(O), and let the network size be A. The network 's performance denotes how accurately it retrieves the correct memorized patterns given a set of input cues, and is defined formally below . A structural lesion of area s (dark shading in Figure 1), causing a functional lesion of area ~s (light shading in Figure 1), will then result In a performance level of approximately P(s) = P(O) [A - (s + ~3)l + Pt:..~3 = P(O) _ (~P~3)/(A _ s) , (1) A-s where Pt:.. denotes the average level of performance over ~3 and ~P = P(O) - Pt:.. . P( s) hence reflects the performance level over the remaining viable parts of the network , discarding the structurally damaged region. Bearing these definitions in mind, a simple analysis shows that the effect of focal lesions is governed by the following rules. = 71-r2 . ~3' the area of Consider a symmetric, circular structural lesion of size s functional damage following such a lesion is then (assuming large lesions and hence VS> d) Rule 1: (2) 38 Eytan Ruppin, James A. Reggia 1.0 .--~----r-------r------.--, 0.8 -k.1 --- k=5 k=25 0.2 0.0 '---~----'"---~-~---'--' 0.0 500.0 1000.0 1500.0 Lesion size Figure 2: Theoretically predicted network performance as a function of a single focal structural lesion's size (area): analytic curves obtained for different k values; A = 1600. and ~ P(O) - Ak..JS , (3) -s for some constant k = yi4;db..P. Thus, the area of functional damage surrounding a single focal structural lesion is proportional to the square root of the structural lesion's area. Some analytic performancejlesioning curves (for various k values) are illustrated in Figure 2. Note the different qualitative shape of these curves as a function of k. Letting x = sjA be the fraction of structural damage, we have pes) P(x) ~ P(O) _ k.JX _1_ 1-x.,jA , (4) that is, the same fraction x of damage results in less performance decrease in larger networks. This surprising result testifies to the possible protective value of having functional 'modular' cortical networks of large size. Expressions 3 and 4 are valid also when the structural lesion has a square shape. To study the effect of the structural lesion's shape, we consider the area b.. 8 [n] of a functional lesion resulting from a rectangular focal lesion of size s = a . b (see Figure 1), where, without loss of generality, n = ajb ~ 1. Then, for large n, we find that the functional damage of a rectangular structural lesion of fixed size increases as its shape is more elongated, following Rule 2: (5) and pes) ~ P(O) _ k.,foS 2(A - s) (6) Patterns of Damage in Neural Networks 39 To study the effect of the number of lesions, consider the area .6. 3 m of a functional lesion composed of m focal rectangular structural lesions (with sides a = n? b), each of area s/m. We find that the functional damage increases with the number offocal sub-lesions (while total structural lesion area is held constant), according to Rule 3: (7) and ~ P(O) _ pes) k.;mns (8) 2(A - s) While Rule 3 presents a lower bound on the functional damage which may actually be significantly larger and involves no approximations, Rule 2 presents an upper bound on the actual functional damage. As we shall show in the next section, the number of lesions actually affects the network performance significantly more than its precise shape. 3 Numerical Simulation Results We now turn to examine the effect of lesions on the performance of an associative memory network via simulations. The goal of these simulations is twofold. First, to examine how accurately the general but approximate theoretical results presented above describe the actual performance degradation in a specific associative network. Second, to compare the effects of focal lesions to those of diffuse ones, as the effect of diffuse damage cannot be described as a limiting case within the framework of our analysis. Our simulations were performed using a standard Tsodyks-Feigelman attractor neural network [8]. This is a Hopfield-like network which has several features which make it more biologically plausible [4], such as low activity and non-zero positive thresholds. In all the experiments, 20 sparse random {O, I} memory patterns (with a fraction of p ~ 1 of l's) were stored in a network of N = 1600 units, placed on a 2-dimensionallattice. The network has spatially organized connectivity, where each unit has 60 incoming connections determined randomly with a Gaussian probability <J>(z) = J1/27rexp( _z2 /2(1'2), where z is the distance between two units in the array. When (1' is small, each unit is connected primarily to its nearby neighbors. As in [4], the cue input patterns are presented via an external field of magnitude e = 0.035, and the noise level is T = 0.005. The performance of the network is measured (over the viable, non-lesioned units) by the standard overlap measure which denotes the similarity between the final state S the network converges to and the memory pattern ~!l that was cued in that trial, defined by m!l(t) = (1 P ~ N )N P I:(~r - p)Si(t) . (9) i=l In all simulations we report the average overlap achieved over 100 trials. We first studied the network's performance at various (1' values. Figure 3a displays how the performance of the network degrades when diffuse structural lesions of increasing size are inflicted upon it (i.e., randomly selected units are clamped to zero), while Figure 3b plots the performance as a function of the size of a single square-shaped focal lesion. As is evident, spatially-organized connectivity enables 40 Eytan Ruppin, James A. Reggia the network to maintain its memory retrieval capacities in face of focal lesions of considerable size. Diffuse lesions are always more detrimental than single focal lesions of identical size. Also plotted in Figure 3b is the analytical curve calculated via expression (3) (with k = 5), which shows a nice fit with the actual performance of the spatially-connected network parametrized by (J' = 1. Concentrating on the study of focal lesions in a spatially-connected network, we adhere to the values (J' = 1 and k = 5 hereafter, and compare the analytical and numerical results. With these values, the analytical curves describing the performance of the network as a function of the fraction of the network lesioned (obtained using expression 4) are similar to the corresponding numerical results. (a) (b) 1.0 r-----,----~-------, 1.0 , -- - - - , - - - - - r - -- - - - , .... ~ .... 0.8 .... ~...: ~.:..-~, - -- ",gna_1 signa =3 "'gna= 10 - - - "'gnam30 ',~''', ~-...~. ; - .. -\ 0.8 0 .6 0 .6 , \ \ \', \ \ \ \ \ \ \ ".-.\ '. \ \ . \ \ , '. \ 0 .4 0.4 0.2 0.2 \ \ \ - - - sigma =1 \ . . . , sigma _3 \ sigma -10 \ - - - sigma =30 " Analytical .osuns, k = 5 " '. \ , \ '.\ , ....................... '0.0 L-_~ 0 .0 _ _'___~_~ 500.0 _ _ _---' 1000.0 Lesion sIZe 1500.0 0.0 ':--~__;:::'::_:_-~-::-:':c_:__---= 0 .0 500.0 1000.0 1500.0 Lesion size Figure 3: Network performance as a function of lesion size: simulation results obtained in four different networks , each characterized by a distinct distribution of spatially-organized connectivity. (a) Diffuse lesions. (b) Focal lesions. To examine Rule 2, a rectangular structural lesion of area s = 300 was induced in the network. As shown in Figure 4a, as the ratio n between the sides is increased, the network's performance further decreases, but this effect is relatively mild. The markedly stronger effect of varying the lesion number (described by Rule 3) is demonstrated in figure 4b, which shows the effect of multiple lesions composed of 2,4,8 and 16 separate focal lesions. For comparison, the performance achieved with a diffuse lesion of similar size is plotted on the 20'th x-ordinate. It is interesting to note that a sufficiently large multiple focal lesion (s = 512) can cause a larger performance decrease than a diffuse lesion of similar size. That is, at some point, when the size of each individual focal lesion becomes small in relation to the spread of each unit's connectivity, our analysis looses its validity, and Rule 3 ceases to hold. Patterns of Damage in Neural Networks (a) 41 (b) 0.90 ,---'-~-"-----'----r-----, <>--<> SlrnuIeIion "'1U~s - . - Analytical reou~ 0.930 0.910 lii <>--tI Sinulation (s D 256) ...... " Sinulation (s ? 512) - - - - Analytic (s =256) - . - Analytic (s ? 512) 0.80 " ~ 0.890 " , " " 0.80 0.870 0.850 '---~-'---'----:"-~~-~-----' 0.0 1.0 2.0 3.0 4.0 5.0 6 .0 Rectangular ratio n 0.50 '----'----"------'-~--'-----' 0 .0 5.0 10.0 15.0 20.0 25.0 No. 01 sub-lesions m Figure 4: Network performance as a function of focal lesion shape (a) and number (b). Both numerical and analytical results are displayed . In Figure 4b, the x-ordinate denotes the number of separate sub-lesions (1,2,4,8,16) , and, for comparison, the performance achieved with a diffuse lesion of similar size is plotted on the 20'th x-ordinate. 4 Discussion We have presented a simple analytical framework for studying the effects of focal lesions on the functioning of spatially organized neural networks . The analysis presented is quite general and a similar approach could be adopted to investigate the effect offocallesions in other neural models. Using this analysis, specific scaling rules have been formulated describing the functional effects of structural focal lesions on memory retrieval performance in associative attractor networks. The functional lesion scales as the square root of the size of a single structural lesion, and the form of the resulting performance curve depends on the impairment span d. Surprisingly, the same fraction of damage results in significantly less performance decrease in larger networks, pointing to their relative robustness. As to the effects of shape and number, elongated structural lesions cause more damage than more symmetrical ones. However, the number of sub-lesions is the most critical factor determining the functional damage and performance decrease in the model. Numerical studies show that in some conditions multiple lesions can damage performance more than diffuse damage, even though the amount of lost innervation is always less in a multiple focal lesion than with diffuse damage. Beyond its computational interest, the study of the effects of focal damage on the performance of neural network models can lead to a better understanding of functional impairments accompanying focal brain lesions. In particular, we are interested in multi-infarct dementia, a frequent cause of dementia (chronic deterioration of cognitive and memory capacities) characterized by a series of multiple, aggregat- 42 Eytan Ruppin, James A. Reggia ing focal lesions. Our results indicate a significant role for the number of infarcts in determining the extent of functional damage and dementia in multi-infarct disease. In our model, multiple focal lesions cause a much larger deficit than their simple 'sum', i.e., a single lesion of equivalent total size. This is consistent with clinical studies that have suggested the main factors related to the prevalence of dementia after stroke to be the infarct number and site, and not the overall infarct size, which is related to the prevalence of dementia in a significantly weaker manner [9, 10]. Our model also offers a possible explanation to the 'multiplicative' interaction that has been postulated to occur between co-existing Alzheimer and multi-infarct dementia [10], and to the role of cortical atrophy in increasing the prevalence of dementia after stroke; in accordance with our model, it is hypothesized that atrophic degenerative changes will lead to an increase in the value of d (and hence of k) and increase the functional damage caused by a lesion of given structural size. This hypothesis, together with a detailed study of the effects of the various network parameters on the value of d, are currently under further investigation. Acknowledgements This research has been supported by a Rothschild Fellowship to Dr. Ruppin and by Awards NS29414 and NS16332 from NINDS. References [1] J. Reggia, R. Berndt, and L. D'Autrechy. Connectionist models in neuropsychology. In Handbook of Neuropsychology, volume 9. 1994, in press. [2] E. Ruppin. Neural modeling of psychiatric disorders. Network: Computation in Neural Systems, 1995. Invited review paper, to appear. [3] J .A. Anderson. Cognitive and psychological computation with neural models. IEEE Trans. on Systems,Man, and Cybernetics, SMC-13(5):799-815, 1983. [4] D. Horn, E. Ruppin, M. Usher, and M. Herrmann. Neural network modeling of memory deterioration in alzheimer's disease. Neural Computation, 5:736-749, 1993. [5] A. M. Thomson and J. Deuchars. Temporal and spatial properties of local circuits in the neocortex. Trends in neuroscience, 17(3):119-126, 1994. [6] J .M. Karlholm. Associative memories with short-range higher order couplings. Neural Networks, 6:409-421, 1993. [7] F.A.W. Wilson, S.P.O Scalaidhe, and P.S. Goldman-Rakic. Dissociation of object and spatial processing domains in primate prefrontal cortex. Science, 260:1955-1958,1993. [8] M.V. Tsodyks and M.V. Feigel'man. The enhanced storage capacity in neural networks with low activity level. Europhys. Lett., 6:101 - 105, 1988. [9] T.K. Tatemichi, M.A. Foulkes, J.P. Mohr, J.R. Hewitt, D. B. Hier, T .R. Price, and P.A. Wolf. Dementia in stroke survivors in the stroke data bank cohort. Stroke, 21:858-866, 1990. [10] T. K. Tatemichi. How acute brain failure becomes chronic: a view of the mechanisms of dementia related to stroke. Neurology, 40:1652-1659, 1990. 

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Phase-Space Learning Fu-Sheng Tsung Chung Tai Ch'an Temple 56, Yuon-fon Road, Yi-hsin Li, Pu-li Nan-tou County, Taiwan 545 Republic of China Garrison W. Cottrell? Institute for Neural Computation Computer Science & Engineering University of California, San Diego La Jolla, California 92093 Abstract Existing recurrent net learning algorithms are inadequate. We introduce the conceptual framework of viewing recurrent training as matching vector fields of dynamical systems in phase space. Phasespace reconstruction techniques make the hidden states explicit, reducing temporal learning to a feed-forward problem. In short, we propose viewing iterated prediction [LF88] as the best way of training recurrent networks on deterministic signals. Using this framework, we can train multiple trajectories, insure their stability, and design arbitrary dynamical systems. 1 INTRODUCTION Existing general-purpose recurrent algorithms are capable of rich dynamical behavior. Unfortunately, straightforward applications of these algorithms to training fully-recurrent networks on complex temporal tasks have had much less success than their feedforward counterparts. For example, to train a recurrent network to oscillate like a sine wave (the "hydrogen atom" of recurrent learning), existing techniques such as Real Time Recurrent Learning (RTRL) [WZ89] perform suboptimally. Williams & Zipser trained a two-unit network with RTRL, with one teacher signal. One unit of the resulting network showed a distorted waveform, the other only half the desired amplitude. [Pea89] needed four hidden units. However, our work demonstrates that a two-unit recurrent network with no hidden units can learn the sine wave very well [Tsu94]. Existing methods also have several other ?Correspondence should be addressed to the second author: gary@cs.ucsd.edu 482 Fu-Sheng Tsung, Garrison W. Cottrell limitations. For example, networks often fail to converge even though a solution is known to exist; teacher forcing is usually necessary to learn periodic signals; it is not clear how to train multiple trajectories at once, or how to insure that the trained trajectory is stable (an attractor). In this paper, we briefly analyze the algorithms to discover why they have such difficulties, and propose a general solution to the problem. Our solution is based on the simple idea of using the techniques of time series prediction as a methodology for recurrent network training. First, by way of introducing the appropriate concepts, consider a system of coupled autonomous l first order network equations: FI (Xl (t), X2(t), . .. , Xn (t)) F2(XI(t), X2(t),? ? ?, Xn(t)) or, in vector notation, X(t) = F(X) where XCt) = (XICt), X2(t),?? ., xn(t)) The phase space is the space of the dependent variables (X), it does not include t, while the state space incorporates t. The evolution of a trajectory X(t) traces out a phase curve, or orbit, in the n-dimensional phase space of X . For low dimensional systems (2- or 3-D), it is easy to visualize the limit sets in the phase space: a fixed point and a limit cycle become a single point and a closed orbit (closed curve), respectively. In the state space they become an infinite straight line and a spiral. F(X) defines the vector field of X, because it associates a vector with each point in the phase space of X whose direction and magnitude determines the movement of that point in the next instant of time (by definition, the tangent vector). 2 ANALYSIS OF CURRENT APPROACHES To get a better understanding of why recurrent algorithms have not been very effective, we look at what happens during training with two popular recurrent learning techniques: RTRL and back propagation through time (BPTT). With each, we illustrate a different problem, although the problems apply equally to each technique. RTRL is a forward-gradient algorithm that keeps a matrix of partial derivatives of the network activation values with respect to every weight. To train a periodic trajectory, it is necessary to teacher-force the visible units [WZ89], i.e., on every iteration, after the gradient has been calculated, the activations of the visible units are replaced by the teacher. To see why, consider learning a pair of sine waves offset by 90?. In phase space, this becomes a circle (Figure la). Initially the network 1 Autonomous means the right hand side of a differential equation does not explicitly reference t, e.g. dx/dt = 2x is autonomous, even though x is a function oft, but dx/dt = 2x+t is not. Continuous neural networks without inputs are autonomous. A nonautonomous system can always be turned into an autonomous system in a higher dimension. Phase-Space Learning 483 a b Figure 1: Learning a pair of sine waves with RTRL learning. (a) without teacher forcing, the network dynamics (solid arrows) take it far from where the teacher (dotted arrows) assumes it is, so the gradient is incorrect. (b) With teacher forcing, the network's visible units are returned to the trajectory. (thick arrows) is at position Xo and has arbitrary dynamics. After a few iterations, it wanders far away from where the teacher (dashed arrows) assumes it to be. The teacher then provides an incorrect next target from the network's current position. Teacher-forcing (Figure 1b), resets the network back on the circle, where the teacher again provides useful information. However, if the network has hidden units, then the phase space of the visible units is just a projection of the actual phase space of the network, and the teaching signal gives no information as to where the hidden units should be in this higherdimensional phase space. Hence the hidden unit states, unaltered by teacher forcing , may be entirely unrelated to what they should be. This leads to the moving targets problem. During training, every time the visible units re-visit a point, the hidden unit activations will differ, Thus the mapping changes during learning. (See [Pin88, WZ89] for other discussions of teacher forcing.) With BPTT, the network is unrolled in time (Figure 2). This unrolling reveals another problem: Suppose in the teaching signal, the visible units' next state is a non-linearly separable function of their current state. Then hidden units are needed between the visible unit layers, but there are no intermediate hidden units in the unrolled network. The network must thus take two time steps to get to the hidden units and back. One can deal with this by giving the teaching signal every other iteration, but clearly, this is not optimal (consider that the hidden units must "bide their time" on the alternate steps).2 The trajectories trained by RTRL and BPTT tend to be stable in simulations of simple tasks [Pea89, TCS90], but this stability is paradoxical. Using teacher forcing, the networks are trained to go from a point on the trajectory, to a point within the ball defined by the error criterion f (see Figure 4 (a)). However, after learning, the networks behave such that from a place near the trajectory, they head for the trajectory (Figure 4 (b)). Hence the paradox. Possible reasons are: 1) the hidden unit moving targets provide training off the desired trajectory, so that if the training is successful, the desired trajectory is stable; 2) we would never consider the training successful if the network "learns" an unstable trajectory; 3) the stable dynamics in typical situations have simpler equations than the unstable dynamics [N ak93]. To create an unstable periodic trajectory would imply the existence of stable regions both inside and outside the unstable trajectory ; dynamically this is 2 At NIPS , 0 delay connections to the hidden units were suggested, which is essentially part of the solution given here. 484 Fu-Sheng Tsung, Garrison W. Cottrell ~------------. Figure 2: A nonlinearly separable mapping must be computed by the hidden units (the leftmost unit here) every other time step. Figure 3: The network used for iterated prediction training. Dashed connections are used after learning. " a b Figure 4: The paradox of attractor learning with teacher forcing. (a) During learning, the network learns to move from the trajectory to a point near the trajectory. (b) After learning, the network moves from nearby points towards the trajectory. more complicated than a simple periodic attractor. In dynamically complex tasks, a stable trajectory may no longer be the simplest solution, and stability could be a problem. In summary, we have pointed out several problems in the RTRL (forward-gradient) and BPTT (backward-gradient) classes of training algorithms: 1. Teacher forcing with hidden units is at best an approximation, and leads to the moving targets problem. 2. Hidden units are not placed properly for some tasks. 3. Stability is paradoxical. 3 PHASE-SPACE LEARNING The inspiration for our approach is prediction training [LF88], which at first appears similar to BPTT, but is subtly different. In the standard scheme, a feedforward network is given a time window, a set of previous points on the trajectory to be learned, as inputs. The output is the next point on the trajectory. Then, the inputs are shifted left and the network is trained on the next point (see Figure 3). Once the network has learned, it can be treated as recurrent by iterating on its own predictions. The prediction network differs from BPTT in two important ways. First, the visible units encode a selected temporal history of the trajectory (the time window) . The point of this delay space embedding is to reconstruct the phase space of the underlying system. [Tak81] has shown that this can always be done for a deterministic system. Note that in the reconstructed phase space, the mapping from one Phase-Space Learning 485 YI+I -------,--........r;.-.-.-. ...... ... ... .'"'"'"'" , ??? a b Figure 5: Phase-space learning. (a) The training set is a sample of the vector field. (b) Phase-space learning network. Dashed connections are used after learning. point to the next (based on the vector field) is deterministic. Hence what originally appeared to be a recurrent network problem can be converted into an entirely feed forward problem. Essentially, the delay-space reconstruction makes hidden states visible, and recurrent hidden units unnecessary. Crucially, dynamicists have developed excellent reconstruction algorithms that not only automate the choices of delay and embedding dimension but also filter out noise or get a good reconstruction despite noise [FS91, Dav92, KBA92]. On the other hand, we clearly cannot deal with non-deterministic systems by this method. The second difference from BPTT is that the hidden units are between the visible units, allowing the network to produce nonlinearly separable transformations of the visible units in a single iteration. In the recurrent network produced by iterated prediction, the sandwiched hidden units can be considered "fast" units with delays on the input/output links summing to 1. Since we are now lear~ing a mapping in phase space, stability is easily ensured by adding additional training examples that converge towards the desired orbit. 3 We can also explicitly control convergence speed by the size and direction of the vectors. Thus, phase-space learning (Figure 5) consists of: (1) embedding the temporal signal to recover its phase space structure, (2) generating local approximations of the vector field near the desired trajectory, and (3) functional approximation of the vector field with a feedforward network. Existing methods developed for these three problems can be directly and independently applied to solve the problem. Since feedforward networks are universal approximators [HSW89], we are assured that at least locally, the trajectory can be represented. The trajectory is recovered from the iterated output of the pre-embedded portion of the visible units. Additionally, we may also extend the phase-space learning framework to also include inputs that affect the output of the system (see [Tsu94] for details).4 In this framework, training multiple attractors is just training orbits in different parts of the phase space, so they simply add more patterns to the training set. In fact, we can now create designer dynamical systems possessing the properties we want, e.g., with combinations of fixed point, periodic, or chaotic attractors. 3The common practice of adding noise to the input in prediction training is just a simple minded way of adding convergence information. 4Principe & Kuo(this volume) show that for chaotic attractors, it is better to treat this as a recurrent net and train using the predictions. 486 Fu-Sheng Tsung, Garrison W. Cottrell 0.5 -0.5 Q -0.5 0 0.5 Figure 6: Learning the van der Pol oscillator. ( a) the training set. (b) Phase space plot of network (solid curve) and teacher (dots). (c) State space plot. As an example, to store any number of arbitrary periodic attractors Zi(t) with periods 11 in a single recurrent network, create two new coordinates for each Zi(t), (Xi(t),Yi(t)) = (sin(*t),cos(*t)), where (Xi,Yi) and (Xj,Yj) are disjoint circles for i 'I j. Then (Xi, Yi, Zi) is a valid embedding of all the periodic attractors in phase space, and the network can be trained. In essence, the first two dimensions form "clocks" for their associated trajectories. 4 SIMULATION RESULTS In this section we illustrate the method by learning the van der Pol oscillator (a much more difficult problem than learning sine waves), learning four separate periodic attractors, and learning an attractor inside the basin of another attractor. 4.1 LEARNING THE VAN DER POL OSCILLATOR The van der Pol equation is defined by: We used the values 0.7, 1, 1 for the parameters a, b, and w, for which there is a global periodic attractor (Figure 6). We used a step size of 0.1, which discretizes the trajectory into 70 points. The network therefore has two visible units. We used two hidden layers with 20 units each, so that the unrolled, feedforward network has a 2-20-20-2 architecture. We generated 1500 training pairs using the vector field near the attractor. The learning rate was 0.01, scaled by the fan-in, momentum was 0.75, we trained for 15000 epochs. The order of the training pairs was randomized. The attractor learned by the network is shown in (Figure 6 (b)). Comparison of the temporal trajectories is shown in Figure 6 (c); there is a slight frequency difference. The average MSE is 0.000136. Results from a network with two layers of 5 hidden units each with 500 data pairs were similar (MSE=0.00034). 4.2 LEARNING MULTIPLE PERIODIC ATTRACTORS [Hop82] showed how to store multiple fixed-point at tractors in a recurrent net. [Bai91] can store periodic and chaotic at tractors by inverting the normal forms of these attractors into higher order recurrent networks. However, traditional recurrent training offers no obvious method of training multiple attractors. [DY89] were able Phase.Space Learning 487 _.0.1. Il.6, 0.63. 0.7 'I'---:.ru"='"""--'O~--:O-=-"---! .I'--::.ru-;---;;--;;-:Oj---! 8 A .ru 0 OJ '1'--::4U~-::-0--;Oj~-! F E 1.--------, 100 1OO D 300 400 ~~ .1 0 50 100 150 1OO ~ 300 H Figure 7: Learning mUltiple attractors. In each case, a 2-20-20-2 network using conjugate gradient is used. Learning 4 attractors: (A) Training set. (B) Eight trajectories of the trained network. (C) Induced vector field of the network. There are five unstable fixed points. (D) State space behavior as the network is "bumped" between attractors. Learning 2 attractors, one inside the other: (E) Training set. (F) Four trajectories ofthe trained network. (G) Induced vector field of the network. There is an unstable limit cycle between the two stable ones. (H) State space behavior with a "bump". to store two limit cycles by starting with fixed points stored in a Hopfield net, and training each fixed point locally to become a periodic attractor. Our approach has no difficulty with multiple attractors. Figure 7 (A-D) shows the result of training four coexisting periodic attractors, one in each quadrant of the two-dimensional phase space. The network will remain in one of the attractor basins until an external force pushes it into another attractor basin. Figure 7 (E-H) shows a network with two periodic attractors, this time one inside the other. This vector field possess an unstable limit cycle between the two stable limit cycles. This is a more difficult task, requiring 40 hidden units, whereas 20 suffice for the previous task (not shown). 5 SUMMARY We have presented a phase space view of the learning process in recurrent nets. This perspective has helped us to understand and overcome some of the problems of using traditional recurrent methods for learning periodic and chaotic attractors. Our method can learn multiple trajectories, explicitly insure their stability, and avoid overfitting; in short, we provide a practical approach to learning complicated temporal behaviors. The phase-space framework essentially breaks the problem into three sub-problems: (1) Embedding a temporal signal to recover its phase space structure, (2) generating local approximations of the vector field near the desired trajectory, and (3) functional approximation in feedforward networks. We have demonstrated that using this method, networks can learn complex oscillations and multiple periodic attractors. 488 Fu-Sheng Tsung, Garrison W. Cottrell Acknowledgements This work was supported by NIH grant R01 MH46899-01A3. Thanks for comments from Steve Biafore, Kenji Doya, Peter Rowat, Bill Hart, and especially Dave DeMers for his timely assistance with simulations. References W. Baird and F. Eeckman. Cam storage of analog patterns and continuous sequences with 3n 2 weights. In R.P. Lippmann, J .E. Moody, and D.S. Touretzky, editors, Advances in Neural Information Processing Systems, volume 3, pages 91-97, 1991. Morgan Kaufmann, San Mateo. [Dav92] M. Davies. Noise reduction by gradient descent. International Journal of Bifurcation and Chaos, 3:113-118, 1992. K. Doya and S. Yoshizawa. Memorizing oscillatory patterns in the analog [DY89] neuron network. In IJCNN, Washington D.C., 1989. IEEE. J.D. Farmer and J.J. Sidorowich. Optimal shadowing and noise reduction. [FS91] Physica D, 47:373-392, 1991. [Hop82] J.J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences, USA, 79, 1982. [HSW89] K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2:359-366, 1989. [KBA92] M.B. Kennel, R. Brown, and H. Abarbanel. Determining embedding dimension for phase-space reconstruction using a geometrical construction. Physical Review A, 45:3403-3411, 1992. [LF88] A. Lapedes and R. Farber. How neural nets work . In D.Z. Anderson, editor, Neural Information Processing Systems, pages 442-456, Denver 1987, 1988. American Institute of Physics, New York. [N ak93] Hiroyuki Nakajima. A paradox in learning trajectories in neural networks. Working paper, Dept. of EE II, Kyoto U., Kyoto, JAPAN, 1993. [Pea89] B.A. Pearlmutter. Learning state space trajectories in recurrent neural networks . Neural Computation, 1:263-269, 1989. [Pin88] F.J . Pineda. Dynamics and architecture for neural computation. Journal of Complexity, 4:216-245, 1988. [Tak81] F. Takens. Detecting strange attractors in turbulence. In D.A. Rand and L.-S. Young, editors, Dynamical Systems and Turbulence, volume 898 of Lecture Notes in Mathematics, pages 366-381, Warwick 1980, 1981. Springer-Verlag, Berlin. [TCS90] F-S. Tsung, G . W. Cottrell, and A. I. Selverston. Some experiments on learning stable network oscillations. In IJCNN, San Diego, 1990. IEEE. [Tsu94] F-S. Tsung. Modelling Dynamical Systems with Recurrent Neural Networks. PhD thesis, University of California, San Diego, 1994. [WZ89] R.J. Williams and D. Zipser. A learning algorithm for continually running fully recurrent neural networks . Neural Computation, 1:270-280, 1989. [Bai91] 

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Synchrony and Desynchrony in Neural Oscillator Networks DeLiang Wang David Terman Department of Computer and Information Science and Center for Cognitive Science The Ohio State University Columbus, Ohio 43210, USA dwang@cis.ohio-state.edu Department of Mathematics The Ohio State University Columbus, Ohio 43210, USA terman@math.ohio-state.edu Abstract An novel class of locally excitatory, globally inhibitory oscillator networks is proposed. The model of each oscillator corresponds to a standard relaxation oscillator with two time scales. The network exhibits a mechanism of selective gating, whereby an oscillator jumping up to its active phase rapidly recruits the oscillators stimulated by the same pattern, while preventing others from jumping up. We show analytically that with the selective gating mechanism the network rapidly achieves both synchronization within blocks of oscillators that are stimulated by connected regions and desynchronization between different blocks. Computer simulations demonstrate the network's promising ability for segmenting multiple input patterns in real time. This model lays a physical foundation for the oscillatory correlation theory of feature binding, and may provide an effective computational framework for scene segmentation and figure/ground segregation. 1 INTRODUCTION A basic attribute of perception is its ability to group elements of a perceived scene into coherent clusters (objects). This ability underlies perceptual processes such as figure/ground segregation, identification of objects, and separation of different objects, and it is generally known as scene segmentation or perceptual organization. Despite the fact 200 DeLiang Wang. David Tennan that humans perform it with apparent ease, the general problem of scene segmentation remains unsolved in the engineering of sensory processing, such as computer vision and auditory processing. Fundamental to scene segmentation is the grouping of similar sensory features and the segregation of dissimilar ones. Theoretical investigations of brain functions and feature binding point to the mechanism of temporal correlation as a representational framework (von der Malsburg, 1981; von der Malsburg and Schneider, 1986). In particular, the correlation theory of von der Malsburg (1981) asserts that an object is represented by the temporal correlation of the fIring activities of the scattered cells coding different features of the object. A natural way of encoding temporal correlation is to use neural oscillations, whereby each oscillator encodes some feature (maybe just a pixel) of an object. In this scheme, each segment (object) is represented by a group of oscillators that shows synchrony (phase-locking) of the oscillations, and different objects are represented by different groups whose oscillations are desynchronized from each other. Let us refer to this form of temporal correlation as oscillatory correlation. The theory of oscillatory correlation has received direct experimental support from the cell recordings in the cat visual cortex (Eckhorn et aI., 1988; Gray et aI., 1989) and other brain regions. The discovery of synchronous oscillations in the visual cortex has triggered much interest from the theoretical community in simulating the experimental results and in exploring oscillatory correlation to solve the problems of scene segmentation. While several demonstrate synchronization in a group of oscillators using local (lateral) connections (Konig and Schillen, 1991; Somers and Kopell, 1993; Wang, 1993, 1995), most of these models rely on long range connections to achieve phase synchrony. It has been pointed out that local connections in reaching synchrony may play a fundamental role in scene segmentation since long-range connections would lead to indiscriminate segmentation (Sporns et aI., 1991; Wang, 1993). There are two aspects in the theory of oscillatory correlation: (1) synchronization within the same object; and (2) desynchronization between different objects. Despite intensive studies on the subject, the question of desynchronization has been hardly addressed. The lack of an effIcient mechanism for de synchronization greatly limits the utility of oscillatory correlation to perceptual organization. In this paper, we propose a new class of oscillatory networks and show that it can rapidly achieve both synchronization within each object and desynchronization between a number of simultaneously presented objects. The network is composed of the following elements: (I) A new model of a basic oscillator; (2) Local excitatory connections to produce phase synchrony within each object; (3) A global inhibitor that receives inputs from the entire network and feeds back with inhibition to produce desynchronization of the oscillator groups representing different objects. In other words, the mechanism of the network consists of local cooperation and global competition. This surprisingly simple neural architecture may provide an elementary approach to scene segmentation and a computational framework for perceptual organization. 2 MODEL DESCRIPTION The building block of this network, a single oscillator i, is defined in the simplest form as a feedback loop between an excitatory unit Xi and an inhibitory unit y( Synchrony and Desynchrony in Neural Oscillator Networks 8 201 , , dxldt= 0 ,, " o. ,, " " ":' . o x -2 2 Figure 1: Nullclines and periodic orbit of a single oscillator as shown in the phase plane. When the oscillator starts at a randomly generated point in the phase plane, it quickly converged to a stable trajectory of a limit cycle. dXi -=3x?dt l dy ? 3 X? I +2-y ,? +p+I?+S? " dt' = e (y(1 + tanh(xl/3) - Yi) Figure 2: Architecture of a two dimensional network with nearest neighbor coupling. The global inhibitor is indicated by the black circle. (la) (lb) where p denotes the amplitude of a Gaussian noise term. Ii represents external stimulation to the oscillator, and Si denotes coupling from other oscillators in the network. The noise term is introduced both to test the robustness of the system and to actively desynchronize different input patterns. The parameter e is chosen to be small. In this case (1), without any coupling or noise, corresponds to a standard relaxation oscillator. The x-nullcline of (1) is a cubic curve, while the y-nullc1ine is a sigmoid function, as shown in Fig. 1. If I > 0, these curves intersect along the middle branch of the cubic, and (1) is oscillatory. The periodic solution alternates between the silent and active phases of near steady state behavior. The parameter yis introduced to control the relative times that the solution spends in these two phases. If I < 0, then the nullc1ines of (1) intersect at a stable fixed point along the left branch of the cubic. In this case the system produces no oscillation. The oscillator model (1) may be interpreted as a model of spiking behavior of a single neuron, or a mean field approximation to a network of excitatory and inhibitory neurons. The network we study here in particular is two dimensional. However, the results can easily be extended to other dimensions. Each oscillator in the network is connected to only its four nearest neighbors, thus forming a 2-D grid. This is the simplest form of local connections. The global inhibitor receives excitation from each oscillator of the grid, and in turn inhibits each oscillator. This architecture is shown in Fig. 2. The intuitive reason why the network gives rise to scene segmentation is the following. When multiple connected objects are mapped onto the grid, local connectivity on the grid will group together the oscillators covered by each object. This grouping will be reflected 202 DeLiang Wang, David Tennan by phase synchrony within each object. The global inhibitor is introduced for desynchronizing the oscillatory responses to different objects. We assume that the coupling term Si in (1) is given by Si = L W ik Soo(xk, 9x ) - W z Soo(z, 9xz ) (2) kEN(i) S (x 9' 00 ,J = __-=-1_ _ 1+ exp[-K(x-e,] (3) where Wik is a connection (synaptic) weight from oscillator k to oscillator i, and N(i) is the set of the neighoring oscillators that connect to i. In this model, NO) is the four immediate neighbors on the 2-D grid, except on the boundaries where N(i) may be either 2 or 3 immediate neighbors. 9x is a threshold (see the sigmoid function of Eq. 3) above which an oscillator can affect its neighbors. W z (positive) is the weight of inhibition from the global inhibitor z, whose activity is defined as (4) where Goo = 0 if Xi < 9zx for every oscillator, and Goo = 1 if Xi ~ 9zx for at least one oscillator i. Hence 9zx represents a threshold. If the activity of every oscillator is below this threshold, then the global inhibitor will not receive any input. In this case z ~ 0 and the oscillators will not receive any inhibition. If, on the other hand, the activity of at least one oscillator is above the threshold 9zx then, the global inhibitor will receive input. In this case z ~ 1, and each oscillator feels inhibition when z is above the threshold 9zx. The parameter l/J determines the rate at which the inhibitor reacts to such stimulation. In summary, once an oscillator is active, it triggers the global inhibitor. This then inhibits the entire network as described in Eq. 1. On the other hand, an active oscillator spreads its activation to its nearest neighbors, again through (1), and from them to its further neighbors. In the next section, we give a number of properties of this system. Besides boundaries, the oscillators on the grid are basically symmetrical. Boundary conditions may cause certain distortions to the stability of synchrous oscillations. Recently, Wang (1993) proposed a mechanism called dynamic normalization to ensure that each oscillator, whether it is in the interior or on a boundary, has equal overall connection weights from its neighbors. The dynamic normalization mechanism is adopted in the present model to form effective connections. For binary images (each pixel being either 0 or 1), the outcome of dynamic normalization is that an effective connection is established between two oscillators if and only if they are neighbors and both of them are activated by external stimulation. The network defined above can readily be applied for segmentation of binary images. For gray-level images (each pixel being in a certain value range), the following slight modification suffices to make the network applicable. An effective connection is established between two oscillators if and only if they are neighbors and the difference of their corresponding pixel values is below a certain threshold. Synchrony and Desynchrony in Neural Oscillator Networks 203 3 ANALYTICAL RESULTS We have formally analyzed the network. Due to space limitations, we can only list the major conclusions without proofs. The interested reader can find the details in Terman and Wang (1994). Let us refer to a pattern as a connected region, and a block be a subset of oscillators stimulated by a given pattern. The following results are about singular solutions in the sense that we formally set E = O. However, as shown in (Terman and Wang, 1994), the results extend to the case E> 0 sufficiently small. Theorem 1. (Synchronization). The parameters of the system can be chosen so that all of the oscillators in a block always jump up simultaneously (synchronize). Moreover, the rate of synchronization is exponential. Theorem 2. (Multiple Patterns) The parameters of the system and a constant T can be chosen to satisfy the following. If at the beginning all the oscillators of the same block synchronize with each other and the temporal distance between any two oscillators belonging to two different blocks is greater than T, then (1) Synchronization within each block is maintained; (2) The blocks activate with a fixed ordering; (3) At most one block is in its active phase at any time. Theorem 3. (Desynchronization) If at the beginning all the oscillators of the system lie not too far away from each other, then the condition of Theorem 2 will be satisfied after some time. Moreover, the time it takes to satisfy the condition is no greater than N cycles, where N is the number of patterns. The above results are true with arbitrary number of oscillators. In summary, the network exhibits a mechanism, referred to as selective gating, which can be intuitively interpreted as follows. An oscillator jumping to its active phase opens a gate to quickly recruit the oscillators of the same block due to local connections. At the same time, it closes the gate to the oscillators of different blocks. Moreover, segmentation of different patterns is achieved very rapidly in terms of oscillation cycles. 4 COMPUTER SIMULATION To illustrate how this network is used for scene segmentation, we have simulated a 2Ox20 oscillator network as defined by (1)-(4). We arbitrarily selected four objects (patterns): two O's, one H, and one I ; and they form the word OHIO . These patterns were simultaneously presented to the system as shown in Figure 3A. Each pattern is a connected region, but no two patterns are connected to each other. All the oscillators stimulated (covered) by the objects received an external input 1=0.2, while the others have 1=-0.02. The amplitude p of the Gaussian noise is set to 0.02. Thus, compared to the external input, a 10% noise is included in every oscillator. Dynamic normalization results in that only two neighboring oscillators stimulated by a single pattern have an effective connection. The differential equations were solved numerically with the following parameter values: E = 0.02, l/J = 3.0; Y= 6.0, f3 = 0.1, K = 50, Ox = -0.5, and 0zx = 0xz = 0.1. The total effective connections were normalized to 6.0. The results described below were robust to considerable changes in the parameters. The phases of all the oscillators on the grid were randomly initialized. 204 DeLiang Wang, David Terman Fig. 3B-3F shows the instantaneous activity (snapshot) of the network at various stages of dynamic evolution. The diameter of each black circle represents the normalized x activity of the corresponding oscillator. Fig. 3B shows a snapshot of the network a few steps after the beginning of the simulation. In Fig. 3B, the activities of the oscillators were largely random. Fig. 3C shows a snapshot after the system had evolved for a short time period. One can clearly seethe effect of grouping and segmentation: all the oscillators belonging to the left 0 were entrained and had large activities. At the same time, the oscillators stimulated by the other three patterns had very small activities. Thus the left 0 was segmented from the rest of the input. A short time later, as shown in Fig. 3D, the oscillators stimulated by the right 0 reached high values and were separated from the rest of the input. Fig. 3E shows another snapshot after Fig. 3D. At this time, pattern I had its turn to be activated and separated from the rest of the input. Finally in Fig. 3F, the oscillators representing H were active and the rest of the input remained silent. This successive "pop-out" of the objects continued in a stable periodic fashion. To provide a complete picture of dynamic evolution, Fig. 30 shows the temporal evolution of each oscillator. Since the oscillators receiving no external input were inactive during the entire simulation process, they were excluded from the display in Fig. 30. The activities of the oscillators stimulated by each object are combined together in the figure. Thus, if they are synchronized, they appear like a single oscillator. In Fig. 30, the four upper traces represent the activities of the four oscillator blocks, and the bottom trace represents the activity of the global inhibitor. The synchronized oscillations within each object are clearly shown within just three cycles of dynamic evolution. 5 DISCUSSION Besides neural plausibility, oscillatory correlation has a unique feature as an computational approach to the engineering of scene segmentation and figure/ground segregation. Due to the nature of oscillations, no single -object can dominate and suppress the perception of the rest of the image permanently. The current dominant object has to give way to other objects being suppressed, and let them have a chance to be spotted. Although at most one object can dominant at any time instant, due to rapid oscillations, a number of objects can be activated over a short time period. This intrinsic dynamic process provides a natural and reliable representation of multiple segmented patterns. The basic principles of selective gating are established for the network with lateral connections beyond nearest neighbors. Indeed, in terms of synchronization, more distant connections even help expedite phase entrainment. In this sense, synchronization with all-to-all connections is an extreme case of our system. With nearest-neighbor connectivity (Fig. 2), any isolated part of an image is considered as a segment. In an noisy image with many tiny regions, segmentation would result in too many small fragments. More distant connections would also provide a solution to this problem. Lateral connections typically take on the form of Gaussian distribution, with the connection strength between two oscillators falling off exponentially. Since global inhibition is superimposed to local excitation, two oscillators positively coupled may be desynchronized if global inhibition is strong enough. Thus, it is unlikely that all objects in an image form a single segment as the result of extended connections. Synchrony and Desynchrony in Neural Oscillator Networks 205 Due to its critical importance for computer vision, scene segmentation has been studied quite extensively. Many techniques have been proposed in the past (Haralick and Shapiro, 1985; Sarkar and Boyer, 1993). Despite these techniques, as pointed out by Haralick and Shapiro (1985), there is no underlying theory of image segmentation, and the techniques tend to be adhoc and emphasize some aspects while ignoring others. Compared to the traditional techniques for scene segmentation, the oscillatory correlation approach offers many unique advantages. The dynamical process is inherently parallel. While conventional computer vision algorithms are based on descriptive criteria and many adhoc heuristics, the network as exemplified in this paper performs computations based on only connections and oscillatory dynamics. The organizational simplicity renders the network particularly feasible for VLSI implementation. Also, continuous-time dynamics allows real time processing, desired by many engineering applications. Acknowledgments DLW is supported in part by the NSF grant IRI-9211419 and the ONR grant NOOOI4-931-0335. DT is supported in part by the NSF grant DMS-9203299LE. References R. Eckhorn, et aI., "Coherent oscillations: A mechanism of feature linking in the visual cortex?" Bioi. Cybem., vol. 60, pp. 121-130, 1988. C.M. Gray, P. Konig, A.K. Engel, and W. Singer, "Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties," Nature, vol. 338, pp. 334-337, 1989. R.M. Haralick and L.G. Shapiro, "Image segmentation techniques," Comput. Graphics Image Process., vol. 29, pp. 100-132, 1985. P. Konig and T.B. Schillen, "Stimulus-dependent assembly formation of oscillatory responses: I. Synchronization," Neural Comput., vol. 3, pp. 155-166, 1991. S. Sarkar and K.L. Boyer, "Perceptual organization in computer vision: a review and a proposal for a classificatory structure," IEEE Trans. Syst. Man Cybern., vol. 23, 382-399, 1993. D. Somers, and N. Kopell, "Rapid synchronization through fast threshold modulation," BioI. Cybern, vol. 68, pp. 393-407, 1993. O. Sporns, G. Tononi, and G.M. Edelman, "Modeling perceptual grouping and figureground segregation by means of active reentrant connections," Proc. Natl. A cad. Sci. USA, vol. 88, pp. 129-133, 1991. D. Terman and D.L. Wang, "Global competition and local cooperation in a network of neural oscillators," Physica D, in press, 1994. C. von der Malsburg, "The correlation theory of brain functions," Internal Report 81-2, Max-Planck-Institut for Biophysical Chemistry, Gottingen, FRG, 1981. C. von der Malsburg and W. Schneider, "A neural cocktail-party processor," Bioi. Cybern., vol. 54, pp. 29-40, 1986. D.L. Wang, "Modeling global synchrony in the visual cortex by locally coupled neural oscillators," Proc. 15th Ann. Conf. Cognit. Sci. Soc., pp. 1058-1063, 1993. D.L. Wang, "Emergent synchrony in locally coupled neural oscillators," IEEE Trans. on Neural Networks, in press, 1995. ? 206 DeLiang Wang, David Terman B A c ???????????????????? .... ..... ????????????? ?????? .... ?....??.?.?.... ??????????????????? , ., ???????????????????? ? ????????????????? ~ ??????? ????????????? ??????????????????? ..... ...?...... .......?.... ...... ???????????????????? ???????? ????? ??? ??? ??????????????????? ???????????????????? . . . .?.. . ... ....... ...?...?... . ?????????? ? ????????? .?.. ? ??????????????? ???.?. ?. .?. ?.?. ? ? ??? ??e? ? ? . D E .... " . ........ . ... . . .... ... .. .... ... ... F ? ....... .. ... . ..... . . .. ....... .... . . .. . . G Figure 3. A An image composed of four patterns which were presented (mapped) to a 20x20 grid of oscillators. B A snapshot of the activities of the oscillator grid at the beginning of dynamic evolution. C A snapshot taken shortly after the beginning. D Shortly after C. E Shortly after D. F Shortly after E. G The upper four traces show the combined temporal activities of the oscillator blocks representing the four patterns, respectively, and the bottom trace shows the temporal activity of the global inhibitor. The simulation took 8,000 integration steps. 

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The Electrotonic Transformation: a Tool for Relating Neuronal Form to Function Nicholas T. Carnevale Department of Psychology Yale University New Haven, CT 06520 Kenneth Y. Tsai Department of Psychology Yale University New Haven, CT 06520 Brenda J. Claiborne Division of Life Sciences University of Texas San Antonio, TX 79285 Thomas H. Brown Department of Psychology Yale University New Haven, CT 06520 Abstract The spatial distribution and time course of electrical signals in neurons have important theoretical and practical consequences. Because it is difficult to infer how neuronal form affects electrical signaling, we have developed a quantitative yet intuitive approach to the analysis of electrotonus. This approach transforms the architecture of the cell from anatomical to electrotonic space, using the logarithm of voltage attenuation as the distance metric. We describe the theory behind this approach and illustrate its use. 1 INTRODUCTION The fields of computational neuroscience and artificial neural nets have enjoyed a mutually beneficial exchange of ideas. This has been most evident at the network level, where concepts such as massive parallelism, lateral inhibition, and recurrent excitation have inspired both the analysis of brain circuits and the design of artificial neural net architectures. Less attention has been given to how properties of the individual neurons or processing elements contribute to network function. Biological neurons and brain circuits have 70 Nicholas Carnevale, Kenneth Y. Tsai, Brenda J. Claiborne, Thomas H. Brown been simultaneously subject to eons of evolutionary pressure. This suggests an essential interdependence between neuronal form and function, on the one hand, and the overall architecture and operation of biological neural nets, on the other. Therefore reverseengineering the circuits of the brain appears likely to reveal design principles that rely upon neuronal properties. These principles may have maximum utility in the design of artificial neural nets that are constructed of processing elements with greater similarity to biological neurons than those which are used in contemporary designs. Spatiotemporal extent is perhaps the most obvious difference between real neurons and processing elements. The processing element of most artificial neural nets is essentially a point in time and space. Its activation level is the instantaneous sum of its synaptic inputs. Of particular relevance to Hebbian learning rules, all synapses are exposed to the same activation level. These simplifications may insure analytical and implementational simplicity, but they deviate sharply from biological reality. Membrane potential, the biological counterpart of activation level, is neither instantaneous nor spatially uniform. Every cell has finite membrane capacitance, and all ionic currents are finite, so membrane potential must lag behind synaptic inputs. Furthermore, membrane capacitance and cytoplasmic resistance dictate that membrane potential will almost never be uniform throughout a living neuron embedded in the circuitry of the brain. The combination of ever-changing synaptic inputs with cellular anatomical and biophysical properties guarantees the existence of fluctuating electrical gradients. Consider the task of building a massively parallel neural net from processing elements with such "nonideal" characteristics. Imagine moreover that the input surface of each processing element is an extensive, highly branched structure over which approximately 10,000 synaptic inputs are distributed. It might be tempting to try to minimize or work around the limitations imposed by device physics. However, a better strategy might be to exploit the computational consequences of these properties by making them part of the design, thereby turning these apparent weaknesses into strengths. To facilitate an understanding of the spatiotemporal dynamics of electrical signaling in neurons, we have developed a new theoretical approach to linear electrotonus and a new way to make practical use of this theory. We present this method and illustrate its application to the analysis of synaptic interactions in hippocampal pyramidal cells. 2 THEORETICAL BACKGROUND Our method draws upon and extends the results of two prior approaches: cable theory and two-port analysis. 2.1 CABLE THEORY The modern use of cable theory in neuroscience began almost four decades ago with the work of RaIl (1977). Much of the attraction of cable theory derives from the conceptual simplicity of the steady-state decay of voltage with distance along an infinite cylindrical cable: V(x) = Voe- xl). where x is physical distance and .4 is the length constant. This exponential relationship makes it useful to define the electrotonic distance X as the The Electronic Transfonnation: A Tool for Relating Neuronal Fonn to Function 7l logarithm of the signal attenuation: X = lnVo/V(x). In an infinite cylindrical cable, electrotonic distance is directly proportional to physical distance: X = x/2 . However, cable theory is difficult to apply to real neurons since dendritic trees are neither infinite nor cylindrical. Because of their anatomical complexity and irregular variations of branch diameter and length, attenuation in neurons is not an exponential function of distance. Even if a cell met the criteria that would allow its dendrites to be reduced to a finite equivalent cylinder (RaIl 1977), voltage attenuation would not bear a simple exponential relationship to X but instead would vary inversely with a hyperbolic function (Jack et a!. 1983). 2.2 TWO-PORT THEORY Because of the limitations and restrictions of cable theory, Carnevale and Johnston (1982) turned to two-port analysis. Among their conclusions, three are most relevant to this discussion. Figure 1: Attenuation is direction-dependent. The first is that signal attenuation depends on the direction of signal propagation. Suppose that i and J are two points in a cell where i is "upstream" from J (voltage is spreading from i to J), and define the voltage attenuation from i to j: A~ = ~ IV). Next suppose that the direction of signal propagation is reversed, so that j is now upstream from i, and define the voltage attenuation A ~ =Vj I~. In general these two *- AV. attenuations will not be equal: A~ 1) )1 They also showed that voltage attenuation in one direction is identical to current attenuation in the opposite direction (Carnevale and Johnston 1982). Suppose current Ii enters the cell at i, and the current that is captured by a voltage clamp at J is Ii' and define the current attenuation A; = Ii j ' Because of the directional reciprocity 11 Similarly, if we interchange the between current and voltage attenuation, A!1) = AV. j1 current entry and voltage clamp sites, the current attenuation ratio would be A )1l = A 1)~ . Finally, they found that charge and DC current attenuation in the same direction are identical (Carnevale and Johnston 1982). Therefore the spread of electrical signals between any two points is completely characterized by the voltage attenuations in both directions. 72 Nicholas Carnevale, Kenneth Y. Tsai, Brenda 1. Claiborne, Thomas H. Brown 2.3 THE ELECTROTONIC TRANSFORMATION The basic idea of the electrotonic transformation is to remap the cell from anatomical space into "electrotonic space," where the distance between points reflects the attenuation of an electrical signal spreading between them. Because of the critical role of membrane potential in neuronal function, it is usually most appropriate to deal with voltage attenuations. 2.3.1 The Distance Metric We use the logarithm of attenuation between points as the distance metric in electrotonic space: Li; = InAij (Brown et a1. 1992, Zador et a1. 1991). To appreciate the utility of this definition, consider voltage spreading from point i to point j , and suppose that k is on the direct path between i and j. The voltage attenuations are AI~ = V,/~ , At = Vk I~, and A& = V, IV; =A~ At . This last equation and our definition of L establish the additive property of electrotonic distance Llj = Lik + LIg" That is, electrotonic distances are additive over a path that has a consistent directIon of signal propagation. This justifies using the logarithm of attenuation as a metric for the electrical separation between points in a cell. At this point several important facts should be noted. First, unlike the electrotonic distance X of classical cable theOI)" our new definition of electrotonic distance L always bears a simple and direct logarithmic relationship to attenuation. Second, because of membrane capacitance, attenuation increases with frequency. Since both steady-state and transient signals are of interest, we evaluate attenuations at several different frequencies . Third, attenuation is direction-dependent and usually asymmetric. Therefore at every frequency of interest, each branch of the cell has two different representations in electrotonic space depending on the direction of signal flow. 2.3.2 Representing a Neuron in Electrotonic Space Since attenuation depends on direction, it is necessary to construct transforms in pairs for each frequency of interest, one for signal spread away from a reference point (Vout ) and the other for spread toward it (Vin ). The soma is often a good choice for the reference point, but any point in the cell could be used, and a different vantage point might be more appropriate for particular analyses. The only difference between using one point i as the reference instead of any other point j is in the direction of signal propagation along the direct path between i and j (dashed arrows in Figure 2). where Vout relative to i is the same as Vin relative to j and vice versa. The directions of signal flow and therefore the attenuations along all other branches of the cell are unchanged. Thus the transforms relative to i and j differ only along the direct path ij, and once the T/~ut and Vin transforms have been created for one reference i, it is easy to assemble the transforms with respect to any other reference j. The Electronic Transformation: A Tool for Relating Neuronal Form to Function 73 .,. .,. Figure 2: Effect of reference point location on direction of signal propagation. We have found two graphical representations of the transform to be particularly useful. "Neuromorphic figures," in which the cell is redrawn so that the relative orientation of branches is preserved (Figures 3 and 4), can be readily compared to the original anatomy for quick, ''big picture" insights regarding synaptic integration and interactions. For more quantitative analyses, it is helpful to plot electrotonic distance from the reference point as a function of anatomical distance (Tsai et al. 1993). 3 COMPUTATIONAL METHODS The voltage attenuations along each segment of the cell are calculated from detailed, accurate morphometric data and the best available experimental estimates of the biophysical properties of membrane and cytoplasm. Any neural simulator like NEURON (Hines 1989) could be used to find the attenuations for the DC Vout transform. The DC Vin attenuations are more time consuming because a separate run must be performed for each of the dendritic terminations. However, the AC attenuations impose a severe computational burden on time-domain simulators because many cycles are required for the response to settle. For example, calculating the DC Vout attenuations in a hippocampal pyramidal cell relative to the soma took only a few iterations on a SUN Sparc 10-40, but more than 20 hours were required for 40 Hz (Tsai et al. 1994). Finding the full set of attenuations for a Vin transform at 40 Hz would have taken almost three months. Therefore we designed an O(N) algorithm that achieves high computational efficiency by operating in the frequency domain and taking advantage of the branched architecture of the cell. In a series of recursive walks through the cell, the algorithm applies Kirchhoff's laws to compute the attenuations in each branch. The electrical characteristics of each segment of the cell are represented by an equivalent T circuit. Rather than "lump" the properties of cytoplasm and membrane into discrete resistances and capacitances, we determine the elements of these equivalent T circuits directly from complex impedance functions that we derived from the impulse response of a finite cable (Tsai et a1. 1994). Since each segment is treated as a cable rather than an isopotential compartment, the resolution of the spatial grid does not affect accuracy, and there is no need to increase the resolution of the spatial grid in order to preserve accuracy as frequency increases. This is an important consideration for hippocampal neurons, which have long membrane time constants and begin to show increased attenuations at frequencies as low as 2 - 5 Hz (Tsai et al. 1994). It also allows us to treat a long unbranched neurite of nearly constant diameter as a single cylinder. Thus runtimes scale linearly with the number of grid points, are independent of frequency, and we can even reduce the number of grid points if the diameters of adjacent 74 Nicholas Carnevale. Kenneth Y. Tsai. Brenda 1. Claiborne. Thomas H. Brown unbranched segments are similar enough. A benchmark of a program that uses our algorithm with NEURON showed a speedup of more than four orders of magnitude without sacrificing accuracy (2 seconds vs. 20 hours to calculate the Vout attenuations at 40 Hz in a CAl pyramidal neuron model with 2951 grid points) (Tsai et al. 1994). 4 RESULTS 4.1 DC TRANSFORMS OF A CAl PYRAMIDAL CELL Figure 3 shows a two-dimensional projection of the anatomy of a rat CA 1 pyramidal neuron (cell 524, left) with neuromorphic renderings of its DC Vout and Vin transforms (middle and right) relative to the soma. The three-dimensional anatomical data were obtained from HRP-filled cells with a computer microscope system as described elsewhere (Rihn and Claiborne 1992, Claiborne 1992). The passive electrical properties used to compute the attenuations were Ri = 200 Ocm, em = 1 J.lF/cm2 (for nonzero frequencies, not shown here) and Rm = 30 kncm2 (Spruston and Johnston 1992). 524 1 Figure 3: CAl pyramidal cell anatomy (cell 524, left) with neuromorphic renderings of Vout (middle) and Vin (right) transforms at DC. The Vout transform is very compact, indicating that voltage propagates from the soma into the dendrites with relatively little attenuation. The basilar dendrites and the terminal branches of the primary apical dendrite are almost invisible, since they are nearly isopotential along their lengths. Despite the fact that the primary apical dendrite has a larger diameter than any of its daughter branches, most of the voltage drop for somatofugal signaling is in the primary apical. Therefore it accounts for almost all of the electrotonic length of the cell in the Vout transform. The Vin transform is far more extensive, but most of the electrotonic length of the cell is now in the basilar and terminal apical branches. This reflects the loading effect of downstream membrane on these thin dendritic branches. 4.2 SYNAPTIC INTERACTIONS The transform can also give clues to possible effects of electrotonic architecture on voltage-dependent forms of associative synaptic plasticity and other kinds of synaptic interactions. Suppose the cell of Figure 3 receives a weak or "student" synaptic input The Electronic Transformation: A Tool for Relating Neuronal Form to Function 75 located 400 J.lm from the soma on the primary apical dendrite, and a strong or "teacher" input is situated 300 J.lm from the soma on the same dendrite. [!] student @ teacher A. cell 524 B. cell 503 Figure 4: Analysis of synaptic interactions. The anatomical arrangement is depicted on the left in Figure 4A ("student" = square, "teacher" = circle). The Vin transform with respect to the student (right figure of this pair) shows that voltage spreads from the teacher to the student synapse with little attenuation, which would favor voltage-dependent associative interactions. Figure 4B shows a different CAl pyramidal cell in which the apical dendrite bifurcates shortly after arising from the soma. Two teacher synapses are indicated, one on the same branch as the student and the other on the opposite branch. The Vin transform with respect to the student (right figure of this pair) shows clearly that the teacher synapse on the same branch is closely coupled to the student, but the other is electrically much more remote and less likely to influence the student synapse. 5. SUMMARY The electrotonic transformation is based on a logical, internally consistent conceptual approach to understanding the propagation of electrical signals in neurons. In this paper we described two methods for graphically presenting the results of the transformation: neuromorphic rendering, and plots of electrotonic distance vs. anatomical distance. Using neuromorphic renderings, we illustrated the electrotonic properties of a previously unreported hippocampal CAl pyramidal neuron as viewed from the soma (cell 524, Figure 3). We also extended the use of the transformation to the study of associative interactions between "teacher" and "student" synapses by analyzing this cell from the viewpoint of a "student" synapse located in the apical dendrites, contrasting this result with a different cell that had a bifurcated primary apical dendrite (cell 503, Figure 4). This demonstrates the versatility of the electrotonic transformation, and shows how it can convey the electrical signaling properties of neurons in ways that are quickly and easily comprehended. This understanding is important for several reasons. First, electrotonus affects the integration and interaction of synaptic inputs, regulates voltage-dependent mechanisms of synaptic plasticity, and influences the interpretation of intracellular recordings. In addition, phylogeny, development, aging, and response to injury and disease are all accompanied by alterations of neuronal morphology, some subtle and some profound. 76 Nicholas Carnevale, Kenneth Y. Tsai, Brenda J. Claiborne, Thomas H. Brown The significance of these changes for brain function becomes clear only if their effects on neuronal signaling are reckoned. Finally, there is good reason to expect that neuronal electrotonus is highly relevant to the design of artificial neural networks. Acknowledgments We thank R.B. Gonzales and M.P. O'Boyle for their contributions to the morphometric analysis, and Z.F. Mainen for assisting in the initial development of graphical rendering. This work was supported in part by ONR, ARPA, and the Yale Center for Theoretical and Applied Neuroscience (CTAN). References Brown, T.H. , Zador, A.M. , Mainen, Z.F. and Claiborne, BJ. Hebbian computations in hippocampal dendrites and spines. In: Single Neuron Computation, eds. McKenna, T., Davis, J. and Zornetzer, S.F., New York, Academic Press, 1992, pp. 81-116. Carnevale, N. T. and Johnston, D.. Electrophysiological characterization of remote chemical synapses. J. Neurophysiol. 47:606-621 , 1982. Claiborne, BJ. The use of computers for the quantitative, three-dimensional analysis of dendritic trees. In: Methods in Neuroscience. Vol. 10: Computers and Computation in the Neurosciences, ed. Conn, P.M., New York, Academic Press, 1992, pp. 315-330. Hines, M. A program for simulation of nerve equations with branching geometries. Internat. J. Bio-Med Computat. 24:55-68, 1989. Rall, W. . Core conductor theory and cable properties of neurons. In: Handbook of Physiology, The Nervous System , ed. Kandel, E.R., Bethesda, MD, Am. Physiol. Soc. , 1977, pp.39-98. Rihn, L.L. and Claiborne, BJ. Dendritic growth and regression in rat dentate granule cells during late postnatal development. Brain Res. Dev. 54(1): 115-24, 1990. Spruston, N. and Johnston, D. Perforated patch-clamp analysis of the passive membrane properties of three classes of hippocampal neurons. J. Neurophysiol. 67 :508529, 1992. Tsai, K.Y., Carnevale, N.T. , Claiborne, BJ. and Brown, T.H. Morphoelectrotonic transforms in three classes of rat hippocampal neurons. Soc. Neurosci. Abst. 19: 1522, 1993. Tsai, K.Y. , Carnevale, N.T. , Claiborne, BJ. and Brown, T.H. Efficient mapping from neuroanatomical to electrotonic space. Network 5:21-46, 1994. Zador, A.M. , Claiborne, BJ. and Brown, T.H. Electrotonic transforms of hippocampal neurons. Soc. Neurosci. Abst. 17: 1515, 1991. 

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Boosting the Performance of RBF Networks with Dynamic Decay Adjustment Michael R. Berthold Forschungszentrum Informatik Gruppe ACID (Prof. D. Schmid) Haid-und-Neu-Strasse 10-14 76131 Karlsruhe, Germany eMail: berthold@fzLde Jay Diamond Intel Corporation 2200 Mission College Blvd. Santa Clara, CA, USA 95052 MS:SC9-15 eMail: jdiamond@mipos3.intel.com Abstract Radial Basis Function (RBF) Networks, also known as networks of locally-tuned processing units (see [6]) are well known for their ease of use. Most algorithms used to train these types of networks, however, require a fixed architecture, in which the number of units in the hidden layer must be determined before training starts. The RCE training algorithm, introduced by Reilly, Cooper and Elbaum (see [8]), and its probabilistic extension, the P-RCE algorithm, take advantage of a growing structure in which hidden units are only introduced when necessary. The nature of these algorithms allows training to reach stability much faster than is the case for gradient-descent based methods. Unfortunately P-RCE networks do not adjust the standard deviation of their prototypes individually, using only one global value for this parameter. This paper introduces the Dynamic Decay Adjustment (DDA) algorithm which utilizes the constructive nature of the P-RCE algorithm together with independent adaptation of each prototype's decay factor. In addition, this radial adjustment is class dependent and distinguishes between different neighbours. It is shown that networks trained with the presented algorithm perform substantially better than common RBF networks. 522 1 Michael R. Berthold, Jay Diamond Introduction Moody and Darken proposed Networks with locally-tuned processing units, which are also known as Radial Basis Functions (RBFs, see [6]). Networks of this type have a single layer of units with a selective response for some range of the input variables. Earn unit has an overall response function, possibly a Gaussian: D_("') (11x - 2riW) X = exp - .J.t,i (1) (Ii Here x is the input to the network, ri denotes the center of the i-th RBF and (Ii determines its standard deviation. The second layer computes the output function for each class as follows: m (2) i=l with m indicating the number of RBFs and Ai being the weight for each RBF. Moody and Darken propose a hybrid training, a combination of unsupervised clustering for the centers and radii of the RBFs and supervised training of the weights. Unfortunately their algorithm requires a fixed network topology, which means that the number of RBFs must be determined in advance. The same problem applies to the Generalized Radial Basis Functions (GRBF), proposed in [12]. Here a gradient descent technique is used to implement a supervised training of the center locations, which has the disadvantage of long training times. In contrast RCE (Restricted Coulomb Energy) Networks construct their architecture dynamically during training (see [7] for an overview). This algorithm was inspired by systems of charged particles in a three-dimensional space and is analogous to the Liapunov equation: 1 m Qi (3) ~ = - L i=l II'"X _ T,...?II 2L l: where ~ is the electrostatic potential induced by fixed particles with charges -Qi and locations ri. One variation of this type of networks is the so called P-RCE network, which attempts to classify data using a probabilistic distribution derived from the training set. The underlying training algorithm for P-RCE is identical to RCE training with gaussian activation functions used in the forward pass to resemble a Probabilistic Neural Network (PNN [10]). PNNs are not suitable for large databases because they commit one new prototype for each training pattern they encounter, effectively becoming a referential memory scheme. In contrast, the P-RCE algorithm introduces a new prototype only when necessary. This occurs when the prototype of a conflicting class misclassifies the new pattern during the training phase. The probabilistic extension is modelled by incrementing the a-priori rate of occurrence for prototypes of the same class as the input vector, therefore weights are only connecting RBFs and an output node of the same class. The recall phase of the P-RCE network is similar to RBFs, except that it uses one global radius for all prototypes and scales each gaussian by the a-priori rate of occurrence: (4) Boosting the Performance of RBF Networks with Dynamic Decay Adjustment 523 Figure 1: This picture shows how a new pattern results in a slightly higher activity for a prototype of the right class than for the conflicting prototype. Using only one threshold, no new prototype would be introduced in this case. where c denotes the class for which the activation is computed, me is the number of prototypes for class c, and R is the constant radius of the gaussian activation functions. The global radius of this method and the inability to recognize areas of conflict, leads to confusion in some areas of the feature space, and therefore non-optimal recognition performance. The Dynamic Decay Adjustment (DDA) algorithm presented in this paper was developed to solve the inherent problems associated with these methods. The constructive part of the P-RCE algorithm is used to build a network with an appropriate number of RBF units, for which the decay factor is computed based on information about neighbours. This technique increases the recognition accuracy in areas of conflict. The following sections explain the algorithm, compare it with others, and examine some simulation results. 2 The Algorithm Since the P-RCE training algorithm already uses an independent area of influence for each RBF, it is relatively straightforward to extract an individual radius. This results, however, in the problem illustrated in figure 1. The new pattern p of class B is properly covered by the right prototype of the same class. However, the left prototype of conflicting class A results in almost the same activation and this leads to a very low confidence when the network must classify the pattern p. To solve this dilemma, two different radii, or thresholds 1 are introduced: a so-called positive threshold (0+), which must be overtaken by an activation of a prototype of the same class so that no new prototype is added, and a negative threshold (0-), which is the upper limit for the activation of conflicting classes. Figure 2 shows an example in which the new pattern correctly results in activations above the positive threshold for the correct class B and below the negative threshold for conflicting class A . This results in better classification-confidence in areas where training IThe conversion from the threshold to the radius is straightforward as long as the activation function is invertible. 524 Michael R. Berthold, Jay Diamond new input pattern (class B) x Figure 2: The proposed algorithm distinguishes between prototypes of correct and conflicting classes and uses different thresholds. Here the level of confidence is higher for the correct classification of the new pattern. patterns did not result in new prototypes. The network is required to hold the following two equations for every pattern x of class c from the training data: 3i : Rf(x) 2:: 8+ Vk :/; c, 1 ~ j ~ mk : Rj(x) (5) < 8- (6) The algorithm to construct a classifier can be extracted partly from the ReE algorithm. The following pseudo code shows what the training for one new pattern x of class c looks like: / / reset weights: FORALL prototypes pf DO Af = 0.0 END FOR / / train one complete epoch FORALL training pattern (x,c) DO: IF 3pi : Ri( x) 2:: 8+ THEN Ai+ = 1.0 ELSE / / "commit": introduce new prototype add new prototype P~c+1 with: ~c+1 =x O'~ +1 = maJ:C {O' : R~ +1 (r7) < 8-} c A~c+1 k#cl\l~J::;mk = 1.0 c mc+= 1 ENDIF / / "shrink": adjust conflicting prototypes FORALL k :/; c, 1 ~ j ~ mk DO O'j = max{O' : Rj(x) < 8-} ENDFOR First, all weights are set to zero because otherwise they would accumulate duplicate information about training patterns. Next all training patterns are presented to the Boos,;ng the Peiformance of RBF Networks with Dynamic Decay Adjustment 525 pIx) pattern class A (1) +2 (2) pIx) pattern class B (3) pattern class A x (4) Figure 3: An example of the DDA- algorithm: (1) a pattern of class A is encountered and a new RBF is created; (2) a training pattern of class B leads to a new prototype for class B and shrinks the radius of the existing RBF of class A; (3) another pattern of class B is classified correctly and shrinks again the prototype of class A; (4) a new pattern of class A introduces another prototype of that class. network. If the new pattern is classified correctly, the weight of the closest prototype is increased; otherwise a new protoype is introduced with the new pattern defining its center. The last step of the algorithm shrinks all prototypes of conflicting classes if their activations are too high for this specific pattern. Running this algorithm over the training data until no further changes are required ensures that equations (5) and (6) hold. The choice of the two new parameters, (J+ and (J- are not as critical as it would initially appear2. For all of the experiments reported, the settings (J+ = 0.4 and (J- = 0.1 were used, and no major correlations of the results to these values were noted. Note that when choosing (J+ = (J- one ends up with an algorithm having the problem mentioned in figure l. Figure 3 shows an example that illustrates the first few training steps of the DDAalgorithm. 3 Results Several well-known databases were chosen to evaluate this algorithm (some can be found in the eMU Neural Network Benchmark Databases (see [13])). The DDA2Theoretically one would expect the dimensionality of the input- space to playa major role for the choice of those parameters 526 Michael R. Berthold, Jay Diamond algorithm was compared against PNN, RCE and P-RCE as well as a classic Multi Layer Perceptron which was trained using a modified Backpropagation algorithm (Rprop, see [9]). The number of hidden nodes of the MLP was optimized manually. In addition an RBF-network with a fixed number of hidden nodes was trained using unsupervised clustering for the center positions and a gradient descent to determine the weights (see [6] for more details). The number of hidden nodes was again optimized manually. ? Vowel Recognition: Speaker independent recognition of the eleven steady state vowels of British English using a specified training set of Linear Predictive Coding (LPC) derived log area ratios (see [3]) resulting in 10 inputs and 11 classes to distinguish. The training set consisted of 528 tokens, with 462 different tokens used to test the network. II performance I #units I #epochs I algorithm Nearest Neighbour 56% 1 ..... 200 MLP (RPRUP) 57% 5 528 PNN 61% ..... 100 RBF 59% 70 27% 125 RCE 3 P-RCE 125 59% 3 DDA-RBF 204 4 65_% ? Sonar Database: Discriminate between sonar signals bounced off a metal cylinder and those bounced off a roughly cylindrical rock (see [4] for more details) . The data has 60 continuous inputs and is separated into two classes. For training and testing 104 samples each were used. algorithm MLP (RPROP) PNN RBF RCE P-RCE DDA-RBF I performance 90.4% 91.3% 90.7% 77.9% 90.4% 93.3% I #units I #epochs I 50 104 80 68 68 68 ..... 250 - ..... 150 3 3 3 ? Two Spirals: This well-known problem is often used to demonstrate the generalization capability of a network (see [5]). The required task involves discriminating between two intertwined spirals. For this paper the spirals were changed slightly to make the problem more demanding. The original spirals radius declines linearly and can be correctly classified by RBF networks with one global radius. To demonstrate the ability of the DDAalgorithm to adjust the radii of each RBF individually, a quadratic decline was chosen for the radius of both spirals (see figure 4) . The training set consisted of 194 points, and the spirals made three complete revolutions. Figure 4 shows both the results of an RBF Network trained with the DDA technique and the same problem solved with a Multi-Layer Perceptron (2-20-20-1) trained using a modified Error Back Propagation algorithm (Rprop, see [9]). Note that in both cases all training points are classified correctly. Boosting the Peifonnance of RBF Networks with Dynamic Decay Adjustment 527 Figure 4: The (quadratic) "two spirals problem" solved by a MLP (left) using Error Back Propagation (after 40000 epochs) and an RBF network (right) trained with the proposed DDA-algorithm (after 4 epochs). Note that all training patterns (indicated by squares vs. crosses) are classified correctly. In addition to these tasks, the BDG-database was used to compare the DDA algorithm to other approaches. This database was used by Waibel et al (see [11]) to introduce the Time Delay Neural Network (TDNN). Previously it has been shown that RBF networks perform equivalently (when using a similar architecture, [1], [2]) with the DDA technique used for training of the RBF units. The BDG task involves distinguishing the three stop consonants "B", "D" and "G". While 783 training sets were used, 749 data sets were used for testing. Each of these contains 15 frames of melscale coefficients, computed from a 10kHz, 12bit converted signal. The final frame frequency was 100Hz. algorithm TDNN TDRBF (P-RCE) TDRBF (DDA) 4 II performance 98.5% 85.2% 98.3% I #epochs I ",50 5 6 Conclusions It has been shown that Radial Basis Function Networks can boost their performance by using the dynamic decay adjustment technique. The algorithm necessary to construct RBF networks based on the RCE method was described and a method to distinguish between conflicting and matching prototypes at the training phase was proposed. An increase in performance was noted, especially in areas of conflict, where standard (P-)RCE did not commit new prototypes. Four different datasets were used to show the performance of the proposed DDAalgorithm. In three of the cases, RBF networks trained with dynamic decay adjustment outperformed known RBF training methods and MLPs. For the fourth task, the BDG-recognition dataset, the TDRBF was able to reach the same level 528 Michael R. Berthold. Jay Diamond of performance as a TDNN. In addition, the new algorithm trains very quickly. Fewer than 6 epochs were sufficient to reach stability for all problems presented. Acknowledgements Thanks go to our supervisors Prof. D. Schmid and Mark Holler for their support and the opportunity to work on this project. References [1] M. R. Berthold: "A Time Delay Radial Basis FUnction Network for Phoneme Recognition" in Proc. of the IEEE International Conference on Neural Networks, 7, p.447D---4473, 1994. [2] M. R. Berthold: "The TDRBF: A Shift Invariant Radial Basis Function Network" in Proc. of the Irish Neural Network Conference, p.7-12, 1994. [3] D. Deterding: "Speaker Normalization for Automatic Speech Recognition" , PhD Thesis, University of Cambridge, 1989. [4] R. Gorman, T. Sejnowski: "Analysis of Hidden Units in a Layered Network Trained to Classify Sonar Targets" in Neural Networks 1, pp.75. [5] K. Lang, M. Witbrock: "Learning to Tell Two Spirals Apart", in Proc. of Connectionist Models Summer School, 1988. [6] J . Moody, C.J. Darken: "Fast Learning in Networks of Locally-Tuned Processing Units" in Neural Computation 1, p.281-294, 1989. [7] M.J. Hudak: "RCE Classifiers: Theory and Practice" in Cybernetics and Systems 23, p.483-515, 1992. [8] D.L. Reilly, L.N. Cooper, C. Elbaum: "A Neural Model for Category Learning" in BioI. Cybernet. 45, p.35-41, 1982. [9] M. Riedmiller, H. Braun: "A Direct Adaptive Method for Faster Backpropagation Learning: The Rprop Algorithm" in Proc. of the IEEE International Conference on Neural Networks, 1, p.586-591 , 1993. [10] D.F. Specht: "Probabilistic Neural Networks" in Neural Networks 3, p.109118,1990. [11] A. Waibel, T. Hanazawa, G. Hinton, K. Shikano, K. Lang: " Phoneme Recognition Using Time-Delay Neural Networks" in IEEE Trans. in Acoustics, Speech and Signal Processing Vol. 37, No. 3, 1989. [12] D. Wettschereck, T. Dietterich: "Improving the Performance of Radial Basis Function Networks by Learning Center Locations" in Advances in Neural Information Processing Systems 4, p.1133- 1140, 1991. [13] S. Fahlman, M. White: "The Carnegie Mellon University Collection of Neural Net Benchmarks" from ftp.cs.cmu .edu in /afs/cs/project/connect/bench. 

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A Charge-Based CMOS Parallel Analog Vector Quantizer Gert Cauwenberghs Johns Hopkins University ECE Department 3400 N. Charles St. Baltimore, MD 21218-2686 gert@jhunix.hcf.jhu.edu Volnei Pedroni California Institute of Technology EE Department Mail Code 128-95 Pasadena, CA 91125 pedroni@romeo.caltech.edu Abstract We present an analog VLSI chip for parallel analog vector quantization. The MOSIS 2.0 J..Lm double-poly CMOS Tiny chip contains an array of 16 x 16 charge-based distance estimation cells, implementing a mean absolute difference (MAD) metric operating on a 16-input analog vector field and 16 analog template vectors. The distance cell including dynamic template storage measures 60 x 78 J..Lm2 ? Additionally, the chip features a winner-take-all (WTA) output circuit of linear complexity, with global positive feedback for fast and decisive settling of a single winner output. Experimental results on the complete 16 x 16 VQ system demonstrate correct operation with 34 dB analog input dynamic range and 3 J..Lsec cycle time at 0.7 mW power dissipation. 1 Introduction Vector quantization (VQ) [1] is a common ingredient in signal processing, for applications of pattern recognition and data compression in vision, speech and beyond. Certain neural network models for pattern recognition, such as Kohonen feature map classifiers [2], are closely related to VQ as well. The implementation of VQ, in its basic form, involves a search among a set of vector templates for the one which best matches the input vector, whereby the degree of matching is quantified by a given vector distance metric. Effi- 780 Gert Cauwenberghs, Volnei Pedroni cient hardware implementation requires a parallel search over the template set and a fast selection and encoding of the "winning" template. The chip presented here implements a parallel synchronous analog vector quantizer with 16 analog input vector components and 16 dynamically stored analog template vectors, producing a 4-bit digital output word encoding the winning template upon presentation of an input vector. The architecture is fully scalable as in previous implementations of analog vector quantizers, e.g. [3,4,5,6], and can be readily expanded toward a larger number of vector components and template vectors without structural modification of the layout. Distinct features of the present implementation include a linear winner-take-all (WTA) structure with globalized positive feedback for fast selection of the winning template, and a mean absolute difference (MAD) metric for the distance estimations, both realized with a minimum amount of circuitry. Using a linear charge-based circuit topology for MAD distance accumulation, a wide voltage range for the analog inputs and templates is achieved at relatively low energy consumption per computation cycle. 2 System Architecture The core of the VQ consists of a 16 x 16 2-D array of distance estimation cells, configured to interconnect columns and rows according to the vector input components and template outputs. Each cell computes in parallel the absolute difference distance between one component x} of the input vector x and the corresponding component yi} of one of the template vectors y, (1) The mean absolute difference (MAD) distance between input and template vectors is accumulated along rows A ? d(X,y') = 1 ~ . 16 ~ Ix} - y'}I, i = 1 ... 16 (2) J=l and presented to the WTA, which selects the single winner kWTA = arg min d(x, f) . i (3) Additional parts are included in the architecture for binary encoding of the winning output, and for address selection to write and refresh the template vectors. 3 VLSI Circuit Implementation The circuit implementation of the major components of the VQ, for MAD distance estimation and WTA selection, is described below. Both MAD distance and WTA cells operate in clocked synchronous mode using a precharge/evaluate scheme in the voltage domain. The approach followed here offers a wide analog voltage range of inputs and templates at low power weak-inversion MOS operation, and a fast and decisive settling of the winning output using a single communication line for global positive feedback. The output encoding and address decoding circuitry are implemented using standard CMOS logic. 781 A Charge-Based CMOS Parallel Analog Vector Quantizer VVRi -r--------------------~-----? VVRi 812 Xj Vref PRE -1 4/2 -r----------~----------------Zi (b) (a) Figure 1: Schematic of distance estimation circuitry. Output precharge circuitry. 3.1 (a) Absolute distance cell. (b) Distance Estimation Cell The schematic of the distance estimation cell, replicated along rows and columns of the VQ array, is shown in Figure 1 (a). The cell contains two source followers, which buffer the input voltage x j and the template voltage yi j . The template voltage is stored dynamically onto Cstore, written or refreshed by activating WRi while the y' j value is presented on the Xj input line. The WRi and WRi signal levels along rows of the VQ array are driven by the address decoder, which selects a single template vector yi to be written to with data presented at the input x when WR is active. Additional lateral transistors connect symmetrically to the source follower outputs x / and yi /- By means of resistive division, the lateral transistors construct the maximum and minimum of xl' and yi / on Zi i HI and Zi j LO, respectively. In particular, when x j is . I HI , . LO muc h Iarger than y' j' the vo tage Z' j approaches x j and the voltage Z' j approaches yi /- By symmetry, the complementary argument holds in case Xj is much smaller than yi j. Therefore, the differential compo~ent of Zi j HI and Zi JLO approximately represents the absolute difference value of xJ and y' j : i Z j with K HI - i LO z j , yi j ') - ~ max(xj, - IXj , - Yi j 'I ~ K . mm(xj IXj - ' i ') ,y Yi j j (4) I, the MOS back gate effect coefficient [7] . The mean absolute difference (MAD) distances (2) are obtained by accumulating con- Gert Cauwenberghs, Volnei Pedroni 782 tributions (4) along rows of cells through capacitive coupling, using the well known technique of correlated double sampling. To this purpose, a coupling capacitor Cc is provided in every cell, coupling its differential output to the corresponding output row line. In the precharge phase, the maximum values Zi j HI are coupled to the output by activating HI, and the output lines are preset to reference voltage Vref by activating PRE, Figure 1 (b). In the evaluate phase, PRE is de-activated, and the minimum values Zi j LO are coupled to the output by activating LO. From (4), the resulting voltage outputs on the floating row lines are given by V, re f - 1 L16 16. - . (Zl . J HI - . LO Zl . J ) (5) J=1 1 Vref- K 16 16~ IXj-/jl. J=1 The last term in (5) corresponds directly to the distance measure d(x, yi) in (2). Notice that the negative sign in (5) could be reversed by interchanging clocks HI and LO, if needed. Since the subsequent WTA stage searches for maximum Zi, the inverted distance metric is in the form needed for VQ. Characteristics of the MAD distance estimation (5), measured directly on the VQ array with uniform inputs x j and templates yi j' are shown in Figure 2. The magnified view in Figure 2 (b) clearly illustrates the effective smoothing of the absolute difference function (4) near the origin, x j ~ yi j' The smoothing is caused by the shift in x/and yi j' due to the conductance of the lateral coupling transistors connected to the source follower outputs in Figure 1 (a), and extends over a voltage range comparable to the thermal voltage kT /q depending on the relative geometry of the transistors and current bias level of the source followers. The observed width of the flat region in Figure 2 spans roughly 60 mY, and shows little variation for bias current settings below 0.5 f-I,A. Tuning of the bias current allows to balance speed and power dissipation requirements, since the output response is slew-rate limited by the source followers. 3.2 Winner-Take-All Circuitry The circuit implementation of the winner-take-all (WTA) function combines the compact sizing and modularity of a linear architecture as in [4,8,9] with positive feedback for fast and decisive output settling independent of signal levels, as in [6,3]. Typical positive feedback structures for WTA operation use a logarithmic tree [6] or a fully interconnected network [3], with implementation complexities of order O(n log n) and O(n2) respectively, n being the number of WTA inputs. The present implementation features an O(n) complexity in a linear structure by means of globalized positive feedback, communicated over a single line. The schematic of the WTA cell, receiving the input Zi and constructing the digital output di through global competition communicated over the COMM line, is shown in Figure 3. The global COMM line is source connected to input transistor Mi and positive feedback transistor Mf, and receives a constant bias current lb (WTA) from Mbl. Locally, the WTA operation is governed by the dynamics of d;' on (parasitic) capacitor Cp' A high pulse A Charge-Based CMOS Parallel Analog Vector Quantizer 783 0.0 -0.12 -0.2 ~ :> -0.14 - -0.4 ..... -0.16 ~ ~I -018 ? l ~I -0.6 ' -N -0.20 -0.8 -0.22 -1.0 ~--~----~------~--~ 1.5 2.0 2.5 -0.2 -0.1 0.0 ! Xj (V) Xj-Yj (V) (a) (b) 0.1 0.2 Figure 2: Distance estimation characteristics, (a) for various values of yi j; (b) magnified view. on RST, resetting d/ to zero, marks the beginning of the WTA cycle. With Mf initially inactive, the total bias current n Ib (WTA) through COMM is divided over all competing WTA cells, according to the relative Zi voltage levels, and each cell fraction is locally mirrored by the Mml-Mm2 pair onto d;', charging C p' The cell with the highest Zi input voltage receives the largest fraction of bias current, and charges Cp at the highest rate. The winning output is detennined by the first d;' reaching the threshold to turn on the corresponding Mf feedback transistor, say i = k. This threshold voltage is given by the source voltage on COMM, common for all cells. The positive feedback of the state dk ' through Mf, which eventually claims the entire fraction of the bias current, enhances and latches the winning output level dk ' to the positive supply and shuts off the remaining losing outputs d;' to zero, i :f:. k. The additional circuitry at the output stage of the cell serves to buffer the binary d;' value at the d; output tenninal. No more than one winner can practically co-exist at equilibrium, by nature of the combined positive feedback and global renonnalization in the WTA competition. Moreover, the output settling times of the winner and losers are fairly independent of the input signal levels, and are given mainly by the bias current level Ib (WTA) and the parasitic capacitance Cpo Tests conducted on a separate 16-element WTA array, identical to the one used on the VQ chip, have demonstrated single-winner WTA operation with response time below 0.5 /-Lsec at less than 2 /-L W power dissipation per cell. 4 Functionality Test To characterize the performance of the entire VQ system under typical real-time conditions, the chip was presented a periodic sequence of 16 distinct input vectors x(i), stored and refreshed dynamically in the 16 template locations y; by circularly incrementing the template address and activating WR at the beginning of every cycle. The test vectors rep- 784 Gert Cauwenberghs, Volnei Pedroni Mm2 2212 Zi -1 8/2 MI 11/2 11/2 di T COMM Mb1 VbWfA 11/2 812 RST Figure 3: Circuit schematic of winner-take-all cell. resent a single triangular pattern rotated over the 16 component indices with single index increments in sequence. The fundamental component xo(i) is illustrated on the top trace of the scope plot in Figure 4. The other components are uniformly displaced in time over one period, by a number of cycles equal to the index, x j (i) = xo(i - j mod 16). Figure 4 also displays the VQ output waveforms in response to the triangular input sequence, with the desired parabolic profile for the analog distance output ZO and the expected alternating bit pattern of the WTA least significant output bit. l The triangle test performed correctly at speeds limited by the instrumentation equipment, and the dissipated power on the chip measures 0.7 mW at 3 f.Lsec cycle time2 and 5 V supply voltage. An estimate for the dynamic range of analog input and template voltages was obtained directly by observing the smallest and largest absolute voltage difference still resolved correctly by the VQ output, uniformly over all components. By tuning the voltage range of the triangular test vectors, the recorded minimum and maximum voltage amplitudes for 5 V supply voltage are Ymin = 87.5 mV and YlII3lt = 4 V, respectively. The estimated analog dynamic range YlII3lt /Ymin is thus 45.7, or roughly 34 dB, per cell. The value obtained for Ymin indicates that the dynamic range is limited mainly by the smoothing of the absolute distance measure characteristic (1) near the origin in Figure 2 (b). We notice that a similar limitation of dynamic range applies to other distance metrics with vanishing slope near the origin as well, the popular mean square error (MSE) formulation in particular. The MSE metric is frequently adopted in VQ implementations using strong inversion MOS circuitry, and offers a dynamic range typically worse than obtained here regardless of implementation accuracy, due to the relatively wide flat region of the MSE distance function near the origin. ITbe voltages on the scope plot are inverted as a consequence of the chip test setup. 2including template write operations A "Charge-Based CMOS Parallel Analog Vector Quantizer 785 Figure 4: Scope plot of VQ waveforms. Top: Analog input Xo. Center: Analog distance output zO. Bottom: Least significant bit of encoded output. Table 1: Technology Supply voltage Power dissipation VQchip Dynamic range inputs, templates Area Features of the VQ chip 2 ~ p-well double-poly CMOS +5V 0.7 mW (3 J.lSec cycle time) 34 dB VQchip 2.2 mm X 2.25 mm distance cell WTAcell 6O~X78~ 76~X80~ 786 Gert Cauwenberghs, Volnei Pedroni 5 Conclusion We proposed and demonstrated a synchronous charge-based CMOS VLSI system for parallel analog vector quantization, featuring a mean absolute difference (MAD) metric, and a linear winner-take-all (WTA) structure with globalized positive feedback. By virtue of the MAD metric, a fairly large (34 dB) analog dynamic range of inputs and templates has been obtained in the distance computations through simple charge-based circuitry. Likewise, fast and unambiguous settling of the WTA outputs, using global competition communicated over a single wire, has been obtained by adopting a compact linear circuit structure to implement the positive feedback WTA function. The resulting structure of the VQ chip is highly modular, and the functional characteristics are fairly consistent over a wide range of bias levels, including the MOS weak inversion and subthreshold regions. This allows the circuitry to be tuned to accommodate various speed and power requirements. A summary of the chip features of the 16 x 16 vector quantizer is presented in Table I. Acknowledgments Fabrication of the CMOS chip was provided through the DARPAINSF MOSIS service. The authors thank Amnon Yariv for stimulating discussions and encouragement. References [1] A. Gersho and RM. Gray, Vector Quantization and Signal Compression, Norwell, MA: Kluwer, 1992. [2] T. Kohonen, Self-Organisation and Associative Memory, Berlin: Springer-Verlag, 1984. [3] Y. He and U. Cilingiroglu, "A Charge-Based On-Chip Adaptation Kohonen Neural Network," IEEE Transactions on Neural Networks, vol. 4 (3), pp 462-469, 1993. [4] J.e. Lee, B.J. Sheu, and W.e. Fang, "VLSI Neuroprocessors for Video Motion Detection," IEEE Transactions on Neural Networks, vol. 4 (2), pp 78-191, 1993. [5] R Tawel, "Real-Time Focal-Plane Image Compression," in Proceedings Data Compression Conference" Snowbird, Utah, IEEE Computer Society Press, pp 401-409, 1993. [6] G.T. Tuttle, S. Fallahi, and A.A. Abidi, "An 8b CMOS Vector NO Converter," in ISSCC Technical Digest, IEEE Press, vol. 36, pp 38-39, 1993. [7] C.A. Mead, Analog VLSI and Neural Systems, Reading, MA: Addison-Wesley, 1989. [8] J. Lazzaro, S. Ryckebusch, M.A. Mahowald, and C.A. Mead, "Winner-Take-All Networks of O(n) Complexity," in Advances in Neural Information Processing Systems, San Mateo, CA: Morgan Kaufman, vol. 1, pp 703-711, 1989. [9] A.G. Andreou, K.A. Boahen, P.O. Pouliquen, A. Pavasovic, RE. Jenkins, and K. Strohbehn, "Current-Mode Subthreshold MOS Circuits for Analog VLSI Neural Systems," IEEE Transactions on Neural Networks, vol. 2 (2), pp 205-213, 1991. 

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Computational structure of coordinate transformations: A generalization study Zoubin Ghahramani zoubin@psyche.mit.edu Daniel M. Wolpert wolpert@psyche.mit.edu Michael I. Jordan jordan@psyche.mit.edu Department of Brain & Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract One of the fundamental properties that both neural networks and the central nervous system share is the ability to learn and generalize from examples. While this property has been studied extensively in the neural network literature it has not been thoroughly explored in human perceptual and motor learning. We have chosen a coordinate transformation system-the visuomotor map which transforms visual coordinates into motor coordinates-to study the generalization effects of learning new input-output pairs. Using a paradigm of computer controlled altered visual feedback, we have studied the generalization of the visuomotor map subsequent to both local and context-dependent remappings. A local remapping of one or two input-output pairs induced a significant global, yet decaying, change in the visuomotor map, suggesting a representation for the map composed of units with large functional receptive fields. Our study of context-dependent remappings indicated that a single point in visual space can be mapped to two different finger locations depending on a context variable-the starting point of the movement. Furthermore, as the context is varied there is a gradual shift between the two remappings, consistent with two visuomotor modules being learned and gated smoothly with the context. 1 Introduction The human central nervous system (CNS) receives sensory inputs from a multitude of modalities, each tuned to extract different forms of information from the 1126 Zoubin Ghahramani, Daniel M. Wolpert, Michael 1. Jordan environment. These sensory signals are initially represented in disparate coordinate systems, for example visual information is represented retinotopically whereas auditory information is represented tonotopically. The ability to transform information between coordinate systems is necessary for both perception and action. When we reach to a visually perceived object in space, for example, the location of the object in visual coordinates must be converted into a representation appropriate for movement, such as the configuration of the arm required to reach the object. The computational structure of this coordinate transformation, known as the visuomotor map, is the focus of this paper. By examining the change in visuomotor coordination under prismatically induced displacement and rotation, Helmholtz (1867/1925) and Stratton (1897a,1897b) pioneered the systematic study of the representation and plasticity of the visuomotor map. Their studies demonstrate both the fine balance between the visual and motor coordinate systems, which is disrupted by such perturbations, and the ability of the visuomotor map to adapt to the displacements induced by the prisms. Subsequently, many studies have further demonstrated the remarkable plasticity of the map in response to a wide variety of alterations in the relationship between the visual and motor system (for reviews see Howard, 1982 and Welch, 1986)-the single prerequisite for adaptation seems to be that the remapping be stable (Welch, 1986). Much less is known, however, about the topological properties of this map. A coordinate transformation such as the visuomotor map can be regarded as a function relating one set of variables (inputs) to another (outputs). For the visuomotor map the inputs are visual coordinates of a desired target and the outputs are the corresponding motor coordinates representing the arm's configuration (e.g. joint angles). The problem of learning a sensorimotor remapping can then be regarded as a function approximation problem. Using the theory of function approximation one can make explicit the correspondence between the representation used and the patterns of generalization that will emerge. Function approximators can predict patterns of generalization ranging from local (look-up tables), through intermediate (CMACs, Albus, 1975; and radial basis functions, Moody and Darken, 1989 ) to global (parametric models). In this paper we examine the representational structure of the visuomotor map through the study of its spatial and contextual generalization properties. In the spatial generalization study we address the question of how pointing changes over the reaching workspace after exposure to a highly localized remapping. Previous work on spatial generalization, in a study restricted to one dimension, has led to the conclusion that the visuomotor map is constrained to generalize linearly (Bedford, 1989). We test this conclusion by mapping out the pattern of generalization induced by one and two remapped points in two dimensions. In the contextual generalization study we examine the question of whether a single point in visual space can be mapped into two different finger locations depending on the context of a movement-the start point. If this context-dependent remapping occurs, the question arises as to how the mapping will generalize as the context is varied. Studies of contextual remapping have previously shown that variables such as eye position (Kohler, 1950; Hay and Pick, 1966; Shelhamer et al., 1991), the feel of prisms (Kravitz, 1972; Welch, 1971) or an auditory tone (Kravitz and Yaffe, 1972), can induce context-dependent aftereffects. The question of how these context- 1127 Computational Structure of Coordinate Transformations dependent maps generalize-which has not been previously explored-reflects on the possible representation of multiple visuomotor maps and their mixing with a context variable. 2 Spatial Generalization To examine the spatial generalization of the visuomotor map we measured the change in pointing behavior subsequent to one- and two-point remappings. In order to measure pointing behavior and to confine subjects to learn limited input-output pairs we used a virtual visual feedback setup consisting of a digitizing tablet to record the finger position on-line and a projection/mirror system to generate a cursor spot image representing the finger position (Figure 1a). By controlling the presence of the cursor spot and its relationship to the finger position, we could both restrict visual feedback of finger position to localized regions of space and introduce perturbations of this feedback. b) a) VGAScreen PnljecIor P_ Ii \ \ /, /0: \ /, /, / ,/ ,l 0 0 0 Finger Position ~~ . Angerf_ o image \ ;(..'.. ~"'" . .. \ \) Rear profection acreen -, ' >'"'!'f'----Vi-rtual-lma-II"Seml-sliverad mirTOr Dl!Jtizjng Tablel Actual Anger Position Actual FIngerPosI1Ion .......... o a 150m '.. -" C)[J' ....., , ? . . d)Q'. ? 1 ? T ? ? e)lZj' " ,t ,, , , , Figure 1. a) Experimental setup. The subjects view the reflected image of the rear projection screen by looking down at the mirror. By matching the screenmirror distance to the mirror-tablet distance all projected images appeared to be in the plane of the finger (when viewed in the mirror) independent of head position. b) The position of the grid of targets relative to the subject. Also shown, for the x-shift condition, is the perceived and actual finger position when pointing to the central training target. The visually perceived finger position is indicated by a cursor spot which is displaced from the actual finger position. c) A schematic showing the perturbation for the x-shift group. To see the cursor spot on the central target the subjects had to place their finger at the position indicated by the tip of the arrow. d) & e) Schematics similar to c) showing the perturbation for the y-shift and two point groups, respectively. In the tradition of adaptation studies (e.g. Welch, 1986), each experimental session consisted of three phases: pre-exposure, exposure, and post-exposure. During the pre- and post-exposure phases, designed to assess the visuomotor map, the subject pointed repeatedly, without visual feedback of his finger position, to a grid of targets over the workspace. As no visual input of finger location was given, no learning of the visuomotor map could occur . During the exposure phase subjects pointed repeatedly to one or two visual target locations, at which we introduced a discrep- 1128 Zoubin Ghahramani, Daniel M. Wolpert, Michael!. Jordan ancy between the actual and visually displayed finger location. No visual feedback of finger position was given except when within 0.5 cm of the target, thereby confining any learning to the chosen input-output pairs. Three local perturbations of the visuomotor map were examined: a 10 cm rightward displacement (x-shift group, Figure lc), 10 cm displacement towards the body (y-shift group, Figure Id), and a displacement at two points, one 10 cm away from, and one 10 cm towards the body (two point group , Figure Ie). For example, for the x-shift displacement the subject had to place his finger 10 cm to the right of the target to visually perceive his finger as being on target (Figure Ib). Separate control subjects, in which the relationship between the actual and visually displayed finger position was left unaltered, were run for both the one- and two-point displacements, resulting in a total of 5 groups with 8 subjects each. 50 -?) 45 .. ,. 40 50 __ .......................... " 4' __ .......................... ..... 40 _ _ ............ _ ..';1.' ] JS >30 30 ........ _ _ _ ............ _ _ ........ --..... ~~ .............. 25 20 1~ 10 45 Cl CJ 41) ~ 45 \ 40 \ I \ 25 t9 20 20 -10 -5 10 15 -10 20 10 ".'i 55 55 50 5(J , , , 45 45 " e 35 ~ >- 30 E- I 40 0. '0 15 20 -10 () x (em) X (I.:m ) c X (em) 35 >- 30 25 20 X (Lm) X ?(,:m) ~ 30 25 25 I' I \ t t 2() t !() 2() t tt tit J f I 2(J G 20 ~ ~ , 10 X (em) I I I I 15 15 -15 ? 10 -5 0 5 10 x (em) 15 20 2.' -15 -10?5 I) 5 11) X (em) 15 211 25 -10 () X (em) Figure 2. Results of the spatial generalization study. The first column shows the mean change in pointing, along with 95% confidence ellipses, for the x-shift, y-shift and two point groups. The second column displays a vector field of changes obtained from the data by Gaussian kernel smoothing. The third column plots the proportion adaptation in the direction of the perturbation. Note that whereas for the x- and yshift groups the lighter shading corresponds to greater adaptation, for the two point group lighter shades correspond to adaptation in the positive y direction and darker shades in the negative y direction. Computational Structure of Coordinate Transfonnations 1129 The patterns of spatial generalization subsequent to exposure to the three local remappings are shown in Figure 2. All three perturbation groups displayed both significant adaptation at the trained points, and significant, through decremented, generalization of this learning to other targets. As expected, the control groups (not shown) did not show any significant changes. The extent of spatial generalization, best depicted by the shaded contour plots in Figure 2, shows a pattern of generalization that decreases with distance away from the trained points. Rather than inducing a single global change in the map, such as a rotation or shear, the two point exposure appears to induce two opposite fields of decaying generalization at the intersection of which there is no change in the visuomotor map. 3 Contextual Generalization The goal of this experiment was first to explore the possibility that multiple visuomotor maps, or modules, could be learned, and if so, to determine how the overall system behaves as the context used in training each module is varied. To achieve this goal, we exposed subjects to context-dependent remappings in which a single visual target location was mapped to two different finger positions depending on the start point of the movement. Pointing to the target from seven different starting points (Figure 3) was assessed before and after an exposure phase. During this exposure phase subjects made repeated movements to the target from starting points 2 and 6 with a different perturbation of the visual feedback depending on the starting point . The form of these context-dependent remappings is shown in Figure 3. For example, for the open x-shift group (Figure 3c), the visual feedback of the finger was displaced to the right for movements from point 2 and to the left from point 6. Therefore the same visual target was mapped to two different finger positions depending on the context of the movement. To test learning of the remapping and generalization to other start points we examined the change in pointing, without visual feedback, to the target from the 7 start points. b) crossed x-shift a) control ~' .... ~ ....'e'- ?t -'? -' t, o 1234567 c) open x-shift ..' ." ........... ~ ...?.?. ......? dlYL\ 1234567 :,., .... o o 1234567 1234567 0 Figure 3. Schematic of the exposure phase in the contextual generalization experiment. Shown are the actual finger path (solid line), the visually displayed finger path (dotted line), the seven start points and the target used in the pre- and postexposure phases. The perturbation introduced depended on whether the movement started form start point 2 or 6. Note that for the three perturbation groups, although the subjects saw a triangle being traced out, the finger took a different path. 1130 Zoubin Ghahramani, Daniel M. Wolpert, Michael I. Jordan The results are shown in Figure 4. Whereas the controls did not show any significant pattern of change, the three other groups showed adaptive, start point dependent, changes in the direction opposite to the perturbation. Thus, for example, the xopen group displayed a pattern of change in the leftward (negative x) direction for movements from the left start points and rightwards for movements from the right start points. Furthermore, as the start point was varied, the change in pointing varied gradually. a) U a control 0--0 croned x?,bltt - - 0.' b) 2.0 a 1.0 !!. !!. " .;J " .S! ;; 0 -0.' ..... >< c. ." -u ....... c. ~v~ -2.' ." , 4 Start point 0.0 , -1.0 2 4 Start point Figure 4. a) Adaptation in the x direction plotted as a function of starting point for the control, crossed x-shift and open x-shift groups (mean and 1 s.e.). b) Adaptation in the y direction for the control and y-shift groups. 4 Discussion Clearly, from the perspective of function approximation theory, the problem of relearning the visuomotor mapping from exposure to one or two input-output pairs is ill-posed. The mapping learned, as measured by the pattern of generalization to novel inputs, therefore reflects intrinsic constraints on the internal representation used. The results from the spatial generalization study suggest that the visuomotor coordinate transformation is internally represented with units with large but localized receptive fields. For example, a neural network model with Gaussian radial basis function units (Moody and Darken, 1989), which can be derived by assuming that the internal constraint in the visuomotor system is a smoothness constraint (Poggio and Girosi, 1989), predicts a pattern of generalization very similar the one experimentally observed (e.g. see Figure 5 for a simulation of the two point generalization experiment).1 In contrast, previously proposed models for the representation of the visuomotor map based on global parametric representations in terms of felt direction of gaze and head position (e.g. Harris, 1965) or linear constraints (Bedford, 1989) do not predict the decaying patterns of Cartesian generalization found. 1 See also Pouget & Sejnowski (this volume) who, based on a related analysis of neurophysiological data from parietal cortex, suggest that a basis function representation may be used in this visuomotor area. Computational Structure of Coordinate Transformations ~ >- 50 , \ 45 \ \ 40 \ \ \ 35 30 25 \ \ ~ \ \ t t ~! I ~ ? 10 ?5 0 X (em) t }~e t t I I I' I' 10 15 20 20 1131 ~ >- I X (em) Figure 5. Simulation of the two point spatial generalization experiment using a radial basis function network with 64 units with 5 cm Gaussian receptive fields. The inputs to the network were the visual coordinates of the target and the outputs were the joint angles for a two-link planar arm to reach the target. The network was first trained to point accurately to the targets , and then, after exposure to the perturbation, its pattern of generalization was assessed. The results from the second study suggest that multiple visuomotor maps can be learned and modulated by a context. A suggestive computational model for how such separate modules can be learned and combined is the mixture-of-experts neural network architecture (Jacobs et al. , 1991). Interpreted in this framework, the gradual effect of varying the context seen in Figure 4 could reflect the output of a gating network which uses context to modulate between two visuomotor maps. However, our results do not rule out models in which a single visuomotor map is parametrized by starting location, such as one based on the coding of locations via movement vectors (Georgopoulos, 1990) . 5 Conclusions The goal of these studies has been to infer the internal constraints in the visuomotor system through the study of its patterns of generalization to local remappings. We have found that local perturbations of the visuomotor map produce global changes, suggesting a distributed representation with large receptive fields. Furthermore, context-dependent perturbations induce changes in pointing consistent with a model of visuomotor learning in which separate maps are learned and gated by the context. The approach taken in this paper provides a strong link between neural network theory and the study of learning in biological systems. Acknowledgements This project was supported in part by a grant from the McDonnell-Pew Foundation , by a grant from ATR Human Information Processing Research Laboratories, by a grant from Siemens Corporation, and by grant N00014-94-1-0777 from the Office of Naval Research. Zoubin Ghahramani and Daniel M. Wolpert are McDonnell-Pew Fellows in Cognitive Neuroscience. Michael I. Jordan is a NSF Presidential Young Investigator. 1132 Zoubin Ghahramani, Daniel M. Wolpert, Michael I. Jordan References Albus, J. (1975) . A new approach to manipulator control: The cerebellar model articulation controller (CMAC). J. of Dynamic Systems, Measurement, and Control, 97:220-227. Bedford, F. (1989). Constraints on learning new mappings between perceptual dimensions. J. of Experimental Psychology: Human Perception and Performance, 15(2):232-248. Georgopoulos, A. (1990). Neurophysiology of reaching. In Jeannerod, M., editor, Attention and performance XIII, pages 227-263 . Lawrence Erlbaum, Hillsdale. Harris, C. (1965). Perceptual adaptation to inverted, reversed, and displaced vision. Psychological Review, 72:419-444. Hay, J. and Pick, H. (1966). Gaze-contingent prism adaptation: Optical and motor factors. J. of Experimental Psychology, 72 :640-648. Howard, 1. (1982). Human visual orientation. Wiley, Chichester, England. Jacobs, R., Jordan, M. , Nowlan, S. , and Hinton, G. (1991) . Adaptive mixture of local experts. Neural Computation, 3:79-87. Kohler, 1. (1950). Development and alterations of the perceptual world: conditioned sensations. Proceedings of the Austrian Academy of Sciences, 227. Kravitz, J. (1972). Conditioned adaptation to prismatic displacement. Perception and Psychophysics, 11:38-42. Kravitz, J . and Yaffe, F . (1972) . Conditioned adaptation to prismatic displacement with a tone as the conditional stimulus. Perception and Psychophysics, 12:305-308. Moody, J. and Darken, C. (1989) . Fast learning in networks of locally-tuned processing units. Neural Computation, 1(2) :281-294. Poggio, T. and Girosi, F. (1989). A theory of networks for approximation and learning. AI Lab Memo 1140, MIT. Pouget, A. and Sejnowski, T. (1994) . Spatial representation in the parietal cortex may use basis functions. In Tesauro, G., Touretzky, D. , and Alspector, J., editors, Advances in Neural Information Processing Systems 7. Morgan Kaufmann. Shelhamer, M., Robinson, D., and Tan, H. (1991) . Context-specific gain switching in the human vestibuloocular reflex. In Cohen, B., Tomko, D., and Guedry, F., editors, Annals Of The New York Academy Of Sciences, volume 656, pages 889-891. New York Academy Of Sciences, New York. Stratton, G. (1897a). Upright vision and the retinal image. Psychological Review, 4:182187. Stra.tton, G. (1897b) . Vision without inversion of the retinal image. Psychological Review, 4:341-360, 463-481. von Helmholtz, H. (1925). Treatise on physiological optics (1867). Optical Society of America, Rochester, New York. Welch, R. (1971). Discriminative conditioning of prism adaptation. Perception and Psychophysics, 10:90-92 . Welch, R. (1986). Adaptation to space perception. In Boff, K., Kaufman, L. , and Thomas, J ., editors, Handbook of perception and performance, volume 1, pages 24- 1-24-45. Wiley-Interscience, New York. 

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Inferring Ground Truth from Subjective Labelling of Venus Images Padhraic Smyth, Usama Fayyad Jet Propulsion Laboratory 525-3660, Caltech, 4800 Oak Grove Drive, Pasadena, CA 91109 Michael Burl, Pietro Perona Department of Electrical Engineering Caltech, MS 116-81, Pasadena, CA 91125 Pierre Baldi* Jet Propulsion Laboratory 303-310, Caltech, 4800 Oak Grove Drive, Pasadena, CA 91109 Abstract In remote sensing applications "ground-truth" data is often used as the basis for training pattern recognition algorithms to generate thematic maps or to detect objects of interest. In practical situations, experts may visually examine the images and provide a subjective noisy estimate of the truth. Calibrating the reliability and bias of expert labellers is a non-trivial problem. In this paper we discuss some of our recent work on this topic in the context of detecting small volcanoes in Magellan SAR images of Venus. Empirical results (using the Expectation-Maximization procedure) suggest that accounting for subjective noise can be quite significant in terms of quantifying both human and algorithm detection performance. 1 Introduction In certain pattern recognition applications, particularly in remote-sensing and medical diagnosis, the standard assumption that the labelling of the data has been * and Division of Biology, California Institute of Technology 1086 Padhraic Smyth. Usama Fayyad. Michael Burl. Pietro Perona. Pierre Baldi carried out in a reasonably objective and reliable manner may not be appropriate. Instead of "ground truth" one may only have the subjective opinion(s) of one or more experts. For example, medical data or image data may be collected off-line and some time later a set of experts analyze the data and produce a set of class labels. The central problem is that of trying to infer the "ground truth" given the noisy subjective estimates of the experts. When one wishes to apply a supervised learning algorithm to the data, the problem is primarily twofold: (i) how to evaluate the relative performance of experts and algorithms, and (ii) how to train a pattern recognition system in the absence of absolute ground truth. In this paper we focus on problem (i), namely the performance evaluation issue, and in particular we discuss the application of a particular modelling technique to the problem of counting volcanoes on the surface of Venus. For problem (ii), in previous work we have shown that when the inferred labels have a probabilistic interpretation, a simple mixture model argument leads to straightforward modifications of various learning algorithms [1]. It should be noted that the issue of inferring ground truth from subjective labels has appeared in the literature under various guises. French [2] provides a Bayesian perspective on the problem of combining multiple opinions. In the field of medical diagnosis there is a significant body of work on latent variable models for inferring hidden "truth" from subjective diagnoses (e.g., see Uebersax [3]). More abstract theoretical models have also been developed under assumptions of specific labelling patterns (e.g., Lugosi [4] and references therein). The contribution of this paper is twofold: (i) this is the first application of latent-variable subjective-rating models to a large-scale image analysis problem as far as we are aware, and (ii) the focus of our work is on the pattern recognition aspect of the problem, i.e., comparing human and algorithmic performance as opposed to simply comparing humans to each other. 2 Background: Automated Detection of Volcanoes in Radar Images of Venus Although modern remote-sensing and sky-telescope technology has made rapid recent advances in terms of data collection capabilities, image analysis often remains a strictly manual process and much investigative work is carried out using hardcopy photographs. The Magellan Venus data set is a typical example: between 1991 and 1994 the Magellan spacecraft transmitted back to earth a data set consisting of over 30,000 high resolution (75m per pixel) synthetic aperture radar (SAR) images of the Venusian surface [5]. This data set is greater than that gathered by all previous planetary missions combined - planetary scientists are literally swamped by data. There are estimated to be on the order of 106 small (less than 15km in diameter) vl.sible volcanoes scattered throughout the 30,000 images [6]. It has been estimated that manually locating all of these volcanoes would require on the order of 10 man-years of a planetary geologist's time to carry out - our experience has been that even a few hours of image analysis severely taxes the concentration abilities of human labellers. From a scientific viewpoint the ability to accurately locate and characterize the Inferring Ground Truth from Subjective Labelling of Venus Images 1087 many volcanoes is a necessary requirement before more advanced planetary geology studies can be carried out: analysis of spatial clustering patterns, correlation with other geologic features, and so forth. From an engineering viewpoint, automation of the volcano detection task presents a significant challenge to current capabilities in computer vision and pattern recognition due to the variability of the volcanoes and the significant background "clutter" present in most of the images. Figure 1 shows a Magellan subimage of size 30km square containing at least 10 small volcanoes. Volcanoes on Venus Figure 1: A 30km x 30km region from the Magellan SAR data, which contains a number of small volcanoes. The purpose of this paper is not to describe pattern recognition methods for volcano detection but rather to discuss some of the issues involved in collecting and calibrating labelled training data from experts. Details of a volcano detection method using matched filtering, SVD projections and a Gaussian classifier are provided in [7]. 3 Volcano Labelling Training examples are collected by having the planetary geologists examine an image on the computer screen and then using a mouse to indicate where they think the volcanoes (if any) are located. Typically it can take from 15 minutes to 1 hour to label an image (depending on how many volcanoes are present), where each image represents a 75km square patch on the surface of Venus. An image may contain on the order of 100 volcanoes, although a more typical number is between 30 and 40. There can be considerable ambiguity in volcano labelling: for the same image, different scientists can produce different label lists, and even the same scientist can produce different lists over time. To address this problem we introduced the notion of having the scientists label training examples into quantized probability 1088 Padhraic Smyth, Usama Fayyad, Michael Burl, Pierro Perona, Pierre Baldi bins or "types", where the probability bins correspond to visually distinguishable sub-categories of volcanoes. In particular, we have used 5 types: (1) summit pits, bright-dark radar pair, and apparent topographic slope, all clearly visible, probability 0.98, (2) only 2 of the 3 criteria of type 1 are visible, probability 0.80, (3) no summit pit visible, evidence of flanks or circular outline, probability 0.60, (4) only a summit pit visible, probability 0.50, (5) no volcano-like features visible, probability 0.0. These subjective probabilities correspond to the mean probability that a volcano exists at a particular location given that it belongs to a particular type and were elicited after considerable discussions with the planetary geologists. Thus, the observed data for each RDI consists of labels l, which are noisy estimates of true "type" t, which in turn is probabilistically related to the hidden event of interest, v, the presence of a volcano: T (1) p(vlD = LP(vlt)p(tID t=1 where T is the number of types (and labels). The subjective probabilities described above correspond to p(vlt): to be able to infer the probability of a volcano given a set of labels l it remains to estimate the p(tlD terms. 4 Inferring the Label-Type Parameters via the EM Procedure We follow a general model for subjective labelling originally proposed by Dawid and Skene [8] and apply it to the image labelling problem: more details on this overall approach are provided in [9]. Let N be the number of local regions of interest (RDl's) in the database (these are 15 pixel square image patches for the volcano application) . For simplicity we consider the case of just a single labeller who labels a given set of regions of interest (RDIs) a number of times - the extension to multiple labellers is straightforward assuming conditional independence of the labellings given the true type. Let nil be the number of times that RDI i is labelled with labell. Let lit denote a binary variable which takes value 1 if the true type of volcano i is t* , and is 0 otherwise. We assume that labels are assigned independently to a given RDI from one labelling to the next, given that the type is known. If the true type t* is known then T p(observed labelslt*, i) ex: II p(lltti!? (2) 1=1 Thus, unconditionally, we have p(observed labels, t*li) ex: )Yit gT(T pet) gP(lltt il , (3) where lit = 1 if t = t* and 0 otherwise. Assuming that each RDI is labelled independently of the others (no spatial correlation in terms of labels), Ng T(T )Yit pet) gp(lltt'l p(observed labels, t;) ex: ~ (4) Inferring Ground Truth from Subjective Labelling of Venus Images 1089 Still assuming that the types t for each ROI are known (the Yit)' the maximum likelihood estimators of p(llt) and p(t) are p(llt) = ~i Yitnil EI Ei Yitnil (5) and (6) From Bayes' rule one can then show that 1 T p(Yit = llobserved data) = C IIp(llt)nilp(t) (7) I where C is a normalization constant. Thus, given the observed data nil and the parameters p(llt) and p(t), one can infer the posterior probabilities of each type via Equation 7. However, without knowing the Yit values we can not infer the parameters p(llt) and p(t). One can treat the Yit as hidden and thus apply the well-known ExpectationMaximization (EM) procedure to find a local maximum of the likelihood function: 1. Obtain some initial estimates of the expected values of Yit, e.g., E[Yitl = ~ Elnil (8) 2. M-step: choose the values of p(llt) and p(t) which maximize the likelihood function (according to Equations 5 and 6), using E[Yitl in place of Yit. 3. E-step: calculate the conditional expectation of Yit, E[Yitldatal = p(Yit = Ildata) (Equation 7). 4. Repeat Steps 2 and 3 until convergence. 5 5.1 Experimental Results Combining Multiple Expert Opinions Labellings from 4 geologists on the 4 images resulted in 269 possible volcanoes (ROIs) being identified. Application of the EM procedure resulted in label-type probability matrices as shown in Table 1 for Labeller C. The diagonal elements provide an indication of the reliability of the labeller. There is significant miscalibration for label 3's: according to the model, a label 3 from Labeller C is most likely to correspond to type 2. The label-type matrices of all 4 labellers (not shown) indicated that the model placed more weight on the conservative labellers (C and D) than the aggressive ones (A and B). The determination of posterior probabilities for each of the ROIs is a fundamental step in any quantitative analysis of the volcano data: p(vlD = E;=1 p(vlt)p(tID where the p(tlD terms are the posterior probabilities of type given labels provided 1090 Padhraic Smyth, Usama Fayyad, Michael Burl, Pietro Perona, Pierre Baldi Table 1: Type-Label Probabilities for Individual Labellers as estimated via the EM Procedure Labell Label 2 Label 3 Label 4 Label 5 Probability(typellabel), Labeller C Type 1 Type 2 Type 3 Type 4 1.000 0.000 0.000 0.000 0.019 0.977 0.004 0.000 0.000 0.667 0.175 0.065 0.000 0.725 0.000 0.042 0.000 0.000 0.389 0.000 Type 5 0.000 0.000 0.094 0.233 0.611 Table 2: 10 ROIs from the database: original scientist labels shown with posterior probabilities estimated via the EM procedure Scientist Labels JD ROI 1 2 3 4 5 6 7 8 9 10 A B 4 1 1 3 3 2 3 2 3 4 4 4 1 1 1 2 1 1 2 4 C 4 4 2 5 3 2 5 4 5 4 D 5 2 2 3 3 4 5 4 3 4 Posterior Probabilities (EM), p(tlD Type 1 Type 2 Type 3 Type 4 Type 5 0.000 0.000 0.000 0.816 0.184 0.000 0.000 0.000 0.991 0.009 0.023 0.000 0.000 0.977 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.536 0.452 0.012 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.999 0.000 0.000 0.000 0.992 0.000 0.008 0.000 0.000 0.000 0.996 0.004 p(vlO 0.408 0.496 0.804 0.600 0.706 0.800 0.600 0.500 0.595 0.498 by the EM procedure, and the p(vlt) terms are the subjective volcano-type probabilities discussed in Section 3.2. As shown in Table 2, posterior probabilities for the volcanoes generally are in agreement with intuition and often correspond to taking the majority vote or the "average" of the C and D labels (the conservative labellers). However some p(vlD estimates could not easily be derived by any simple averaging or voting scheme, e.g., see ROIs 3, 5 and 7 in the table. 5.2 Experiment on Comparing Human and Algorithm Performance The standard receiver operating characteristic (ROC) plots detections versus false alarms [10] . The ROCs shown here differ in two significant ways [11] : (1) the false alarm axis is normalized relative to the number of true positives (necessary since the total number of possible false alarms is not well defined for object detection in images), and (2) the reference labels used in scoring are probabilistic: a detection "scores" p(v) on the detection axis and 1 - p( v) on the false alarm axis. Inferring Ground Truth from Subjective Labelling of Venus Images 100 100 Mean Detection Rate [%) 80 Mean Detection Rate [%) 80 60 60 40 40 20 1091 -SVD Algorithm -t( Scientist A -0 Scientist B -- Scientist C .+ ScientistD 0~_ _~_ _~~======7 o 20 40 60 80 Mean False Alarm Rate (expressed as % of total number of volcanoes) -SVD Algorithm -t( Scientist A -0 Scientist B -- Scientist C .+ ScientistD 20 40 60 80 Mean False Alarm Rate (expressed as % of total number of volcanoes) Figure 2: Modified ROCs for both scientists and algorithms: (a) without the labelling or type uncertainty, (b) with full uncertainty model factored in. As before, data came from 4 images, and there were 269 labelled local regions. The SVD-Gaussian algorithm was evaluated in cross-validation mode (train on 3 images, test on the 4th) and the results combined. The first ROC (Figure 2(a)) does not take into account either label-type or type-volcano probabilities, i.e., the reference list (for algorithm training and overall evaluation) is a consensus list (2 scientists working together) where labels 1,2,3,4 are ignored and all labelled items are counted equally as volcanoes. The individual labellers and algorithm are then scored in the standard "non-weighted" ROC fashion. This curve is optimistic in terms of depicting the accuracy of the detectors since it ignores the underlying probabilistic nature of the labels. Even with this optimistic curve, volcano labelling is relatively inaccurate by either man or machine. Figure 2(b) shows a weighted ROC: for each of 4 scientists the probabilistic "reference labels" were derived via the EM procedure as in Table 2 from the other 3 scientists, and the detections of each scientist were scored according to each such reference set. Performance of the algorithm (the SVD-Gaussian method) was evaluated relative to the EM-derived label estimates of all 4 scientists. Accounting for all of the uncertainty in the data results in a more realistic, if less flattering, set of performance characteristics. The algorithm's performance degrades more than the scientist's performance (for low false alarms rates compared to Figure 2(a)) when the full noise model is used. The algorithm is estimating the posterior probabilities of volcanoes rather poorly and the complete uncertainty model is more sensitive to this fact. This is a function of the SVD feature space rather than the Gaussian classification model. 1092 6 Padhraic Smyth, Usama Fayyad, Michael Burl, Pietro Perona, Pierre Baldi Conclusion Ignoring subjective uncertainty in image labelling can lead to significant overconfidence in terms of performance estimation (for both humans and machines). For the volcano detection task a simple model for uncertainty in the class labels provided insight into the performance of both human and algorithmic detectors. An obvious extension of the maximum likelihood framework outlined here is a Bayesian approach [12]: accounting for parameter uncertainty in the model given the limited amount of training data available is worth investigating. Acknowledgements The research described in this paper was carried out by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration and was supported in part by ARPA under grant number NOOOl4-92-J-1860. References 1. P. Smyth, "Learning with probabilistic supervision," in Computational Learning Theory and Natural Learning Systems 3, T. Petcshe, M. Kearns, S . Hanson, R. Rivest (eds), Cambridge, MA: MIT Press, to appear. 2. S. French, "Group consensus probability distributions: a critical survey," in Bayesian Statistics 2, J. M. Bernardo, M . H. DeGroot, D. V. Lindley, A. F. M. Smith (eds.), Elsevier Science Publishers, North-Holland, pp.183-202, 1985. 3. J . S. Uebersax, "Statistical modeling of expert ratings on medical treatment appropriateness," J. Amer. Statist. Assoc., vol.88, no.422, pp.421-427, 1993. 4. G. Lugosi, "Learning with an unreliable teacher," Pattern Recognition, vol. 25, no.1, pp.79-87. 1992. 5. Science, special issue on Magellan data, April 12, 1991. 6. J . C. Aubele and E . N. Slyuta, "Small domes on Venus: characteristics and origins," in Earth, Moon and Planets, 50/51, 493-532, 1990. 7. M. C. Burl, U. M . Fayyad, P . Perona, P . Smyth, and M. P . Burl, "Automating the hunt for volcanoes on Venus," in Proceedings of the 1994 Computer Vision and Pattern Recognition Conference: CVPR-94, Los Alamitos, CA: IEEE Computer Society Press, pp.302-309, 1994. 8. A. P . Dawid and A. M. Skene, "Maximum likelihood estimation of observer error-rates using the EM algorithm," Applied Statistics, vol.28, no.1, pp.2G-28, 1979. 9. P. Smyth, M. C . Burl, U. M. Fayyad, P . Perona, 'Knowledge discovery in large image databases: dealing with uncertainties in ground truth,' in Knowledge Discovery in Databases 2, U. M. Fayyad, G. Piatetsky-Shapiro, P . Smyth, R. Uthurasamy (eds.), AAAI/MIT Press, to appear, 1995. 10. M. S. Chesters, "Human visual perception and ROC methodology in medical imaging," Phys. Med. BioI., vol.37, no.7, pp.1433-1476, 1992. 11. M. C. Burl, U. M . Fayyad, P. Perona, P . Smyth, "Automated analysis of radar imagery of Venus: handling lack of ground truth," in Proceedings of the IEEE Conference on Image Processing, Austin, November 1994. 12. W . Buntine, "Operations for learning with graphical models," Journal of Artificial Intelligence Research, 2, pp.159-225, 1994. 

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305 ALVINN: AN AUTONOMOUS LAND VEHICLE IN A NEURAL NETWORK Dean A. Pomerleau Computer Science Department Carnegie Mellon University Pittsburgh, PA 15213 ABSTRACT ALVINN (Autonomous Land Vehicle In a Neural Network) is a 3-layer back-propagation network designed for the task of road following. Currently ALVINN takes images from a camera and a laser range finder as input and produces as output the direction the vehicle should travel in order to follow the road. Training has been conducted using simulated road images. Successful tests on the Carnegie Mellon autonomous navigation test vehicle indicate that the network can effectively follow real roads under certain field conditions. The representation developed to perfOIm the task differs dramatically when the networlc is trained under various conditions, suggesting the possibility of a novel adaptive autonomous navigation system capable of tailoring its processing to the conditions at hand. INTRODUCTION Autonomous navigation has been a difficult problem for traditional vision and robotic techniques, primarily because of the noise and variability associated with real world scenes. Autonomous navigation systems based on traditional image processing and pattern recognition techniques often perform well under certain conditions but have problems with others. Part of the difficulty stems from the fact that the processing performed by these systems remains fixed across various driving situations. Artificial neural networks have displayed promising performance and flexibility in other domains characterized by high degrees of noise and variability, such as handwritten character recognition [Jackel et al., 1988] [Pawlicki et al., 1988] and speech recognition [Waibel et al., 1988]. ALVINN (Autonomous Land Vehicle In a Neural Network) is a connectionist approach to the navigational task of road following. Specifically, ALVINN is an artificial neural network designed to control the NAVLAB, the Carnegie Mellon autonomous navigation test vehicle. NETWORK ARCmTECTURE ALVINN's current architecture consists of a single hidden layer back-propagation network 306 Pomerleau Road Intensity Feedback Unit 45 Direction Output Units 8x32 Range Finder Input Retina 30x32 Video Input Retina Figure 1: ALVINN Architecture (See Figure 1). The input layer is divided into three sets of units: two "retinas" and a single intensity feedback unit. The two retinas correspond to the two forms of sensory input available on the NAVLAB vehicle; video and range information. The first retina, consisting of 3002 units, receives video camera input from a road scene. The activation level of each unit in this retina is proportional to the intensity in the blue color band of the corresponding patch of the image. The blue band of the color image is used because it provides the highest contrast between the road and the non-road. The second retina, consisting of 8x32 units, receives input from a laser range finder. The activation level of each unit in this retina is proportional to the proximity of the corresponding area in the image. The road intensity feedback unit indicates whether the road is lighter or darker than the non-road in the previous image. Each of these 1217 input units is fully connected to the hidden layer of 29 units, which is in tum fully connected to the output layer. The output layer consists of 46 units, divided into two groups. The first set of 45 units is a linear representation of the tum curvature along which the vehicle should travel in order to head towards the road center. The middle unit represents the "travel straight ahead" condition while units to the left and right of the center represent successively sharper left and right turns. The network is trained with a desired output vector of all zeros except for a "hill" of activation centered on the unit representing the correct tum curvature, which is the curvature which would bring the vehicle to the road center 7 meters ahead of its current position. More specifically, the desired activation levels for ALVlNN: An Autonomous Land Vehicle in a Neural Network Real Road Image Simulated Road Image Figure 2: Real and simulated road images the nine units centered around the correct tum curvature unit are 0.10, 0.32, 0.61, 0.89, 1.00,0.89,0.61,0.32 and 0.10. During testing, the tum curvature dictated by the network is taken to be the curvature represented by the output unit with the highest activation level. The final output unit is a road intensity feedback unit which indicates whether the road is lighter or darker than the non-road in the current image. During testing, the activation of the output road intensity feedback unit is recirculated to the input layer in the style of Jordan [Jordan, 1988] to aid the network's processing by providing rudimentary infonnation concerning the relative intensities of the road and the non-road in the previous image. TRAINING AND PERFORMANCE Training on actual road images is logistically difficult, because in order to develop a general representation, the network must be presented with a large number of training exemplaIS depicting roads under a wide variety of conditions. Collection of such a data set would be difficult, and changes in parameters such as camera orientation would require collecting an entirely new set of road images. To avoid these difficulties we have developed a simulated road generator which creates road images to be used as training exemplars for the network. Figure 2 depicts the video images of one real and one artificial road. Although not shown in Figure 2, the road generator also creates corresponding simulated range finder images. At the relatively low resolution being used it is difficult to distinguish between real and simulated roads. NetwoIk: training is performed using these artificial road "snapshots" and the Warp back- 307 ? 308 Pomerleau Figure 3: NAVLAB, the eMU autonomous navigation test vehicle. propagation simulator described in [Pomerleau et al., 1988]. Training involves first creating a set of 1200 road snapshots depicting roads with a wide variety of retinal orientations and positions, under a variety of lighting conditions and with realistic noise levels. Back-propagation is then conducted using this set of exemplars until only asymptotic performance improvements appear likely. During the early stages of training, the input road intensity unit is given a random activation level. This is done to prevent the network from merely learning to copy the activation level of the input road intensity unit to the output road intensity unit, since their activation levels should almost always be identical because the relative intensity of the road and the non-road does not often change between two successive images. Once the network has developed a representation that uses image characteristics to determine the activation level for the output road intensity unit, the network is given as input whether the road would have been darker or lighter than the non-road in the previous image. Using this extra information concerning the relative brightness of the road and the non-road, the network is better able to determine the correct direction for the vehicle to trave1. After 40 epochs of training on the 1200 simulated road snapshots, the network correctly dictates a tum curvature within two units of the correct answer approximately 90% of the time on novel simulated road images. The primary testing of the ALVINN's performance has been conducted on the NAVLAB (See Figure 3). The NAVLAB is a modified Chevy van equipped with 3 Sun computers, a Warp, a video camera, and a laser range finder, which serves as a testbed for the CMU autonomous land vehicle project [Thorpe et al., 1987]. Performance of the network to date is comparable to that achieved by the best traditional vision-based autonomous navigation algorithm at CMU under the limited conditions tested. Specifically, the network can accurately drive the NAVLAB at a speed of 1/2 meter per second along a 400 meter path through a wooded ALVINN: An Autonomous Land Vehicle in a Neural Network Weights to Direction Output Units t~li'C jjllil Weight to Output Feedback Unit D Weights from Video Camera Retina Road 1 Edges Weight from Input Feedback Unit n Weight from Bias Unit ? Weights from Range Finder Retina Excitatory Periphery Connections Inhibitory Central Connections Figure 4: Diagram of weights projecting to and from a typical hidden unit in a network trained on roads with a fixed width. The schematics on the right are aids for interpretation. area of the CMU campus under sunny fall conditions. Under similar conditions on the same course, the ALV group at CMU has recently achieved similar driving accuracy at a speed of one meter per second by implementing their image processing autonomous navigation algorithm on the Watp computer. In contrast, the ALVINN network is currently simulated using only an on-boani Sun computer, and dramatic speedups are expected when tests are perfonned using the Watp. NETWORK REPRESENTATION The representation developed by the network to perfonn the road following task depends dramatically on the characteristics of the training set. When trained on examples of roads with a fixed width, the network develops a representations consisting of overlapping road filters. Figure 4 is a diagram of the weights projecting to and from a single hidden unit . in such a network. As indicated by the weights to and from the feedback units, this hidden unit expects the road to be lighter than the non-road in the previous image and supports the road being lighter than the non-road in the current image. More specifically, the weights from the 309 310 Pomerleau Weights to Direction Output Units 1::tllII:UIll t k? Weight to Output Feedback Unit - mDIRoad ~ N on-road ~ Weights from Video Camera Retina . ' : ' .' : .. Weight from Input Feedback Unit . : ' : .' 11111111 1111111 I[ Weight from Bias Unit 111111111111 ? .: . Weights from Range Finder Retina Figure 5: Diagram of weights projecting to and from a typical hidden unit in a network trained on roads with different widths. video camera retina support the interpretation that this hidden unit is a filter for two light roads, one slightly left and the other slightly right or center (See schematic to the right of the weights from the video retina in Figure 4). This interpretation is also supported by the weights from the range finder retina, which has a much wider field of view than the video camera. This hidden unit is excited if there is high range activity (Le. obstacles) in the periphery and inhibited if there is high range activity in the central region of the scene where this hidden unit expects the road to be (See schematic to the right of the weights from the range finder retina in Figure 4). Finally, the two road filter interpretation is reflected in the weights from this hidden unit to the direction output units. Specifically, this hidden unit has two groups of excitatory connections to the output units, one group dictating a slight left turn and the other group dictating a slight right turn. Hidden units which act as filters for 1 to 3 roads are the representation structures most commonly developed when the network is trained on roads with a fixed width. The network develops a very different representation when trained on snapshots with widely varying road widths. A typical hidden unit from this type of representation is depicted in figure 5. One important feature to notice from the feedback weights is that this unit is filtering for a road which is darlcer than the non-road. More importantly, it is evident from the video camera retina weights that this hidden unit is a filter solely for the left edge of the road (See schematic to the right of the weights from the range finder ALVINN: An Autonomous Land Vehicle in a Neural Network retina in Figure 5). This hidden unit supports a rather wide range of travel directions. This is to be expected, since the correct travel direction for a road with an edge at a particular location varies substantially depending on the road's width. This hidden unit would cooperate with hidden units that detect the right road edge to determine the correct travel direction in any particular situation. DISCUSSION AND EXTENSIONS The distinct representations developed for different circumstances illustrate a key advantage provided by neural networks for autonomous navigation. Namely, in this paradigm the data, not the programmer, determines the salient image features crucial to accurate road navigation. From a practical standpoint, this data responsiveness has dramatically sped ALVINN's development. Once a realistic artificial road generator was developed, back-propagation producted in half an hour a relatively successful road following system. It took many months of algorithm development and parameter tuning by the vision and autonomous navigation groups at CMU to reach a similar level of performance using traditional image processing and pattern recognition techniques. More speculatively, the flexibility of neural network representations provides the possibility of a very different type of autonomous navigation system in which the salient sensory features are determined for specific driving conditions. By interactively training the network on real road images taken as a human drives the NAVLAB, we hope to develop a system that adapts its processing to accommodate current circumstances. This is in contrast with other autonomous navigation systems at CMU [Thorpe et al., 1987] and elsewhere [Dunlay & Seida, 1988] [Dickmanns & Zapp, 1986] [Kuan et al., 1988]. Each of these implementations has relied on a fixed, highly structured and therefore relatively inflexible algorithm for finding and following the road, regardless of the conditions at hand. There are difficulties involved with training "on-the-fly" with real images. If the network is not presented with sufficient variability in its training exemplars to cover the conditions it is likely to encounter when it takes over driving from the human operator, it will not develop a sufficiently robust representation and will perform poorly. In addition, the network must not solely be shown examples of accurate driving, but also how to recover (i.e. return to the road center) once a mistake has been made. Partial initial training on a variety of simulated road images should help eliminate these difficulties and facilitate better performance. Another important advantage gained through the use of neural networks for autonomous navigation is the ease with which they assimilate data from independent sensors. The current ALVINN implementation processes data from two sources, the video camera and the laser range finder. During training, the network discovers how information from each source relates to the task, and weights each accordingly. As an example, range data is in some sense less important for the task of road following than is the video data. The range data contains information concerning the position of obstacles in the scene, but nothing explicit about the location of the road. As a result, the range data is given less significance in the representation, as is illustrated by the relatively small 311 312 Pomerleau magnitude weights from the range finder retina in the weight diagrams. Figures 4 and 5 illustrate that the range finder connections do correlate with the connections from the video camera, and do contribute to choosing the correct travel direction. Specifically, in both figures, obstacles located outside the area in which the hidden unit expects the road to be located increase the hidden unit's activation level while obstacles located within the expected road boundaries inhibit the hidden unit. However the contributions from the range finger connections aren't necessary for reasonable performance. When ALVINN was tested with normal video input but an obstacle-free range finder image as constant input, there was no noticeable degradation in driving performance. Obviously under off-road driving conditions obstacle avoidance would become much more important and hence one would expect the range finder retina to playa much more significant role in the network's representation. We are currently working on an off-road version of ALVINN to test this hypothesis. Other current directions for this project include conducting more extensive tests of the network's performance under a variety of weather and lighting conditions. These will be crucial for making legitimate performance comparisons between ALVINN and other autonomous navigation techniques. We are also working to increase driving speed by implementing the network simulation on the on-board Warp computer. Additional extensions involve exploring different network architectures for the road following task. These include 1) giving the network additional feedback information by using Elman's [Elman, 1988] technique of recirculating hidden activation levels, 2) adding a second hidden layer to facilitate better internal representations, and 3) adding local connectivity to give the network a priori knowledge of the two dimensional nature of the input In the area of planning, interesting extensions include stopping for, or planning a path around, obstacles. One area of planning that clearly needs work is dealing sensibly with road forks and intersections. Currently upon reaching a fork, the network may output two widely discrepant travel directions, one for each choice. The result is often an oscillation in the dictated travel direction and hence inaccurate road following. Beyond dealing with individual intersections, we would eventually like to integrate a map into the system to enable global point-to-point path planning. CONCLUSION More extensive testing must be performed before definitive conclusions can be drawn concerning the peiformance of ALVINN versus other road followers. We are optimistic concerning the eventual contributions neural networks will make to the area of autonomous navigation. But perhaps just as interesting are the possibilities of contributions in the other direction. We hope that exploring autonomous navigation, and in particular some of the extensions outlined in this paper, will have a significant impact on the field of neural networks. We certainly believe it is important to begin researching and evaluating neural networks in real world situations, and we think autonomous navigation is an interesting application for such an approach. ALVINN: An Autonomous Land Vehicle in a Neural Network Acknowledgements This work would not have been possible without the input and support provided by Dave Touretzky, Joseph Tebelskis, George Gusciora and the CMU Warp group, and particularly Charles Thorpe, Till Crisman, Martial Hebert, David Simon, and rest of the CMU ALV group. This research was supported by the Office of Naval Research under Contracts NOOOI4-87-K-0385, NOOOI4-87-K-0533 and NOOOI4-86-K-0678, by National Science Foundation Grant EET-8716324, by the Defense Advanced Research Projects Agency (DOD) monitored by the Space and Naval Warfare Systems Command under Contract NOOO39-87-C-0251, and by the Strategic Computing Initiative of DARPA, through ARPA Order 5351, and monitored by the U.S. Army Engineer Topographic Laboratories under contract DACA76-85-C-0003 titled "Road Following". References [Dickmanns & Zapp, 1986] Dickmanns, E.D., Zapp, A. (1986) A curvature-based scheme for improving road vehicle guidance by computer vision. "Mobile Robots", SPIE-Proc. Vol. 727, Cambridge, MA. [Elman, 1988] Elman, J.L, (1988) Finding structure in time. Technical report 8801. Center for Research in Language, University of California, San Diego. [Dunlay & Seida, 1988] Dunlay, R.T., Seida, S. (1988) Parallel off-road perception processing on the ALV. Proc. SPIE Mobile Robot Conference, Cambridge MA. [Jackel et al., 1988] Jackel, L.D., Graf, H.P., Hubbard, W., Denker, J.S., Henderson, D., Guyon, 1. (1988) An application of neural net chips: Handwritten digit recognition. Proceedings of IEEE International Conference on Neural Networks, San Diego, CA. [Jordan, 1988] Jordan, M.l. (1988) Supervised learning and systems with excess degrees of freedom. COINS Tech. Report 88-27, Computer and Infolll1ation Science, University of Massachusetts, Amherst MA. [Kuan et al., 1988] Kuan, D., Phipps, G. and Hsueh, A.-C. Autonomous Robotic Vehicle Road Following. IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 10, Sept. 1988. [pawlicki et al., 1988] Pawlicki, T.E, Lee, D.S., Hull, J.J., Srihari, S.N. (1988) Neural network models and their application to handwritten digit recognition. Proceedings of IEEE International Conference on Neural Networks, San Diego, CA. [pomerleau et al., 1988] Pomerleau, D.A., Gusciora, G.L., Touretzky, D.S., and Kung, H.T. (1988) Neural network simulation at Waq> speed: How we got 17 million connections per second. Proceedings of IEEE International Conference on Neural Networks, San Diego, CA. [Th0IJle et al., 1987] Thorpe, c., Herbert, M., Kanade, T., Shafer S. and the members of the Strategic Computing Vision Lab (1987) Vision and navigation for the Carnegie Mellon NAVLAB. Annual Revi~ of Computer Science Vol. 11, Ed. Joseph Traub, Annual Reviews Inc., Palo Alto, CA. [Waibel et al., 1988] Waibel, A, Hanazawa, T., Hinton, G., Shikano, K., Lang, K. (1988) Phoneme recognition: Neural Networks vs. Hidden Markov Models. Proceedings from Int. Conf. on Acoustics, Speech and Signal Processing, New York, New York. 313 

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Using Voice Transformations to Create Additional Training Talkers for Word Spotting Eric I. Chang and Richard P. Lippmann MIT Lincoln Laboratory Lexington, MA 02173-0073, USA eichang@sst.ll.mit.edu and rpl@sst.ll.mit.edu Abstract Speech recognizers provide good performance for most users but the error rate often increases dramatically for a small percentage of talkers who are "different" from those talkers used for training. One expensive solution to this problem is to gather more training data in an attempt to sample these outlier users. A second solution, explored in this paper, is to artificially enlarge the number of training talkers by transforming the speech of existing training talkers. This approach is similar to enlarging the training set for OCR digit recognition by warping the training digit images, but is more difficult because continuous speech has a much larger number of dimensions (e.g. linguistic, phonetic, style, temporal, spectral) that differ across talkers. We explored the use of simple linear spectral warping to enlarge a 48-talker training data base used for word spotting. The average detection rate overall was increased by 2.9 percentage points (from 68.3% to 71.2%) for male speakers and 2.5 percentage points (from 64.8% to 67.3%) for female speakers. This increase is small but similar to that obtained by doubling the amount of training data. 1 INTRODUCTION Speech recognizers, optical character recognizers, and other types of pattern classifiers used for human interface applications often provide good performance for most users. Performance is often, however, low and unacceptable for a small percentage of "outlier" users who are presumably not represented in the training data. One expensive solution to this problem is to obtain more training data in the hope of including users from these outlier 876 Eric I. Chang, Richard P. Lippmann classes. Other approaches already used for speech recognition are to use input features and distance metrics that are relatively invariant to linguistically unimportant differences between talkers and to adapt a recognizer for individual talkers. Talker adaptation is difficult for word spotting and with poor outlier users because the recognition error rate is high and talkers often can not be prompted to recite standard phrases that can be used for adaptation. An alternative approach, that has not been fully explored for speech recognition, is to artificially expand the number of training talkers using voice transformations. Transforming the speech of one talker to make it sound like that of another is difficult because speech varies across many difficult-to-measure dimensions including linguistic, phonetic, duration, spectra, style, and accent. The transformation task is thus more difficult than in optical character recognition where a small set of warping functions can be successfully applied to character images to enlarge the number of training images (Drucker, 1993). This paper demonstrates how a transformation accomplished by warping the spectra of training talkers to create more training data can improve the performance of a whole-word word spotter on a large spontaneous-speech data base. 2 BASELINE WORD SPOTTER A hybrid radial basis function (RBF) - hidden Markov model (HMM) keyword spotter has been developed over the past few years that provides state-of-the-art performance for a whole-word word spotter on the large spontaneous-speech credit-card speech corpus. This system spots 20 target keywords, includes one general filler class, and uses a Viterbi decoding backtrace as described in (Chang, 1994) to backpropagate errors over a sequence of input speech frames. This neural network word spotter is trained on target and background classes, normalizes target outputs using the background output, and thresholds the resulting score to generate putative hits, as shown in Figure 1. Putative hits in this figure are input patterns which generate normalized scores above a threshold. The performance of this, and other spotting systems, is analyzed by plotting a detection versus false alarm rate curve. This curve is generated by adjusting the classifier output threshold to allow few or many putative hits. The figure of merit (FOM) is defined as the average keyword detection rate when the false alarm rate ranges from 1 to 10 false alarms per keyword per hour. The previous best FOM for this word spotter is 67.8% when trained using 24 male talkers and tested on 11 male talkers, and 65.9% when trained using 24 female talkers and tested on 11 female talkers. The overall FOM for all talkers is 66.3%. ATIVE HITS CONTINUOUS SPEECH INPUT Figure 1: NEURAL NETWORK WORDSPOTTER Block diagram of neural network word spotter. Using Voice Transfonnations to Create Additional Training Talkers for Word Spotting 877 3 TALKER VARIABILITY FOM scores of test talkers vary over a wide range. When training on 48 talkers and then performing testing on 22 talkers from the 70 conversations in the NIST Switchboard credit card database, the FOM ofthe test talkers varies from 16.7% to 100%. Most talkers perform well above 50%, but there are two female talkers with FOM's of 16.7% and 21.4%. The low FOM for individual speakers indicates a lack of training data with voice qualities that are similar to these test speakers. 4 CREATING MORE TRAINING DATA USING VOICE TRANSFORMATIONS Talker adaptation is difficult for word spotting because error rates are high and talkers often can not be prompted to verify adaptation phrases. Our approach to increasing performance across talkers uses voice transformation techniques to generate more varied training examples of keywords as shown in Figure 2. Other researchers have used talker transformation techniques to produce more natural synthesized speech (lwahashi, 1994, Mizuno, 1994), but using talker transformation techniques to generate more training data is novel. We have implemented a new voice transformation technique which utilizes the Sinusoidal Transform Analysis/Synthesis System (STS) described in (Quatieri, 1992). This technique attempts to transform one talker's speech pattern to that of a different talker. The STS generates a 512 point spectral envelope of the input speech 100 times a second and also separates pitch and voicing information. Separation of vocal tract characteristic and pitch information has allowed the implementation of pitch and time transformations in previous work (Quatieri, 1992). The system has been modified to generate and accept a spectral en- VOICE TRANSFORMATION SYSTEM ORIGINAL SPEECH TRANSFORMED SPEECH Figure 2: Generating more training data by artificially transforming original speech training data. 878 Eric I. Chang, Richard P. Lippmann velo.pe file fro.m an input speech sample. We info.nnally explo.red different techniques to' transfo.nn the spectral envelo.pe to' generate mo.re varied training examples by listening to' transfo.nned speech. This resulted in the fo.llo.wing algo.rithm that transfo.nns a talker's vo.ice by scaling the spectral envelo.pe o.f training talkers. Training co.nversatio.ns are upsampled fro.m 8000 Hz to' 10,000 Hz to' be co.mpatible with existing STS co.ding so.ftware. 1. 2. The STS system pro.cesses the upsampled files and generates a 512 point spectral envelo.pe o.f the input speech wavefo.rm at a frame rate o.f 100 frames a seco.nd and with a windo.w length o.f appro.ximately 2.5 times the length o.f each pitch perio.d. 3. A new spectral envelo.pe is generated by linearly expanding o.r co.mpressing the spectral axis. Each spectral po.int is identified by its index, ranging fro.m 0 to' 511. To. transfo.rm a spectral pro.file by 2, the new spectral value at frequency f is generated by averaging the spectral values aro.und the o.riginal spectral pro.file at frequency o.f 0.5 f The transfo.nnatio.n process is illustrated in Figure 3. In this figure, an o.riginal spectral envelo.pe is being expanded by two.. The spectral value at index 150 is thus transfo.rmed to. spectral index 300 in the new envelo.pe and the o.riginal spectral info.rmatio.n at high frequencies is lo.st. 4. The transfo.rmed spectral value is used to. resynthesize a speech wavefo.rm using the vocal tract excitatio.n info.nnatio.n extracted fro.m the o.riginal file. Vo.ice transfo.rmatio.n with the STS co.der allo.ws listening to. transfo.rmed speech but requires lo.ng co.mputatio.n. We simplified o.ur appro.ach to' o.ne o.f mo.difying the spectral scale in the spectral do.main directly within a mel-scale filterbank analysis pro.gram. The inco.ming speech sample is pro.cessed with an FFT to' calculate spectral magnitudes. Then spectral magnitudes are linearly transfo.rmed. Lastly mel-scale filtering is perfo.rmed with 10 linearly spaced filters up to' 1000 Hz and lo.garithmically spaced filters fro.m 1000 Hz up. A co.sine transfo.rm is then used to' generate mel-scaled cepstral values that are used by the wo.rdspo.tter. Much faster pro.cessing can be achieved by applying the spectral transfo.rmatio.n as part o.f the filterbank analysis. Fo.r example, while perfo.rming spectral transfo.nnatio.n using the STS algo.rithm takes up to' approximately 10 times real time, spectral transfo.rmatio.n within the mel-scale filterbank pro.gram can be acco.mplished within 1110 real time o.n a Sparc 10 wo.rkstatio.n. The rapid pro.cessing rate allo.ws o.n-line spectral transfo.rmatio.n. 5 WORD SPOTTING EXPERIMENTS Linear warping in the spectral do.main, which is used in the above algo.rithm, is co.rrect when the vo.cal tract is mo.delled as a series o.f lo.ssless aco.ustic tubes and the excitatio.n so.urce is at o.ne end o.f the vo.cal tract (Wakita, 1977). Wakita sho.wed that if the vo.cal tract is mo.delled as a series o.f equal length, lo.ssless, and co.ncatenated aco.ustic tubes, then the ratio. o.f the areas between the tubes determines the relative reso.nant frequencies o.f the vo.cal tract, while the o.verall length o.fthe vo.cal tract linearly scales fo.rmant frequencies. Preliminary research was co.nducted using linear scaling with spectral ratio.s ranging fro.m 0.6 to. 1.8 to. alter test utterances. After listening to. the STS transfo.rmed speech and also. o.bserving Using Voice Transformations to Create Additional Training Talkers for Word Spotting Original Spectral Envelope 879 III I I o 150 511 " Transformed Spectral Envelope o 300 511 Spectral Value Index Figure 3: An example of the spectral transformation algorithm where the original spectral envelope frequency scale is expanded by 2. spectrograms of the transformed speech, it was found that speech transformed using ratios between 0.9 and 1.1 are reasonably natural and can represent speech without introducing artifacts. Using discriminative training techniques such as FOM training carries the risk of overtraining the wordspotter on the training set and obtaining results that are poor on the testing set. To delay the onset of overtraining, we artificially transformed each training set conversation during each epoch using a different random linear transformation ratio. The transformation ratio used for each conversation is calculated using the following formula: ratio == a + N (0,0.06), where a is the transformation ratio that matches each training speaker to the average of the training set speakers, and N is a normally distributed random variable with a mean of 0.0 and standard deviation of 0.06. For each training conversation, the long term averages of formant frequencies for formant 1, 2, and 3 are calculated. A least square estimation is then performed to match the formant frequencies of each training set conversation to the group average formant frequencies. The transformation equation is described below: 880 Eric I. Chang, Richard P. Lippmann 100 ? 90 ? ? ? t ? :--! ~ 80 o - a- ?? .,. ? ? NORMAL 50 Figure 4: o 234 5 678 9 10 NUMBER OF EPOCHS OF FOM TRAINING Average detection accuracy (FOM) for the male training and testing set versus the number of epochs of FOM training. The transform ratio for each individual conversation is calculated to improve the naturalness of the transformed speech. In preliminary experiments, each conversation was transformed with fixed ratios of 0.9, 0.95, 1.05, and 1.1. However, for a speaker with already high formant frequencies, pushing the formant frequencies higher may make the transformed speech sound unnatural. By incorporating the individual formant matching ratio into the transformation ratio, speakers with high formant frequencies are not transformed to very high frequencies and speakers with low formant frequencies are not transformed to even lower formant frequency ranges. Male and female conversations from the NIST credit card database were used separately to train separate word spotters. Both the male and the female partition of data used 24 conversations for training and 11 conversations for testing. Keyword occurrences were extracted from each training conversation and used as the data for initialization of the neural network word spotter. Also, each training conversation was broken up into sentence length segments to be used for embedded reestimation in which the keyword models are joined with the filler models and the parameters of all the models are jointly estimated. After embedded reestimation, Figure of Merit training as described in (Lippmann. 1994) was performed for up to 10 epochs. During each epoch. each training conversation is transformed using a transform ratio randomly generated as described above. The performance of the word spotter after each iteration of training is evaluated on both the training set and the testing set. 6 WORD SPOTTING RESULTS Training and testing set FOM scores for the male speakers and the female speakers are shown in Figure 4 and Figure 5 respectively. The x axis plots the number of epochs of FOM training where each epoch represents presenting all 24 training conversations once. The FOM for word spotters trained with the normal training conversations and word spotters Using Voice Transformations to Create Additional Training Talkers for Word Spotting 881 100 90 -?j ~ - =I - -. " - ? .- au ? .. - ra- ". TRAIN ? 80 60 ~= ? \ 50 o NORMAL 23456789 10 NUMBER OF EPOCHS OF FOM TRAINING Figure 5: Average detection accuracy (FOM) for the female training and testing set versus the number of iterations of FOM training. trained with artificially expanded training conversations are shown in each plot. After the first epoch, the FOM improves significantly. With only the original training conversations (normal), the testing set FOM rapidly levels off while the training set FOM keeps on improving. When the training conversations are artificially expanded, the training set FOM is below the training set FOM from the normal training set due to more difficult training data. However, the testing set FOM continues to improve as more epochs of FOM training are performed. When comparing the FOM of wordspotters trained on the two sets of data after ten epochs of training, the FOM for the expanded set was 2.9 percentage points above the normal FOM for male speakers and 2.5 percentage points above the normal FOM for female speakers. For comparison, Carlson has reported that for a high performance word spotter on this database, doubling the amount of training data typically increases the FOM by 2 to 4 percentage points (Carlson, 1994). 7 SUMMARY Lack of training data has always been a constraint in training speech recognizers. This research presents a voice transformation technique which increases the variety among training talkers. The resulting more varied training set provided up to 2.9 percentage points of improvement in the figure of merit (average detection rate) of a high performance word spotter. This improvement is similar to the increase in performance provided by doubling the amount of training data (Carlson, 1994). This technique can also be applied to other speech recognition systems such as continuous speech recognition, talker identification, and isolated speech recognition. 882 Eric I. Chang. Richard P. Lippmcmn ACKNOWLEDGEMENT This work was sponsored by the Advanced Research Projects Agency. The views expressed are those of the authors and do not reflect the official policy or position of the U.S. Government. We wish to thank Tom Quatieri for providing his sinusoidal transform analysis/synthesis system. BIBLIOGRAPHY B. Carlson and D. Seward. (1994) Diagnostic Evaluation and Analysis of Insufficient and Task-Independent Training Data on Speech Recognition. In Proceedings Speech Research Symposium XlV, Johns Hopkins University. E. Chang and R. Lippmann. (1994) Figure of Merit Training for Detection and Spotting. In Neural Information Processing Systems 6, G. Tesauro, J. Cohen, and J. Alspector, (Eds.), Morgan Kaufmann: San Mateo, CA. H. Drucker, R. Schapire, and P. Simard. (1993) Improving Performance in Neural Networks Using a Boosting Algorithm. In Neural Information Processing Systems 5, S. Hanson, J. Cowan, and C. L. Giles, (Eds.), Morgan Kaufmann: San Mateo, California. N. Iwahashi and Y. Sagisaka. (1994) Speech Spectrum Transformation by Speaker Interpolation. In Proceedings International Conference on Acoustics Speech and Signal Processing, Vol. 1,461-464. R. Lippmann, E. Chang & C. Jankowski. (1994) Wordspotter Training Using Figure-ofMerit Back Propagation. In Proceedings of International Conference on Acoustics Speech and Signal Processing, Vol. 1,389-392. H. Mizuno and M. Abe. (1994) Voice Conversion Based on Piecewise Linear Conversion Rules of Formant Frequency and Spectrum Tilt. In Proceedings International Conference Oil Acoustics Speech and Signal Processing, Vol. 1,469-472. T. Quatieri and R. McAulay. (1992) Shape Invariant Time-Scale and Pitch Modification of Speech. In IEEE Trans. Signal Processing, vol 40, no 3. pp. 497-510. Hisashi Wakita. (1977) Normalization of Vowels by Vocal-Tract Length and Its Application to Vowel Identification. In IEEE Trans. Acoustics, Speech, and Signal Processing, vol. ASSP-25, No.2., pp. 183-192. 

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Reinforcement Learning Algorithm for Partially Observable Markov Decision Problems Tommi Jaakkola tommi@psyche.mit.edu Satinder P. Singh singh@psyche.mit.edu Michael I. Jordan jordan@psyche.mit.edu Department of Brain and Cognitive Sciences, BId. E10 Massachusetts Institute of Technology Cambridge, MA 02139 Abstract Increasing attention has been paid to reinforcement learning algorithms in recent years, partly due to successes in the theoretical analysis of their behavior in Markov environments. If the Markov assumption is removed, however, neither generally the algorithms nor the analyses continue to be usable. We propose and analyze a new learning algorithm to solve a certain class of non-Markov decision problems. Our algorithm applies to problems in which the environment is Markov, but the learner has restricted access to state information. The algorithm involves a Monte-Carlo policy evaluation combined with a policy improvement method that is similar to that of Markov decision problems and is guaranteed to converge to a local maximum. The algorithm operates in the space of stochastic policies, a space which can yield a policy that performs considerably better than any deterministic policy. Although the space of stochastic policies is continuous-even for a discrete action space-our algorithm is computationally tractable. 346 1 Tommi Jaakkola, Satinder P. Singh, Michaell. Jordan INTRODUCTION Reinforcement learning provides a sound framework for credit assignment in unknown stochastic dynamic environments. For Markov environments a variety of different reinforcement learning algorithms have been devised to predict and control the environment (e.g., the TD(A) algorithm of Sutton, 1988, and the Q-Iearning algorithm of Watkins, 1989). Ties to the theory of dynamic programming (DP) and the theory of stochastic approximation have been exploited, providing tools that have allowed these algorithms to be analyzed theoretically (Dayan, 1992; Tsitsiklis, 1994; Jaakkola, Jordan, & Singh, 1994; Watkins & Dayan, 1992). Although current reinforcement learning algorithms are based on the assumption that the learning problem can be cast as Markov decision problem (MDP), many practical problems resist being treated as an MDP. Unfortunately, if the Markov assumption is removed examples can be found where current algorithms cease to perform well (Singh, Jaakkola, & Jordan, 1994). Moreover, the theoretical analyses rely heavily on the Markov assumption. The non-Markov nature of the environment can arise in many ways. The most direct extension of MDP's is to deprive the learner of perfect information about the state of the environment. Much as in the case of Hidden Markov Models (HMM's), the underlying environment is assumed to be Markov, but the data do not appear to be Markovian to the learner. This extension not only allows for a tractable theoretical analysis, but is also appealing for practical purposes. The decision problems we consider here are of this type. The analog of the HMM for control problems is the partially observable Markov decision process (POMDP; see e.g., Monahan, 1982). Unlike HMM's, however, there is no known computationally tractable procedure for POMDP's. The problem is that once the state estimates have been obtained, DP must be performed in the continuous space of probabilities of state occupancies, and this DP process is computationally infeasible except for small state spaces. In this paper we describe an alternative approach for POMDP's that avoids the state estimation problem and works directly in the space of (stochastic) control policies. (See Singh, et al., 1994, for additional material on stochastic policies.) 2 PARTIAL OBSERVABILITY A Markov decision problem can be generalized to a POMDP by restricting the state information available to the learner. Accordingly, we define the learning problem as follows. There is an underlying MDP with states S {SI, S2, ... , SN} and transition probability pOII! " the probability of jumping from state S to state s' when action a is taken in state s. For every state and every action a (random) reward is provided to the learner. In the POMDP setting, the learner is not allowed to observe the state directly but only via messages containing information about the state. At each time step t an observable message mt is drawn from a finite set of messages according to an unknown probability distribution P(mlst) 1. We assume that the learner does = 1 For simplicity we assume that this distribution depends only on the current state. The analyses go through also with distributions dependent on the past states and actions Reinforcement Learning Algorithm for Markov Decision Problems 347 not possess any prior information about the underlying MDP beyond the number of messages and actions. The goal for the learner is to come up with a policy-a mapping from messages to actions-that gives the highest expected reward. As discussed in Singh et al. (1994), stochastic policies can yield considerably higher expected rewards than deterministic policies in the case of POMDP's. To make this statement precise requires an appropriate technical definition of "expected reward," because in general it is impossible to find a policy, stochastic or not, that maximizes the expected reward for each observable message separately. We take the timeaverage reward as a measure of performance, that is, the total accrued reward per number of steps taken (Bertsekas, 1987; Schwartz, 1993). This approach requires the assumption that every state of the underlying controllable Markov chain is reachable. In this paper we focus on a direct approach to solving the learning problem. Direct approaches are to be compared to indirect approaches, in which the learner first identifies the parameters of the underlying MDP, and then utilizes DP to obtain the policy. As we noted earlier, indirect approaches lead to computationally intractable algorithms. Our approach can be viewed as providing a generalization of the direct approach to MDP's to the case of POMDP's. 3 A MONTE-CARLO POLICY EVALUATION Advantages of Monte-Carlo methods for policy evaluation in MDP's have been reviewed recently (Barto and Duff, 1994). Here we present a method for calculating the value of a stochastic policy that has the flavor of a Monte-Carlo algorithm. To motivate such an approach let us first consider a simple case where the average reward is known and generalize the well-defined MDP value function to the POMDP setting. In the Markov case the value function can be written as (cf. Bertsekas, 1987): N V(s) = N-oo~ lim ~ E{R(st, Ut) - Risl = s} (1) t=l where St and at refer to the state and the action taken at the tth step respectively. This form generalizes easily to the level of messages by taking an additional expectation: (2) V(m) = E {V(s)ls -+ m} where s -+ m refers to all the instances where m is observed in sand E{?ls -+ m} is a Monte-Carlo expectation. This generalization yields a POMDP value function given by (3) V(m) = P(slm)V(s) I: ,Em in which P(slm) define the limit occupancy probabilities over the underlying states for each message m. As is seen in the next section value functions of this type can be used to refine the currently followed control policy to yield a higher average reward. Let us now consider how the generalized value functions can be computed based on the observations. We propose a recursive Monte-Carlo algorithm to effectively compute the averages involved in the definition of the value function. In the simple 348 Tommi Jaakkola, Satinder P. Singh, Michael I. Jordan case when the average payoff is known this algorithm is given by f3t(m) Xt(m) Xt(m) (1 - Kt(m) htf3t-l(m) + Kt(m) lIt(m) Xt(m) (1- Kt(m))lIt-l(m) + f3t(m)[R(st, at) - (4) R] (5) where Xt(m) is the indicator function for message m, Kt(m) is the number of times m has occurred, and 'Yt is a discount factor converging to one in the limit. This algorithm can be viewed as recursive averaging of (discounted) sample sequences of different lengths each of which has been started at a different occurrence of message m . This can be seen by unfolding the recursion, yielding an explicit expression for lit (m). To this end, let tk denote the time step corresponding to the ph occurrence of message m and for clarity let Rt = R(st, Ut) - R for every t. Using these the recursion yields: 1 lIt(m) = Kt(m) [ Rtl + rl,l R t1 +1 + ... + r1,t-tl Rt (6) where we have for simplicity used rk ,T to indicate the discounting at the Tth step in the kth sequence. Comparing the above expression to equation 1 indicates that the discount factor has to converge to one in the limit since the averages in V(s) or V(m) involve no discounting . To address the question of convergence of this algorithm let us first assume a constant discounting (that is, 'Yt 'Y < 1). In this case, the algorithm produces at best an approximation to the value function. For large K(m) the convergence rate by which this approximate solution is found can be characterized in terms of the bias and variance. This gives Bias{V(m)} Q( (1 - r)-l / K(m) and Var{V(m)} Q( (1 r)-2 / K(m) where r = Ehtk-tk-l} is the expected effective discounting between observations. Now, in order to find the correct value function we need an appropriate way of letting 'Yt - 1 in the limit. However, not all such schedules lead to convergent algorithms; setting 'Yt = 1 for all t , for example, would not. By making use of the above bounds a feasible schedule guaranteeing a vanishing bias and variance can be found . For instance, since 'Y > 7 we can choose 'Yk(m) = 1 - K(m)I/4. Much faster schedules are possible to obtain by estimating r. = Let us now revise the algorithm to take into account the fact that the learner in fact has no prior knowledge of the average reward. In this case the true average reward appearing in the above algorithm needs to be replaced with an incrementally updated estimate R t - l . To improve the effect this changing estimate has on the values we transform the value function whenever the estimate is updated. This transformation is given by Xt(m) (1 - Kt(m) )Ct-1(m) + f3t(m) (7) (8) lIt(m) - Ct(m)(Rt - Rt-l) and, as a result, the new values are as if they had been computed using the current estimate of the average reward. Reinforcement Learning Algorithm for Markov Decision Problems 349 To carry these results to the control setting and assign a figure of merit to stochastic policies we need a quantity related to the actions for each observed message. As in the case of MDP's, this is readily achieved by replacing m in the algorithm just described by (m, a). In terms of equation 6, for example, this means that the sequences started from m are classified according to the actions taken when m is observed. The above analysis goes through when m is replaced by (m, a), yielding "Q-values" on the level of messages: (9) In the next section we show how these values can be used to search efficiently for a better policy. 4 POLICY IMPROVEMENT THEOREM Here we present a policy improvement theorem that enables the learner to search efficiently for a better policy in the continuous policy space using the "Q-values" Q(m, a) described in the previous section. The theorem allows the policy refinement to be done in a way that is similar to policy improvement in a MDP setting. Theorem 1 Let the current stochastic policy 1I"(alm) lead to Q-values Q1r(m, a) on the level of messages. For any policy 11"1 (aim) define J1r 1 (m) = L: 11"1 (alm)[Q1r(m, a) - V 1r (m)] a The change in the average reward resulting from changing the current policy according to 1I"(alm) -+ (1- {)1I"(alm) + {1I"1(alm) is given by ~R1r = {L: p1r(m)J1r 1 (m) + O({2) m where p1r (m) are the occupancy probabilities for messages associated with the current policy. The proof is given in Appendix. In terms of policy improvement the theorem can be interpreted as follows . Choose the policy 1I"1(alm) such that 1 J1r (m) = max[Q1r(m, a) a V 1r (m)] (10) If now J1r 1 (m) > 0 for some m then we can change the current policy towards 11"1 and expect an increase in the average reward as shown by the theorem . The { factor suggests local changes in the policy space and the policy can be refined until m~l J1r\ m) 0 for all m which constitutes a local maximum for this policy improvement method. Note that the new direction 1I"1(alm) in the policy space can be chosen separately for each m. = 5 THE ALGORITHM Based on the theoretical analysis presented above we can construct an algorithm that performs well in a POMDP setting. The algorithm is composed of two parts: First, 350 Tommi Jaakkola, Satinder P. Singh, Michael I. Jordan Q(m, a) values-analogous to the Q-values in MDP-are calculated via a MonteCarlo approach. This is followed by a policy improvement step which is achieved by increasing the probability of taking the best action as defined by Q(m,a). The new policy is guaranteed to yield a higher average reward (see Theorem 1) as long as for somem max[Q(m, a) - V(m)] > 0 (11) a This condition being false constitutes a local maximum for the algorithm. Examples illustrating that this indeed is a local maximum can be found fairly easily. In practice, it is not feasible to wait for the Monte-Carlo policy evaluation to converge but to try to improve the policy before the convergence. The policy can be refined concurrently with the Monte-Carlo method according to 1I"(almn) -+ 1I"(almn) + ?[Qn(m n, a) - Vn(m n)] (12) with normalization. Other asynchronous or synchronous on-online updating schemes can also be used. Note that if Qn(m, a) = Q(m, a) then this change would be statistically equivalent to that of the batch version with the concomitant guarantees of giving a higher average reward. 6 CONCLUSIONS In this paper we have proposed and theoretically analyzed an algorithm that solves a reinforcement learning problem in a POMDP setting, where the learner has restricted access to the state of the environment. As the underlying MDP is not known the problem appears to the learner to have a non-Markov nature. The average reward was chosen as the figure of merit for the learning problem and stochastic policies were used to provide higher average rewards than can be achieved with deterministic policies. This extension from MDP's to POMDP's greatly increases the domain of potential applications of reinforcement learning methods. The simplicity of the algorithm stems partly from a Monte-Carlo approach to obtaining action-dependent values for each message. These new "Q-values" were shown to give rise to a simple policy improvement result that enables the learner to gradually improve the policy in the continuous space of probabilistic policies. The batch version of the algorithm was shown to converge to a local maximum. We also proposed an on-line version of the algorithm in which the policy is changed concurrently with the calculation of the "Q-values." The policy improvement of the on-line version resembles that of learning automata. APPENDIX Let us denote the policy after the change by 11"!. Assume first that we have access to Q1I"(s, a), the Q-values for the underlying MDP, and to p1l"< (slm), the occupancy probabilities after the policy refinement. Define J(m, 11"!, 11"!, 11") =L 1I"!(alm) L p1l"< (slm)[Q1I"(s, a) - V1I"(s)] (13) where we have used the notation that the policies on the left hand side correspond to the policies on the right respectively. To show how the average reward depends 351 Reinforcement Learning Algorithm for Markov Decision Problems on this quantity we need to make use of the following facts. The Q-values for the underlying MDP satisfy (Bellman's equation) Q1r(s, a) = R(s, a) - R1r + L:P~3' V 1r (s') (14) " In addition, 2:0 7r(alm)Q1r(s, a) = V 1r (s), implying that J(m, 7r!, 7r!, 7r f ) = 0 (see eq. 13). These facts allow us to write J(m, 7r f , 7r f , 7r) - J(m, 7r f , 7r f , 7r f ) L: 7r f (alm) L: p 1r ?slm)[Q1r(s, a) - V 1r (s) - Q1r< (s, a) + V 1r <(s)] (15) 3 By weighting this result for each class by p 1r < (m) and summing over the messages the probability weightings for the last two terms become equal and the terms cancel. This procedure gives us R1r < - R1r = L:P1r?m)J(m, 7r f ,7r f , 7r) (16) m This result does not allow the learner to assess the effect of the policy refinement on the average reward since the JO term contains information not available to the learner. However, making use of the fact that the policy has been changed only slightly this problem can be avoided. As 7r! is a policy satisfying maxmo l7r f (alm) -7r(alm)1 ::; (, it can then be shown that there exists a constant C such that the maximum change in P(slm), pes), P(m) is bounded by Cf. Using these bounds and indicating the difference between 7r! and 7r dependent quantities by ~ we get L:[7r(alm) + ~7r(alm)] L)P1r(slm) + ~p1r(slm)][Q1r(s, a) - V 1r (s)] o o 3 o 3 where the second equality follows from third from the bounds stated earlier. 2:0 7r(alm)[Q1r(s, a) - V1r(s)] = 0 and the The equation characterizing the change in the average reward due to the policy change (eq. 16) can be now rewritten as follows: R1r< - R1r = L: p 1r <(m)J(m, 7r f , 7r, 7r) + 0?(2) m 352 Tommi Jaakkola, Satinder P. Singh, Michael I. Jordan m a where the bounds (see above) have been used for Vir' (m) - p1r(m). This completes 0 the proof. Acknowledgments The authors thank Rich Sutton for pointing out errors at early stages of this work. This project was supported in part by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, by a grant from Siemens Corporation and by grant NOOOI4-94-1-0777 from the Office of Naval Research. Michael I. Jordan is a NSF Presidential Young Investigator. References Barto, A., and Duff, M. (1994). Monte-Carlo matrix inversion and reinforcement learning. In Advances of Neural Information Processing Systems 6, San Mateo, CA, 1994. Morgan Kaufmann. Bertsekas, D. P. (1987). Dynamic Programming: Deterministic and Stochastic Models. Englewood Cliffs, NJ: Prentice-Hall. Dayan, P. (1992). The convergence of TD(A) for general A. Machine Learning, 8, 341-362. Jaakkola, T., Jordan M.I., and Singh, S. P. (1994). On the convergence of stochastic iterative Dynamic Programming algorithms. Neural Computation 6, 1185-1201. Monahan, G. (1982). A survey of partially observable Markov decision processes. Management Science, 28, 1-16. Singh, S. P., Jaakkola, T., Jordan, M.1. (1994). Learning without state estimation in partially observable environments. In Proceedings of the Eleventh Machine Learning Conference. Sutton, R. S. (1988). Learning to predict by the methods of temporal differences. Machine Learning, 3, 9-44. Schwartz, A. (1993). A reinforcement learning method for maximizing un discounted rewards. In Proceedings of the Tenth Machine Learning Conference. Tsitsiklis J. N. (1994). Asynchronous stochastic approximation and Q-Iearning. Machine Learning 16, 185-202. 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-Iearning. Machine Learning, 8, 279-292. 

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Hyperparameters, Evidence and Generalisation for an Unrealisable Rule Glenn Marion and David Saad glennyGed.ac.uk, D.SaadGed.ac.uk Department of Physics, University of Edinburgh, Edinburgh, EH9 3JZ, U.K. Abstract Using a statistical mechanical formalism we calculate the evidence, generalisation error and consistency measure for a linear perceptron trained and tested on a set of examples generated by a non linear teacher. The teacher is said to be unrealisable because the student can never model it without error. Our model allows us to interpolate between the known case of a linear teacher, and an unrealisable, nonlinear teacher. A comparison of the hyperparameters which maximise the evidence with those that optimise the performance measures reveals that, in the non-linear case, the evidence procedure is a misleading guide to optimising performance. Finally, we explore the extent to which the evidence procedure is unreliable and find that, despite being sub-optimal, in some circumstances it might be a useful method for fixing the hyperparameters. 1 INTRODUCTION The analysis of supervised learning or learning from examples is a major field of research within neural networks. In general, we have a probabilistic 1 teacher, which maps an N dimensional input vector x to output Yt(x) according to some distribution P(Yt I x). We are supplied with a data set v= ({Yt(x lJ ), xlJ} : J.' = l..p) generated from P(Yt I x) by independently sampling the input distribution, P(x), p times. One attempts to optimise a model mapping (a student), parameterised by lThis accommodates teachers with deterministic output corrupted by noise. 256 Glenn Marion, David Saad some vector w, with respect to the underlying teacher. The training error Ew (V) is some measure of the difference between the student and the teacher outputs over the set V. Simply minimising the training error leads to the problem of over-fitting. In order to make successful predictions out-with the set V it is essential to have some prior preference for particular rules. Occams razor is an expression of our preference for the simplest rules which account for the data. Clearly Ew(V) is an unsatisfactory performance measure since it is limited to the training examples. Very often we are interested in the students ability to model a random example drawn from P(Yt I x)P(x), but not necessarily in the training set, one measure of this performance is the generalisation error. It is also desirable to predict, or estimate, the level of this error. The teacher is said to be an unrealisable rule, for the student in question, if the minimum generalisation error is non-zero. One can consider the Supervised Learning Paradigm within the context of Bayesian Inference. In particular MacKay [MacKay 92(a)] advocates the evidence procedure as a 'principled' method which, in some situations, does seem to improve performance [Thodberg 93]. However, in others, as MacKay points out the evidence procedure can be misleading [MacKay 92(b )]. In this paper we do not seek to comment on the validity of of the evidence procedure as an approximation to Hierarchical Bayes (see for example [Wolpert and Strauss 94]). Rather, we ask which performance measures do we seek to optimise and under what conditions will the evidence procedure optimise them? Theoretical results have been obtained for a linear percept ron trained on data produced by a linear perceptron [Bruce and Saad 94]. They suggest that the evidence procedure is a useful guide to optimising the learning algorithm's performance. In what follows we examine the evidence procedure for the case of a linear perceptron learning a non linear teacher. In the next section we review the Bayesian scheme and define the evidence and the relevant performance measures. In section 3 we introduce our student and teacher and discuss the calculation. Finally, in section 4 we examine the extent to which the evidence procedure optimises performance. 2 BAYESIAN FORMALISM 2.1 THE EVIDENCE If we take Ew(V) to be the usual sum squared error and assume that our data is corrupted by Gaussian noise with variance 1/2/3 then the probability, or likelihood, ofthe data(V) being produced given the model wand /3 is P(D 1/3, w) ex: e-~Ew(1)). In order to incorporate Occams Razor we also assume a prior distribution on the teacher rules, that is, we believe a priori in some rules more strongly than others. Specifically we believe that pew I ,) ex: e-"'(C(w). MUltiplying the likelihood by the prior we obtain the post training or student distribution 2 P( w I V", /3) ex: e-~Ew(1))-''YC(w). It is clear that the most probable model w? is given by minimising the composite cost function /3Ew(V)+,C(w) with respect to the weights (w). This formalises the trade off between fitting the data and minimising student complexity. In this sense the Bayesian viewpoint coincides with the usual backprop standpoint. 2Integrating this over f3 and 'Y gives us the posterior P(w I 1?. Hyperparameters, Evidence and Generalisation for an Unrealisable Rule 257 In fact, it should be noted that stochastic minimisation can also give rise to the same post training distribution [Seung et aI92). The parameters (3 and, are known as the hyperparameters. Here we consider C(w) = wtw in which case, is termed the weight decay. The evidence is the normalisation constant in the above expression for the post training distribution. P('D I 'Y,(3) = Jn dWjP('D I (3, w)P(w I,) J That is, the probability of the data set ('D) given the hyperparameters. The evidence procedure fixes the hyperparameters to the values that maximise this probability. 2.2 THE PERFORMANCE MEASURES Many performance measures have been introduced in the literature (See e.g., [Krogh and Hertz 92) and [Seung et aI92)) . Here, we consider the squared difference between the average (over the post training distribution) of the student output (y.(x)}w and that of the teacher, Yt(x) , averaged over all possible test questions and teacher outputs, P(Yt, x) and finally over all possible sets of data, 'D. fg = ((Yt(x) - (Y. (x?)w ?}P(X,Yf).'l) This is equivalent to the generalisation error given by Krogh and Hertz. Another factor we can consider is the variance of the output over the student distribution ({y.(x) - (y.(x)}wP}w,P(x)' This gives us a measure of the confidence we should have in our post training distribution and could possibly be calculated if we could estimate the input distribution P(x). Here we extend Bruce and Saad's definition [Bruce and Saad 94] of the consistency measure Dc to include unrealisable = IiIIlp_oo fg, rules by adding the asymptotic error fr: Dc = ({y.(x) - (y.(x)}w}2}w,p(x),'P - fg + fr;' We regard Dc = 0 as optimal since then the variance over our student distribution is an accurate prediction of the decaying part of the generalisation error. We can consider both these performance measures as objective functions measuring the students ability to mimic the underlying teacher. Clearly, they can only be calculated in theory and perhaps, estimated in practice. In contrast, the evidence is only a function of our assumptions and the data and the evidence procedure is, therefore, a practical method of setting the hyperparameters. 3 THE MODEL In our model the student is simply a linear perceptron. The output for an input vector xl' is given by = w .xl' / v'N. The examples, against which the student is trained and tested, are produced by sampling the input distribution, P(x) and then generating outputs from the distribution, Y: P(Yt I x) = t 0=1 P(y~ I x, O)P(x I O)PA 2:0=1 P(O)P(x I 0) 258 Glenn Marion, David Saad -1.0 I I I -0.6 0.0 0.6 1.0 ? Figure 1: A 2-teacher in 1D : The average output (Yt}P(yl%') (i) for Dw = 0 , (ii) for Dw > 0 (0'%'1 = 0'%',) and (iii) with Dw > 0 (0'%'1 ?- 0'%,,) . where P(Yt I x, n) <X exp([Yt - wn.xF /20'2), P(x I n) is N(an,O',;o) 3 and PA is chosen such that I:~=l PA=1. Thus, each component in the sum is a linear perceptron, whose output is corrupted by Gaussian noise of variance 0'2, and we refer to this teacher as an n-teacher. In what follows , for simplicity, we consider a two teacher (n=2) with an = O. The parameter Dw = Jv Iw 1-w 2 12 and the input distribution determine the form of the teacher. This is shown in Figure 1. which displays the average output of a 2-teacher with one dimensional input vector. For 0'Xl =0'X2' Dw controls the variance about a linear mean output, and for fixed O'XI ?- 0'%'2' Dw controls the nonlinearity of the teacher. In the latter case, in the large N limit the variance of P(Yt I x) is zero. We can now explicitly write the evidence and perform the integration over the student parameters (over weights) . Taking the logarithm of the resulting expression leads to In P(1) I >" ,13) = - N 1(1) where the 1 is analogous to a free energy in statistical physics. 1 >.. - 1(1) = -In 2 11' a 13 1 1 1 + -In - + -lndetg + -ln211' + -P 'g'kPk - e 2 11' 2N 2 N J J and, n gjk1 = L Afk + >"Ojk p a=- n=1 N Here we are using the convention that summations are implied where repeated indices occur. 3Where N(x, 0'2) denotes a normal distribution with mean x and variance 0'2 . Hyperparameters, Evidence and Generalisation for an Un realisable Rule 259 The performance measures for this model are 2 {g = (~'x PA{w?w? - Oc = where, Nt (trg}'V - u2 (Wj}w = Pl:gl:j ' {g and 2w?(Wj}w + (Wj}!}}'V + {r; u;eff = PAu;o In order to pursue the calculation we consider the average of I(V) over all possible data sets just as, earlier, we defined our performance measures as averages over all data sets. This is some what artificial as we would normally be able to calculate I(V) and be interested in the generalisation error for our learning algorithm given a particular instance of the data. However, here we consider the thermodynamic limit (i.e., N,p - 00 s.t. 0 = piN = const.) in which, due to our sampling assumptions, the behaviours for typical examples of V coincide with that of the average. Details of the calculation will be published else where [Marion and Saad 95]. 4 RESULTS AND DISCUSSION We can now examine the evidence and the performance measures for our unlearnable problem. We note that in two limits we recover the learnable, linear teacher, case. Specifically if the probability of picking one of the component teachers is zero or if both component teacher vectors are aligned. In what follows we set Pi = P~ and normalise the components of the teacher such that Iwol = l. Firstly let us consider the performance measures. The asymptotic value of both {g and loci for large 0 is PiP~u;lu;:lDwlu;eff' This is the minimum generalisation error attainable and reflects the effective noise level due to the mismatch between student and teacher. We note here that the generalisation error is a function of ~ rather than f3 and 'Y independently. Figure 2a shows the generalisation error plotted against o. The addition of unlearn ability (Dw > 0) has a similar effect to the addition of noise on the examples. The appearance of the hump can be easily understood; If there is no noise or ~ is large enough then there is a steady reduction in {g. However, if this is not so then for small 0 the student learns this effective noise and the generalisation error increases with o . As the student gets more examples the effects of the noise begin to average out and the student starts to learn the rule. The point at which the generalisation error starts to decrease is influenced by the effective noise level and the prior constraint. Figure 2b shows the absolute value of the consistency measure v's 0 for non-optimal f3. Again we see that unlearn ability acts as an effective noise. For a few examples with ~ small or with large effective noise the student distribution is narrowed until the Oc is zero. However, the generalisation error is still increasing (as described above) and loci increases to a local maximum, it then asymptotically tends to { ,q . If there is no noise or ~ is large enough then loci steadily reduces as the number of examples increases. We now examine the evidence procedure. Firstly we define f3ev ( 'Y) and 'Yev (f3) to be the hyperparameters which maximise the evidence. The evidence procedure 260 Glenn Marion, David Saad 4,-------------------, 3 l ,I I l I 4 ,r,, (Ui) 3 ".. : ""'... " 1 ,~.. ............ . ... -... ""_ . . . . --..... _- J 18 2 (ii) I .....................................?... (i) O~---r~-,----~--~ o 1 2 (iii) --- 0 0 3 1 2 3 tl " (b) Consistency Measure (a) Generalisation error Figure 2: The performance measures: Graph a shows (g for finite lambda. a(i) and a(ii) are the learnable case with noise in the latter case. a(iii) shows that the effect of adding unlearn ability is qualitatively the same as adding noise. Graph b. shows the modulus of the consistency error v's a. Curves b(i) and b(ii) are the learnable case without and with noise respectively. Curve b(iii) is an unlearnable case with the same noise level. picks the point in hyperparameter space where these curves coincide. We denote the asymptotic values of 13ev(-y) and 'Yev(13) in the limit of large a by 1300 and 'Yoo respectively. In the linear case (Dw = 0) the evidence procedure assignments of the hyperparameters (for finite a) coincide with 1300 and 'Yoo and also optimise (g and 6c in agreement with [Bruce and Saad 94] . This is shown in Figure 3a where we plot the 13 which optimises the evidence (13ev) , the consistency measure (136 c) and the generalisation error (13!g) versus 'Y. The point at which the three curves coincide is the point in the 13-'Y plane identified by the evidence procedure. However, we note here that, if one of the hyperparameters is poorly determined then maximising the evidence with respect to the other is a misleading guide to optimising performance even in the linear case. The results for an unrealisable rule in the linear regime (Dw > 0, lrXI = lrX:l) are similar to the learnable case but with an increased noise due to the unlearn ability. The evidence procedure still optimises performance. In the non-linear regime (Dw > 0 , lrXI ?- lrX:l) the evidence procedure fails to minimise either performance measure. This is shown in Figure 3b where the evidence procedure point does not lie on 13!g ('Y) or 136 c (-y). Indeed, its hyperparameter assignments do not coincide with 1300 and 'Yoo but are a dependent. How badly does the evidence procedure fail? We define the percentage degradation in generalisation performance as I'\, = 100 * 9 (Aev) - (;Pt) / (;pt. Where Aev is the evidence procedure assignment and (;pt is the optimal generalisation error with respect to A. This is plotted in Figure 4a. We also define 1'\,6 = 100* 16c(Aev)1 /(g(A ev ). This measures the error in using the variance of the ?( Hyperparameters, Evidence and Generalisation for an Unrealisable Rule 1.0.....---------..,. ../ .....?.. 0.8- /Jopt. o.e ./... - ... _ ............ ", (i) ,. ". ~-"":: " '--1' 0.4. "", \" 0.2 (Ii) ". ".::r- \..(~) 0.0 I I 0.2 0.4. I .., o.e (a) Linear Case 0.5.....-----------, /Jopt. 0.3 (1) \~~"-/~ . (11),','/~; ,.' I 0.8 ,_ _ _.... 0.2 0.1 ,',',,, 0.0 261 1.0 0.0-f---'T"---,---r---1 0.0 0.6 1.0 2.0 .., (b) Non-Linear Case Figure 3: The evidence procedure:Optimal f3 v's /. In both graphs for (i) the evidence(f3ev), (ii) the generalisation error (f3f g ) and (iii) the consistency measure (f36J . The point which the evidence procedure picks in the linear case is that where all three curves coincide, whereas in the non linear case it coincides only with f3ev . post training distribution to estimate the generalisation error as a percentage of the generalisation error itself. Examples of this quantity are plotted in Figure 4b. There are three important points to note concerning I'\, and 1'\,6 . Firstly, the larger the deviation from a linear rule the greater is the error. Secondly, that it is the magnitude of the effective noise due to unlearnability relative to the real noise which determines this error. In other words , if the real noise is large enough to swamp the non-linearity of the rule then the evidence procedure will not be very misleading . Finally, the magnitude of the error for relatively large deviations from linearity is only a few percent and thus the evidence procedure might well be a reasonable, if not optimal, method for setting the hyperparameters. However, clearly it would be preferable to improve our student space to enable it to model the teacher . 5 CONCLUSION We have examined the generalisation error, the consistency measure and the evidence procedure within a model which allows us to interpolate between a learnable and an unlearnable scenario. We have seen that the unlearnability acts like an effective noise on the examples. Furthermore, we have seen that for a linear student the evidence procedure breaks down, in that it fails to optimise performance, when the teacher output is non-linear. However, even for relatively large deviations of the teacher from linearity the evidence procedure is close to optimal. Bayesian methods, such as the evidence procedure, are based on the assumption that the student or hypothesis space contains the teacher generating the data. In our case, in the non-linear regime, this is clearly not true and so it is perhaps not surprising that the evidence procedure is sub-optimal. Whether or not such a breakdown of the evidence procedure is a generic feature of a mismatch between the hypothesis space and the teacher is a matter for further study. 262 Glenn Marion, David Saad f\ 0.4-/J \\ J 0.3- J IC .' ! 0.1-1 0.0 (i) 3 (iii) \ \ " ",. , .,.., \ IC, / 2 " " ... ~.,... . . .................. --- _ ..-. 0.0 4 (Ii) \ \ 0.2- _. ,, ,, ,, , 5 o~,-----------------~ 0.5 t.o ----2.0 (I) (Ii) (iii) \ ,.0\0., "\ , ' .. !, ,I ! \\ V 0 0 """ "1 . .... - 3 2 Cl (a) ., .., 4 Cl (b) Figure 4: The relative degradation in performance compared to the optimal when using the evidence procedure to set the hyperparameters. Graph (a) shows the percentage degradation in generalisation performance K, ? a(i) has Dw = 1 with the real noise level u 1. a(ii) has this noise level reduced to u 0.1 and a(iii) has increased non-linearity, Dw = 3, and u = 1. Graph (b) shows the error made in predicting the generalisation error from the variance of the post training distribution as a percentage of the generalisation error itself, "'6 . b(i) and b(ii) have the same parameter values as a(i) and a(ii), whilst b(iii) has Dw = 3 and u = 0.1 = = Acknowledgments We are very grateful to Alastair Bruce and Peter Sollich for useful discussions. GM is supported by an E.P.S.R.C. studentship. References Bruce, A.D. and Saad, D. (1994) Statistical mechanics of hypothesis evaluation. J. of Phys. A: Math. Gen. 27:3355-3363 Krogh, A. and Hertz, J. (1992) Generalisation in a linear perceptron in the presence of noise. J. of Phys. A: Math. Gen. 25:1135-1147 MacKay, D.J.C. (1992a) Bayesian interpolation. Neural Compo 4:415-447 MacKay, D.J.C. (1992b) A practical Bayesian framework for backprop networks. Neural Compo 4:448-472 Marion, G. and Saad, D. (1995) A statistical mechanical analysis of a Bayesian inference scheme for an unrealisable rule. To appear in J. of Phys. A: Math. Gen. Seung, H. S, Sompolinsky, H., Tishby, N. (1992) Statistical mechanics of learning from examples. Phys. Rev. A, 45:6056-6091 Thodberg, H.H. (1994) Bayesian backprop in action:pruning, ensembles, error bars and application to spectroscopy. Advances in Neural Information Processing Systems 6:208-215. Cowan et al.(Eds.), Morgan Kauffmann, San Mateo, CA Wolpert, D. H and Strauss, C. E. M. (1994) What Bayes has to say about the evidence procedure. To appear in Maximum entropy and Bayesian methods. G . Heidbreder (Ed.), Kluwer. 

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A Non-linear Information Maximisation Algorithm that Performs Blind Separation. Anthony J. Bell tonylOsalk.edu Terrence J. Sejnowski terrylOsalk.edu Computational Neurobiology Laboratory The Salk Institute 10010 N. Torrey Pines Road La Jolla, California 92037-1099 and Department of Biology University of California at San Diego La Jolla CA 92093 Abstract A new learning algorithm is derived which performs online stochastic gradient ascent in the mutual information between outputs and inputs of a network. In the absence of a priori knowledge about the 'signal' and 'noise' components of the input, propagation of information depends on calibrating network non-linearities to the detailed higher-order moments of the input density functions. By incidentally minimising mutual information between outputs, as well as maximising their individual entropies, the network 'factorises' the input into independent components. As an example application, we have achieved near-perfect separation of ten digitally mixed speech signals. Our simulations lead us to believe that our network performs better at blind separation than the HeraultJ utten network, reflecting the fact that it is derived rigorously from the mutual information objective. 468 1 Anthony J. Bell, Terrence J. Sejnowski Introduction Unsupervised learning algorithms based on information theoretic principles have tended to focus on linear decorrelation (Barlow & Foldiak 1989) or maximisation of signal-to-noise ratios assuming Gaussian sources (Linsker 1992). With the exception of (Becker 1992), there has been little attempt to use non-linearity in networks to achieve something a linear network could not. Non-linear networks, however, are capable of computing more general statistics than those second-order ones involved in decorrelation, and as a consequence they are capable of dealing with signals (and noises) which have detailed higher-order structure. The success of the 'H-J' networks at blind separation (Jutten & Herault 1991) suggests that it should be possible to separate statistically independent components, by using learning rules which make use of moments of all orders. This paper takes a principled approach to this problem, by starting with the question of how to maximise the information passed on in non-linear feed-forward network. Starting with an analysis of a single unit, the approach is extended to a network mapping N inputs to N outputs. In the process, it will be shown that, under certain fairly weak conditions, the N ---. N network forms a minimally redundant encoding ofthe inputs, and that it therefore performs Independent Component Analysis (ICA) . 2 Information maximisation The information that output Y contains about input X is defined as: = H(Y) - I(Y, X) H(YIX) (1) where H(Y) is the entropy (information) in the output, while H(YIX) is whatever information the output has which didn't come from the input. In the case that we have no noise (or rather, we don't know what is noise and what is signal in the input), the mapping between X and Y is deterministic and H(YIX) has its lowest possible value of -00 . Despite this, we may still differentiate eq.l as follows (see [5]): 8 8 8w I(Y, X) = 8w H(Y) (2) Thus in the noiseless case, the mutual information can be maximised by maximising the entropy alone. 2.1 One input, one output. Consider an input variable, x, passed through a transforming function, g( x), to produce an output variable, y, as in Fig.2.1(a). In the case that g(x) is monotonically increasing or decreasing (ie: has a unique inverse), the probability density function (pdf) of the output fy(y) can be written as a function of the pdfofthe input fx(x), (Papoulis, eq. 5-5): f (y) = fx(x) y 8y/8x (3) A Non-Linear Information Maximization Algorithm That Performs Blind Separation -1: 469 The entropy of the output, H(y), is given by: H(y) = -E[lnfy(y)] = fy(y)lnfy(y)dy (4) where E[.] denotes expected value. Substituting eq.3 into eq.4 gives H(y) = E [In ~:] (5) - E[lnfx(x)] The second term on the right may be considered to be unaffected by alterations in a parameter, w, determining g(x). Therefore in order to maximise the entropy of y by changing w, we need only concentrate on maximising the first term, which is the average log of how the input affects the output. This can be done by considering the 'training set' of x's to approximate the density fx(x), and deriving an 'online', stochastic gradient descent learning rule: ~W ex oH OW = ~ (In OY) = (oy) -1 ~ (oy) OW OX OX OW ox (6) In the case of the logistic transfer function y = (1 + e- U )-l , u = wx + Wo in which the input is multiplied by a weight wand added to a bias-weight wo, the terms above evaluate as: wY(l - y) (7) y(l - y)(l + wX(l - 2y)) (8) Dividing eq.8 by eq.7 gives the learning rule for the logistic function, as calculated from the general rule of eq.6: 1 ex - ~w W + x(l - 2y) (9) Similar reasoning leads to the rule for the bias-weight: ~wo ex 1- 2y (10) The effect of these two rules can be seen in Fig. 1a. For example, if the input pdf fx(x) was gaussian, then the ~wo-rule would centre the steepest part of the sigmoid curve on the peak of fx(x), matching input density to output slope, in a manner suggested intuitively by eq.3. The ~w-rule would then scale the slope of the sigmoid curve to match the variance of fx (x). For example, narrow pdfs would lead to sharply-sloping sigmoids. The ~w-rule is basically anti-Hebbian 1 , with an anti-decay term. The anti-Hebbian term keeps y away from one uninformative situation: that of y being saturated to 0 or 1. But anti-Hebbian rules alone make weights go to zero, so the anti-decay term (l/w) keeps y away from the other uninformative situation: when w is so small that y stays around 0.5. The effect of these two balanced forces is to produce an output pdf fy(y) which is close to the flat unit distribution, which is the maximum entropy distribution for a variable 11? y = tanh(wx + wo) then ~w ex !;; - 2xy 470 Anthony 1. Bell, Terrence 1. Sejnowski 4 ~---' 0 --~---4"""",""::~---- X y (a) ----~~~--~----X ox (b) Figure 1: (a) Optimal information flow in sigmoidal neurons (Schraudolph et al 1992). Input x having density function fx(x), in this case a gaussian, is passed through a non-linear function g(x). The information in the resulting density, fy(Y) depends on matching the mean and variance of x to the threshold and slope of g( x). In (b) fy(y) is plotted for different values of the weight w. The optimal weight, Wopt transmits most information. bounded between 0 and 1. Fig. 1b illustrates a family of these distributions, with the highest entropy one occuring at Wopt . A rule which maximises information for one input and one output may be suggestive for structures such as synapses and photoreceptors which must position the gain of their non-linearity at a level appropriate to the average value and size of the input fluctuations. However, to see the advantages of this approach in artificial neural networks, we now analyse the case of multi-dimensional inputs and outputs. 2.2 N inputs, N outputs. Consider a network with an input vector x, a weight matrix Wand a monotonically transformed output vector y = g(Wx + wo). Analogously to eq.3, the multivariate probability density function of y can be written (Papoulis, eq. 6-63): f ( ) = fx(x) y y (11) IJI where IJI is the absolute value ofthe Jacobian of the transformation. The Jacobian is the determinant of the matrix of partial derivatives: [ 8Xl ~ 8x n ~] ~ 8Xl ~ 8xn J = det: : (12) The derivation proceeds as in the previous section except instead of maximising In(8y/8x), now we maximise In IJI. For sigmoidal units, y = (1 + e-U)-l, U = A Non-Linear Information Maximization Algorithm That Performs Blind Separation 471 Wx + Wo, the resulting learning rules are familiar in form: D. W ex: [WTrl + x(1 - D.wo ex: 1- 2y 2y? (13) (14) except that now x, y, Wo and 1 are vectors (1 is a vector of ones), W is a matrix, and the anti-Hebbian term has become an outer product. The anti-decay term has generalised to the inverse of the transpose of the weight matrix. For an individual weight, Wij, this rule amounts to: cof Wij D.wij ex: det W + xj(l - 2Yi) (15) where cof Wij, the cofactor of Wij, is (-1 )i+j times the determinant of the matrix obtained by removing the ith row and the jth column from W. This rule is the same as the one for the single unit mapping, except that instead of W = 0 being an unstable point of the dynamics, now any degenerate weight matrix is unstable, since det W = 0 if W is degenerate. This fact enables different output units, Yi, to learn to represent different components in the input. When the weight vectors entering two output units become too similar, det W becomes small and the natural dynamic of learning causes these weight vectors to diverge. This effect is mediated by the numerator, cof Wij. When this cofactor becomes small, it indicates that there is a degeneracy in the weight matrix of the rest of the layer (ie: those weights not associated with input Xj or output Yi). In this case, any degeneracy in W has less to do with the specific weight Wij that we are adjusting. 3 Finding independent components - blind separation Maximising the information contained in a layer of units involves maximising the entropy of the individual units while minimising the mutual information (the redundancy) between them. Considering two units: (16) For I(Yl, Y2) to be zero, Yl and Y2 must be statistically independent of each other, so that f Y1 Y2(Yl, Y2) = fYl (ydfY2(Y2). Achieving such a representation is variously called factorial code learning, redundancy reduction (Barlow 1961, Atick 1992), or independent component analysis (ICA), and in the general case of continuously valued variables of arbitrary distributions, no learning algorithm has been shown to converge to such a representation. Our method will converge to a minimum redundancy, factorial representation as long as the individual entropy terms in eq.16 do not override the redundancy term, making an I(Yl, Y2) = 0 solution sub-optimal. One way to ensure this cannot occur is if we have a priori knowledge of the general form of the pdfs of the independent components. Then we can tailor our choice of node-function to be optimal for transmitting information about these components. For example, unit distributions require piecewise linear node-functions for highest H(y), while the more common gaussian forms require roughly sigmoidal curves. Once we have chosen our nodefunctions appropriately, we can be sure that an output node Y cannot have higher 472 Anthony 1. Bell, Terrence 1. Sejnowski- slt--_---<x\._ _---4!'.~ o o o )<----~ unknown mixing process 1'-----..:?0 BUND SEPARATION (learnt weights) WA '" after learning: 1-4.091 0.13 0.09 0.07 1-2.921 0.00 0.02 -0.02 -0.06 0.02 0.03 0.00 -0.07 0.14 1-3.501 -0.07 -0.01 0.02 -0.06 -0.08 1-2.2ij 11.971 0.02 -0.01 0.04 Figure 2: (a) In blind separation, sources, s, have been linearly scrambled by a matrix, A, to form the inputs to the network, x. We must recover the sources at our output y, by somehow inverting the mapping A with our weight matrix W. The problem: we know nothing about A or the sources. (b) A successful 'unscrambling' occurs when WA is a 'permutation' matrix. This one resulted from separating five speech signals with our algorithm. entropy by representing some combination of independent components than by representing just one. When this condition is satisfied for all output units, the residual goal, of minimising the mutual information between the outputs, will dominate the learning. See [5] for further discussion of this. With this caveat in mind, we turn to the problem of blind separation, (Jutten & Herault 1991), illustrated in Fig.2. A set of sources, Sl, ... , SN, (different people speaking, music, noise etc) are presumed to be mixed approximately linearly so that all we receive is N superpositions of them, Xl, ... , X N, which are input to our single-layer information maximisation network. Providing the mixing matrix, A, is non-singular then the original sources can be recovered if our weight matrix, W, is such that W A is a 'permutation' matrix containing only one high value in each row and column. Unfortunately we know nothing about the sources or the mixing matrix. However, if the sources are statistically independent and non-gaussian, then the information in the output nodes will be maximised when each output transmits one independent component only. This problem cannot be solved in general by linear decorrelation techniques such as those of (Barlow & Foldicik 1989) since second-order statistics can only produce symmetrical decorrelation matrices. We have tested the algorithm in eq.13 and eq.14 on digitally mixed speech signals, and it reliably converges to separate the individual sources. In one example, five separately-recorded speech signals from three individuals were sampled at 8kHz. Three-second segments from each were linearly mixed using a matrix of random values between 0.2 and 4. Each resulting mixture formed an incomprehensible babble. Time points were generated at random, and for each, the corresponding five mixed values were entered into the network, and weights were altered according to eq.13 and eq.14. After on the order of 500,000 points were presented, the network A Non-Linear Information Maximization Algorithm That Performs Blind Separation 473 had converged so that WA was the matrix shown in Fig.2b. As can be seen from the permutation structure of this matrix, on average, 95% of each output unit is dedicated to one source only, with each unit carrying a different source. Any residual interference from the four other sources was inaudible. We have not yet performed any systematic studies on rates of convergence or existence of local minima. However the algorithm has converged to separate N independent components in all our tests (for 2 ~ N ~ 10). In contrast, we have not been able to obtain convergence of the H-J network on our data set for N > 2. Finally, the kind of linear static mixing we have been using is not equivalent to what would be picked up by N microphones positioned around a room. However, (Platt & Faggin 1992) in their work on the H-J net, discuss extensions for dealing with time-delays and non-static filtering, which may also be applicable to our methods. 4 Discussion The problem of Independent Component Analysis (ICA) has become popular recently in the signal processing community, partly as a result ofthe success ofthe H-J network. The H-J network is identical to the linear decorrelation network of (Barlow & Foldicik 1989) except for non-linearities in the anti-Hebb rule which normally performs only decorrelation. These non-linearities are chosen somewhat arbitrarily in the hope that their polynomial expansions will yield higher-order cross-cumulants useful in converging to independence (Comon et aI, 1991). The H-J algorithm lacks an objective function, but these insights have led (Comon 1994) to propose minimising mutual information between outputs (see also Becker 1992). Since mutual information cannot be expressed as a simple function of the variables concerned, Comon expands it as a function of cumulants of increasing order. In this paper, we have shown that mutual information, and through it, ICA, can be tackled directly (in the sense of eq.16) through a stochastic gradient approach. Sigmoidal units, being bounded, are limited in their 'channel capacity' . Weights transmitting information try, by following eq.13, to project their inputs to where they can make a lot of difference to the output, as measured by the log of the Jacobian of the transformation. In the process, each set of statistically 'dependent' information is channelled to the same output unit. The non-linearity is crucial. If the network were just linear, the weight matrix would grow without bound since the learning rule would be: (17) This reflects the fact that the information in the outputs grows with their variance. The non-linearity also supplies the higher-order cross-moments necessary to maximise the infinite-order expansion of the information. For example, when y = tanh( u), the learning rule has the form D.. W ex: [wT] -1 - 2yxT , from which we can write that the weights stabilise (or (D..W) = 0) when 1= 2(tanh(u)uT ). Since tanh is an odd function, its series expansion is of the form tanh( u) = L: j bj u 2P+1 , the bj being coefficients, and thus this convergence criterion amounts to the condition L:i,j bijp(U;P+1Uj) = 0 for all output unit pairs i 1= j, for p = 0,1,2,3 ..., 474 Anthony J. Bell, Terrence J. Sejnowski and for the coefficients bijp coming from the Taylor series expansion of the tanh function. These and other issues are covered more completely in a forthcoming paper (Bell & Sejnowski 1995). Applications to blind deconvolution (removing the effects of unknown causal filtering) are also described, and the limitations of the approach are discussed. Acknowledgements We are much indebted to Nici Schraudolph, who not only supplied the original idea in Fig.l and shared his unpublished calculations [13], but also provided detailed criticism at every stage of the work. Much constructive advice also came from Paul Viola and Alexandre Pouget. References [1] Atick J.J. 1992. Could information theory provide an ecological theory of sensory processing, Network 3, 213-251 [2] Barlow H.B. 1961. Possible principles underlying the transformation of sensory messages, in Sensory Communication, Rosenblith W.A. (ed), MIT press [3] Barlow H.B. & Foldicik P. 1989. Adaptation and decorrelation in the cortex, in Durbin R. et al (eds) The Computing Neuron, Addison-Wesley [4] Becker S. 1992. An information-theoretic unsupervised learning algorithm for neural networks, Ph.D. thesis, Dept. of Compo Sci., Univ. of Toronto [5] Bell A.J. & Sejnowski T.J. 1995. An information-maximisation approach to blind separation and blind deconvolution, Neural Computation, in press [6] Comon P., Jutten C. & Herault J. 1991. Blind separation of sources, part II: problems statement, Signal processing, 24, 11-21 [7] Comon P. 1994. Independent component analysis, a new concept? Signal processing, 36, 287-314 [8] Hopfield J.J. 1991. Olfactory computation and object perception, Proc. N atl. Acad. Sci. USA, vol. 88, pp.6462-6466 [9] Jutten C. & Herault J. 1991. Blind separation of sources, part I: an adaptive algorithm based on neuromimetic architecture, Signal processing, 24, 1-10 [10] Linsker R. 1992. Local synaptic learning rules suffice to maximise mutual information in a linear network, Neural Computation, 4, 691-702 [11] Papoulis A. 1984. Probability, random variables and stochastic processes, 2nd edition, McGraw-Hill, New York [12] Platt J .C. & Faggin F. 1992. Networks for the separation of sources that are superimposed and delayed, in Moody J.E et al (eds) Adv. Neur. Inf. Proc. Sys. 4, Morgan-Kaufmann [13] Schraudolph N.N., Hart W .E. & Belew R.K. 1992. Optimal information flow in sigmoidal neurons, unpublished manuscript 

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Anatomical origin and computational role of diversity in the response properties of cortical neurons Kalanit Grill Spectort Shimon Edelmant Rafael Malacht Depts of tApplied Mathematics and Computer Science and tN eurobiology The Weizmann Institute of Science Rehovot 76100, Israel {kalanit.edelman. malach }~wisdom . weizmann .ac.il Abstract The maximization of diversity of neuronal response properties has been recently suggested as an organizing principle for the formation of such prominent features of the functional architecture of the brain as the cortical columns and the associated patchy projection patterns (Malach, 1994). We show that (1) maximal diversity is attained when the ratio of dendritic and axonal arbor sizes is equal to one, as found in many cortical areas and across species (Lund et al., 1993; Malach, 1994), and (2) that maximization of diversity leads to better performance in systems of receptive fields implementing steerable/shiftable filters, and in matching spatially distributed signals, a problem that arises in many high-level visual tasks. 1 Anatomical substrate for sampling diversity A fundamental feature of cortical architecture is its columnar organization, manifested in the tendency of neurons with similar properties to be organized in columns that run perpendicular to the cortical surface. This organization of the cortex was initially discovered by physiological experiments (Mouncastle, 1957; Hubel and Wiesel, 1962), and subsequently confirmed with the demonstration of histologically defined columns. Tracing experiments have shown that axonal projections throughout the cerebral cortex tend to be organized in vertically aligned clusters or patches. In particular, intrinsic horizontal connections linking neighboring cortical sites, which may extend up to 2 - 3 mm, have a striking tendency to arborize selectively in preferred sites, forming distinct axonal patches 200 - 300 J.lm in diameter. Recently, it has been observed that the size of these patches matches closely the average diameter of individual dendritic arbors of upper-layer pyramidal cells 118 Kalanit Grill Spector, Shimon Edelman, Rafael Malach .... 2&00 10(0 : : &00 , / '" ! .._._.r::~ ._. ~._ . _._;~ \ ~ ~ ~ ~ ~ ,*,*".",p.d ~ hm pa~ ) N BO 80 100 '0 0:2 0" 0S 0.8 t 1 :2 1 .. 1S t IS r?? o betiIJMn I'IWI'ot'I .. d ptIId! Figure 1: Left: histograms of the percentage of patch-originated input to the neurons, plotted for three values of the ratio r between the dendritic arbor and the patch diameter (0,5, 1.0, 2.0). The flattest histogram is obtained for r = 1.0 Right: the di versity of neuronal properties (as defined in section 1) vs. r. The maximum is attained for r = 1.0, a value compatible with the anatomical data. (see Malach, 1994, for a review). Determining the functional significance of this correlation, which is a fundamental property that holds throughout various cortical areas and across species (Lund et al., 1993), may shed light on the general principles of operation of the cortical architecture. One such driving principle may be the maximization of diversity of response properties in the neuronal population (Malach, 1994). According to this hypothesis, matching the sizes of the axonal patches and the dendritic arbors causes neighboring neurons to develop slightly different functional selectivity profiles, resulting in an even spread of response preferences across the cortical population, and in an improvement of the brain's ability to process the variety of stimuli likely to be encountered in the environment. 1 To test the effect of the ratio between axonal patch and dendritic arbor size on the diversity of the neuronal population, we conducted computer simulations based on anatomical data concerning patchy projections (Rockland and Lund, 1982; Lund et al., 1993; Malach, 1992; Malach et al., 1993). The patches were modeled by disks, placed at regular intervals of twice the patch diameter, as revealed by anatomical labeling. Dendritic arbors were also modeled by disks, whose radii were manipulated in different simulations. The arbors were placed randomly over the axonal patches, at a density of 10,000 neurons per patch. We then calculated the amount of patchrelated information sampled by each neuron, defined to be proportional to the area of overlap of the dendritic tree and the patch. The results of the calculations for three 1 Necessary conditions for obtaining dendritic sampling diversity are that dendritic arbors cross freely through column borders, and that dendrites which cross column borders sample with equal probability from patch and inter-patch compartments. These assumptions were shown to be valid in (Malach, 1992; Malach, 1994). Diversity in the Response Properties of Cortical Neurons 119 values of the ratio of patch and arbor diameters appear in Figure 1. The presence of two peaks in the histogram obtained with the arbor/patch ratio r = 0.5 indicates that two dominant groups are formed in the population, the first receiving most of its input from the patch, and the second - from the inter-patch sources. A value of r = 2.0, for which the dendritic arbors are larger than the axonal patch size, yields near uniformity of sampling properties, with most of the neurons receiving mostly patch-originated input , as apparent from the single large peak in the histogram. To quantify the notion of diversity, we defined it as diversity "'< I~; I > -1, where n(p) is the number of neurons that receive p percent of their inputs from the patch, and < . > denotes average over p. Figure 1, right, shows that diversity is maximized when the size of the dendritic arbors matches that of the axonal patches, in accordance with the anatomical data. This result confirms the diversity maximization hypothesis stated in (Malach, 1994). 2 Orientation tuning: a functional manifestation of sam pIing diversity The orientation columns in VI are perhaps the best-known example of functional architecture found in the cortex (Bubel and Wiesel, 1962). Cortical maps obtained by optical imaging (Grinvald et al., 1986) reveal that orientation columns are patchy rather then slab-like: domains corresponding to a single orientation appear as a mosaic of round patches, which tend to form pinwheel-like structures. Incremental changes in the orientation of the stimulus lead to smooth shifts in the position of these domains. We hypothesized that this smooth variation in orientation selectivity found in VI originates in patchy projections, combined with diversity in the response properties of neurons sampling from these projections. The simulations described in the rest of this section substantiate this hypothesis. Computer simulations. The goal of the simulations was to demonstrate that a limited number of discretely tuned elements can give rise to a continuum of responses. We did not try to explain how the original set of discrete orientations can be formed by projections from the LGN to the striate cortex; several models for this step can be found in the literature (Bubel and Wiesel, 1962; Vidyasagar, 1985).2 In setting the size of the original discrete orientation columns we followed the notion of a point image (MacIlwain, 1986), defined as the minimal cortical separation of cells with non-overlapping RFs. Each column was tuned to a specific angle, and located at an approximately constant distance from another column with the same orientation tuning (we allowed some scatter in the location of the RFs). The RFs of adjacent units with the same orientation preference were overlapping, and the amount of overlap 2ln particular, it has been argued (Vidyasagar, 1985) that the receptive fields at the output of the LGN are already broadly tuned for a small number of discrete orientations (possibly just horizontal and vertical), and that at the cortical level the entire spectrum of orientations is generated from the discrete set present in the geniculate projection. 120 Kalanit Grill Spector, Shimon Edelman, Rafael Malach 3< 007 32 OOS 30 OOs II 0 28 ! 00' 1 ! . 26 g 003 .. 24 ? 00' 22 001 0 0 20 .0 20 30 '" runber ~ shlttlng fIlters So SO 70 '?0 .0 20 30 '" nunber cI shttng tlMlI'1 so . Figure 2: The effects of (independent) noise in the basis RFs and in the steering/shifting coefficients. Left: the approximation error vs. the number of basis RFs used in the linear combination. Right: the signal to noise ratio vs. the number of basis RFs. The SNR values were calculated as 10 loglO (signal energy/noise energy). Adding RFs to the basis increases the accuracy of the resultant interpolated RF. was determined by the number of RFs incorporated into the network. The preferred orientations were equally spaced at angles between 0 and 1r. The RFs used in the simulations were modeled by a product of a 2D Gaussian G 1 , centered at rj, with orientation selectivity G 2, and optimal angle Oi: G(r, rj, 0, Oi) = G1(r, rj)G 2(O, Oi). According to the recent results on shiftable/steerable filters (Simon celli et al., 1992), a RF located at and tuned to the orientation ,po can be obtained by a linear combination of basis RFs, as follows: ro G(r, ro, 0, ,po) M-IN-I L L bj(ro)ki(,po)G(r,rj,O,Oi) j=O i=O M-I N-I 2: bj(ro)G1(r, rj) 2: ki(,po)G2(O,Oi) j=O (1 ) i=O From equation 1 it is clear that the linear combination is equivalent to an outer product of the shifted and the steered RFs, with {ki(,pO)}~~1 and {bj(ro)}~~l denoting the steering and shifting coefficients, respectively. Because orientation and localization are independent parameters, the steering coefficients can be calculated separately from the shifting coefficients. The number of steering coefficients depends on the polar Fourier bandwidth of the basis RF, while the number of steering filters is inversely proportional to the basis RF size (Grill-Spector et al., 1995). In the presence of noise this minimal basis has to be extended (see Figure 2). The results ofthe simulation for several RF sizes are shown in Figure 3, left. As expected, the number of basis RFs required to approximate a desired RF is inversely proportional to the 121 Diversity in the Response Properties of Cortical Neurons The dependency oI1he nlnber d RF9 a1 the venene. '. ,5 2 van.,08 25 35 Figure 3: Left: error of the steering/shifting approximation for several basis RF sizes. Right: the number of basis RFs required to achieve a given error for different sizes of the basis RFs. The dashed line is the hyperbola num RFs x size = const. size of the basis RFs (Figure 3, right). Steerability and biological considerations. The anatomical finding that the columnar "borders" are freely crossed by dendritic and axonal arbors (Malach, 1992), and the mathematical properties of shiftable/steerable filters outlined above suggest that the columnar architecture in VI provides a basis for creating a continuum of RF properties, rather that being a form of organizing RFs in discrete bins. Computationally, this may be possible if the input to neurons is a linear combination of outputs of several RFs, as in equation 1. The anatomical basis for this computation may be provided by intrinsic cortical connections. It is known that long-range (I"V 1mm) connections tend to link cells with like orientation preference, while the short-range (I"V 400 J.lm) connections are made to cells of diverse orientation preferences (Malach et al., 1993). We suggest that the former provide the inputs necessary to shift the position of the desired RF, while the latter participate in steering the RF to an arbitrary angle (see Grill-Spector et al., 1995, for details). 3 Matching with patchy connections Many visual tasks require matching between images taken at different points in space (as in binocular stereopsis) or time (as in motion processing). The first and foremost problem faced by a biological system in solving these tasks is that the images to be compared are not represented as such anywhere in the system: instead of images, there are patterns of activities of neurons, with RFs that are overlapping, are not located on a precise grid, and are subject to mixing by patchy projections in each successive stage of processing. In this section, we show that a system composed of scattered RFs with smooth and overlapping tuning functions can, as a matter of 122 Kalanit Grill Spector, Shimon Edelman, Rafael Malach fact, perform matching precisely by allowing patchy connections between domains. Moreover, the weights that must be given to the various inputs that feed a RF carrying out the match are identical to the coefficients that would be generated by a learning algorithm required to capture a certain well-defined input-output relationship from pairs of examples. DOMAIN A Figure 4: Unit C receives patchy input from areas A and B which contain receptors with overlapping RFs. Consider a unit C, sampling two domains A and B through a Gaussian-profile dendritic patch equal in size to that of the axonal arbor of cells feeding A and B (Figure 4). The task faced by unit C is to determine the degree to which the activity patterns in domains A and B match. Let <Pjp be the response of the j'th unit in A to an input x-;': A. . = exp{ - ( .. Xp - ")2 Xj 20'2 'l'JP } (2) where xj be the optimal pattern to which the j'th unit is tuned (the response Bjp of a unit in B is of similar form). If, for example, domains A and B contain orientation selective cells, then xj would be the optimal combination of orientation and location of a bar stimulus. For simplicity we assume that all the RFs are of the same size 0', that unit C samples the same number of neurons N from both domains, and that the input from each domain to unit C is a linear combination of the responses of the units in each area. The input to C from domain A, with x-;, presented to the system is then: N Ain =L j=1 aj<pjp (3) Diversity in the Response Properties of Cortical Neurons 123 The problem is to find coefficients {aj} and {bj } such that on a given set of inputs {x-;} the outputs of domains A and B will match. We define the matching error as follows: Em = t (~a,~,p -~b'8'p)' (4) Proposition 1 The desired coefficients, minimizing Em, can be generated by an algorithm trained to learn an input/output mapping from a set of examples. This proposition can be proved by taking the derivative of Em with respect to the coefficients (Grill-Spector et al., 1995). Learning here can be carried out by radial basis function (RBF) approximation (Poggio and Girosi, 1990), which is particularly suitable for our purpose, because its basis functions can be regarded as multidimensional Gaussian RFs. 4 Summary Our results show that maximal diversity of neuronal response properties is attained when the ratio of dendritic and axonal arbor sizes is equal to 1, a value found in many cortical areas and across species (Lund et al., 1993; Malach, 1994). Maximization of diversity also leads to better performance in systems of receptive fields implementing steerablejshiftable filters, which may be necessary for generating the seemingly continuous range of orientation selectivity found in VI, and in ma.tching spatially distributed signals. This cortical organization principle may, therefore, have the double advantage of accounting for the formation of the cortical columns and the associated patchy projection patterns, and of explaining how systems of receptive fields can support functions such as the generation of precise response tuning from imprecise distributed inputs, and the matching of distributed signals, a problem that arises in visual tasks such as stereopsis, motion processing, and recognition. References Grill-Spector, K. , Edelman, S., and Malach, R. (1995). Anatomical origin and computational role of diversity in the response properties of cortical neurons. In Aertsen, A., editor, Brain Theory: biological basis and computational theory of vision. Elsevier. in press. Grinvald, A., Lieke, T., Frostigand, R., Gilbert, C. , and Wiesel, T. (1986). Functional architecture of the cortex as revealed by optical imaging of intrinsic signals. Nature, 324:361-364. Rubel, D. and Wiesel, T . (1962) . Receptive fields, binocular interactions and functional architecture in the cat's visual cortex. Journal of Physiology, 160:106-154. 124 Kalanit Grill Spector, Shimon Edelman, Rafael Malach Lund, J., Yoshita, S., and Levitt , J. (1993). Comparison of intrinsic connections in different areas of macaque cerebral cortex. Cerebral Cortex, 3:148- 162. MacIlwain, J. (1986). Point images in the visual system: new interest in an old idea. Trends in Neurosciences, 9:354- 358 . Malach, R. (1992) . Dendritic sampling across processing streams in monkey striate cortex. Journal of Comparative Neurobiology, 315:305-312. Malach, R. (1994). Cortical columns as devices for maximizing neuronal diversity. Trends in Neurosciences, 17:101- 104. Malach, R., Amir, Y., Harel, M., and Grinvald, A. (1993) . Relationship between intrinsic connections and functional architecture,revealed by optical imaging and in vivo targeted biocytine injections in primate striate cortex. Proceedings of the National Academy of Science, USA, 90:10469- 10473. Mouncastle, V. (1957). Modality and topographic properties of single neurons of cat's somatic sensory cortex. Journal of Neurophysiology, 20:408-434 . Poggio, T. and Girosi, F. (1990). Regularization algorithms for learning that are equivalent to multilayer networks . Science, 247:978- 982. Rockland, K. and Lund, J. (1982). Widespread periodic intrinsic connections in the tree shrew visual cortex. Science, 215: 1532-1534. Simoncelli, E., Freeman, W., Adelson, E., and Heeger, D. (1992). Shiftable multiscale transformations. IEEE Transactions on Information Theory, 38:587-607. Vidyasagar, T. (1985). Geniculate orientation biases as cartesian coordinates for cortical orientation detectors. In Models for the visual cortex, pages 390- 395 . Wiley, New York. 

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Factorial Learning by Clustering Features Joshua B. Tenenbaum and Emanuel V. Todorov Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 {jbt.emo}~psyche . mit.edu Abstract We introduce a novel algorithm for factorial learning, motivated by segmentation problems in computational vision, in which the underlying factors correspond to clusters of highly correlated input features. The algorithm derives from a new kind of competitive clustering model, in which the cluster generators compete to explain each feature of the data set and cooperate to explain each input example, rather than competing for examples and cooperating on features, as in traditional clustering algorithms. A natural extension of the algorithm recovers hierarchical models of data generated from multiple unknown categories, each with a different, multiple causal structure. Several simulations demonstrate the power of this approach. 1 INTRODUCTION Unsupervised learning is the search for structure in data. Most unsupervised learning systems can be viewed as trying to invert a particular generative model of the data in order to recover the underlying causal structure of their world. Different learning algorithms are then primarily distinguished by the different generative models they embody, that is, the different kinds of structure they look for. Factorial learning, the subject of this paper, tries to find a set of independent causes that cooperate to produce the input examples. We focus on strong factorial learning, where the goal is to recover the actual degrees of freedom responsible for generating the observed data, as opposed to the more general weak approach, where the goal 562 Joshua B. Tenenbaum, Emmanuel V. Todorov Figure 1: A simple factorial learning problem. The learner observes an articulated hand in various configurations, with each example specified by the positions of 16 tracked features (shown as black dots). The learner might recover four underlying factors, corresponding to the positions of the fingers, each of which claims responsiblity for four features of the data set. is merely to recover some factorial model that explains the data efficiently. Strong factorial learning makes a claim about the nature of the world, while weak factorial learning only makes a claim about the nature of the learner's representations (although the two are clearly related). Standard subspace algorithms, such as principal component analysis, fit a linear, factorial model to the input data, but can only recover the true causal structure in very limited situations, such as when the data are generated by a linear combination of independent factors with significantly different variances (as in signal-from-noise separation). Recent work in factorial learning suggests that the general problem of recovering the true, multiple causal structure of an arbitrary, real-world data set is very difficult, and that specific approaches must be tailored to specific, but hopefully common, classes of problems (Foldiak, 1990; Saund, 1995; Dayan and Zemel, 1995). Our own interest in multiple cause learning was motivated by segmentation problems in computational vision, in which the underlying factors correspond ideally to disjoint clusters of highly correlated input features. Examples include the segmentation of articulated objects into functionally independent parts, or the segmentation of multiple-object motion sequences into tracks of individual objects. These problems, as well as many other problems of pattern recognition and analysis, share a common set of constraints which makes factorial learning both appropriate and tractable. Specifically, while each observed example depends on some combination of several factors, anyone input feature always depends on only one such factor (see Figure 1). Then the generative model decomposes into independent sets of functionally grouped input features, or functional parts (Tenenbaum, 1994). In this paper, we propose a learning algorithm that extracts these functional parts. The key simplifying assumption, which we call the membership constmint, states that each feature belongs to at most one functional part, and that this membership is constant over the set of training examples. The membership constraint allows us to treat the factorial learning problem as a novel kind of clustering problem. The cluster generators now compete to explain each feature of the data set and cooperate to explain each input example, rather than competing for examples and cooperating on features, as in traditional clustering systems such as K-means or mixture models. The following sections discuss the details of the feature clustering algorithm for extracting functional parts, a simple but illustrative example, and extensions. In particular, we demonstrate a natural way to relax the strict membership constraint and thus learn hierarchical models of data generated from multiple unknown categories, each with a different multiple causal structure. Factorial Learning by Clustering Features 2 563 THE FEATURE CLUSTERING ALGORITHM Our algorithm for extracting functional parts derives from a statistical mechanics formulation of the soft clustering problem (inspired by Rose, Gurewitz, and Fox, 1990; Hinton and Zemel, 1994). We take as input a data set {xt}, with I examples of J real-valued features. The best K-cluster representation of these J features is given by an optimal set of cluster parameters, {Ok}, and an optimal set of assignments, {Pi/.J. The assignment Pjk specifies the probability of assigning feature j to cluster k, and depends directly on Ejk = 2:i(Xj'l - fj~)2, the total squared difference (over the I training examples) between the observed feature values x}il and cluster k's predictions fj~. The parameters Ok define cluster k's generative model, and thus determine the predictions fj~(Ok). If we limit functional parts to clusters of linearly correlated features, then the appropriate generative model has fj~ = WjkYkil + Uj, with cluster parameters Ok = {Ykil , Wjk, Uj} to be estimated. That is, for each example i, part k predicts the value of input feature j as a linear function of some part-specific factor Ykil (such as finger position in Figure 1). For the purposes of this paper, we assume zero-mean features and ignore the Uj terms. Then Ejk = 2:i(Xj'l - WjkYkil )2. The optimal cluster parameters and assignments can now be found by maximizing the complete log likelihood of the data given the K-cluster representation, or equivalently, in the framework of, statistical mechanics, by minimizing the free energy F =E - 1 -H (3 1 = LLPjk(Ejk + -logpjk) (3 k j (1) subject to the membership constraints, 2:k pjk = 1, (\lj). Minimizing the energy, E = L j LPjkEjk, (2) k reduces the expected reconstruction error, leading to more accurate representations. Maximizing the entropy, (3) H = - LLPjklogPjk, j k distributes responsibility for each feature across many parts, thus decreasing the independence of the parts and leading to simpler representations (with fewer degrees of freedom). In line with Occam's Razor, minimizing the energy-entropy tradeoff finds the representation that, at a particular temperature 1/(3, best satisfies the conflicting requirements of low error and low complexity. We minimize the free energy with a generalized EM procedure (Neal and Hinton, 1994), setting derivatives to zero and iterating the resulting update equations: e-(3Ejk Pjk = Ykil 2:k e-(3Ejk LPjkWjkXj'l (4) (5) j Wjk = Lxj'lYtl. (6) 564 Joshua B. Tenenbaum. Emmanuel V. Todorov This update procedure assumes a normalization step Ykil = Ykil /(Eil (y(l)2)1/2 in each iteration, because without some additional constraint on the magnitudes of l (or Wjk), inverting the generative model fj~ = WjkYkil is an ill-posed problem. yt This algorithm maps naturally onto a simple network architecture. The hidden unit activities, representing the part-specific factors Ykil , are computed from the observations via bottom-up weights PjkWjk, normalized, and multiplied by top-down weights Wjk to generate the network's predictions fj~. The weights adapt according to a hybrid learning rule, with Wjk determined by a Hebb rule (as in subspace learning algorithms), and pjk determined by a competitive, softmax function of the reconstruction error Ejk (as in soft mixture models). xyl 3 LEARNING A HIERARCHY OF PARTS The following simulation illustrates the algorithm's behavior on a simple, part segmentation task. The training data consist of 60 examples with 16 features each, representing the horizontal positions of 16 points on an articulated hand in various configurations (as in Figure 1). The data for this example were generated by a hierarchical, random process that produced a low correlation between all 16 features, a moderate correlation between the four features on each finger, and a high correlation between the two features on each joint (two joints per finger). To fully explain this data set, the algorithm should be able to find a corresponding hierarchy of increasingly complex functional part representations. To evaluate the network's representation of this data set, we inspect the learned weights PjkWjk, which give the total contribution of feature j to part k in (5) . In Figure 2, these weights are plotted for several different values of /3, with gray boxes indicating zero weights, white indicating strong positive weights, and black indicating strong negative weights. The network was configured with K = 16 part units, to ensure that all potential parts could be found. When fewer than K distinct parts are found, some of the cluster units have identical parameters (appearing as identical columns in Figure 2). These results were generated by deterministic annealing, starting with /3 ? 1, and perturbing the weights slightly each time /3 was increased, in order to break symmetries. Figure 2 shows that the number of distinct parts found increases with /3, as more accurate (and more complex) representations become favored. In (4), we see that /3 controls the number of distinct parts via the strength of the competition for features. At /3 = 0, every part takes equal responsibility for every feature. Without competition, there can be no diversity, and thus only one distinct part is discovered at low /3, corresponding to the whole hand (Figure 2a). As /3 increases, the competition for features gets stiffer, and parts split into their component subparts. The network finds first four distinct parts (with four features each), corresponding to individual fingers (Figure 2c), and then eight distinct parts (with two features each), corresponding to individual joints (Figure 2d). Figure 2b shows an intermediate representation, with something between one and four parts. Four distinct columns are visible, but they do not cleanly segregate the features. Figure 3 plots the decrease in mean reconstruction error (expressed by the energy E) Factorial Learning by Clustering Features (a) 565 ~=1 ~= (b) 100 ~ ~~+-~~+-~~~~~~ ~ ~~+-r+~+-~-r~r+-r~ ~ r4~+-r+~4=P+-r~r+~~ Part k Part k (e) ~= 1000 ~=2oo00 (d) Part k Part k Figure 2: A hierarchy of functional part representations, parameterized by {3. F - - - e - - - - - -___ b d 2 3 4 log ~ Figure 3: A phase diagram distinguishes true parts (a, c, d) from spurious ones (b). 566 Joshua B. Tenenbaum. Emmanuel V. Todorov as (3 increases and more distinct parts emerge. Notice that within the three stable phases corresponding to good part decompositions (Figures 2a, 2c, 2d), E remains practically constant over wide variations in (3. In contrast, E varies rapidly at the boundaries between phases, where spurious part structure appears (Figure 2b). In general, good representations should lie at stable points of this phase diagram, where the error-complexity tradeoff is robust. Thus the actual number of parts in a particular data set, as well as their hierarchical structure, need not be known in advance, but can be inferred from the dynamics of learning. 4 LEARNING MULTIPLE CATEGORIES Until this point, we have assumed that each feature belongs to at most one part over the entire set of training examples, and tried to find the single K -part model that best explains the data as a whole. But the notion that a single model must explain the whole data set is quite restrictive. The data may contain several categories of examples, each characterized by a different pattern of feature correlations, and then we would like to learn a set of models, each capturing the distinctive part structure of one such category. Again we are motivated by human vision, which easily recognizes many categories of motion defined by high-level patterns of coordinated part movement, such as hand gestures and facial expressions. If we know which examples belong to which categories, learning multiple models is no harder than learning one, as in the previous section. A separate model can be fit to each category m of training examples, and the weights P'j'k wjk are frozen to produce a set of category templates. However, if the category identities are unknown, we face a novel kind of hierarchical learning task. We must simultaneously discover the optimal clustering of examples into categories, as well as the optimal clustering of features into parts within each category. We can formalize this hierarchical clustering problem as minimizing a familiar free energy, (7) in which gim. specifies the probability of assigning example i to category m, and Tim. is the associated cost. This cost is itself the free energy of the mth K -part model on the ith example, Tim = L LP'j'k(Ejk' j + _(31 logp'j'k) , (8) k in which P'j'k specifies the probability of assigning feature j to part k within category m, and Ejk' = (xj - Wjky~m.)2 is the usual reconstruction error from Section 2. This algorithm was tested on a data set of 256 hand configurations with 20 features each (similar to those in Figure 1), in which each example expresses one of four possible "gestures", i.e. patterns of feature correlation. As Table 1 indicates, the five features on each finger are highly correlated across the entire data set, while variable correlations between the four fingers distinguish the gesture categories. Note that a single model with four parts explains the full variance of the data just as well as the actual four-category generating process. However, most of the data Factorial Learning by Clustering Features 567 Table 1: The 20 features are grouped into either 2, 3, or 4 functional parts. Examples 1- 64 65 - 127 128 - 192 193 - 256 No. of parts 2 3 3 4 1 1 1-5 1-5 Part composition 10 11----20 10 11-15 16--20 6-10 11 20 6--10 11-15 16--20 can also be explained by one of several simpler models, making the learner's task a challenging balancing act between accuracy and simplicity. Figure 4 shows a typical representation learned for this data set. The algorithm was configured with M = 8 category models (each with K = 8 parts), but only four distinct categories of examples are found after annealing on a (holding {3 constant), and their weights prk wjk are depicted in Figure 4a. Each category faithfully captures one of the actual generating categories in Table 1, with the correct number and composition of functional parts. Figure 4b depicts the responsibility gi'm that each learned category m takes for each feature i. Notice the inevitable effect of a bias towards simpler representations. Many examples are misassigned relative to Table 1, when categories with fewer degrees of freedom than their true generating categories can explain them almost as accurately. 5 CONCLUSIONS AND FUTURE DIRECTIONS The notion that many data sets are best explained in terms of functionally independent clusters of correlated features resonates with similar proposals of Foldiak (1990), Saund (1995), Hinton and Zemel (1994), and Dayan and Zemel (1995). Our approach is unique in actually formulating the learning task as a clustering problem and explicitly extracting the functional parts of the data. Factorial learning by clustering features has three principal advantages. First, the free energy cost function for clustering yields a natural complexity scale-space of functional part representations, parameterized by {3. Second, the generalized EM learning algorithm is simple and quick, and maps easily onto a network architecture. Third, by nesting free energies, we can seamlessly compose objective functions for quite complex, hierarchical unsupservised learning problems, such as the multiple category, multiple part mixture problem of Section 4. The primary limitation of our approach is that when the generative model we assume does not in fact apply to the data, the algorithm may fail to recover any meaningful structure. In ongoing work, we are pursuing a more flexible generative model that allows the underlying causes to compete directly for arbitrary featureexample pairs ij, rather than limiting competition only to features j, as in Section 2, or only to examples i, as in conventional mixture models, or segregating competition for examples and features into hierarchical stages, as in Section 4. Because this introduces many more degrees of freedom, robust learning will require additional constraints, such as temporal continuity of examples or spatial continuity of features. 568 Joshua B. Tenenbaum, Emmanuel V. Todorov (a) Category 1 category 2 category 3 category 4 Part k Part k Part k Part k (b) 1 E 521 ..--------~..~w.~? Cl (*) 31------~????? . . . . .. 64 128 Example i 192 256 Figure 4: Learning multiple categories, each with a different part structure. Acknowledgements Both authors are Howard Hughes Medical Institute Predoctoral Fellows. We thank Whitman Richards, Yair Weiss, and Stephen Gilbert for helpful discussions. References Dayan, P. and Zemel, R. S. (1995). Computation, in press. Competition and multiple cause models. Neural Foldiak, P. (1990). Forming sparse representations by local anti-hebbian learning. Biological Cybernetics 64, 165-170. Hinton, G. E . and Zemel, R. S. (1994). Autoencoders, minimum description length and Helmholtz free energy. In J. D. Cowan, G. Tesauro, & J. Alspector (eds.), Advances in Neural Information Processing Systems 6. San Mateo, CA: Morgan Kaufmann, 3-10. Neal, R. M., and Hinton, G. E. (1994). A new view of the EM algorithm that justifies incremental and other variants. Rose, K., Gurewitz, F., and Fox, G. (1990). Statistical mechanics and phase transitions in clustering. Physical Review Letters 65, 945-948. Saund, E. (1995). A mUltiple cause mixture model for unsupervised learning. Neural Computation 7,51-71. Tenenbaum, J . (1994) . Functional parts. In A. Ram & K. Eiselt (eds.), Proceedings of the Sixteenth Annual Conference of the Cognitive Science Society. Hillsdale, N J: Lawrence Erlbaum, 864-869. 

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Predicting the Risk of Complications in Coronary Artery Bypass Operations using Neural Networks Richard P. Lippmann, Linda Kukolich MIT Lincoln Laboratory 244 Wood Street Lexington, MA 02173-0073 Dr. David Shahian Lahey Clinic Burlington, MA 01805 Abstract Experiments demonstrated that sigmoid multilayer perceptron (MLP) networks provide slightly better risk prediction than conventional logistic regression when used to predict the risk of death, stroke, and renal failure on 1257 patients who underwent coronary artery bypass operations at the Lahey Clinic. MLP networks with no hidden layer and networks with one hidden layer were trained using stochastic gradient descent with early stopping. MLP networks and logistic regression used the same input features and were evaluated using bootstrap sampling with 50 replications. ROC areas for predicting mortality using preoperative input features were 70.5% for logistic regression and 76.0% for MLP networks. Regularization provided by early stopping was an important component of improved perfonnance. A simplified approach to generating confidence intervals for MLP risk predictions using an auxiliary "confidence MLP" was developed. The confidence MLP is trained to reproduce confidence intervals that were generated during training using the outputs of 50 MLP networks trained with different bootstrap samples. 1 INTRODUCTION In 1992 there were roughly 300,000 coronary artery bypass operations perfonned in the United States at a cost of roughly $44,000 per operation. The $13.2 billion total cost of these operations is a significant fraction of health care spending in the United States. This has led to recent interest in comparing the quality of cardiac surgery across hospitals using risk-adjusted procedures and large patient populations. It has also led to interest in better assessing risks for individual patients and in obtaining improved understanding of the patient and procedural characteristics that affect cardiac surgery outcomes. 1056 Richard P. Lippmann, Linda Kukolich, David Shahian INPUT FEATURES ~ -... -.... ... ~ -~ SELECT I-FEATURES I-AND REPLACE MISSING FEATURES r-r-- CLASSIFY -... CONFIDENCE NETWORK ... ~ RISK PROBABILITY CONFIDENCE INTERVAL Figure 1. Block diagram of a medical risk predictor. This paper describes a experiments that explore the use of neural networks to predict the risk of complications in coronary artery bypass graft (CABO) surgery. Previous approaches to risk prediction for bypass surgery used linear or logistic regression or a Bayesian approach which assumes input features used for risk prediction are independent (e.g. Edwards, 1994; Marshall, 1994; Higgins, 1992; O'Conner, 1992). Neural networks have the potential advantages of modeling complex interactions among input features, of allowing both categorical and continuous input features, and of allowing more flexibility in fitting the expected risk than a simple linear or logistic function. 2 RISK PREDICTION AND FEATURE SELECTION A block diagram of the medical risk prediction system used in these experiments is shown in Figure 1. Input features from a patient's medical record are provided as 105 raw inputs, a smaller subset of these features is selected, missing features in this subset are replaced with their most likely values from training data, and a reduced input feature vector is fed to a classifier and to a "confidence network". The classifier provides outputs that estimate the probability or risk of one type of complication. The confidence network provides upper and lower bounds on these risk estimates. Both logistic regression and multilayer sigmoid neural network (MLP) classifiers were evaluated in this study. Logistic regression is the most common approach to risk prediction. It is structurally equivalent to a feed-forward network with linear inputs and one output unit with a sigmoidal nonlinearity. Weights and offsets are estimated using a maximum likelihood criterion and iterative "batch" training. The reference logistic regression classifier used in these experiments was implemented with the SPlus glm function (Mathsoft, 1993) which uses iteratively reweighted least squares for training and no extra regularization such as weight decay. Multilayer feed-forward neural networks with no hidden nodes (denoted single-layer MLPs) and with one hidden layer and from 1 to 10 hidden nodes were also evaluated as implemented using LNKnet pattern classification software (Lippmann, 1993). An MLP committee classifier containing eight members trained using different initial random weights was also evaluated. All classifiers were evaluated using a data base of 1257 patients who underwent coronary artery bypass surgery from 1990 to 1994. Classifiers were used to predict mortality, postoperative strokes, and renal failure. Predictions were made after a patient's medical history was obtained (History), after pre-surgical tests had been performed (Post-test), immediately before the operation (preop), and immediately after the operation (Postop). Bootstrap sampling (Efron, 1993) was used to assess risk prediction accuracy because there were so few Predicting the Risk of Complications in Coronary Artery Bypass Operations 1057 patients with complications in this data base. The number of patients with complications was 33 or 2.6% for mortality, 25 or 2.0% for stroke, and 21 or 1.7% for renal failure. All experiments were performed using 50 bootstrap training sets where a risk prediction technique is trained with a bootstrap training set and evaluated using left-out patterns. NComplications NHigh mSTORY Age COPD (Chronic Obs. Pul. Disease) % True Hits 27/674 71126 5.6% 8nl 61105 6/21 11.3% 5.7% 26.6% 211447 111115 10/127 7/64 4.7% 6.6% 7.9% 10.9% 121113 91184 10.6% 4.9% 4.0% POST?TEST Pulmonary Ventricular Congestion X-ray Cardiomegaly X-ray Pulmonary Edema PREOP NTG (Nitroglycerin) IABP (Intraaortic Balloon Pump) Urgency Status MI When POSTOP Blood Used (Packed Cells) Perfusion Time Figure 2. Features selected to predict mortality. The initial set of 105 raw input features included binary (e.g. MalelFemale), categorical (e.g. MI When: none, old, recent, evolving), and continuous valued features (e.g. Perfusion Time, Age). There were many missing and irrelevant features and all features were only weakly predictive. Small sets of features were selected for each complication using the following procedures: (1) Select those 10 to 40 features experience and previous studies indicate are related to each complication, (2) Omit features if a univariate contingency table analysis shows the feature is not important, (3) Omit features that are missing for more than 5% of patients, (4) Order features by number of true positives, (5) Omit features that are similar to other features keeping the most predictive, and (7) Add features incrementally as a patient's hospital interaction progresses. This resulted in sets of from 3 to 11 features for the three complications. Figure 2 shows the 11 features selected to predict mortality. The first column lists the features, the second column presents a fraction equal to the number of complications when the feature was "high" divided by the number of times this feature was "high" (A threshold was assigned for continuous and categorical features that provided good separation), and the last column is the second column expressed as a percentage. Classifiers were provided identical sets of input features for all experiments. Continuous inputs to all classifiers were normalized to have zero mean and unit variance, categorical inputs ranged from -(D-1)/2 to (D-1)/2 in steps of 1.0, where D is the number of categories, and binary inputs were -0.5 or 0.5 . 3 PERFORMANCE COMPARISONS Risk prediction was evaluated by plotting and computing the area under receiver operating characteristic (ROC) curves and also by using chi-square tests to determine how accurately classifiers could stratify subjects into three risk categories. Automated experiments were performed using bootstrap sampling to explore the effect of varying the training step size J058 Richard P. Lippmann. Linda Kukolich, David Shahian 100 ~ 80 :~ ~ c(/) 60 Q) en ~ 40 " I HISTORY (68.6%) ?fl. 20 o o 20 40 60 80 100 0 20 40 60 % FALSE ALARMS (100 - Specificity) 80 100 Figure 3. Fifty preoperative bootstrap ROCs predicting mortality using an MLP classifier with two hidden nodes and the average ROC (left), and average ROCS for mortality using history, preoperative, and postoperative features (right). from 0.005 to 0.1; of using squared-error, cross-entropy, and maximum likelihood cost functions; of varying the number of hidden nodes from 1 to 8; and of stopping training after from 5 to 40 epochs. ROC areas varied little as parameters were varied. Risk stratification, which measures how well classifier outputs approximate posterior probabilities, improved substantially with a cross-entropy cost function (instead of squared error), with a smaller stepsize (0.01 instead of 0.05 or 0.1) and with more training epochs (20 versus 5 or 10). An MLP classifier with two hidden nodes provided good overall performance across complications and patient stages with a cross-entropy cost function, a stepsize of 0.01, momentum of 0.6, and stochastic gradient descent stopping after 20 epochs. A single-layer MLP provided good performance with similar settings, but stopping after 5 epochs. These settings were used for all experiments. The left side of Figure 3 shows the 50 bootstrap ROCs created using these settings for a two-hidden-node MLP when predicting mortality with preoperative features and the ROC created by averaging these curves. There is a large variability in these ROes due to the small amount of training data. The ROC area varies from 67% to 85% (cr=4.7) and the sensitivity with 20% false alarms varies from 30% to 79%. Similar variability occurs for other complications. The right side of Figure 3 shows average ROCs for mortality created using this MLP with history, preoperative, and postoperative features. As can be seen, the ROC area and prediction accuracy increases from 68.6% to 79.2% as more input features become available. Figure 4 shows ROC areas across all complications and patient stages. Only three and two patient stages are shown for stroke and renal failure because no extra features were added at the missing stages for these complications. ROC areas are low for all complications and range from 62% to 80%. ROC areas are highest using postoperative features, lowest using only history features, and increase as more features are added. ROC areas are highest for mortality (68 to 80%) and lower for stroke (62 to 71 %) and renal failure (62 to 67% ).The MLP classifier with two hidden nodes (MLP) always provided slightly higher ROC areas than logistic regression. The average increase with the MLP classifier was 2.7 percentage Predicting the Risk of Complications in Coronary Artery Bypass Operations 100 90 III Logistic MORTALITY -80 C ~ 70 ~ 60 8a: 1059 [3 Single?Layer MLP ? MLP ? MLP?Commillee 50 40 30 HISTORY POSTTEST PRE?OP PATIENT STAGE POST?OP 100 90 - 80 C ? w 70 100 STROKE 90 80 I=2a I=2a 70 ~ 60 8a: RENAL FAILURE 60 50 50 40 40 30 30 HISTORY POSTTEST PATIENT STAGE POST?OP HISTORY POST?OP PATIENT STAGE Figure 4. ROC areas across all complications and patient stages for logistic regression, single-layer MLP classifier, two-layer MLP classifier with two hidden nodes, and a committee classifier containing eight two-layer MLP classifiers trained using different random starting weights. points (the increase ranged from 0.3 to 5.5 points). The single-layer MLPclassifier also provided good performance. The average ROC area with the single-layer MLP was only 0.6 percentage points below that of the MLP with two hidden nodes. The committee using eight two-layer MLP classifiers performed no better than an individual two-layer MLP classifier. Classifier outputs were used to bin or stratify each patient into one of four risk levels (05%, 5-10%, and 10-100%) by treating the output as an estimate of the complication posterior probability. Figure 5 shows the accuracy of risk stratification for the MLP classifier for all complications. Each curve was obtained by averaging 50 individual curves obtained using bootstrap sampling as with the ROC curves. Individual curves were obtained by placing each patient into one of the three risk bins based on the MLP output. The x's represent the average MLP output for all patients in each bin. Open squares are the true percentage of patients in each bin who experienced a complication. The bars represent ?2 binomial deviations about the true patient percentages. Risk prediction is accurate if the x's are close to the squares and within the confidence intervals. As can be seen, risk prediction is accurate and close to the actual number of patients who experienced complications. It is difficult, however, to assess risk prediction given the limited numbers of patients in the two highest bins. For example, in Figure 5, the median number of patients with complications was only 2 out of 20 in the middle bin and 2 out of 13 in the upper bin. Good and similar risk stratification, as measured by a chi-square test, was provided by all classifiers. Differences between classifier predictions and true patient percentages were small and not statistically significant. 1060 Richard P. Lippmann, Linda Kukolich, David Shahian ~ ~------------------------------~ MORTALITY 0- PATIENT COUNT X - MLP OUTPUT 30 10 o ____.M____________________________ 40 ~ ~--------------------------r_--_, RENAL FAILURE 30 20 10 o ~ __ _w~ 0-5 ________ ~ __________ 5?10 ~ __ ~ 10-100 BIN PROBABILITY RANGE ('?o) Figure 5. Accuracy of MLP risk stratification for three complications using preoperative features. Open squares are true percentages of patients in each bin with a complication, x's are MLP predictions, bars represent ?2 binomial standard deviation confidence intervals. 4 CONFIDENCE MLP NETWORKS Estimating the confidence in the classification decision produced by a neural network is a critical issue that has received relatively little study. Not being able to provide a confidence measure makes it difficult for physicians and other professionals to accept the use of complex networks. Bootstrap sampling (Efron, 1993) was selected as an approach to generate confidence intervals for medical risk prediction because 1) It can be applied to any type of classifier, 2) It measures variability due to training algorithms, implementation differences, and limited training data, and 3) It is simple to implement and apply. As shown in the top half of Figure 6, 50 bootstrap sets of training data are created from the original training data by resampling with replacement. These bootstrap training sets are used to train 50 bootstrap MLP classifiers using the same architecture and training procedures that were selected for the risk prediction MLP. When a pattern is fed into these classifiers, their outputs provide an estimate of the distribution of the output of the risk prediction MLP. Lower and upper confidence bounds for any input are obtained by sorting these outputs and selecting the 10% and 90% cumulative levels. It is computationally expensive to have to maintain and query 50 bootstrap MLPs whenever confidence bounds are desired. A simpler approach is to train a single confidence MLP to replicate the confidence bounds predicted by the 50 bootstrap MLPs, as shown in the bot- Predicting the Risk of Complications in Coronary Artery Bypass Operations 1061 OUTPUT STATISTICS UPPER LIMIT / INPUT PATTERN RISK PREDICTION MLP CONFIDENCE MLP LOWER LIMIT Figure 6. A confidence MLP trained using 50 bootstrap MLPs produces upper and lower confidence bounds for a risk prediction MLP. tom half of Figure 6. The the confidence MLP is fed the input pattern and the output of the risk prediction MLP and produces at its output the confidence intervals that would have been produced by 50 bootstrap MLPs. The confidence MLP is a mapping or regression network that replaces the 50 bootstrap networks. It was found that confidence networks with one hidden layer, two hidden nodes, and a linear output could accurately reproduce the upper and lower confidence intervals created by 50 bootstrap two-layer MLP networks. The confidence network outputs were almost always within ?15% of the actual bootstrap bounds. Upper and lower bounds produced by these confidence networks for all patients using preoperative features predicting mortality are show in Figure 7. Bounds are high (? 10 percentage points) when the complication risk is near 20% and drop to lower values (?0.4 percentage points) when the risk is near 1%. This relatively simple approach makes it possible to create and replicate confidence intervals for many types of classifiers. 5 SUMMARY AND FUTURE PLANS MLP networks provided slightly better risk prediction than conventional logistic regression when used to predict the risk of death, stroke, and renal failure on 1257 patients who underwent coronary artery bypass operations. Bootstrap sampling was required to compare approaches and regularization provided by early stopping was an important component of improved performance. A simplified approach to generating confidence intervals for MLP risk predictions using an auxiliary "confidence MLP" was also developed. The confidence MLP is trained to reproduce the confidence bounds that were generated during training by 50 MLP networks trained using bootstrap samples. Current research is validating these results using larger data sets, exploring approaches to detect outlier patients who are so different from any training patient that accurate risk prediction is suspect, developing approaches to explaining which input features are important for an individual patient, and determining why MLP networks provide improved performance. Richard P. Lippmann, Linda Kukolich, David Shahian 1062 ~r------.------.-------r------' T 30 T '# ... T ... TT 025 I- J;.... 5 :J20 T T UPPER W .. (,) Z 15 W Q u:: o(,) Z 10 5 COMPLICATION RISK% Figure 7. Upper and lower confidence bounds for all patients and preoperative mortality risk predictions calculated using two MLP confidence networks. ACKNOWLEDGMENT This work was sponsored by the Department of the Air Force. The views expressed are those of the authors and do not reflect the official policy or position of the U.S. Government. We wish to thank Stephanie Moisakis and Anne Nilson at the Lahey Clinic and Yuchun Lee at Lincoln Laboratory for assistance in organizing and preprocessing the data. BIBLIOGRAPHY F. Edwards, R. Clark, and M. Schwartz. (1994) Coronary Artery Bypass Grafting: The Society of Thoracic Surgeons National Database Experience. In Annals Thoracic Surgery, Vol. 57, 12-19. Bradley Efron and Robert J. Tibshirani. (1993) An Introduction to the Bootstrap. Monographs on Statistics and Applied Probability 57, New York: Chapman and Hall (1993). T. Higgins, F. Estafanous, et. al. (1992) Stratification of Morbidity and Mortality Outcome by Preoperative Risk Factors in Coronary Artery Bypass Patients. In Journal of the American Medical Society, Vol. 267, No. 17,2344-2348. R. Lippmann, L. Kukolich, and E. Singer. (1993) LNKnet: Neural Network, Machine Learning, and Statistical Software for Pattern Classification. In Lincoln Laboratory Journal, Vol. 6, No.2, 249-268. Marshall Guillenno, Laurie W. Shroyer, et al. (1994) Bayesian-Logit Model for Risk Assessment in Coronary Artery Bypass Grafting, In Annals Thoracic Surgery, Vol. 57, 1492-5000. G. O'Conner, S. Plume, et. al. (1992) Multivariate Prediction of In-Hospital Mortality Associated with Coronary Artery Bypass Surgery. In Circulation, Vol. 85, No.6, 21102118. Statistical Sciences. (1993) S-PLUS Guide to Statistical and Mathematical Analyses, Version 3.2, Seattle: StatSci, a division of MathSoft, Inc. 

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An Analog Neural Network Inspired by Fractal Block Coding Fernando J. Pineda Andreas G. Andreou The Applied Physics Laboratory The Johns Hopkins University Johns Hokins Road Laurel, MD 20723-6099 Dept. of Electrical & Computer Engineering The Johns Hopkins University 34th & Charles St. Baltimore, MD 21218 Abstract We consider the problem of decoding block coded data, using a physical dynamical system. We sketch out a decompression algorithm for fractal block codes and then show how to implement a recurrent neural network using physically simple but highly-nonlinear, analog circuit models of neurons and synapses. The nonlinear system has many fixed points, but we have at our disposal a procedure to choose the parameters in such a way that only one solution, the desired solution, is stable. As a partial proof of the concept, we present experimental data from a small system a 16-neuron analog CMOS chip fabricated in a 2m analog p-well process. This chip operates in the subthreshold regime and, for each choice of parameters, converges to a unique stable state. Each state exhibits a qualitatively fractal shape. 1. INTRODUCTION Sometimes, a nonlinear approach is the simplest way to solve a linear problem. This is true when computing with physical dynamical systems whose natural operations are nonlinear. In such cases it may be expensive, in terms of physical complexity, to linearize the dynamics. For example in neural computation active ion channels have highly non linear input-output behaviour (see Hille 1984). Another example is 796 Fernando Pineda. Andreas G. Andreou subthreshold CMOS VLSI technology 1. In both examples the physics that governs the operation of the active devices, gives rise to gain elements that have exponential transfer characteristics. These exponentials result in computing structures with non-linear dynamics. It is therefore worthwhile, from both scientific and engineering perspectives, to investigate the idea of analog computation by highly non-linear components. This paper, explores an approach for solving a specific linear problem with analog circuits that have nonlinear transfer functions. The computational task considered here is that of fractal block code decompression (see e.g. Jacquin, 1989). The conventional approach to decompressing fractal codes is essentially an excercise in solving a high-dimenional sparse linear system of equations by using a relaxation algorithm. The relaxation algorithm is performed by iteratively applying an affine transformation to a state vector. The iteration yields a sequence of state vectors that converges to a vector of decoded data. The approach taken in this paper is based on the observation that one can construct a physically-simple nonlinear dyanmical system whose unique stable fixed point coincides with the solution of the sparse linear system of equations. In the next section we briefly summarize the basic ideas behind fractal block coding. This is followed by a description of an analog circuit with physically-simple nonlinear neurons. We show how to set the input voltages for the network so that we can program the position of the stable fixed point. Finally , we present experimental results obtained from a test chip fabricated in a 2mm CMOS process. 2. FRACTAL BLOCK CODING IN A NUTSHELL Let the N-dimensional state vector I represent a one dimensional curve sampled on N points. An affine transformation of this vector is simply a transformation of the form I' = WI+B , where W is an NxN -element matrix and B is an N-component vector. This transformation can be iterated to produce a sequence of vectors I(O)... . ,I(n). The sequence converges to a unique final state 1* that is independent of the initial state 1(0) if the maximum eigenvalue A.max of the matrix W satisfies Amax <1. The uniqueness of the final state implies that to transmit the state r to a receiver, we can either transmit r directly, In or we can transmit Wand B and let the receiver perform the iteration to generate the latter case we say that Wand B constitute an encoding of the state 1*. For this encoding to be useful, the amount of data needed to transmit Wand B must be less than the amount of data needed to transmit r This is the case when Wand B are sparse and parameterized and when the total number of bits needed to transmit these parameters is less than the total number of bits needed to transmit the uncompressed state r. r Fractal block coding is a special case of the above approach. It amounts to choosing a lWe consider subthreshold analog VLSI., (Mead 1989; Andreou and Boahen, 1994). A simple subthreshold model is ~ = I~nfet) exp(K'Vgb)( exp( -vsb) -exp( -Vdb?) for NFETS, where 1C - 0.67 and I~ t) = 9.7 x 10-18 A. The voltage differences Vgb, ,vsb,and Vdb are in units of the thermal voltafje, Vth= 0.025V. We use a corresponding expression for PFETs of the from Ids = I~pfe exp( -K'Vgb)( exp(vsb) - exp(vdb?) where I~Pfet) =3.8xl0- 18 A. An Analog Neural Network Inspired by Fractal Block Coding 797 blocked structure for the matrix W. This structure forces large-scale features to be mapped into small-scale features. The result is a steady state r that represents a curve with self similar (actually self affine) features. As a concrete example of such a structure, consider the following transformation of the state I. for O~ i ~ N -1 2 (1) r i = w R12i - N + bR for N 2: ~ i ~ N-l This transformation has two blocks. The transformation of the first N/2 components of I depend on the parameters W Land b L while the transformation of the second N/2 components depend on the parameters WR, and bR . Consequently just four parameters completely specify this transformation. This transformation can be expressed as a single affine transformation as follows: wL /' 0 I'N/2-1 /' N12 /' N-l = 10 wL 1 N/2-1 1 WR wR bL + bL N/2 bR I N- 1 bR (2) The top and bottom halves of I I depend on the odd and even components of I respectively. This subsampling causes features of size I to be mapped into features of size 112. A subsampled copy of the state I with transformed intensities is copied into the top half of 1'. Similarly, a subsampled copy of the state I with transformed intensities is copied into the bottom half of 1'. If this transformation is iterated, the sequence of transformed vectors will converge provided the eigenvalues determined by WL and WR are all less than one (i.e. WL and WR < 1). Although this toy example has just four free parameters and is thus too trivial to be useful for actual compression applications, it does suffice to generate state vectors with fractal properties since at steady state, the top and bottom halves of I' differ from the entire curve by an affine transformation. In this paper we will not describe how to solve the inverse problem which consists of finding a parameterized affine transformation that produces a given final state T. We note, however, that it is a special (and simpler) case of the recurrent network training problem, since the problem is linear, has no hidden units and has only one fixed point. The reader is refered to (Pineda, 1988) or. for a least squares algorithm in the context of neural nets or to (Monroe and Dudbridge, 1992) for a least squares algorithm in the context of coding. 3. A CMOS NEURAL NETWORK MODEL Now that we have described the salient aspects of the fractal decompression problem, we tum to the problem of implementing an analog neural network whose nonlinear dynamics converges to the same fixed point as the linear system. Nonlinearity arises because we 798 Fernando Pineda, Andreas G. Andreou make no special effort to linearize the gain elements (controlled conductances and transconductances) of the implementation medium. In this section we first describe a simple neuron. Then we analyze the dynamics of a network composed of such neurons. Finally we describe how to program the fixed point in the actual physical network. 3.1 The analog Neuron \\(~t) woulgll ) like to create a neuron model that calculates the transformation I =al + b . Consider the circuit shown in figure 1. This has three functional sections which compute by adding and subtracting currents and. where voltages are "log" coded; this is the essence of the "current-mode" aproach in circuit design (Andreou et.al. 1994). The first section, receives an input voltage from a presynaptic neuron, converts it into a current I(in), and multiplies it by a weight a. The second section adds and subtracts the bias current b. The last section converts the output current into an output voltage and transmits it to the next neuron in the network. Since the transistors have exponential transfer characteristics, this voltage is logarithmically coded. The parameters a and b are set by external voltages. Theyarameter a, is set by a single external voltage Va while the bias parameter b = br-) - b( + is set by two external voltages vb (+) and vb H . Two voltages are used for b to account for both positive and negative bIas values since b(-?0 and b( +?0 . '{,(+) r---------, r---- -, r-----I I I I I I I I I I I I I I I I I I I I I I I I (ou I (in) I I I I I I II I : I I I I I (out) I I ~ I I I~--~---+----~~r--;--~--~V I I I I I I I I I I I I I I I I I I I I I I I I L ______ -' .I I I l : aiin~ V L ________ ------ ~-) Figure 1. The analog neuron has three sections. To derive the dynamical equations of the neuron, it is neccesary to add up all the currents and invoke Kirchoffs current law, which requires that lout ) _al in ) +b(+) -b(-) = Ic . (3) If we now assume a simple subthreshold model for the behavior of the FET's and PFETs in the neuron, we can obtain the following expression for the current across the capacitor: Q dl out ) lout) dt --- =1 c (4) An Analog Neural Network Inspired by Fractal Block Coding 799 where Q = Cl1cVth determines the characteristic time scale of the neuron 2. It immediately follows from the last two expressions that the dynamics of a single neuron is determined by the equation Q dJCout) = _/(out) (I(out) _ al(in) _ b). (5) dt Where b = M-) - M+) . This equation appears to have a quadratic nonlinearity on the r.h.s. In fact, the noninearity is even more complicated since, the cooeficients a, M+) and b( -) are not constants, but depend on I(out) (through v(out). Application of the simple subthreshold model, results in a multiplier gain that is a function of v( out) (and hence tout)) as well as Va . It is given by a( va' v<out?) = 2exp(- v~ {sinh( v~ - va) -sinh( v~ _v(out? J (6) Similarly, the currents b(+) and b H are given by and b( +) = I~fpet) exp( KVb(+) )( 1- exp( _v(out?) (7.a) b(-) = I~nfet) exp(KVb (_) )(I_exp(_v(Out?) (7.b) == vdd - va . 3.2 Network dynamics and Stability considerations respectively, where va With these results we conclude that, a network of neurons, in which each neuron receives input from only one other neuron, would have a dynamical equation of the form d['! = -1?(/? - a? ( I? ) 1 ?(?) - b?) Q_ dt !! ! I J! I (8) where the connectivity of the network is determined by the function j( i) . The fixed points of these highly nonlinear equations occur when the r.h.s. of (8) vanishes. This can only happen if either Ii = Oor if (Ii - aJj(i) - bi ) = 0 for each i. The local stability of each of these fixed points follows by examining the eigenvalues (A.) of the corresponding jacobian. The expression for the jacobian at a general point I is J'!k i = dF dlk =-Q[(/.! - a.J.(?) - b?)8? I J I I Ik + [,(1a~1I J.(.)I I - b~)8'k I I a.J.8 '(')k] . I I J I (9) Where the partial derivatives, a'i and b'j are with respect to Ii. At a fixed point the jacobian takes the form J ik { bi 8ik = Q -/i [ (1- a[lj(i) - b[)8ik - ai8j(i)k] if if Ii =0 (Ii - ailj(i) - b i ) = O? (10) 2C represents the total gate capacitance from all the transistors connected to the horizontal line of the neuron. For the 2J..l analog proc~s~, the gate capacitance is aprroximately 0.5 fF/J..l2 so a 10J..l x 10J..l FET has a charactenstlc charge of Q =2.959 x 10- 4 Coulombs at room temperature. 800 Fernando Pineda, Andreas G. Andreou There are two cases of interest. The first case is when no neurons have zero output. This is the "desired solution." In this case, the jacobian specializes to J ik =-QIi [(1-aiIj(i) -b[)Oik -aiOj(i)k]' (11) Where, from (6) and (7), it can be shown that the partial derivatives, a'i and b'i are both non-positive. It immediately follows, from Gerschgorin's theorem, that a sufficient condition that the eigenvalues be negative and that the fixed point be stable, is that lail <l. The second case is when at least one of the neurons has zero output. We call these fixed points the "spurious solutions." In this case some of the eigenvalues are very easy to calculate because terms of the form (bi -).) ,where Ii = 0, can be factored from the expression for det(J-.?J). Thus some eigenvalues can be made positive by making some of the bi positive. Accordingly, if all the bi satisfy bi >0 , some of the eigenvalues will necessarily be positive and the spurious solutions will be unstable. To summarize the above discussion, we have shown that by choosing bi >0 and lail <1 for all i, we can make the desired fixed point stable and the spurious fixed points unstable. Note that a sufficient condition for bi >0 is if b~ +) =O. It remains to show that the system must converge to the desired fixed point, i.e. that the system cannot oscillate or wander chaotically. To do this we consider the connectivity of the network we implemented in our test chip. This is shown schematically in figure 2. The first eight neurons receive input from the odd numbered neurons while the second eight neurons receive input from the even numbered neurons. The neurons on the lefthand side all share the weight, WL, while the neurons on the right share the weight WR. By tracing the connections, we find that there are two independent loops of neurons: loop #1 = {0,8,12,14,IS,7,3,1} and loop #2 = {2,9,4,1O,13,6,1l,S}. Figure 2. The connection topology for the test chip is determined by the matrix of equation (1). The neurons are labeled 0-15. By inspecting each loop, we see that it passes through either the left or right hand range an even number of times. Hence, if there are any inhibitory weights in a loop, there must be an even number of them. This is the "even loop criterion", and it suffices to prove that the network is globally asymptotically stable, (Hirsch, 1987). 3.3. Programming the fixed point The nonlinear circuit of the previous section converges to a fixed point which is the solution of the following system of transcendental equations * -bi(-) (Ii* ,vb<-?-O Ii* -ai(li* ,va)Ij(i) (12) An Analog Neural Network Inspired by Fractal Block Coding 801 where the coefficients ai and bi are given by equations (6) and (7b) respectively. Similarly, the iterated affine transformations converge to the solution of the following linear equations * - Bj Ii* - A/j(j) =0 (13) where the coefficients {Ai ,Bi } and the connectionsj(i) are obtained by solving the approximate inverse problem with the additional constraints that bi >0 and lail <1 for all i,. The requirement that the fixed points of the two systems be identical results in the conditions Aj = aj(lj* ,va) ) Bj -- b(-)(I* j j ,Vb(-) (14) These equations can be solved for the required input voltages Va, and v b (-). Thus we are able to construct a nonlinear dynamical system that converges to the same fixed point as a linear system. For this programming method to work, of course, the subthreshold model we have used to characterize the network must accurately model the physical properties of the neural network. 4. PRELIMINARY RESULTS As a first step towards realizing a working system, we fabricated a Tiny chip containing 16 neurons arranged in two groups of eight. The topology is the same as shown in figure 2. The neurons are similar to those in figure 1 except that the bias term in each block of 8 neurons has the form b = kb( -) + (7 - k )b( -) , where O::;k::;7 is the label of a particular neuron within a block. This form increases the complexity of the neurons, but also allows us to represent ramps more easily (see figure 3). We fabricated the chip through MOSIS in a 2~m p-well CMOS process. A switching layer allows us to change the connection topology at run-time. One of the four possible configurations corresponds to the toplogy of figure 2. Six external voltages {Va ,V H ' Vi) H ' Va ,Vb H ' Vi)H }parameterize the fixed points of the network. These are confrolfM blpote~tioIdeters~ There is multiplexing circuitry included on the chip that selects which neuron output is to be amplified by a sense-amp and routed off-chip. The neurons can be addressed individually by a 4-bit neuron address. The addressing and analog-to-digital conversion is performed by a Motorolla 68HCIIAI microprocessor. We have operated the chip at 5volts and at 2.6 volts. Figure 3. shows the scanned steady state output of one of the test chips for a particular choice of input parameters with vdd =5 volts. The curve in figure 3. exhibits the qualitatively self-similar features of a recursively generated object. We are able to see three generations of a ramp. At 2.5 volts we see a very similar curve. We find that the chip draws 16.3 ~ at 2.5 volts. This corresponds to a steady state power dissipation of 411lW. Simulations indicate that the chip is operating in the subthreshold regime when Vdd = 2.5 volts. Simulations also indicate that the chip settles in less than one millisecond. We are unable to perform quantitiative measurements with the first chip because of several layout errors. On the other hand, we have experimentally verified that the network is indeed stable and that network produces qualitative fractals. We explored the parameter space informatlly. At no time did we encounter anything but the desired solutions. Fernando Pineda, Andreas G. Andreou 802 O~--~~~------~--==:L----~----~----~- o 2 4 6 8 Neuron label 10 12 14 Figure 3 D/A output for chip #3 for a particular set of input voltages. We have already fabricated a larger design without the layout problems of the prototype. This second design has 32 pixeles and a richer set of permitted topologies. We expect to make quantitative measurements with this second design. In particular we hope to use it to decompress an actual block code. Acknowledgements The work described here is funded by APL !R&D as well as a grant from the National Science Foundation ECS9313934, Paul Werbos is the monitor. The authors would like to thank Robert Jenkins, Kim Strohbehn and Paul Furth for many useful conversations and suggestions. References Andreou, A.G. and Boahen, K.A. Neural Information Processing I: The Current-Mode approach, Analog VLSI: Signal and Information Processing, (eds: M Ismail and T. Fiez) MacGraw-Hill Inc., New York. Chapter 6 (1994). Hille, B., Ionic Channels of Excitable Membranes, Sunderland, MA, Sinauer Associates Inc. (1984). Hirsch, M. ,Convergence in Neural Nets, Proceedings of the IEEE ICNN, San Diego, CA, (1987). Jacquin, A. E., A Fractal Theory of iterated Markov operators with applications to digital image coding, Ph.D. Dissertation, Georgia Institute of Technology (1989). Mead, c., Analog VLSI and Neural Systems, Addison Wesley, (1989) Monroe, D.M. and Dudbridge, F. Fractal block coding of images, Electronics Letters, 28, pp. 1053-1055, (1992). Pineda, F.J., Dynamics and Architecuture for Neural Computation, Journal of Complexity, 4, 216-245 (1988). 

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A Lagrangian Formulation For Optical Backpropagation Training In Kerr-Type Optical Networks James E. Steck Mechanical Engineering Wichita State University Wichita, KS 67260-0035 Steven R. Skinner Electrical Engineering Wichita State University Wichita, KS 67260-0044 Alvaro A. Cruz-Cabrara Electrical Engineering Wichita State University Wichita, KS 67260-0044 Elizabeth C. Behrman Physics Department Wichita State University Wichita, KS 67260-0032 Abstract A training method based on a form of continuous spatially distributed optical error back-propagation is presented for an all optical network composed of nondiscrete neurons and weighted interconnections. The all optical network is feed-forward and is composed of thin layers of a Kerrtype self focusing/defocusing nonlinear optical material. The training method is derived from a Lagrangian formulation of the constrained minimization of the network error at the output. This leads to a formulation that describes training as a calculation of the distributed error of the optical signal at the output which is then reflected back through the device to assign a spatially distributed error to the internal layers. This error is then used to modify the internal weighting values. Results from several computer simulations of the training are presented, and a simple optical table demonstration of the network is discussed. Elizabeth C. Behrman 772 1 KERR TYPE MATERIALS Kerr-type optical networks utilize thin layers of Kerr-type nonlinear materials, in which the index of refraction can vary within the material and depends on the amount of light striking the material at a given location. The material index of refraction can be described by: n(x)=no+nzI(x), where 110 is the linear index of refraction, ~ is the nonlinear coefficient, and I(x) is the irradiance of a applied optical field as a function of position x across the material layer (Armstrong, 1962). This means that a beam of light (a signal beam carrying information perhaps) passing through a layer of Kerr-type material can be steered or controlled by another beam of light which applies a spatially varying pattern of intensity onto the Kerr-type material. Steering of light with a glass lens (having constance index of refraction) is done by varying the thickness of the lens (the amount of material present) as a function of position. Thus the Kerr effect can be loosely thought of as a glass lens whose geometry and therefore focusing ability could be dynamically controlled as a function of position across the lens. Steering in the Kerr material is accomplished by a gradient or change in the material index of refraction which is created by a gradient in applied light intensity. This is illustrated by the simple experiment in Figure 1 where a small weak probe beam is steered away from a straight path by the intensity gradient of a more powerful pump beam. lex) Pump I~ > /-.. . x Figure 1: Light Steering In Kerr Materials 2 OPTICAL NETWORKS USING KERR MATERIALS The Kerr optical network, shown in Figure 2, is made up of thin layers of the Kerr- type nonlinear medium separated by thick layers of a linear medium (free space) (Skinner, 1995). The signal beam to be processed propagates optically in a direction z perpendicular to the layers, from an input layer through several alternating linear and nonlinear layers to an output layer. The Kerr material layers perform the nonlinear processing and the linear layers serve as connection layers. The input (l(x)) and the weights (W\(x),W2(x) ... Wn(x)) are irradiance fields applied to the Kerr type layers, as functions of lateral position x, thus varying the A Lagrangian Formulation for Optical Backpropasation 773 refractive index profile of the nonlinear medium. Basically, the applied weight irradiences steer the signal beam via the Kerr effect discussed above to produce the correct output. The advantage of this type of optical network is that both neuron processing and weighted connections are achieved by uniform layers of the Kerr material. The all optical nature eliminates the need to physically construct neurons and connections on an individual basis. O(x,y) ? Plane Wave (Eo) Figure 2: Kerr Optical Neural Network Architecture If E;(ex) is the light entering the itlt nonlinear layer at lateral position ex, then the effect of the nonlinear layer is given by (1) where W;( ex) is the applied weight field. Transmission of light at lateral location ex at the beginning of the itlt linear layer to location p just before the i+ 1tit nonlinear layer is given by where c 3 ko =-- I 2!:lLI (2) OPTICAL BACK-PROPAGATION TRAINING Traditional feed-forward artificial neural networks composed of a finite number of discrete neurons and weighted connections can be trained by many techniques. Some of the most successful techniques are based upon the well known training method called backpropagation which results from minimizing the network output error, with respect to the network weights by a gradient descent algorithm. The optical network is trained using a form of continuous optical back-propagation which is developed for a nondiscrete network. Gradient descent is applied to minimize the error over the entire output region of the optical network. This error is a continuous distribution of error calculated over the output region. 774 Elizabeth C. Behrman Optical back-propagation is a specific technique by which this error distribution is optically propagated backward through the linear and nonlinear optical layers to produce error signals by which the light applied to the nonlinear layers is modified. Recall that this applied light Wi controls what serves as connection "weights" in the optical network. Optical backpropagation minimizes the error Lo over an output region 0 0 > a subdomain of the fmal or nth layer of the network, ~ where = 'Y r O(u'fJ '(uliu )c o (3) subject to the constraint that the propagated light, Ei( ex), satisfies the equations of forward propagation (1) and (2). O(P) = En+I(P) and is the network output, y is a scaling factor on the output intensity. Lo then is the squared error between the desired output value D and the average intensity 10 of the output distribution O( P). This constrained minimization problem is posed in a Lagrange formulation similar to the work of (Ie Cunn, 1988) for conventional feedforward networks and (pineda, 1987) for conventional recurrent networks; the difference being that for the optical network of this paper the Electric field E and the Lagrange multiplier are complex and also continuous in the spatial variable thus requiring the Lagrangian below. A Lagrangian is defmed as; L = 4, + + :t fA; t u ) [ EI+I(U) - fFI~)~ Ie -jctP ... )z 0. 0. JA/+~U) [Ei+~U) it. - 0. - fF~~)~/e-jC~13-?)Z dP ] ax (4) dP r ax 0. Taking the variation ofL with respect to E i, the Lagrange multipliers Ai, and using gradient descent to minimize L with respect to the applied weight fields Wi gives a set of equations that amount to calculating the error at the output and propagating the error optically backwards through the network. The pertinent results are given below. The distributed assigrunent of error on the output field is calculated by A1f+1(R.) = ... ~0 'Y ' (R.) [ D - 10 ] ... (5) This error is then propagated back through the nth or final linear optical layer by the equation ?c 6 (~) = ~ " 1t r z A + (u) e -jC,/..13-u) dx ) Co " 1 (6) which is used to update the "weight" light applied to the nth nonlinear layer. Optical backpropagation, through the ith nonlinear layer (giving AlP? followed by the linear layer (giving ~i-I(P? is performed according to the equations A Lagrangian Formulation for Optical Backpropagation 775 (7) This gives the error signal ~j'I(P) used to update the "weight" light distribution Wj.I(P) applied to the i-I nonlinear layer. The "weights" are updated based upon these errors according to the gradient descent rule Wi-(~) = w/,/d(P) +l'lt~)ktPNLin2W/"t~) 2 IM[ ~(~) 6,(~) e-~ANL.nZ<lw,ClId(p)f.IE.(I\)I'>] (8) where ,,;CP) is a learning rate which can be, but usually is not a function of layer number i and spatial position p. Figure 3 shows the optical network (thick linear layers and thin nonlinear layers) with the unifonn plane wave Eo, the input signal distribution I, forward propagation signals EI E2 .. . En' the weighting light distributions at the nonlinear layers WI W 2 .. , W n. Also shown are the error signal An+1 at the output and the back-propagated error signals ~n ... ~2 ~I for updating the nonlinear layers. Common nonlinear materials exist for which the material constants are such that the second term in the first of Equations 7 becomes small. Ignoring this second term gives an approximate fonn of optical backpropagation which amounts to calculating the error at the output of the network and then reversing its direction to optically propagate this error backward through the device. This can be easily seen by comparing Equations 6 and 7 (with the second tenn dropped) for optical back-propagation of the output error An with Equations I and 2 for the forward propagation of the signal E j. This means that the optical back-propagation training calculations potentially can be implemented in the same physical device as the forward network calculations. Equation (8) then becomes; Wi-(~) + = Wio/d(~) (2t'l,(~'>kot:.NL"2) w//d(~) [ (Et~) At~? - ~t~) ~(~)r] (9) which may be able to be implemented optically. 4 SIMULATION RESULTS To prove feasibility, the network was then trained and tested on several benchmark classification problems, two of which are discussed here. More details on these and other simulations of the optical network can be found in (Skinner, 1995). In the first (Using Nworks, 1991), iris species were classified into one of three categories: Setosa, Versicolor or Virginica. Classification was based upon length and width of the sepals and Elizabeth C. Behrman 776 petals. The network consisted of an input self-defocusing layer with an applied irradiance field which was divided into 4 separate Gaussian distributed input regions 25 microns in width followed by a linear layer. This pattern is repeated for 4 more groups composed of a nonlinear layer (with applied weights) followed by a linear layer. The final linear layer has three separate output regions 10 microns wide for binary classification as to species. The nonlinear layers were all 20 microns thick with n2=-.05 and the linear layers were 100 microns thick. The wavelength of applied light was 1 micron and the width of the network was 512 microns discretized into 512 pixels. This network was trained on a set of 50 training pairs to produce correct classification of all 50 training pairs. The network was then used to classify 50 additional pairs of test data which were not used in the training phase. The network classified 46 of these correctly for a 92 % accuracy level which is comparable to a standard feedforward network with discrete output region sigmoidal neurons. -\ \.. In the second problem, we tested the performance of the network on a set of data from a dolphin sonar discrimination experiment (Roitblat, 1991). In this study a dolphin was presented with one of three of different types objects (a tube, a sphere, and a cone), allowed to echolocate, and rewarded for choosing the correct one from a comparison array. The Fourier transforms of his click echoes, in the form of average amplitudes in each of 30 frequency bins, were then used as inputs for a neural network. Nine nonlinear layers were used along with 30 input regions and 3 \1l'Y). ? ? output plane tttttttttttT ALn ttl O(x,y) ?n(x,y) ? ? ? ? ? ? ? ? ? I .ili'Ln Wn(x,y) En(x,y) t t t t. t t t. t t t t ? 01 (X,y) ? ? ? ? ??? ? t?. t t t W1 (x,y) El(X,y) ttttttttttt I .ili'L1 T ALo ...----_ _----,1 ~ ~1 ____________I(_X_,y_)____________ Eo .ili'Lo ttttttttttt Plane Wave Figure 3: Optical Network Forward Data and Backward Error DataFlow A Lagrangian Formulatio1l for Optical Backpropagation 777 output regions, the remainder of the network physical parameters were the same as above for the iris classification. Half the data (13 sets of clicks) was used to train the network, with the other half of the data (14 sets) used to test the training. After training, classification of the test data set was 100% correct. 5 EXPERIMENTAL RESULTS As a proof of the concept, the optical neural network was constructed in the laboratory to be trained to perform various logic functions. Two thermal self-defocusing layers were used, one for the input and the other for a single layer of weighting. The nonlinear coefficient of the index of refraction (nJ was measured to be -3xlO'" cm21W. The nonlinear layers had a thickness (~NLo and ~NL.) of 630llm and were separated by a distance (~Lo) of 15cm. The output region was 100llm wide and placed 15cm (~L.) behind the weighting layer. The experiment used HeNe laser light to provide the input plane wave and the input and weighting irradiances. The spatial profiles of the input and weighting layers were realized by imaging a LCD spatial light modulator onto the respective nonlinear layers. The inputs were two bright or dark regions on a Gaussian input beam producing the intensity profile: whereIo= 12.5 mW/cm2, leo = 900llm, Xo = 600Ilffi, K. = 400Ilffi, and Qo and Q. are the logic inputs taking on a value of zero or one. The weight profile W.(x) = Ioexp[-(xIKo)2][1 +w.(x)] where w.(x) can range from zero to one and is found through training using an algorithm which probed the weighting mask in order update the training weights. Table 1 shows the experimental results for three different logic gates. Given is the normalized output before and after training. The network was trained to recognize a logic zero for a normalized output ~ 0.9 and a logic one or a normalized output ~ 1.1. An output value greater than I is considered a logic one and an output value less than one is a logic zero. RME is the root mean error. 6 CONCLUSIONS Work is in progress to improve the logic gate results by increasing the power of propagating signal beam as well as both the input and weighting beams. This will effectively increase the nonlinear processing capability of the network since a higher power produces more nonlinear effect. Also, more power will allow expansion of all of the beams thereby increasing the effective resolution of the thermal materials. Thisreduces the effect of heat transfer within the material which tends to wash out or diffuse benificial steep gradients in temperature which are what produce the gradients in the index of refraction. In addition, the use of photorefractive crystals for optical weight storage shows promise for being able to optically phase conjugate and backpropagate the output errror as well as implement the weight update rule for all optical network training. This appears to be simpler than optical networks using volume hologram weight storage because the Kerr network requires only planar hologram storage. 778 Elizabeth C. Behrman Inputs Start Finish NOR Change Output AND Start Finish Change Output Start XNOR Finish Change Output 0 0 1.001 1.110 .109 0 1 .802 .884 .998 .082 0 1.092 .757 -.241 .855 -.237 0 0 .998 1.084 .086 .880 .933 1 1 .053 0 1 0 .698 1 1 Rl\1E .807 .896 7.3% .089 0 1.440 -7.3% 1.124 0 -.254 0 .893 -.316 -16.4% 1 .994 7.3% .928 .035 1.073 .079 2.7% -4.6% 0 1 .772 .074 0 1.148 .894 0 16.4% Table 1: Preliminary Experimental Logic Gate Results References Armstrong, J.A., Bloembergen, N., Ducuing, J., and Pershan, P.S., (1962) "Interactions Between Light Waves in a Nonlinear Dielectric", Physical Review, Vol. 127, pp. 1918-1939. Ie Cun, Yann, (1988) itA Theoretical Framework for Back-Propagation", Proceedings of the 1988 Connectionist Models Summer School, Morgan Kaufmann, pp. 21-28. Pineda, F.J., (1987) "Generalization of backpropagation to recurrent and higher order neural networks", Proceedings of IEEE Conference on Neural information Processing Systems, November 1987, IEEE Press. Roitblat., Moore, Nachtigall, and Penner, (1991) "Natural dolphin echo recognition using an integrator gateway network," in Advances in Neural Processing Systems 3 Morgan Kaufmann, San Mateo, CA, 273-281. Skinner, S.R, Steck, J.E., Behnnan, E .C., (1995) "An Optical Neural Network Using Kerr Type Nonlinear Materials", To Appear in Applied Optics. "Using Nworks, (1991) An Extended Tutorial for NeuralWorks Professional /lIPlus and NeuralWorks Explorer, NeuralWare, Inc. Pittsburgh, PA, pg. UN-18. 

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Learning Many Related Tasks at the Same Time With Backpropagation Rich Caruana School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 caruana@cs.cmu.edu Abstract Hinton [6] proposed that generalization in artificial neural nets should improve if nets learn to represent the domain's underlying regularities . Abu-Mustafa's hints work [1] shows that the outputs of a backprop net can be used as inputs through which domainspecific information can be given to the net . We extend these ideas by showing that a backprop net learning many related tasks at the same time can use these tasks as inductive bias for each other and thus learn better. We identify five mechanisms by which multitask backprop improves generalization and give empirical evidence that multi task backprop generalizes better in real domains. 1 INTRODUCTION You and I rarely learn things one at a time, yet we often ask our programs to-it must be easier to learn things one at a time than to learn many things at once. Maybe not. The things you and I learn are related in many ways . They are processed by the same sensory apparatus, controlled by the same physical laws, derived from the same culture, ... Perhaps it is the similarity between the things we learn that helps us learn so well. What happens when a net learns many related functions at the same time? Will the extra information in the teaching signal of the related tasks help it learn better? Section 2 describes five mechanisms that improve generalization in backprop nets trained simultaneously on related tasks. Section 3 presents empirical results from a road-following domain and an object-recognition domain where backprop with multiple tasks improves generalization 10-40%. Section 4 briefly discusses when and how to use multitask backprop . Section 5 cites related work and Section 6 outlines directions for future work. 658 2 Rich Caruana MECHANISMS OF MULTITASK BACKPROP We identified five mechanisms that improve generalization in backprop nets trained simultaneously on multiple related tasks. The mechanisms all derive from the summing of error gradient terms at the hidden layer from the different tasks. Each exploits a different relationship between the tasks. 2.1 Data Amplification Data amplification is an effective increase in sample size due to extra information in the training signal of related tasks. There are two types of data amplification. 2.1.1 Statistical Data Amplification Statistical amplification, occurs when there is noise in the training signals. Consider two tasks, T and T', with independent noise added to their training signals, that both benefit from computing a feature F of the inputs. A net learning both T and T' can, if it recognizes that the two tasks share F, use the two training signals to learn F better by averaging F through the noise. The simplest case is when T T', i.e., when the two outputs are independently corrupted versions of the same signal. = 2.1.2 Blocking Data Amplification The 2nd form of data amplification occurs even if there is no noise. Consider two tasks, T and T', that use a common feature F computable from the inputs, but each uses F for different training patterns. A simple example is T = A OR F and T' = NOT(A) OR F. T uses F when A = 0 and provides no information about F when A = 1. Conversely, T' provides information about F only when A = 1. A net learning just T gets information about F only on training patterns for which A 0, but is blocked when A = 1. But a net learning both T and T' at the same time gets information about F on every training pattern; it is never blocked. It does not see more training patterns, it gets more information for each pattern. If the net learning both tasks recognizes the tasks share F, it will see a larger sample of F. Experiments with blocked functions like T and T' (where F is a hard but learnable function of the inputs such as parity) indicate backprop does learn common subfeatures better due to the larger effective sample size. = 2.2 Attribute Selection Consider two tasks, T and T', that use a common subfeature F. Suppose there are many inputs to the net, but F is a function of only a few of the inputs. A net learning T will,_if there is limited training data and/or significant noise, have difficulty distinguishing inputs relevant to F from those irrelevant to it. A net learning both T and T', however, will better select the attributes relevant to F because data amplification provides better training signals for F and that allows it to better determine which inputs to use to compute F. (Note: data amplification occurs even when there is no attribute selection problem. Attribute selection is a consequence of data amplification that makes data amplification work better when a selection problem exists.) We detect attribute selection by looking for connections to relevant inputs that grow stronger compared to connections for irrelevant inputs when multiple tasks are trained on the net. Learning Many Related Tasks at the Same Time with Backpropagation 2.3 659 Eavesdropping Consider a feature F, useful to tasks, T and T', that is easy to learn when learning T, but difficult to learn when learning T' because T' uses F in a more complex way. A net learning T will learn F, but a net learning just T' may not. If the net learning T' also learns T, T' can eavesdrop on the hidden layer learned for T (e.g., F) and thus learn better. Moreover, once the connection is made between T' and the evolving representation for F, the extra information from T' about F will help the net learn F better via the other mechanisms. The simplest case of eavesdropping is when T = F. Abu-Mostafa calls these catalytic hints[l]. In this case the net is being told explicitly to learn a feature F that is useful to the main task. Eavesdropping sometimes causes non-monotonic generalization curves for the tasks that eavesdrop on other tasks. This happens when the eavesdropper begins to overtrain, but then finds something useful learned by another task, and begins to perform better as it starts using this new information. 2.4 Representation Bias Because nets are initialized with random weights, backprop is a stochastic search procedure; multiple runs rarely yield identical nets. Consider the set of all nets (for fixed architecture) learnable by backprop for task T. Some of these generalize better than others because they better "represent" the domain's regularities. Consider one such regularity, F, learned differently by the different nets. Now consider the set of all nets learnable by backprop for another task T' that also learns regularity F. If T and T' are both trained on one net and the net recognizes the tasks share F, search will be biased towards representations of F near the intersection of what would be learned for T or T' alone. We conjecture that representations of F near this intersection often better capture the true regularity of F because they satisfy more than one task from the domain. Representations of F Findable by Backprop A form of representation bias that is easier to experiment with occurs when the representations for F sampled by the two tasks are different minima. Suppose there are two minima, A and B, a net can find for task T. Suppose a net learning task T' also has two minima, A and C. Both share the minima at A (i.e., both would perform well if the net entered that region of weight space), but do not overlap at Band C. We ran two experiments. In the first, we selected the minima so that nets trained on T alone are equally likely to find A or B, and nets trained on T' alone are equally likely to find A or C. Nets trained on both T and T' usually fall into A for both tasks. 1 Tasks prefer hidden layer representations that other tasks prefer. In the second experiment we selected the minima so that T has a strong preference lIn these experiments the nets have sufficient capacity to find independent minima for the tasks. They are not forced to share the hidden layer representations. But because the initial weights are random, they do initially share the hidden layer and will separate the tasks (i.e., use independent chunks of the hidden layer for each task) only if learning causes them to. 660 Rich Caruana for B over A: a net trained on T always falls into B. T', however, still has no preference between A or C. When both T and T' are trained on one net, T falls into B as expected: the bias from T' is unable to pull it to A. Surprisingly, T' usually falls into C, the minima it does not share with T! T creates a "tide" in the hidden layer representation towards B that flows away from A. T' has no preference for A or C, but is subject to the tide created by T. Thus T' usually falls into C; it would have to fight the tide from T to fall into A. Tasks prefer NOT to use hidden layer representations that other tasks prefer NOT to use. 2.5 How the Mechanisms are Related The "tide" mentioned while discussing representation bias results from the aggregation of error gradients from multiple tasks at the hidden layer. It is what makes the five mechanisms tick. It biases the search trajectory towards better performing regions of weight space. Because the mechanisms arise from the same underlying cause, it easy for them to act in concert. Their combined effect can be substantial. Although the mechanisms all derive from gradient summing, they are not the same. Each emphasizes a different relationship between tasks and has different effects on what is learned. Changes in architecture, representation, and the learning procedure affect the mechanisms in different ways. One particularly noteworthy difference between the mechanisms is that if there are minima, representation bias affects learning even with infinite sample size. The other mechanisms work only with finite sample size: data amplification (and thus attribute selection) and eavesdropping are beneficial only when the sample size is too small for the training signal for one task to provide enough information to the net for it to learn good models. 3 EMPIRICAL RESULTS Experiments on carefully crafted test problems verify that each of the mechanisms can work. 2 These experiments, however, do not indicate how effective multitask backprop is on real problems: tweaking the test problems alters the size of the effects. Rather than present results for contrived problems, we present a more convincing demonstration of multi task backprop by testing it on two realistic domains. 3.1 1D-ALVINN ID-ALVINN uses a road image simulator developed by Pomerleau. It was modified to generate I-D road images comprised of a single 32-pixel horizontal scan line instead of the original 2-D 30x32-pixel image. This was done to speed learning to allow thorough experimentation. ID-ALVINN retains much of the complexity of the original 2-D domain-the complexity lost is road curvature and that due to the smaller input (960 pixels vs. 32 pixels). The principle task in ID-ALVINN is to predict steering direction. Eight additional tasks were used for multitask backprop: ? ? ? ? whether the road is one or two lanes location of left edge of road location of road center intensity of region bordering road ? ? ? ? location of centerline (2-lane roads only) location of right edge of road intensity of road surface intensity of centerline (2-lane roads only) 2We have yet to determine how to directly test the hypothesis that representations for F in the intersection of T and T' perform better. Testing this requires interpreting representations learned for real tasks; experiments on test problems demonstrate only that search is biased towards the intersection, not that the intersection is the right place to be. Learning Many Related Tasks at the Same Time with Backpropagation 661 Table 1 shows the performance of single and multitask backprop (STB and MTB, respectively) on 1D-ALVINN using nets with one hidden layer. The MTB net has 32 inputs, 16 hidden units, and 9 outputs. The 36 STB nets have 32 inputs, 2, 4, 8 or 16 hidden units, and 1 output. A similar experiment using nets with 2 hidden layers containing 2, 4, 8, 16, or 32 hidden units per layer for STB and 32 hidden units per layer for MTB yielded comparable results. The size of the MTB nets is not optimized in either experiment. Table 1: Performance of STB and MTB with One Hidden Layer on 1D-ALVINN II TASK 1 or 2 Lanes Left Edge Right Edge Line Center Road Center Road Greylevel Edge Greylevel Line Greylevel Steering ROOT-MEAN SQUARED ERROR ON TEST SET Single Task Backprop MTB % Change % Change Best STB Mean STB 16HU 2HU T4HU I 8HU f 16HU .201 .209 .207 .178 .156 14.1 27.4 .069 .071 .073 .073 .062 11.3 15.3 .076 .062 .058 .056 .051 9.8 23.5 .152 .152 .151 .153 .152 0.7 0.8 .038 .037 .039 .042 .034 8.8 14.7 .055 .055 .054 .038 42.1 43.4 .054 .038 .037 .038 .039 .038 -2.6 0.0 .054 .054 .054 .054 0.0 0.0 .054 .069 .093 .087 .072 .058 19.0 38.4 The entries under the STB and MTB headings are the peak generalization error for nets of the specified size. The italicized STB entries are the STB runs that yielded best performance. The last two columns compare STB and MTB. The first is the percent difference between MTB and the best STB run. Positive percentages indicate MTB performs better. This test is biased towards STB because it compares a single run of MTB on an unoptimized net size with several independent runs of STB that use different random seeds and are able to find near-optimal net size. The last column is the percent difference between MTB and the average STB. Note that on the important steering task, MTB outperforms STB 20-40%. 3.2 ID-DOORS To test multitask backprop on a real problem, we created an object recognition domain similar in some respects to 1D-ALVINN. In 1D-DOORS the main tasks are to locate doorknobs and recognize door types (single or double) in images of doors collected with a robot-mounted camera. Figure 1 shows several door images. As with 1D-ALVINN, the problem was simplified by using horizontal stripes from image, one for the green channel and one for the blue channel. Each stripe is 30 pixels wide (accomplished by applying Gaussian smoothing to the original 150 pixel wide image) and occurs at the vertical location of the doorknob. 10 tasks were used: ? ? ? ? ? horizontal location of doorknob horizontal location of doorway center horizontal location of left door jamb width of left door jamb horizontal location of left edge of door ? ? ? ? ? single or double door width of doorway horizontal location of right door jamb width of right door jamb horizontal location of right edge of door The difficulty of 1D-DOORS precludes running as exhaustive a set of experiments as with 1D-ALVINN: runs were done only for the two most important and difficult tasks: doorknob location and door type. STB was tested on nets using 6, 24, and 96 hidden units. MTB was tested on a net with 120 hidden units. The results 662 Rich Caruana Figure 1: Sample Doors from the ID-DOORS Domain are in Table 2. STB generalizes 35-45% worse than MTB on these tasks. Less thorough experiments on the other eight tasks suggest MTB probably always yields performance equal to or better than STB. Table 2: Performance of STB and MTB on ID-DOORS. RMS ERROR ON TEST SET TASK Doorknob Loc Door Type 4 DISCUSSION In our experience, multitask backprop usually generalizes better than single task backprop. The few cases where STB has been better is on simpler tasks, and there the difference between MTB and STB was small. Multitask backprop appears to provide the most benefit on hard tasks. MTB also usually learns in fewer epochs than STB. When all tasks must be learned, MTB is computationally more efficient than training single nets. When few tasks are important, however, STB is usually more efficient (but also less accurate). Tasks do not always learn at the same rate. It is important to watch the training curve of each MTB task individually and stop training each task when its performance peaks. The easiest way to do this to take a snapshot of the net when performance peaks on a task of interest. MTB does not mean one net should be used to predict all tasks, only that all tasks should be trained on one net so they may benefit each other. Do not treat tasks as one task just because they are being trained on one net! Balancing tasks (e.g., using different learning rates for different outputs) sometimes helps tasks learn at similar rates, thus maximizing the potential benefits of MTB. Also, because the training curves for MTB are often more complex due to interactions between tasks (MTB curves are frequently multimodal), it is important to train MTB nets until all tasks appear to be overtraining. Restricting the capacity of MTB nets to force sharing or prevent overtraining usually hurts performance instead of helping it. MTB does not depend on restricted net capacity. We created the extra tasks in ID-ALVINN and ID-DOORS specifically because we thought they would improve performance on the important tasks. Multitask backprop can be used in other ways. Often the world gives us related tasks to learn. For example, the Calendar Apprentice System (CAP)[4] learns to predict the Location, Time_Of _Day, Day_Of _Week, and Duration of the meetings it schedules. These tasks are functions of the same data, share many common features, Learning Many Related Tasks at the Same Time with Backpropagation 663 and would be easy to learn together. Sometimes the world gives us related tasks in mysterious ways. For example, in a medical domain we are examining where the goal is to predict illness severity, half of the lab tests are cheap and routinely measured before admitting a patient (e.g., blood pressure, pulse, age). The rest are expensive tests requiring hospitalization. Users tell us it would be useful to predict if the severity of the illness warrants admission (and further testing) using just the pre-admission tests. Rather than ignore the most diagnostically useful information in the database, we use the expensive tests as additional tasks the net must learn. They are not very predictable from the simple pre-admission tests, but providing them to the net as outputs helps it learn illness severity better. Multitask backprop is one way of providing to a net information that at run time would only be available in the future. The training signals are needed only for the training set because they are outputs-not inputs-to the net. 5 RELATED WORK Training nets with many outputs is not new; NETtalk [9] used one net to learn phonemes and stress. This approach was natural for NETtalk where the goal was to control a synthesizer that needed both phoneme and stress commands at the same time. No analysis, however, was made of the advantages of using one net for all the tasks3 , and the different outputs were not treated as independent tasks. For example, the NETtalk stress task overtrains badly long before the phoneme task is learned well, but NETtaik did not use different snapshots of the net for different tasks. NETtalk also made no attempt to balance tasks so that they would learn at a similar rate, or to add new tasks that might improve learning but which would not be useful for controlling the synthesizer. Work has been done on serial transfer between nets [8]. Improved learning speed was reported, but not improved generalization. The key difficulties with serial transfer are that it is difficult to scale to many tasks, it is hard to prevent catastrophic interference from erasing what was learned previously, the learning sequence must be defined manually, and serial learning precludes mutual benefit between tasks. This work is most similar to catalytic hints [1][10] where extra tasks correspond to important learnable features of a main task. This work extends hints by showing that tasks can be related in more diverse ways, by expanding the class of mechanisms responsible for multitask backprop, by showing that capacity restriction is not an important mechanism for multitask backprop [2], and by demonstrating that creating many new related tasks may be an efficient way of providing domain-specific inductive bias to backprop nets. 6 FUTURE WORK We used vanilla backprop to show the benefit of training many related tasks on one net. Additional techniques may enhance the effects. Regularization and incremental net growing procedures might improve performance by promoting sharing without restricting capacity. New techniques may also be necessary to enhance the benefit of multitask backprop. Automatic balancing of task learning rates would make MTB easier to use. It would also be valuable to know when the different MTB mechanisms are working-they might be useful in different kinds of domains and might benefit from different regularization or balancing techniques. Finally, although MTB usually seems to help and rarely hurts, the only way to know it 3S ee [5] for evidence that NETtalk is harder to learn using separate nets. 664 Rich Caruana helps is to try it. It would be better to have a predictive theory of how tasks should relate to benefit MTB , particularly if new tasks are to be created only to provide a multitask benefit for the other important tasks in the domain. 7 SUMMARY Five mechanisms that improve generalization performance on nets trained on multiple related tasks at the same time have been identified. These mechanisms work without restricting net capacity or otherwise reducing the net's VC-dimension. Instead, they exploit backprop's ability to combine the error terms for related tasks into an aggregate gradient that points towards better underlying represen tations. Multitask backprop was tested on a simulated domain, ID-ALVINN, and on a real domain, ID-DOORS . It improved generalization performance on hard tasks in these domains 20-40% compared with the best performance that could be obtained from multiple trials of single task backprop. Acknowledgements Thanks to Tom Mitchell, Herb Simon, Dean Pomerleau, Tom Dietterich, Andrew Moore, Dave Touretzky, Scott Fahlman, Sebastian Thrun, Ken Lang, and David Zabowski for suggestions that have helped shape this work. This research is sponsored in part by the Advanced Research Projects Agency (ARPA) under grant no. F33615-93-1-1330. References [1] Y.S. Abu-Mostafa, "Learning From Hints in Neural Networks," Journal of Complexity 6:2, pp. 192-198,1989. [2] Y.S. Abu-Mostafa, "Hints and the VC Dimension," Neural Computation, 5:2, 1993. [3] R. Caruana, "Multitask Connectionist Learning," Proceedings of the 1993 Connectionist Models Summer School, pp. 372-379, 1993. [4] L. Dent, J. Boticario, J. McDermott, T. Mitchell, and D. Zabowski, "A Personal Learning Apprentice," Proceedings of 1992 National Conference on Artificial Intelligence, 1992. [5] T.G. Dietterich, H. Hild, and G. Bakiri, "A Comparative Study of ID3 and Backpropagation for English Text-to-speech Mapping," Proceedings of the Seventh International Conference on Artificial Intelligence, pp. 24-31, 1990. [6] G.E. Hinton, "Learning Distributed Representations of Concepts," Proceedings of the Eight International Conference of The Cognitive Science Society, pp. 112, 1986. [7] D.A. Pomerleau, "Neural Network Perception for Mobile Robot Guidance," Carnegie Mellon University: CMU-CS-92-115, 1992. [8] L.Y. Pratt, J. Mostow, and C.A. Kamm, "Direct Transfer of Learned Information Among Neural Networks," Proceedings of AAAI-91, 1991. [9] T.J. Sejnowski and C.R. Rosenberg, "NETtalk: A Parallel Network that Learns to Read Aloud ," John Hopkins: JHU/EECS-8'6/01, 1986. [10] S.C. Suddarth and A.D.C. Holden, "Symbolic-neural Systems and the Use of Hints for Developing Complex Systems," International Journal of MaxMachine Studies 35:3, pp. 291-311, 1991. 

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Non-linear Prediction of Acoustic Vectors Using Hierarchical Mixtures of Experts S.R.Waterhouse A.J.Robinson Cambridge University Engineering Department, Trumpington St ., Cambridge, CB2 1PZ, England. Tel: [+44] 223 332800, Fax: [+44] 223 332662, Email: srwlO01.ajr@eng.cam.ac.uk URL: http://svr-www.eng.cam.ac.ukr srw1001 Abstract In this paper we consider speech coding as a problem of speech modelling. In particular, prediction of parameterised speech over short time segments is performed using the Hierarchical Mixture of Experts (HME) (Jordan & Jacobs 1994). The HME gives two advantages over traditional non-linear function approximators such as the Multi-Layer Percept ron (MLP); a statistical understanding of the operation of the predictor and provision of information about the performance of the predictor in the form of likelihood information and local error bars. These two issues are examined on both toy and real world problems of regression and time series prediction. In the speech coding context, we extend the principle of combining local predictions via the HME to a Vector Quantization scheme in which fixed local codebooks are combined on-line for each observation. 1 INTRODUCTION We are concerned in this paper with the application of multiple models, specifically the Hierarchical Mixtures of Experts, to time series prediction, specifically the problem of predicting acoustic vectors for use in speech coding. There have been a number of applications of multiple models in time series prediction. A classic example is the Threshold Autoregressive model (TAR) which was used by Tong & 836 S. R. Waterhouse, A. J. Robinson Lim (1980) to predict sunspot activity. More recently, Lewis, Kay and Stevens (in Weigend & Gershenfeld (1994)) describe the use of Multivariate and Regression Splines (MARS) to the prediction of future values of currency exchange rates. Finally, in speech prediction, Cuperman & Gersho (1985) describe the Switched Inter-frame Vector Prediction (SIVP) method which switches between separate linear predictors trained on different statistical classes of speech. The form of time series prediction we shall consider in this paper is the single step prediction fI(t) of a future quantity y(t) , by considering the previous samples. This may be viewed as a regression problem over input-output pairs {x t), y(t)}~ where x(t) is the lag vector (y(t-I), y(t-2), ... , y(t- p ?. We may perform this regression using standard linear models such as the Auto-Regressive (AR) model or via nonlinear models such as connectionist feed-forward or recurrent networks. The HME overcomes a number of problems associated with traditional connectionist models via its architecture and statistical framework. Recently, Jordan & Jacobs (1994) and Waterhouse & Robinson (1994) have shown that via the EM algorithm and a 2nd order optimization scheme known as Iteratively Reweighted Least Squares (IRLS), the HME is faster than standard Multilayer Perceptrons (MLP) by at least an order of magnitude on regression and classification tasks respectively. Jordan & Jacobs also describe various methods to visualise the learnt structure of the HME via 'deviance trees' and histograms of posterior probabilities. In this paper we provide further examples of the structural relationship of the trained HME and the input-output space in the form of expert activation plots. In addition we describe how the HME can be extended to give local error bars or measures of confidence in regression and time series prediction problems. Finally, we describe the extension of the HME to acoustic vector prediction, and a VQ coding scheme which utilises likelihood information from the HME. c: 2 HIERARCHICAL MIXTURES OF EXPERTS The HME architecture (Figure 1) is based on the principle of 'divide and conquer' in which a large, hard to solve problem is broken up into many, smaller, easier to solve problems. It consists of a series of 'expert networks' which are trained on different parts of the input space. The outputs of the experts are combined by a 'gating network' which is trained to stochastically select the expert which is performing best at solving a particular part of the problem. The operation of the HME is as follows: the gating networks receive the input vectors x(t) and produce as outputs probabilities P(mi/.x(t), 7'/j) for each local branch mj of assigning the current input to the different branches, where T/j are the gating network parameters. The expert networks sit at the leaves of the tree and each output a vector flJt) given input vector x(t) and parameters Bj . These outputs are combined in a weighted sum by P(mjlX<t), T/j) to give the overall output vector for this region. This procedure continues recursively upwards to the root node. In time series prediction, each expert j is a linear single layer network with the form: flY) = B; x (t) where B; is matrix and form to an AR model. x(t) is the lag vector discussed earlier, which is identical in Non-Linear Prediction of Acoustic Vectors Using Hierarchical Mixtures of Experts x x x 837 x Figure 1: The Hierarchical Mixture of Experts. 2.1 Error bars via HME Since each expert is an AR model, it follows that the output of each expert y(t) is the expected value of the observations y(t) at each time t. The conditional likelihood of yet) given the input and expert mj is P(y(t) Ix (t), mj, Bj) = 12:Cj I exp ( - ~ (y - yy?)T Cj(y - yjt))) where Cj is the covariance matrix for expert mj which is updated during training as: C = _1_ "'" h(t)(y(r) _ y(t)l (y(t) _ y~t)) J ' " h~t) L..J J ] ] L.Jt J t where hy) are the posterior probabilities I of each expert mj' Taking the moments of the overall likelihood of the HME gives the output of the HME as the conditional expected value of the target output yct), yet) = E(yct)lxct) , 0, M) = 2: P(mjlxct), l1j)E(y(t)lx ct), ej,mj) =2: gY)iJ/t), j j Where M represents the overall HME model and Taking the second central moment of yct) gives, C = = e the overall set of parameters. I E?y(t) - yy?)2 xct), 0, M) 2: P(mJlx(t), l1j)E?y(t) 2: gjt)(Cj + yjt). iJj(t)T), yjt))2I x (t), ej, mj) j = j lSee (Jordan & Jacobs 1994) for a fuller discussion of posterior probabilities and likelihoods in the context of the HME. 838 S. R. Waterhouse, A. J. Robinson which gives, in a direct fashion, the covariance of the output given the input and the model. If we assume that the observations are generated by an underlying model, which generates according to some function f(x(t)) and corrupted by zero mean normally distributed noise n(x) with constant covariance 1:, then the covariance of y(t) is given by, V(y(t)) =V(t o) + 1:, so that the covariance computed by the method above, V(y(t)) , takes into account the modelling error as well as the uncertainty due to the noise. Weigend & Nix (1994) also calculate error bars using an MLP consisting of a set of tanh hidden units to estimate the conditional mean and an auxiliary set of tanh hidden units to estimate the variance, assuming normally distributed errors. Our work differs in that there is no assumption of normality in the error distribution, rather that the errors of the terminal experts are distributed normally, with the total error distribution being a mixture of normal distributions. 3 SIMULATIONS In order to demonstrate the utility of our approach to variance estimation we consider one toy regression problem and one time series prediction problem. 3.1 Toy Problem: Computer generated data 2,------..--...., 0 . 08,------~-..., 1.5 O. OS N ~ fO.04 ~ -0.02 2 x ~-0.5 -1 -0.2 -1.5 -0.4 0.8 Q) go.S -2 -O.S -2.5 -0.8 -3 '------'------' o 2 x .~ ~0.4 -1 ' - - - - - - ' - - - - - " o 2 x x 2 Figure 2: Performance on the toy data set of a 5 level binary HME. (a) training set (dots) and underlying function f(x) (solid), (b) underlying function (solid) and prediction y(x) (dashed), (c) squared deviation of prediction from underlying function, (d) true noise variance (solid) and variance of prediction (dashed). By way of comparison, we used the same toy problem as Weigend & Nix (1994) which consists of 1000 training points and 10000 separate evaluation points from Non-Linear Prediction of Acoustic Vectors Using Hierarchical Mixtures of Experts 839 the function g(x) where g(x) consists of a known underlying function f(x) corrupted by normally distributed noise N(O, (J2(X)) , f(x) = sin(2.5x) x sin(l. 5x), (J2(x) = 0.01 + O. 25 x [1 - sin(2. 5x)f. As can be seen by Figure 2, the HME has learnt to approximate both the underlying function and the additive noise variance. The deviation of the estimated variance from the "true" noise variance may be due to the actual noise variance being lower than the maximum denoted by the solid line at various points. 3.2 Sunspots 1920 1930 1940 1950 1960 1970 1980 Year 8 6 t 8.4 )( w 2 . . -.-- . ? ? ? ? ? .. ? ... II ? l- ? - ? I ? - ? - .. - ? -? ? ? ? OL-------~--------~--------~--------L-------~--------~ 1920 1930 1940 1950 1960 1970 1980 Year Figure 3: Performance on the Sunspots data set. (a) Actual Values (x) and predicted values (0) with error bars. (b) Activation of the expert networks; bars wide in the vertical axis indicate strong activation. Notice how expert 7 concentrates on the lulls in the series while expert 2 deals with the peaks. I METHOD I MLP TAR HME NMSE ' Train 1700-1920 0.082 0.097 0.061 Test 1921-1955 1956-1979 0.086 0.35 0.097 0.28 0.089 0.27 Table 1: Results of single step prediction on the Sunspots data set using a mixture of 7 experts (104 parameters) and a lag vector of 12 years. NMSE' is the NMSE normalised by the variance of the entire record 1700 to 1979. 840 S. R. Waterhouse, A. J. Robinson The Sunspots2 time series consists of yearly sunspot activity from 1700 to 1979 and was first tackled using connectionist models by Weigend, Huberman & Rumelhart (1990) who used a 12-8-1 MLP (113 parameters) . Prior to this work, the TAR was used by Tong (1990). Our results, which were obtained using a random leave 10% out cross validation method, are shown in Table 1. We are considering only single step prediction on this problem, which involves prediction of the next value based on a set of previous values of the time series. Our results are evaluated in terms of Normalised Mean Squared Error (NMSE) (Weigend et al. 1990), which is defined as the ratio of the variance of the prediction on the test set to the variance of the test set itself. The HME outperforms both the TAR and the MLP on this problem, and additionally provides both information about the structure of the network after training via the expert activation plot and error bars of the predictions, as shown in Figure 3. Further improvements may be possible by using likelihood information during cross validation so that a joint optimisation of overall error and variance is achieved. 4 SPEECH CODING USING HME In the standard method of Linear Predictive Coding (LPC) (Makhoul 1975), speech is parametrised into a set of vectors of duration one frame (around 10 ms). Whilst simple scalar quantization of the LPC vectors can achieve bit rates of around 2400 bits per second (bps), Yong, Davidson & Gersho (1988) have shown that simple linear prediction of Line Spectral Pairs (LSP) (Soong & Juang 1984) vectors followed by Vector Quantization (VQ) (Abut , Gray & Rebolledo 1984) of the error vectors can yield bit rates of around 800 bps. In this paper we describe a speech coding framework which uses the HME in two stages. Firstly, the HME is used to perform prediction of the acoustic vectors. The error vectors are then quantized efficiently by using a VQ scheme which utilises the likelihood information derived from the HME . 4.1 Mixing VQ codebooks ia Gating networks In a VQ scheme using a Euclidean distance measure , there is an implicit assumption that the inputs follow a Gaussian probability density function (pdf). This is satisfied if we quantize the residuals from a linear predictor , but not the residuals from an HME which follow a mixture of Gaussians pdf. A more efficient method is therefore to generate separate VQ code books for each expert in the HME and combine them via the priors on each expert from the gating networks. The code book for the overall residual vectors on the test set is then generated at each time dynamically by choosing the first D x gjt) codes, where D is the size of the expert codebooks and gY) is the prior on each expert. 2 Available via anonymous DataSunspots.Yearly ftp at fip.cs.colorado.edu III jpub jTime-Series as Non-Linear Prediction of Acoustic Vectors Using Hierarchical Mixtures of Experts 4.2 841 Results of Speech Coding Evaluations Initial experiments were performed using 23 Mel scale log energy frequency bins as acoustic vectors and using single variances Cj = (J;I as expert network covariance matrices. The results of training over 100 ,000 frames and evaluation over a further 100 ,000 frames on the Resource Management (RM) corpus are shown in Table 2 and Figure 4 which shows the good specialisation of the HME in this problem. METHOD Prediction Train 12.07 18.1 20.20 Linear 1 level HME 2 level HME Gain (dB) Test 10.95 15.55 16.39 Table 2: Prediction of Acoustic Vectors using linear prediction and binary branching HMEs with 1 and 2 levels. Prediction gain (Cuperman & Gersho 1985) is the ratio of the signal variance to prediction error variance. (a) Spectogram 20 !!! ::3 c;; 15 Cf ~ 10 ::3 8 < . 5 o 10 20 30 40 50 60 70 80 90 100 Frame (b) Gating Network Decisions 71--- - - - I - -__.- I 6~----. . . .~.---~--_ _--------_=--~~--? ~5~~---------1~---.I---1IHI~I~?--. . . .~.~.----~4~~~----------------~~~~---------- w3~~---------~-----. .~__------_.----------------- - 2.-.---- - -- - - -11-1...---11---1- --11-1 ? ? I II . ....- I ? 1~--------------~~~~--------------------------------~ 10 20 30 40 50 60 70 80 90 100 Frame Figure 4: The behaviour of a mixture of 7 experts at predicting Mel-scale log energy frequency bins over 100 16ms fram es. The top figure is a spectrogram of the speech and the lower figure is an expert activation plot, showmg the gating network decisions. We have conducted further experiments using LSPs and cepstrals as acoustic vectors , and using diagonal expert network covariance matrices, on a very large speech corpus. However, initial experiments show only a small improvement in gain over a single linear predictor and further investigation is underway. We have also coded acoustic vectors using 8 bits per frame with frame lengths of 12.5 ms, passing power, pitch and degree of voicing as side band information , without appreciable distortion over simple LPC coding. A full system will include prediction of all acoustic 842 S. R. Waterhouse, A. J. Robinson parameters and we anticipate further reductions on this initial figure with future developments. 5 CONCLUSION The aim of speech coding is the efficient coding of the speech signal with little perceptual loss. This paper has described the use of the HME for acoustic vector prediction. We have shown that the HME can provide improved performance over a linear predictor and in addition it provides a time varying variance for the prediction error. The decomposition of the linear prediction problem into a solution via a mixture of experts also allows us to construct a VQ codebook on the fly by mixing the codebooks of the various experts. We expect that the direct computation of the time varying nature of the prediction accuracy will find many applications. Within the acoustic vector prediction problem we would like to exploit this information by exploring the continuum between the fixed bit rate coder described here and a variable bit rate coder that produces constant spectral distortion. Acknowledgements This work was funded in part by Hewlett Packard Laboratories, UK. Steve Waterhouse is supported by an EPSRC Research Students hip and Tony Robinson was supported by a EPSRC Advanced Research Fellowship. References Abut, H., Gray, R. M. & Rebolledo, G. (1984), 'Vector quantization of speech and speechlike waveforms', IEEE Transactions on Acoustics, Speech, and Signal Processing. Cuperman, V. & Gersho, A. (1985), 'Vector predictive coding of speech at 16 kblt/s', IEEE Transactions on Communications COM-33, 685-696. Jordan, M. I. & Jacobs, R. A. (1994), 'Hierarchical Mixtures of Experts and the EM algorithm', Neural Computation 6, 181-214. Makhoiil, J. (1975), 'Linear prediction: A tutorial review', Proceedings of the IEEE 63(4) 561-580. Soong~ F. k. & Juang, B. H. (1984), Line spectrum pair (LSP) and speech data compressIon. Tong, H. (1990), Non-linear Time Series: a dynamical systems approach, Oxford Universi~y Press. Tong, H. & Lim, K. (1980), 'Threshold autoregression, limit cycles and cyclical data', Journal of Royal Statistical Society. Waterhouse, S. R. & Robinson, A. J. (1994), Classification using hierarchical mixtures of experts, in 'IEEE Workshop on Neural Networks for SigI!al Processing'. Weigend, A. S. & Gershenfeld, N. A. (1994), Time Series Prediction: Forecasting the Future and Understanding the Past) Addison-Wesley. Weigend, A. S. & Nix, D. A. (1994), Predictions with confidence intervals (local error bars), Technical Report CU-CS-724-94, Department of Computer Science and Institute of Coznitive Science, University of Colorado, Boulder, CO 80309-0439. Weigend, A. S., Huberman, B. A. & Rumelhart, D. E. (1990), 'Predicting the future: a connectionist approach', International Journal of Neural Systems 1, 193-209. Yong, M., Davidson, G. & Gersho, A. (1988), Encoding of LPC spectral parameters using switched-adaptive interframe vector prediction, in 'Proceedings of the IEEE International Conference on Acoustics Speech, and Signal Processing', pp. 402-405. 

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