Daily Papers

In this work, a method of random parameters generation for randomized learning of a single-hidden-layer feedforward neural network is proposed. The method firstly, randomly selects the slope angles of the hidden neurons activation functions from an interval adjusted to the target function, then randomly rotates the activation functions, and finally distributes them across the input space. For complex target functions the proposed method gives better results than the approach commonly used in practice, where the random parameters are selected from the fixed interval. This is because it introduces the steepest fragments of the activation functions into the input hypercube, avoiding their saturation fragments.

We consider the problem of learning a target function corresponding to a single hidden layer neural network, with a quadratic activation function after the first layer, and random weights. We consider the asymptotic limit where the input dimension and the network width are proportionally large. Recent work [Cui & al '23] established that linear regression provides Bayes-optimal test error to learn such a function when the number of available samples is only linear in the dimension. That work stressed the open challenge of theoretically analyzing the optimal test error in the more interesting regime where the number of samples is quadratic in the dimension. In this paper, we solve this challenge for quadratic activations and derive a closed-form expression for the Bayes-optimal test error. We also provide an algorithm, that we call GAMP-RIE, which combines approximate message passing with rotationally invariant matrix denoising, and that asymptotically achieves the optimal performance. Technically, our result is enabled by establishing a link with recent works on optimal denoising of extensive-rank matrices and on the ellipsoid fitting problem. We further show empirically that, in the absence of noise, randomly-initialized gradient descent seems to sample the space of weights, leading to zero training loss, and averaging over initialization leads to a test error equal to the Bayes-optimal one.

The study of universal approximation properties (UAP) for neural networks (NN) has a long history. When the network width is unlimited, only a single hidden layer is sufficient for UAP. In contrast, when the depth is unlimited, the width for UAP needs to be not less than the critical width w^*_{min}=max(d_x,d_y), where d_x and d_y are the dimensions of the input and output, respectively. Recently, cai2022achieve shows that a leaky-ReLU NN with this critical width can achieve UAP for L^p functions on a compact domain K, i.e., the UAP for L^p(K,R^{d_y}). This paper examines a uniform UAP for the function class C(K,R^{d_y}) and gives the exact minimum width of the leaky-ReLU NN as w_{min}=max(d_x+1,d_y)+1_{d_y=d_x+1}, which involves the effects of the output dimensions. To obtain this result, we propose a novel lift-flow-discretization approach that shows that the uniform UAP has a deep connection with topological theory.

There has been a recent surge of interest in modeling neural networks (NNs) as Gaussian processes. In the limit of a NN of infinite width the NN becomes equivalent to a Gaussian process. Here we demonstrate that for an ensemble of large, finite, fully connected networks with a single hidden layer the distribution of outputs at initialization is well described by a Gaussian perturbed by the fourth Hermite polynomial for weights drawn from a symmetric distribution. We show that the scale of the perturbation is inversely proportional to the number of units in the NN and that higher order terms decay more rapidly, thereby recovering the Edgeworth expansion. We conclude by observing that understanding how this perturbation changes under training would reveal the regimes in which the Gaussian process framework is valid to model NN behavior.

While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data. However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important optimization and generalization errors. To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset. A DeepONet consists of two sub-networks, one for encoding the input function at a fixed number of sensors x_i, i=1,dots,m (branch net), and another for encoding the locations for the output functions (trunk net). We perform systematic simulations for identifying two types of operators, i.e., dynamic systems and partial differential equations, and demonstrate that DeepONet significantly reduces the generalization error compared to the fully-connected networks. We also derive theoretically the dependence of the approximation error in terms of the number of sensors (where the input function is defined) as well as the input function type, and we verify the theorem with computational results. More importantly, we observe high-order error convergence in our computational tests, namely polynomial rates (from half order to fourth order) and even exponential convergence with respect to the training dataset size.

There is some theoretical evidence that deep neural networks with multiple hidden layers have a potential for more efficient representation of multidimensional mappings than shallow networks with a single hidden layer. The question is whether it is possible to exploit this theoretical advantage for finding such representations with help of numerical training methods. Tests using prototypical problems with a known mean square minimum did not confirm this hypothesis. Minima found with the help of deep networks have always been worse than those found using shallow networks. This does not directly contradict the theoretical findings---it is possible that the superior representational capacity of deep networks is genuine while finding the mean square minimum of such deep networks is a substantially harder problem than with shallow ones.

Recent works have suggested that finite Bayesian neural networks may sometimes outperform their infinite cousins because finite networks can flexibly adapt their internal representations. However, our theoretical understanding of how the learned hidden layer representations of finite networks differ from the fixed representations of infinite networks remains incomplete. Perturbative finite-width corrections to the network prior and posterior have been studied, but the asymptotics of learned features have not been fully characterized. Here, we argue that the leading finite-width corrections to the average feature kernels for any Bayesian network with linear readout and Gaussian likelihood have a largely universal form. We illustrate this explicitly for three tractable network architectures: deep linear fully-connected and convolutional networks, and networks with a single nonlinear hidden layer. Our results begin to elucidate how task-relevant learning signals shape the hidden layer representations of wide Bayesian neural networks.

The commitment to single-precision floating-point arithmetic is widespread in the deep learning community. To evaluate whether this commitment is justified, the influence of computing precision (single and double precision) on the optimization performance of the Conjugate Gradient (CG) method (a second-order optimization algorithm) and RMSprop (a first-order algorithm) has been investigated. Tests of neural networks with one to five fully connected hidden layers and moderate or strong nonlinearity with up to 4 million network parameters have been optimized for Mean Square Error (MSE). The training tasks have been set up so that their MSE minimum was known to be zero. Computing experiments have disclosed that single-precision can keep up (with superlinear convergence) with double-precision as long as line search finds an improvement. First-order methods such as RMSprop do not benefit from double precision. However, for moderately nonlinear tasks, CG is clearly superior. For strongly nonlinear tasks, both algorithm classes find only solutions fairly poor in terms of mean square error as related to the output variance. CG with double floating-point precision is superior whenever the solutions have the potential to be useful for the application goal.

This paper presents a novel technique based on gradient boosting to train the final layers of a neural network (NN). Gradient boosting is an additive expansion algorithm in which a series of models are trained sequentially to approximate a given function. A neural network can also be seen as an additive expansion where the scalar product of the responses of the last hidden layer and its weights provide the final output of the network. Instead of training the network as a whole, the proposed algorithm trains the network sequentially in T steps. First, the bias term of the network is initialized with a constant approximation that minimizes the average loss of the data. Then, at each step, a portion of the network, composed of J neurons, is trained to approximate the pseudo-residuals on the training data computed from the previous iterations. Finally, the T partial models and bias are integrated as a single NN with T times J neurons in the hidden layer. Extensive experiments in classification and regression tasks, as well as in combination with deep neural networks, are carried out showing a competitive generalization performance with respect to neural networks trained with different standard solvers, such as Adam, L-BFGS, SGD and deep models. Furthermore, we show that the proposed method design permits to switch off a number of hidden units during test (the units that were last trained) without a significant reduction of its generalization ability. This permits the adaptation of the model to different classification speed requirements on the fly.

Humans are remarkably data-efficient when adapting to new unseen conditions, like driving a new car. In contrast, modern robotic control systems, like neural network policies trained using Reinforcement Learning (RL), are highly specialized for single environments. Because of this overfitting, they are known to break down even under small differences like the Simulation-to-Reality (Sim2Real) gap and require system identification and retraining for even minimal changes to the system. In this work, we present RAPTOR, a method for training a highly adaptive foundation policy for quadrotor control. Our method enables training a single, end-to-end neural-network policy to control a wide variety of quadrotors. We test 10 different real quadrotors from 32 g to 2.4 kg that also differ in motor type (brushed vs. brushless), frame type (soft vs. rigid), propeller type (2/3/4-blade), and flight controller (PX4/Betaflight/Crazyflie/M5StampFly). We find that a tiny, three-layer policy with only 2084 parameters is sufficient for zero-shot adaptation to a wide variety of platforms. The adaptation through In-Context Learning is made possible by using a recurrence in the hidden layer. The policy is trained through a novel Meta-Imitation Learning algorithm, where we sample 1000 quadrotors and train a teacher policy for each of them using Reinforcement Learning. Subsequently, the 1000 teachers are distilled into a single, adaptive student policy. We find that within milliseconds, the resulting foundation policy adapts zero-shot to unseen quadrotors. We extensively test the capabilities of the foundation policy under numerous conditions (trajectory tracking, indoor/outdoor, wind disturbance, poking, different propellers).

We revisit the one-shot Neural Architecture Search (NAS) paradigm and analyze its advantages over existing NAS approaches. Existing one-shot method, however, is hard to train and not yet effective on large scale datasets like ImageNet. This work propose a Single Path One-Shot model to address the challenge in the training. Our central idea is to construct a simplified supernet, where all architectures are single paths so that weight co-adaption problem is alleviated. Training is performed by uniform path sampling. All architectures (and their weights) are trained fully and equally. Comprehensive experiments verify that our approach is flexible and effective. It is easy to train and fast to search. It effortlessly supports complex search spaces (e.g., building blocks, channel, mixed-precision quantization) and different search constraints (e.g., FLOPs, latency). It is thus convenient to use for various needs. It achieves start-of-the-art performance on the large dataset ImageNet.

Developing Intelligent Systems involves artificial intelligence approaches including artificial neural networks. Here, we present a tutorial of Deep Neural Networks (DNNs), and some insights about the origin of the term "deep"; references to deep learning are also given. Restricted Boltzmann Machines, which are the core of DNNs, are discussed in detail. An example of a simple two-layer network, performing unsupervised learning for unlabeled data, is shown. Deep Belief Networks (DBNs), which are used to build networks with more than two layers, are also described. Moreover, examples for supervised learning with DNNs performing simple prediction and classification tasks, are presented and explained. This tutorial includes two intelligent pattern recognition applications: hand- written digits (benchmark known as MNIST) and speech recognition.

Our proposed deeply-supervised nets (DSN) method simultaneously minimizes classification error while making the learning process of hidden layers direct and transparent. We make an attempt to boost the classification performance by studying a new formulation in deep networks. Three aspects in convolutional neural networks (CNN) style architectures are being looked at: (1) transparency of the intermediate layers to the overall classification; (2) discriminativeness and robustness of learned features, especially in the early layers; (3) effectiveness in training due to the presence of the exploding and vanishing gradients. We introduce "companion objective" to the individual hidden layers, in addition to the overall objective at the output layer (a different strategy to layer-wise pre-training). We extend techniques from stochastic gradient methods to analyze our algorithm. The advantage of our method is evident and our experimental result on benchmark datasets shows significant performance gain over existing methods (e.g. all state-of-the-art results on MNIST, CIFAR-10, CIFAR-100, and SVHN).

Few algorithms for supervised training of spiking neural networks exist that can deal with patterns of multiple spikes, and their computational properties are largely unexplored. We demonstrate in a set of simulations that the ReSuMe learning algorithm can be successfully applied to layered neural networks. Input and output patterns are encoded as spike trains of multiple precisely timed spikes, and the network learns to transform the input trains into target output trains. This is done by combining the ReSuMe learning algorithm with multiplicative scaling of the connections of downstream neurons. We show in particular that layered networks with one hidden layer can learn the basic logical operations, including Exclusive-Or, while networks without hidden layer cannot, mirroring an analogous result for layered networks of rate neurons. While supervised learning in spiking neural networks is not yet fit for technical purposes, exploring computational properties of spiking neural networks advances our understanding of how computations can be done with spike trains.

This paper demonstrates that a single-layer neural network using Parametric Rectified Linear Unit (PReLU) activation can solve the XOR problem, a simple fact that has been overlooked so far. We compare this solution to the multi-layer perceptron (MLP) and the Growing Cosine Unit (GCU) activation function and explain why PReLU enables this capability. Our results show that the single-layer PReLU network can achieve 100\% success rate in a wider range of learning rates while using only three learnable parameters.

We present a neural network architecture and training method designed to enable very rapid training and low implementation complexity. Due to its training speed and very few tunable parameters, the method has strong potential for applications requiring frequent retraining or online training. The approach is characterized by (a) convolutional filters based on biologically inspired visual processing filters, (b) randomly-valued classifier-stage input weights, (c) use of least squares regression to train the classifier output weights in a single batch, and (d) linear classifier-stage output units. We demonstrate the efficacy of the method by applying it to image classification. Our results match existing state-of-the-art results on the MNIST (0.37% error) and NORB-small (2.2% error) image classification databases, but with very fast training times compared to standard deep network approaches. The network's performance on the Google Street View House Number (SVHN) (4% error) database is also competitive with state-of-the art methods.

Despite their great success, there is still no comprehensive theoretical understanding of learning with Deep Neural Networks (DNNs) or their inner organization. Previous work proposed to analyze DNNs in the Information Plane; i.e., the plane of the Mutual Information values that each layer preserves on the input and output variables. They suggested that the goal of the network is to optimize the Information Bottleneck (IB) tradeoff between compression and prediction, successively, for each layer. In this work we follow up on this idea and demonstrate the effectiveness of the Information-Plane visualization of DNNs. Our main results are: (i) most of the training epochs in standard DL are spent on {\emph compression} of the input to efficient representation and not on fitting the training labels. (ii) The representation compression phase begins when the training errors becomes small and the Stochastic Gradient Decent (SGD) epochs change from a fast drift to smaller training error into a stochastic relaxation, or random diffusion, constrained by the training error value. (iii) The converged layers lie on or very close to the Information Bottleneck (IB) theoretical bound, and the maps from the input to any hidden layer and from this hidden layer to the output satisfy the IB self-consistent equations. This generalization through noise mechanism is unique to Deep Neural Networks and absent in one layer networks. (iv) The training time is dramatically reduced when adding more hidden layers. Thus the main advantage of the hidden layers is computational. This can be explained by the reduced relaxation time, as this it scales super-linearly (exponentially for simple diffusion) with the information compression from the previous layer.

Deep learning sometimes appears to work in unexpected ways. In pursuit of a deeper understanding of its surprising behaviors, we investigate the utility of a simple yet accurate model of a trained neural network consisting of a sequence of first-order approximations telescoping out into a single empirically operational tool for practical analysis. Across three case studies, we illustrate how it can be applied to derive new empirical insights on a diverse range of prominent phenomena in the literature -- including double descent, grokking, linear mode connectivity, and the challenges of applying deep learning on tabular data -- highlighting that this model allows us to construct and extract metrics that help predict and understand the a priori unexpected performance of neural networks. We also demonstrate that this model presents a pedagogical formalism allowing us to isolate components of the training process even in complex contemporary settings, providing a lens to reason about the effects of design choices such as architecture & optimization strategy, and reveals surprising parallels between neural network learning and gradient boosting.

We study the problem of learning a single neuron with respect to the L_2^2-loss in the presence of adversarial label noise. We give an efficient algorithm that, for a broad family of activations including ReLUs, approximates the optimal L_2^2-error within a constant factor. Our algorithm applies under much milder distributional assumptions compared to prior work. The key ingredient enabling our results is a novel connection to local error bounds from optimization theory.

The wide application of pre-trained models is driving the trend of once-for-all training in one-shot neural architecture search (NAS). However, training within a huge sample space damages the performance of individual subnets and requires much computation to search for an optimal model. In this paper, we present PreNAS, a search-free NAS approach that accentuates target models in one-shot training. Specifically, the sample space is dramatically reduced in advance by a zero-cost selector, and weight-sharing one-shot training is performed on the preferred architectures to alleviate update conflicts. Extensive experiments have demonstrated that PreNAS consistently outperforms state-of-the-art one-shot NAS competitors for both Vision Transformer and convolutional architectures, and importantly, enables instant specialization with zero search cost. Our code is available at https://github.com/tinyvision/PreNAS.

The canonical deep learning approach for learning requires computing a gradient term at each layer by back-propagating the error signal from the output towards each learnable parameter. Given the stacked structure of neural networks, where each layer builds on the representation of the layer below, this approach leads to hierarchical representations. More abstract features live on the top layers of the model, while features on lower layers are expected to be less abstract. In contrast to this, we introduce a new learning method named NoProp, which does not rely on either forward or backwards propagation. Instead, NoProp takes inspiration from diffusion and flow matching methods, where each layer independently learns to denoise a noisy target. We believe this work takes a first step towards introducing a new family of gradient-free learning methods, that does not learn hierarchical representations -- at least not in the usual sense. NoProp needs to fix the representation at each layer beforehand to a noised version of the target, learning a local denoising process that can then be exploited at inference. We demonstrate the effectiveness of our method on MNIST, CIFAR-10, and CIFAR-100 image classification benchmarks. Our results show that NoProp is a viable learning algorithm which achieves superior accuracy, is easier to use and computationally more efficient compared to other existing back-propagation-free methods. By departing from the traditional gradient based learning paradigm, NoProp alters how credit assignment is done within the network, enabling more efficient distributed learning as well as potentially impacting other characteristics of the learning process.

The success of deep learning is due in large part to our ability to solve certain massive non-convex optimization problems with relative ease. Though non-convex optimization is NP-hard, simple algorithms -- often variants of stochastic gradient descent -- exhibit surprising effectiveness in fitting large neural networks in practice. We argue that neural network loss landscapes often contain (nearly) a single basin after accounting for all possible permutation symmetries of hidden units a la Entezari et al. 2021. We introduce three algorithms to permute the units of one model to bring them into alignment with a reference model in order to merge the two models in weight space. This transformation produces a functionally equivalent set of weights that lie in an approximately convex basin near the reference model. Experimentally, we demonstrate the single basin phenomenon across a variety of model architectures and datasets, including the first (to our knowledge) demonstration of zero-barrier linear mode connectivity between independently trained ResNet models on CIFAR-10. Additionally, we identify intriguing phenomena relating model width and training time to mode connectivity. Finally, we discuss shortcomings of the linear mode connectivity hypothesis, including a counterexample to the single basin theory.

Can we automatically design a Convolutional Network (ConvNet) with the highest image classification accuracy under the runtime constraint of a mobile device? Neural architecture search (NAS) has revolutionized the design of hardware-efficient ConvNets by automating this process. However, the NAS problem remains challenging due to the combinatorially large design space, causing a significant searching time (at least 200 GPU-hours). To alleviate this complexity, we propose Single-Path NAS, a novel differentiable NAS method for designing hardware-efficient ConvNets in less than 4 hours. Our contributions are as follows: 1. Single-path search space: Compared to previous differentiable NAS methods, Single-Path NAS uses one single-path over-parameterized ConvNet to encode all architectural decisions with shared convolutional kernel parameters, hence drastically decreasing the number of trainable parameters and the search cost down to few epochs. 2. Hardware-efficient ImageNet classification: Single-Path NAS achieves 74.96% top-1 accuracy on ImageNet with 79ms latency on a Pixel 1 phone, which is state-of-the-art accuracy compared to NAS methods with similar constraints (<80ms). 3. NAS efficiency: Single-Path NAS search cost is only 8 epochs (30 TPU-hours), which is up to 5,000x faster compared to prior work. 4. Reproducibility: Unlike all recent mobile-efficient NAS methods which only release pretrained models, we open-source our entire codebase at: https://github.com/dstamoulis/single-path-nas.

Neural networks are trained by choosing an architecture and training the parameters. The choice of architecture is often by trial and error or with Neural Architecture Search (NAS) methods. While NAS provides some automation, it often relies on discrete steps that optimize the architecture and then train the parameters. We introduce a novel neural network training framework that fundamentally transforms the process by learning architecture and parameters simultaneously with gradient descent. With the appropriate setting of the loss function, it can discover sparse and compact neural networks for given datasets. Central to our approach is a multi-scale encoder-decoder, in which the encoder embeds pairs of neural networks with similar functionalities close to each other (irrespective of their architectures and weights). To train a neural network with a given dataset, we randomly sample a neural network embedding in the embedding space and then perform gradient descent using our custom loss function, which incorporates a sparsity penalty to encourage compactness. The decoder generates a neural network corresponding to the embedding. Experiments demonstrate that our framework can discover sparse and compact neural networks maintaining a high performance.

Despite their widespread success, the application of deep neural networks to functional data remains scarce today. The infinite dimensionality of functional data means standard learning algorithms can be applied only after appropriate dimension reduction, typically achieved via basis expansions. Currently, these bases are chosen a priori without the information for the task at hand and thus may not be effective for the designated task. We instead propose to adaptively learn these bases in an end-to-end fashion. We introduce neural networks that employ a new Basis Layer whose hidden units are each basis functions themselves implemented as a micro neural network. Our architecture learns to apply parsimonious dimension reduction to functional inputs that focuses only on information relevant to the target rather than irrelevant variation in the input function. Across numerous classification/regression tasks with functional data, our method empirically outperforms other types of neural networks, and we prove that our approach is statistically consistent with low generalization error. Code is available at: https://github.com/jwyyy/AdaFNN.

We introduce an algorithm where the individual bits representing the weights of a neural network are learned. This method allows training weights with integer values on arbitrary bit-depths and naturally uncovers sparse networks, without additional constraints or regularization techniques. We show better results than the standard training technique with fully connected networks and similar performance as compared to standard training for convolutional and residual networks. By training bits in a selective manner we found that the biggest contribution to achieving high accuracy is given by the first three most significant bits, while the rest provide an intrinsic regularization. As a consequence more than 90\% of a network can be used to store arbitrary codes without affecting its accuracy. These codes may be random noise, binary files or even the weights of previously trained networks.

Backpropagation is the standard method for achieving state-of-the-art accuracy in neural network training, but it often imposes high memory costs and lacks biological plausibility. In this paper, we introduce the Mono-Forward algorithm, a purely local layerwise learning method inspired by Hinton's Forward-Forward framework. Unlike backpropagation, Mono-Forward optimizes each layer solely with locally available information, eliminating the reliance on global error signals. We evaluated Mono-Forward on multi-layer perceptrons and convolutional neural networks across multiple benchmarks, including MNIST, Fashion-MNIST, CIFAR-10, and CIFAR-100. The test results show that Mono-Forward consistently matches or surpasses the accuracy of backpropagation across all tasks, with significantly reduced and more even memory usage, better parallelizability, and a comparable convergence rate.

Our understanding of the generalization capabilities of neural networks (NNs) is still incomplete. Prevailing explanations are based on implicit biases of gradient descent (GD) but they cannot account for the capabilities of models from gradient-free methods nor the simplicity bias recently observed in untrained networks. This paper seeks other sources of generalization in NNs. Findings. To understand the inductive biases provided by architectures independently from GD, we examine untrained, random-weight networks. Even simple MLPs show strong inductive biases: uniform sampling in weight space yields a very biased distribution of functions in terms of complexity. But unlike common wisdom, NNs do not have an inherent "simplicity bias". This property depends on components such as ReLUs, residual connections, and layer normalizations. Alternative architectures can be built with a bias for any level of complexity. Transformers also inherit all these properties from their building blocks. Implications. We provide a fresh explanation for the success of deep learning independent from gradient-based training. It points at promising avenues for controlling the solutions implemented by trained models.

In recent years, deep learning has been a revolution in the field of machine learning, for computer vision in particular. In this approach, a deep (multilayer) artificial neural network (ANN) is trained in a supervised manner using backpropagation. Huge amounts of labeled examples are required, but the resulting classification accuracy is truly impressive, sometimes outperforming humans. Neurons in an ANN are characterized by a single, static, continuous-valued activation. Yet biological neurons use discrete spikes to compute and transmit information, and the spike times, in addition to the spike rates, matter. Spiking neural networks (SNNs) are thus more biologically realistic than ANNs, and arguably the only viable option if one wants to understand how the brain computes. SNNs are also more hardware friendly and energy-efficient than ANNs, and are thus appealing for technology, especially for portable devices. However, training deep SNNs remains a challenge. Spiking neurons' transfer function is usually non-differentiable, which prevents using backpropagation. Here we review recent supervised and unsupervised methods to train deep SNNs, and compare them in terms of accuracy, but also computational cost and hardware friendliness. The emerging picture is that SNNs still lag behind ANNs in terms of accuracy, but the gap is decreasing, and can even vanish on some tasks, while the SNNs typically require much fewer operations.

In this work, we demonstrate that a simple two-layer neural network with standard activation functions can learn an arbitrary word operation in any finite group, provided sufficient width is available and exhibits grokking while doing so. To explain the mechanism by which this is achieved, we reframe the problem as that of learning a particular 3-tensor, which we show is typically of low rank. A key insight is that low-rank implementations of this tensor can be obtained by decomposing it along triplets of basic self-conjugate representations of the group and leveraging the fusion structure to rule out many components. Focusing on a phenomenologically similar but more tractable surrogate model, we show that the network is able to find such low-rank implementations (or approximations thereof), thereby using limited width to approximate the word-tensor in a generalizable way. In the case of the simple multiplication word, we further elucidate the form of these low-rank implementations, showing that the network effectively implements efficient matrix multiplication in the sense of Strassen. Our work also sheds light on the mechanism by which a network reaches such a solution under gradient descent.

There has been a large literature of neural architecture search, but most existing work made use of heuristic rules that largely constrained the search flexibility. In this paper, we first relax these manually designed constraints and enlarge the search space to contain more than 10^{160} candidates. In the new space, most existing differentiable search methods can fail dramatically. We then propose a novel algorithm named Gradual One-Level Differentiable Neural Architecture Search (GOLD-NAS) which introduces a variable resource constraint to one-level optimization so that the weak operators are gradually pruned out from the super-network. In standard image classification benchmarks, GOLD-NAS can find a series of Pareto-optimal architectures within a single search procedure. Most of the discovered architectures were never studied before, yet they achieve a nice tradeoff between recognition accuracy and model complexity. We believe the new space and search algorithm can advance the search of differentiable NAS.

This paper presents a novel joint neural networks approach to address the challenging one-shot object recognition and detection tasks. Inspired by Siamese neural networks and state-of-art multi-box detection approaches, the joint neural networks are able to perform object recognition and detection for categories that remain unseen during the training process. Following the one-shot object recognition/detection constraints, the training and testing datasets do not contain overlapped classes, in other words, all the test classes remain unseen during training. The joint networks architecture is able to effectively compare pairs of images via stacked convolutional layers of the query and target inputs, recognising patterns of the same input query category without relying on previous training around this category. The proposed approach achieves 61.41% accuracy for one-shot object recognition on the MiniImageNet dataset and 47.1% mAP for one-shot object detection when trained on the COCO dataset and tested using the Pascal VOC dataset. Code available at https://github.com/cjvargasc/JNN recog and https://github.com/cjvargasc/JNN detection/

Deep learning has revolutionized industries like computer vision, natural language processing, and speech recognition. However, back propagation, the main method for training deep neural networks, faces challenges like computational overhead and vanishing gradients. In this paper, we propose a novel instant parameter update methodology that eliminates the need for computing gradients at each layer. Our approach accelerates learning, avoids the vanishing gradient problem, and outperforms state-of-the-art methods on benchmark data sets. This research presents a promising direction for efficient and effective deep neural network training.

Deep feedforward and recurrent networks have achieved impressive results in many perception and language processing applications. This success is partially attributed to architectural innovations such as convolutional and long short-term memory networks. The main motivation for these architectural innovations is that they capture better domain knowledge, and importantly are easier to optimize than more basic architectures. Recently, more complex architectures such as Neural Turing Machines and Memory Networks have been proposed for tasks including question answering and general computation, creating a new set of optimization challenges. In this paper, we discuss a low-overhead and easy-to-implement technique of adding gradient noise which we find to be surprisingly effective when training these very deep architectures. The technique not only helps to avoid overfitting, but also can result in lower training loss. This method alone allows a fully-connected 20-layer deep network to be trained with standard gradient descent, even starting from a poor initialization. We see consistent improvements for many complex models, including a 72% relative reduction in error rate over a carefully-tuned baseline on a challenging question-answering task, and a doubling of the number of accurate binary multiplication models learned across 7,000 random restarts. We encourage further application of this technique to additional complex modern architectures.

It is not fully understood why adversarial examples can deceive neural networks and transfer between different networks. To elucidate this, several studies have hypothesized that adversarial perturbations, while appearing as noises, contain class features. This is supported by empirical evidence showing that networks trained on mislabeled adversarial examples can still generalize well to correctly labeled test samples. However, a theoretical understanding of how perturbations include class features and contribute to generalization is limited. In this study, we provide a theoretical framework for understanding learning from perturbations using a one-hidden-layer network trained on mutually orthogonal samples. Our results highlight that various adversarial perturbations, even perturbations of a few pixels, contain sufficient class features for generalization. Moreover, we reveal that the decision boundary when learning from perturbations matches that from standard samples except for specific regions under mild conditions. The code is available at https://github.com/s-kumano/learning-from-adversarial-perturbations.

A significant weakness of most current deep Convolutional Neural Networks is the need to train them using vast amounts of manu- ally labelled data. In this work we propose a unsupervised framework to learn a deep convolutional neural network for single view depth predic- tion, without requiring a pre-training stage or annotated ground truth depths. We achieve this by training the network in a manner analogous to an autoencoder. At training time we consider a pair of images, source and target, with small, known camera motion between the two such as a stereo pair. We train the convolutional encoder for the task of predicting the depth map for the source image. To do so, we explicitly generate an inverse warp of the target image using the predicted depth and known inter-view displacement, to reconstruct the source image; the photomet- ric error in the reconstruction is the reconstruction loss for the encoder. The acquisition of this training data is considerably simpler than for equivalent systems, requiring no manual annotation, nor calibration of depth sensor to camera. We show that our network trained on less than half of the KITTI dataset (without any further augmentation) gives com- parable performance to that of the state of art supervised methods for single view depth estimation.

Neural networks can approximate complex functions, but they struggle to perform exact arithmetic operations over real numbers. The lack of inductive bias for arithmetic operations leaves neural networks without the underlying logic necessary to extrapolate on tasks such as addition, subtraction, and multiplication. We present two new neural network components: the Neural Addition Unit (NAU), which can learn exact addition and subtraction; and the Neural Multiplication Unit (NMU) that can multiply subsets of a vector. The NMU is, to our knowledge, the first arithmetic neural network component that can learn to multiply elements from a vector, when the hidden size is large. The two new components draw inspiration from a theoretical analysis of recently proposed arithmetic components. We find that careful initialization, restricting parameter space, and regularizing for sparsity is important when optimizing the NAU and NMU. Our proposed units NAU and NMU, compared with previous neural units, converge more consistently, have fewer parameters, learn faster, can converge for larger hidden sizes, obtain sparse and meaningful weights, and can extrapolate to negative and small values.

While neural networks are used for classification tasks across domains, a long-standing open problem in machine learning is determining whether neural networks trained using standard procedures are optimal for classification, i.e., whether such models minimize the probability of misclassification for arbitrary data distributions. In this work, we identify and construct an explicit set of neural network classifiers that achieve optimality. Since effective neural networks in practice are typically both wide and deep, we analyze infinitely wide networks that are also infinitely deep. In particular, using the recent connection between infinitely wide neural networks and Neural Tangent Kernels, we provide explicit activation functions that can be used to construct networks that achieve optimality. Interestingly, these activation functions are simple and easy to implement, yet differ from commonly used activations such as ReLU or sigmoid. More generally, we create a taxonomy of infinitely wide and deep networks and show that these models implement one of three well-known classifiers depending on the activation function used: (1) 1-nearest neighbor (model predictions are given by the label of the nearest training example); (2) majority vote (model predictions are given by the label of the class with greatest representation in the training set); or (3) singular kernel classifiers (a set of classifiers containing those that achieve optimality). Our results highlight the benefit of using deep networks for classification tasks, in contrast to regression tasks, where excessive depth is harmful.

Neuroimage processing tasks like segmentation, reconstruction, and registration are central to the study of neuroscience. Robust deep learning strategies and architectures used to solve these tasks are often similar. Yet, when presented with a new task or a dataset with different visual characteristics, practitioners most often need to train a new model, or fine-tune an existing one. This is a time-consuming process that poses a substantial barrier for the thousands of neuroscientists and clinical researchers who often lack the resources or machine-learning expertise to train deep learning models. In practice, this leads to a lack of adoption of deep learning, and neuroscience tools being dominated by classical frameworks. We introduce Neuralizer, a single model that generalizes to previously unseen neuroimaging tasks and modalities without the need for re-training or fine-tuning. Tasks do not have to be known a priori, and generalization happens in a single forward pass during inference. The model can solve processing tasks across multiple image modalities, acquisition methods, and datasets, and generalize to tasks and modalities it has not been trained on. Our experiments on coronal slices show that when few annotated subjects are available, our multi-task network outperforms task-specific baselines without training on the task.

We propose a novel deep network structure called "Network In Network" (NIN) to enhance model discriminability for local patches within the receptive field. The conventional convolutional layer uses linear filters followed by a nonlinear activation function to scan the input. Instead, we build micro neural networks with more complex structures to abstract the data within the receptive field. We instantiate the micro neural network with a multilayer perceptron, which is a potent function approximator. The feature maps are obtained by sliding the micro networks over the input in a similar manner as CNN; they are then fed into the next layer. Deep NIN can be implemented by stacking mutiple of the above described structure. With enhanced local modeling via the micro network, we are able to utilize global average pooling over feature maps in the classification layer, which is easier to interpret and less prone to overfitting than traditional fully connected layers. We demonstrated the state-of-the-art classification performances with NIN on CIFAR-10 and CIFAR-100, and reasonable performances on SVHN and MNIST datasets.

Deep learning has revolutionized computer vision, but it achieved its tremendous success using deep network architectures which are mostly hand-crafted and therefore likely suboptimal. Neural Architecture Search (NAS) aims to bridge this gap by following a well-defined optimization paradigm which systematically looks for the best architecture, given objective criterion such as maximal classification accuracy. The main limitation of NAS is however its astronomical computational cost, as it typically requires training each candidate network architecture from scratch. In this paper, we aim to alleviate this limitation by proposing a novel training-free proxy for image classification accuracy based on Fisher Information. The proposed proxy has a strong theoretical background in statistics and it allows estimating expected image classification accuracy of a given deep network without training the network, thus significantly reducing computational cost of standard NAS algorithms. Our training-free proxy achieves state-of-the-art results on three public datasets and in two search spaces, both when evaluated using previously proposed metrics, as well as using a new metric that we propose which we demonstrate is more informative for practical NAS applications. The source code is publicly available at http://www.github.com/ondratybl/VKDNW

Backpropagation, which uses the chain rule, is the de-facto standard algorithm for optimizing neural networks nowadays. Recently, Hinton (2022) proposed the forward-forward algorithm, a promising alternative that optimizes neural nets layer-by-layer, without propagating gradients throughout the network. Although such an approach has several advantages over back-propagation and shows promising results, the fact that each layer is being trained independently limits the optimization process. Specifically, it prevents the network's layers from collaborating to learn complex and rich features. In this work, we study layer collaboration in the forward-forward algorithm. We show that the current version of the forward-forward algorithm is suboptimal when considering information flow in the network, resulting in a lack of collaboration between layers of the network. We propose an improved version that supports layer collaboration to better utilize the network structure, while not requiring any additional assumptions or computations. We empirically demonstrate the efficacy of the proposed version when considering both information flow and objective metrics. Additionally, we provide a theoretical motivation for the proposed method, inspired by functional entropy theory.

At the heart of foundation models is the philosophy of "more is different", exemplified by the astonishing success in computer vision and natural language processing. However, the challenges of optimization and inherent complexity of transformer models call for a paradigm shift towards simplicity. In this study, we introduce VanillaNet, a neural network architecture that embraces elegance in design. By avoiding high depth, shortcuts, and intricate operations like self-attention, VanillaNet is refreshingly concise yet remarkably powerful. Each layer is carefully crafted to be compact and straightforward, with nonlinear activation functions pruned after training to restore the original architecture. VanillaNet overcomes the challenges of inherent complexity, making it ideal for resource-constrained environments. Its easy-to-understand and highly simplified architecture opens new possibilities for efficient deployment. Extensive experimentation demonstrates that VanillaNet delivers performance on par with renowned deep neural networks and vision transformers, showcasing the power of minimalism in deep learning. This visionary journey of VanillaNet has significant potential to redefine the landscape and challenge the status quo of foundation model, setting a new path for elegant and effective model design. Pre-trained models and codes are available at https://github.com/huawei-noah/VanillaNet and https://gitee.com/mindspore/models/tree/master/research/cv/vanillanet.

Deep neural networks have recently been shown to achieve highly competitive performance in many computer vision tasks due to their abilities of exploring in a much larger hypothesis space. However, since most deep architectures like stacked RNNs tend to suffer from the vanishing-gradient and overfitting problems, their effects are still understudied in many NLP tasks. Inspired by this, we propose a novel multi-layer RNN model called densely connected bidirectional long short-term memory (DC-Bi-LSTM) in this paper, which essentially represents each layer by the concatenation of its hidden state and all preceding layers' hidden states, followed by recursively passing each layer's representation to all subsequent layers. We evaluate our proposed model on five benchmark datasets of sentence classification. DC-Bi-LSTM with depth up to 20 can be successfully trained and obtain significant improvements over the traditional Bi-LSTM with the same or even less parameters. Moreover, our model has promising performance compared with the state-of-the-art approaches.

Delivering malware covertly and evasively is critical to advanced malware campaigns. In this paper, we present a new method to covertly and evasively deliver malware through a neural network model. Neural network models are poorly explainable and have a good generalization ability. By embedding malware in neurons, the malware can be delivered covertly, with minor or no impact on the performance of neural network. Meanwhile, because the structure of the neural network model remains unchanged, it can pass the security scan of antivirus engines. Experiments show that 36.9MB of malware can be embedded in a 178MB-AlexNet model within 1% accuracy loss, and no suspicion is raised by anti-virus engines in VirusTotal, which verifies the feasibility of this method. With the widespread application of artificial intelligence, utilizing neural networks for attacks becomes a forwarding trend. We hope this work can provide a reference scenario for the defense on neural network-assisted attacks.

We investigate methods for combining multiple self-supervised tasks--i.e., supervised tasks where data can be collected without manual labeling--in order to train a single visual representation. First, we provide an apples-to-apples comparison of four different self-supervised tasks using the very deep ResNet-101 architecture. We then combine tasks to jointly train a network. We also explore lasso regularization to encourage the network to factorize the information in its representation, and methods for "harmonizing" network inputs in order to learn a more unified representation. We evaluate all methods on ImageNet classification, PASCAL VOC detection, and NYU depth prediction. Our results show that deeper networks work better, and that combining tasks--even via a naive multi-head architecture--always improves performance. Our best joint network nearly matches the PASCAL performance of a model pre-trained on ImageNet classification, and matches the ImageNet network on NYU depth prediction.

To reduce the overwhelming size of Deep Neural Networks (DNN) teacher-student methodology tries to transfer knowledge from a complex teacher network to a simple student network. We instead propose a novel method called the teacher-class network consisting of a single teacher and multiple student networks (i.e. class of students). Instead of transferring knowledge to one student only, the proposed method transfers a chunk of knowledge to each student. Our students are not trained for problem-specific logits, they are trained to mimic knowledge (dense representation) learned by the teacher network thus the combined knowledge learned by the class of students can be used to solve other problems as well. The proposed teacher-class architecture is evaluated on several benchmark datasets such as MNIST, Fashion MNIST, IMDB Movie Reviews, CAMVid, CIFAR-10 and ImageNet on multiple tasks including image classification, sentiment classification and segmentation. Our approach outperforms the state of-the-art single student approach in terms of accuracy as well as computational cost while achieving 10-30 times reduction in parameters.

Many deployed learned models are black boxes: given input, returns output. Internal information about the model, such as the architecture, optimisation procedure, or training data, is not disclosed explicitly as it might contain proprietary information or make the system more vulnerable. This work shows that such attributes of neural networks can be exposed from a sequence of queries. This has multiple implications. On the one hand, our work exposes the vulnerability of black-box neural networks to different types of attacks -- we show that the revealed internal information helps generate more effective adversarial examples against the black box model. On the other hand, this technique can be used for better protection of private content from automatic recognition models using adversarial examples. Our paper suggests that it is actually hard to draw a line between white box and black box models.

We conjecture that hidden state vectors corresponding to individual input tokens encode information sufficient to accurately predict several tokens ahead. More concretely, in this paper we ask: Given a hidden (internal) representation of a single token at position t in an input, can we reliably anticipate the tokens that will appear at positions geq t + 2? To test this, we measure linear approximation and causal intervention methods in GPT-J-6B to evaluate the degree to which individual hidden states in the network contain signal rich enough to predict future hidden states and, ultimately, token outputs. We find that, at some layers, we can approximate a model's output with more than 48% accuracy with respect to its prediction of subsequent tokens through a single hidden state. Finally we present a "Future Lens" visualization that uses these methods to create a new view of transformer states.

Deep learning has been successful in automating the design of features in machine learning pipelines. However, the algorithms optimizing neural network parameters remain largely hand-designed and computationally inefficient. We study if we can use deep learning to directly predict these parameters by exploiting the past knowledge of training other networks. We introduce a large-scale dataset of diverse computational graphs of neural architectures - DeepNets-1M - and use it to explore parameter prediction on CIFAR-10 and ImageNet. By leveraging advances in graph neural networks, we propose a hypernetwork that can predict performant parameters in a single forward pass taking a fraction of a second, even on a CPU. The proposed model achieves surprisingly good performance on unseen and diverse networks. For example, it is able to predict all 24 million parameters of a ResNet-50 achieving a 60% accuracy on CIFAR-10. On ImageNet, top-5 accuracy of some of our networks approaches 50%. Our task along with the model and results can potentially lead to a new, more computationally efficient paradigm of training networks. Our model also learns a strong representation of neural architectures enabling their analysis.

We present a fast, fully parameterizable GPU implementation of Convolutional Neural Network variants. Our feature extractors are neither carefully designed nor pre-wired, but rather learned in a supervised way. Our deep hierarchical architectures achieve the best published results on benchmarks for object classification (NORB, CIFAR10) and handwritten digit recognition (MNIST), with error rates of 2.53%, 19.51%, 0.35%, respectively. Deep nets trained by simple back-propagation perform better than more shallow ones. Learning is surprisingly rapid. NORB is completely trained within five epochs. Test error rates on MNIST drop to 2.42%, 0.97% and 0.48% after 1, 3 and 17 epochs, respectively.

The generation of transferable adversarial perturbations typically involves training a generator to maximize embedding separation between clean and adversarial images at a single mid-layer of a source model. In this work, we build on this approach and introduce Neuron Attack for Transferability (NAT), a method designed to target specific neuron within the embedding. Our approach is motivated by the observation that previous layer-level optimizations often disproportionately focus on a few neurons representing similar concepts, leaving other neurons within the attacked layer minimally affected. NAT shifts the focus from embedding-level separation to a more fundamental, neuron-specific approach. We find that targeting individual neurons effectively disrupts the core units of the neural network, providing a common basis for transferability across different models. Through extensive experiments on 41 diverse ImageNet models and 9 fine-grained models, NAT achieves fooling rates that surpass existing baselines by over 14\% in cross-model and 4\% in cross-domain settings. Furthermore, by leveraging the complementary attacking capabilities of the trained generators, we achieve impressive fooling rates within just 10 queries. Our code is available at: https://krishnakanthnakka.github.io/NAT/

In the artificial neuron, I replace the dot product with the weighted Lehmer mean, which may emulate different cases of a generalized mean. The single neuron instance is replaced by a multiplet of neurons which have the same averaging weights. A group of outputs feed forward, in lieu of the single scalar. The generalization parameter is typically set to a different value for each neuron in the multiplet. I further extend the concept to a multiplet taken from the Gini mean. Derivatives with respect to the weight parameters and with respect to the two generalization parameters are given. Some properties of the network are investigated, showing the capacity to emulate the classical exclusive-or problem organically in two layers and perform some multiplication and division. The network can instantiate truncated power series and variants, which can be used to approximate different functions, provided that parameters are constrained. Moreover, a mean case slope score is derived that can facilitate a learning-rate novelty based on homogeneity of the selected elements. The multiplet neuron equation provides a way to segment regularization timeframes and approaches.

The ability to perform pixel-wise semantic segmentation in real-time is of paramount importance in mobile applications. Recent deep neural networks aimed at this task have the disadvantage of requiring a large number of floating point operations and have long run-times that hinder their usability. In this paper, we propose a novel deep neural network architecture named ENet (efficient neural network), created specifically for tasks requiring low latency operation. ENet is up to 18times faster, requires 75times less FLOPs, has 79times less parameters, and provides similar or better accuracy to existing models. We have tested it on CamVid, Cityscapes and SUN datasets and report on comparisons with existing state-of-the-art methods, and the trade-offs between accuracy and processing time of a network. We present performance measurements of the proposed architecture on embedded systems and suggest possible software improvements that could make ENet even faster.

Backpropagation is still the de facto algorithm used today to train neural networks. With the exponential growth of recent architectures, the computational cost of this algorithm also becomes a burden. The recent PEPITA and forward-only frameworks have proposed promising alternatives, but they failed to scale up to a handful of hidden layers, yet limiting their use. In this paper, we first analyze theoretically the main limitations of these approaches. It allows us the design of a forward-only algorithm, which is equivalent to backpropagation under the linear and orthogonal assumptions. By relaxing the linear assumption, we then introduce FOTON (Forward-Only Training of Orthogonal Networks) that bridges the gap with the backpropagation algorithm. Experimental results show that it outperforms PEPITA, enabling us to train neural networks of any depth, without the need for a backward pass. Moreover its performance on convolutional networks clearly opens up avenues for its application to more advanced architectures. The code is open-sourced at https://github.com/p0lcAi/FOTON .

Self-Supervised Learning (SSL) has been shown to learn useful and information-preserving representations. Neural Networks (NNs) are widely applied, yet their weight space is still not fully understood. Therefore, we propose to use SSL to learn hyper-representations of the weights of populations of NNs. To that end, we introduce domain specific data augmentations and an adapted attention architecture. Our empirical evaluation demonstrates that self-supervised representation learning in this domain is able to recover diverse NN model characteristics. Further, we show that the proposed learned representations outperform prior work for predicting hyper-parameters, test accuracy, and generalization gap as well as transfer to out-of-distribution settings.

No free lunch theorems for supervised learning state that no learner can solve all problems or that all learners achieve exactly the same accuracy on average over a uniform distribution on learning problems. Accordingly, these theorems are often referenced in support of the notion that individual problems require specially tailored inductive biases. While virtually all uniformly sampled datasets have high complexity, real-world problems disproportionately generate low-complexity data, and we argue that neural network models share this same preference, formalized using Kolmogorov complexity. Notably, we show that architectures designed for a particular domain, such as computer vision, can compress datasets on a variety of seemingly unrelated domains. Our experiments show that pre-trained and even randomly initialized language models prefer to generate low-complexity sequences. Whereas no free lunch theorems seemingly indicate that individual problems require specialized learners, we explain how tasks that often require human intervention such as picking an appropriately sized model when labeled data is scarce or plentiful can be automated into a single learning algorithm. These observations justify the trend in deep learning of unifying seemingly disparate problems with an increasingly small set of machine learning models.

While depth tends to improve network performances, it also makes gradient-based training more difficult since deeper networks tend to be more non-linear. The recently proposed knowledge distillation approach is aimed at obtaining small and fast-to-execute models, and it has shown that a student network could imitate the soft output of a larger teacher network or ensemble of networks. In this paper, we extend this idea to allow the training of a student that is deeper and thinner than the teacher, using not only the outputs but also the intermediate representations learned by the teacher as hints to improve the training process and final performance of the student. Because the student intermediate hidden layer will generally be smaller than the teacher's intermediate hidden layer, additional parameters are introduced to map the student hidden layer to the prediction of the teacher hidden layer. This allows one to train deeper students that can generalize better or run faster, a trade-off that is controlled by the chosen student capacity. For example, on CIFAR-10, a deep student network with almost 10.4 times less parameters outperforms a larger, state-of-the-art teacher network.

One-shot Neural architecture search (One-shot NAS) has been proposed as a time-efficient approach to obtain optimal subnet architectures and weights under different complexity cases by training only once. However, the subnet performance obtained by weight sharing is often inferior to the performance achieved by retraining. In this paper, we investigate the performance gap and attribute it to the use of uniform sampling, which is a common approach in supernet training. Uniform sampling concentrates training resources on subnets with intermediate computational resources, which are sampled with high probability. However, subnets with different complexity regions require different optimal training strategies for optimal performance. To address the problem of uniform sampling, we propose ShiftNAS, a method that can adjust the sampling probability based on the complexity of subnets. We achieve this by evaluating the performance variation of subnets with different complexity and designing an architecture generator that can accurately and efficiently provide subnets with the desired complexity. Both the sampling probability and the architecture generator can be trained end-to-end in a gradient-based manner. With ShiftNAS, we can directly obtain the optimal model architecture and parameters for a given computational complexity. We evaluate our approach on multiple visual network models, including convolutional neural networks (CNNs) and vision transformers (ViTs), and demonstrate that ShiftNAS is model-agnostic. Experimental results on ImageNet show that ShiftNAS can improve the performance of one-shot NAS without additional consumption. Source codes are available at https://github.com/bestfleer/ShiftNAS.

Neural Module Networks, originally proposed for the task of visual question answering, are a class of neural network architectures that involve human-specified neural modules, each designed for a specific form of reasoning. In current formulations of such networks only the parameters of the neural modules and/or the order of their execution is learned. In this work, we further expand this approach and also learn the underlying internal structure of modules in terms of the ordering and combination of simple and elementary arithmetic operators. Our results show that one is indeed able to simultaneously learn both internal module structure and module sequencing without extra supervisory signals for module execution sequencing. With this approach, we report performance comparable to models using hand-designed modules.

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the i-th neuron in a nonlinear operator layer is defined by mathcal O_i(u) = sigmaleft( sum_j mathcal W_{ij} u + mathcal B_{ij}right). Here, mathcal W_{ij} denotes the bounded linear operator connecting j-th input neuron to i-th output neuron, and the bias mathcal B_{ij} takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).

Training a neural network is a monolithic endeavor, akin to carving knowledge into stone: once the process is completed, editing the knowledge in a network is nearly impossible, since all information is distributed across the network's weights. We here explore a simple, compelling alternative by marrying the representational power of deep neural networks with the flexibility of a database. Decomposing the task of image classification into image similarity (from a pre-trained embedding) and search (via fast nearest neighbor retrieval from a knowledge database), we build a simple and flexible visual memory that has the following key capabilities: (1.) The ability to flexibly add data across scales: from individual samples all the way to entire classes and billion-scale data; (2.) The ability to remove data through unlearning and memory pruning; (3.) An interpretable decision-mechanism on which we can intervene to control its behavior. Taken together, these capabilities comprehensively demonstrate the benefits of an explicit visual memory. We hope that it might contribute to a conversation on how knowledge should be represented in deep vision models -- beyond carving it in ``stone'' weights.

One-shot neural architecture search (NAS) substantially improves the search efficiency by training one supernet to estimate the performance of every possible child architecture (i.e., subnet). However, the inconsistency of characteristics among subnets incurs serious interference in the optimization, resulting in poor performance ranking correlation of subnets. Subsequent explorations decompose supernet weights via a particular criterion, e.g., gradient matching, to reduce the interference; yet they suffer from huge computational cost and low space separability. In this work, we propose a lightweight and effective local intrinsic dimension (LID)-based method NAS-LID. NAS-LID evaluates the geometrical properties of architectures by calculating the low-cost LID features layer-by-layer, and the similarity characterized by LID enjoys better separability compared with gradients, which thus effectively reduces the interference among subnets. Extensive experiments on NASBench-201 indicate that NAS-LID achieves superior performance with better efficiency. Specifically, compared to the gradient-driven method, NAS-LID can save up to 86% of GPU memory overhead when searching on NASBench-201. We also demonstrate the effectiveness of NAS-LID on ProxylessNAS and OFA spaces. Source code: https://github.com/marsggbo/NAS-LID.

Major winning Convolutional Neural Networks (CNNs), such as AlexNet, VGGNet, ResNet, GoogleNet, include tens to hundreds of millions of parameters, which impose considerable computation and memory overhead. This limits their practical use for training, optimization and memory efficiency. On the contrary, light-weight architectures, being proposed to address this issue, mainly suffer from low accuracy. These inefficiencies mostly stem from following an ad hoc procedure. We propose a simple architecture, called SimpleNet, based on a set of designing principles, with which we empirically show, a well-crafted yet simple and reasonably deep architecture can perform on par with deeper and more complex architectures. SimpleNet provides a good tradeoff between the computation/memory efficiency and the accuracy. Our simple 13-layer architecture outperforms most of the deeper and complex architectures to date such as VGGNet, ResNet, and GoogleNet on several well-known benchmarks while having 2 to 25 times fewer number of parameters and operations. This makes it very handy for embedded systems or systems with computational and memory limitations. We achieved state-of-the-art result on CIFAR10 outperforming several heavier architectures, near state of the art on MNIST and competitive results on CIFAR100 and SVHN. We also outperformed the much larger and deeper architectures such as VGGNet and popular variants of ResNets among others on the ImageNet dataset. Models are made available at: https://github.com/Coderx7/SimpleNet

Despite the remarkable capabilities of deep neural networks in image recognition, the dependence on activation functions remains a largely unexplored area and has yet to be eliminated. On the other hand, Polynomial Networks is a class of models that does not require activation functions, but have yet to perform on par with modern architectures. In this work, we aim close this gap and propose MONet, which relies solely on multilinear operators. The core layer of MONet, called Mu-Layer, captures multiplicative interactions of the elements of the input token. MONet captures high-degree interactions of the input elements and we demonstrate the efficacy of our approach on a series of image recognition and scientific computing benchmarks. The proposed model outperforms prior polynomial networks and performs on par with modern architectures. We believe that MONet can inspire further research on models that use entirely multilinear operations.

Neural architecture search (NAS) has shown promising results discovering models that are both accurate and fast. For NAS, training a one-shot model has become a popular strategy to rank the relative quality of different architectures (child models) using a single set of shared weights. However, while one-shot model weights can effectively rank different network architectures, the absolute accuracies from these shared weights are typically far below those obtained from stand-alone training. To compensate, existing methods assume that the weights must be retrained, finetuned, or otherwise post-processed after the search is completed. These steps significantly increase the compute requirements and complexity of the architecture search and model deployment. In this work, we propose BigNAS, an approach that challenges the conventional wisdom that post-processing of the weights is necessary to get good prediction accuracies. Without extra retraining or post-processing steps, we are able to train a single set of shared weights on ImageNet and use these weights to obtain child models whose sizes range from 200 to 1000 MFLOPs. Our discovered model family, BigNASModels, achieve top-1 accuracies ranging from 76.5% to 80.9%, surpassing state-of-the-art models in this range including EfficientNets and Once-for-All networks without extra retraining or post-processing. We present ablative study and analysis to further understand the proposed BigNASModels.

Understanding neural networks is challenging in part because of the dense, continuous nature of their hidden states. We explore whether we can train neural networks to have hidden states that are sparse, discrete, and more interpretable by quantizing their continuous features into what we call codebook features. Codebook features are produced by finetuning neural networks with vector quantization bottlenecks at each layer, producing a network whose hidden features are the sum of a small number of discrete vector codes chosen from a larger codebook. Surprisingly, we find that neural networks can operate under this extreme bottleneck with only modest degradation in performance. This sparse, discrete bottleneck also provides an intuitive way of controlling neural network behavior: first, find codes that activate when the desired behavior is present, then activate those same codes during generation to elicit that behavior. We validate our approach by training codebook Transformers on several different datasets. First, we explore a finite state machine dataset with far more hidden states than neurons. In this setting, our approach overcomes the superposition problem by assigning states to distinct codes, and we find that we can make the neural network behave as if it is in a different state by activating the code for that state. Second, we train Transformer language models with up to 410M parameters on two natural language datasets. We identify codes in these models representing diverse, disentangled concepts (ranging from negative emotions to months of the year) and find that we can guide the model to generate different topics by activating the appropriate codes during inference. Overall, codebook features appear to be a promising unit of analysis and control for neural networks and interpretability. Our codebase and models are open-sourced at https://github.com/taufeeque9/codebook-features.

In recent years, deep neural networks (DNNs) achieved unprecedented performance in many low-level vision tasks. However, state-of-the-art results are typically achieved by very deep networks, which can reach tens of layers with tens of millions of parameters. To make DNNs implementable on platforms with limited resources, it is necessary to weaken the tradeoff between performance and efficiency. In this paper, we propose a new activation unit, which is particularly suitable for image restoration problems. In contrast to the widespread per-pixel activation units, like ReLUs and sigmoids, our unit implements a learnable nonlinear function with spatial connections. This enables the net to capture much more complex features, thus requiring a significantly smaller number of layers in order to reach the same performance. We illustrate the effectiveness of our units through experiments with state-of-the-art nets for denoising, de-raining, and super resolution, which are already considered to be very small. With our approach, we are able to further reduce these models by nearly 50% without incurring any degradation in performance.

Training neural networks involves solving large-scale non-convex optimization problems. This task has long been believed to be extremely difficult, with fear of local minima and other obstacles motivating a variety of schemes to improve optimization, such as unsupervised pretraining. However, modern neural networks are able to achieve negligible training error on complex tasks, using only direct training with stochastic gradient descent. We introduce a simple analysis technique to look for evidence that such networks are overcoming local optima. We find that, in fact, on a straight path from initialization to solution, a variety of state of the art neural networks never encounter any significant obstacles.

Recurrent neural network is a powerful model that learns temporal patterns in sequential data. For a long time, it was believed that recurrent networks are difficult to train using simple optimizers, such as stochastic gradient descent, due to the so-called vanishing gradient problem. In this paper, we show that learning longer term patterns in real data, such as in natural language, is perfectly possible using gradient descent. This is achieved by using a slight structural modification of the simple recurrent neural network architecture. We encourage some of the hidden units to change their state slowly by making part of the recurrent weight matrix close to identity, thus forming kind of a longer term memory. We evaluate our model in language modeling experiments, where we obtain similar performance to the much more complex Long Short Term Memory (LSTM) networks (Hochreiter & Schmidhuber, 1997).

Deep neural networks are widely used for classification. These deep models often suffer from a lack of interpretability -- they are particularly difficult to understand because of their non-linear nature. As a result, neural networks are often treated as "black box" models, and in the past, have been trained purely to optimize the accuracy of predictions. In this work, we create a novel network architecture for deep learning that naturally explains its own reasoning for each prediction. This architecture contains an autoencoder and a special prototype layer, where each unit of that layer stores a weight vector that resembles an encoded training input. The encoder of the autoencoder allows us to do comparisons within the latent space, while the decoder allows us to visualize the learned prototypes. The training objective has four terms: an accuracy term, a term that encourages every prototype to be similar to at least one encoded input, a term that encourages every encoded input to be close to at least one prototype, and a term that encourages faithful reconstruction by the autoencoder. The distances computed in the prototype layer are used as part of the classification process. Since the prototypes are learned during training, the learned network naturally comes with explanations for each prediction, and the explanations are loyal to what the network actually computes.

While many unsupervised learning models focus on one family of tasks, either generative or discriminative, we explore the possibility of a unified representation learner: a model which uses a single pre-training stage to address both families of tasks simultaneously. We identify diffusion models as a prime candidate. Diffusion models have risen to prominence as a state-of-the-art method for image generation, denoising, inpainting, super-resolution, manipulation, etc. Such models involve training a U-Net to iteratively predict and remove noise, and the resulting model can synthesize high fidelity, diverse, novel images. The U-Net architecture, as a convolution-based architecture, generates a diverse set of feature representations in the form of intermediate feature maps. We present our findings that these embeddings are useful beyond the noise prediction task, as they contain discriminative information and can also be leveraged for classification. We explore optimal methods for extracting and using these embeddings for classification tasks, demonstrating promising results on the ImageNet classification task. We find that with careful feature selection and pooling, diffusion models outperform comparable generative-discriminative methods such as BigBiGAN for classification tasks. We investigate diffusion models in the transfer learning regime, examining their performance on several fine-grained visual classification datasets. We compare these embeddings to those generated by competing architectures and pre-trainings for classification tasks.

Finding a code to unravel the population of neural responses that leads to a distinct animal behavior has been a long-standing question in the field of neuroscience. With the recent advances in machine learning, it is shown that the hierarchically Deep Neural Networks (DNNs) perform optimally in decoding unique features out of complex datasets. In this study, we utilize the power of a DNN to explore the computational principles in the mammalian brain by exploiting the Neuropixel data from Allen Brain Institute. We decode the neural responses from mouse visual cortex to predict the presented stimuli to the animal for natural (bear, trees, cheetah, etc.) and artificial (drifted gratings, orientated bars, etc.) classes. Our results indicate that neurons in mouse visual cortex encode the features of natural and artificial objects in a distinct manner, and such neural code is consistent across animals. We investigate this by applying transfer learning to train a DNN on the neural responses of a single animal and test its generalized performance across multiple animals. Within a single animal, DNN is able to decode the neural responses with as much as 100% classification accuracy. Across animals, this accuracy is reduced to 91%. This study demonstrates the potential of utilizing the DNN models as a computational framework to understand the neural coding principles in the mammalian brain.

A multilayer perceptron can behave as a generative classifier by applying bidirectional learning (BL). It consists of training an undirected neural network to map input to output and vice-versa; therefore it can produce a classifier in one direction, and a generator in the opposite direction for the same data. The learning process of BL tries to reproduce the neuroplasticity stated in Hebbian theory using only backward propagation of errors. In this paper, two novel learning techniques are introduced which use BL for improving robustness to white noise static and adversarial examples. The first method is bidirectional propagation of errors, which the error propagation occurs in backward and forward directions. Motivated by the fact that its generative model receives as input a constant vector per class, we introduce as a second method the hybrid adversarial networks (HAN). Its generative model receives a random vector as input and its training is based on generative adversarial networks (GAN). To assess the performance of BL, we perform experiments using several architectures with fully and convolutional layers, with and without bias. Experimental results show that both methods improve robustness to white noise static and adversarial examples, and even increase accuracy, but have different behavior depending on the architecture and task, being more beneficial to use the one or the other. Nevertheless, HAN using a convolutional architecture with batch normalization presents outstanding robustness, reaching state-of-the-art accuracy on adversarial examples of hand-written digits.

Modern deep neural networks (DNNs) represent a formidable challenge for theorists: according to the commonly accepted probabilistic framework that describes their performance, these architectures should overfit due to the huge number of parameters to train, but in practice they do not. Here we employ results from replica mean field theory to compute the generalisation gap of machine learning models with quenched features, in the teacher-student scenario and for regression problems with quadratic loss function. Notably, this framework includes the case of DNNs where the last layer is optimised given a specific realisation of the remaining weights. We show how these results -- combined with ideas from statistical learning theory -- provide a stringent asymptotic upper bound on the generalisation gap of fully trained DNN as a function of the size of the dataset P. In particular, in the limit of large P and N_{out} (where N_out is the size of the last layer) and N_out ll P, the generalisation gap approaches zero faster than 2 N_out/P, for any choice of both architecture and teacher function. Notably, this result greatly improves existing bounds from statistical learning theory. We test our predictions on a broad range of architectures, from toy fully-connected neural networks with few hidden layers to state-of-the-art deep convolutional neural networks.

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

Trainable layers such as convolutional building blocks are the standard network design choices by learning parameters to capture the global context through successive spatial operations. When designing an efficient network, trainable layers such as the depthwise convolution is the source of efficiency in the number of parameters and FLOPs, but there was little improvement to the model speed in practice. This paper argues that simple built-in parameter-free operations can be a favorable alternative to the efficient trainable layers replacing spatial operations in a network architecture. We aim to break the stereotype of organizing the spatial operations of building blocks into trainable layers. Extensive experimental analyses based on layer-level studies with fully-trained models and neural architecture searches are provided to investigate whether parameter-free operations such as the max-pool are functional. The studies eventually give us a simple yet effective idea for redesigning network architectures, where the parameter-free operations are heavily used as the main building block without sacrificing the model accuracy as much. Experimental results on the ImageNet dataset demonstrate that the network architectures with parameter-free operations could enjoy the advantages of further efficiency in terms of model speed, the number of the parameters, and FLOPs. Code and ImageNet pretrained models are available at https://github.com/naver-ai/PfLayer.

The weight matrix (WM) of a neural network (NN) is its program. The programs of many traditional NNs are learned through gradient descent in some error function, then remain fixed. The WM of a self-referential NN, however, can keep rapidly modifying all of itself during runtime. In principle, such NNs can meta-learn to learn, and meta-meta-learn to meta-learn to learn, and so on, in the sense of recursive self-improvement. While NN architectures potentially capable of implementing such behaviour have been proposed since the '90s, there have been few if any practical studies. Here we revisit such NNs, building upon recent successes of fast weight programmers and closely related linear Transformers. We propose a scalable self-referential WM (SRWM) that learns to use outer products and the delta update rule to modify itself. We evaluate our SRWM in supervised few-shot learning and in multi-task reinforcement learning with procedurally generated game environments. Our experiments demonstrate both practical applicability and competitive performance of the proposed SRWM. Our code is public.

We present a framework for using transformer networks as universal computers by programming them with specific weights and placing them in a loop. Our input sequence acts as a punchcard, consisting of instructions and memory for data read/writes. We demonstrate that a constant number of encoder layers can emulate basic computing blocks, including embedding edit operations, non-linear functions, function calls, program counters, and conditional branches. Using these building blocks, we emulate a small instruction-set computer. This allows us to map iterative algorithms to programs that can be executed by a looped, 13-layer transformer. We show how this transformer, instructed by its input, can emulate a basic calculator, a basic linear algebra library, and in-context learning algorithms that employ backpropagation. Our work highlights the versatility of the attention mechanism, and demonstrates that even shallow transformers can execute full-fledged, general-purpose programs.

Neural network classifiers have become the de-facto choice for current "pre-train then fine-tune" paradigms of visual classification. In this paper, we investigate k-Nearest-Neighbor (k-NN) classifiers, a classical model-free learning method from the pre-deep learning era, as an augmentation to modern neural network based approaches. As a lazy learning method, k-NN simply aggregates the distance between the test image and top-k neighbors in a training set. We adopt k-NN with pre-trained visual representations produced by either supervised or self-supervised methods in two steps: (1) Leverage k-NN predicted probabilities as indications for easy vs. hard examples during training. (2) Linearly interpolate the k-NN predicted distribution with that of the augmented classifier. Via extensive experiments on a wide range of classification tasks, our study reveals the generality and flexibility of k-NN integration with additional insights: (1) k-NN achieves competitive results, sometimes even outperforming a standard linear classifier. (2) Incorporating k-NN is especially beneficial for tasks where parametric classifiers perform poorly and / or in low-data regimes. We hope these discoveries will encourage people to rethink the role of pre-deep learning, classical methods in computer vision. Our code is available at: https://github.com/KMnP/nn-revisit.

These handouts are designed for people who is just starting involved with the topic artificial neural networks. We show how it works a single artificial neuron (McCulloch & Pitt model), mathematically and graphically. We do explain the delta rule, a learning algorithm to find the neuron weights. We also present some examples in MATLAB/Octave. There are examples for classification task for lineal and non-lineal problems. At the end, we present an artificial neural network, a feed-forward neural network along its learning algorithm backpropagation. ----- Estos apuntes est\'an dise\~nados para personas que por primera vez se introducen en el tema de las redes neuronales artificiales. Se muestra el funcionamiento b\'asico de una neurona, matem\'aticamente y gr\'aficamente. Se explica la Regla Delta, algoritmo deaprendizaje para encontrar los pesos de una neurona. Tambi\'en se muestran ejemplos en MATLAB/Octave. Hay ejemplos para problemas de clasificaci\'on, para problemas lineales y no-lineales. En la parte final se muestra la arquitectura de red neuronal artificial conocida como backpropagation.

Despite their massive size, successful deep artificial neural networks can exhibit a remarkably small difference between training and test performance. Conventional wisdom attributes small generalization error either to properties of the model family, or to the regularization techniques used during training. Through extensive systematic experiments, we show how these traditional approaches fail to explain why large neural networks generalize well in practice. Specifically, our experiments establish that state-of-the-art convolutional networks for image classification trained with stochastic gradient methods easily fit a random labeling of the training data. This phenomenon is qualitatively unaffected by explicit regularization, and occurs even if we replace the true images by completely unstructured random noise. We corroborate these experimental findings with a theoretical construction showing that simple depth two neural networks already have perfect finite sample expressivity as soon as the number of parameters exceeds the number of data points as it usually does in practice. We interpret our experimental findings by comparison with traditional models.

In recent years, supervised learning with convolutional networks (CNNs) has seen huge adoption in computer vision applications. Comparatively, unsupervised learning with CNNs has received less attention. In this work we hope to help bridge the gap between the success of CNNs for supervised learning and unsupervised learning. We introduce a class of CNNs called deep convolutional generative adversarial networks (DCGANs), that have certain architectural constraints, and demonstrate that they are a strong candidate for unsupervised learning. Training on various image datasets, we show convincing evidence that our deep convolutional adversarial pair learns a hierarchy of representations from object parts to scenes in both the generator and discriminator. Additionally, we use the learned features for novel tasks - demonstrating their applicability as general image representations.

In this paper, we introduce a novel layer designed to be used as the output of pre-trained neural networks in the context of classification. Based on Associative Memories, this layer can help design Deep Neural Networks which support incremental learning and that can be (partially) trained in real time on embedded devices. Experiments on the ImageNet dataset and other different domain specific datasets show that it is possible to design more flexible and faster-to-train Neural Networks at the cost of a slight decrease in accuracy.

Deep learning employs multi-layer neural networks trained via the backpropagation algorithm. This approach has achieved success across many domains and relies on adaptive gradient methods such as the Adam optimizer. Sequence modeling evolved from recurrent neural networks to attention-based models, culminating in the Transformer architecture. Transformers have achieved state-of-the-art performance in natural language processing (for example, BERT and GPT-3) and have been applied in computer vision and computational biology. However, theoretical understanding of these models remains limited. In this paper, we examine the mathematical foundations of deep learning and Transformers and present a novel theoretical result. We review key concepts from linear algebra, probability, and optimization that underpin deep learning, and we analyze the multi-head self-attention mechanism and the backpropagation algorithm in detail. Our main contribution is a universal approximation theorem for Transformers: we prove that a single-layer Transformer, comprising one self-attention layer followed by a position-wise feed-forward network with ReLU activation, can approximate any continuous sequence-to-sequence mapping on a compact domain to arbitrary precision. We provide a formal statement and a complete proof. Finally, we present case studies that demonstrate the practical implications of this result. Our findings advance the theoretical understanding of Transformer models and help bridge the gap between theory and practice.

Training state-of-the-art, deep neural networks is computationally expensive. One way to reduce the training time is to normalize the activities of the neurons. A recently introduced technique called batch normalization uses the distribution of the summed input to a neuron over a mini-batch of training cases to compute a mean and variance which are then used to normalize the summed input to that neuron on each training case. This significantly reduces the training time in feed-forward neural networks. However, the effect of batch normalization is dependent on the mini-batch size and it is not obvious how to apply it to recurrent neural networks. In this paper, we transpose batch normalization into layer normalization by computing the mean and variance used for normalization from all of the summed inputs to the neurons in a layer on a single training case. Like batch normalization, we also give each neuron its own adaptive bias and gain which are applied after the normalization but before the non-linearity. Unlike batch normalization, layer normalization performs exactly the same computation at training and test times. It is also straightforward to apply to recurrent neural networks by computing the normalization statistics separately at each time step. Layer normalization is very effective at stabilizing the hidden state dynamics in recurrent networks. Empirically, we show that layer normalization can substantially reduce the training time compared with previously published techniques.

Existing analyses of the expressive capacity of Transformer models have required excessively deep layers for data memorization, leading to a discrepancy with the Transformers actually used in practice. This is primarily due to the interpretation of the softmax function as an approximation of the hardmax function. By clarifying the connection between the softmax function and the Boltzmann operator, we prove that a single layer of self-attention with low-rank weight matrices possesses the capability to perfectly capture the context of an entire input sequence. As a consequence, we show that one-layer and single-head Transformers have a memorization capacity for finite samples, and that Transformers consisting of one self-attention layer with two feed-forward neural networks are universal approximators for continuous permutation equivariant functions on a compact domain.

We report the effects of replacing the scaled dot-product (within softmax) attention with the negative-log of Euclidean distance. This form of attention simplifies to inverse distance weighting interpolation. Used in simple one hidden layer networks and trained with vanilla cross-entropy loss on classification problems, it tends to produce a key matrix containing prototypes and a value matrix with corresponding logits. We also show that the resulting interpretable networks can be augmented with manually-constructed prototypes to perform low-impact handling of special cases.

We introduce the concept of scalable neural network kernels (SNNKs), the replacements of regular feedforward layers (FFLs), capable of approximating the latter, but with favorable computational properties. SNNKs effectively disentangle the inputs from the parameters of the neural network in the FFL, only to connect them in the final computation via the dot-product kernel. They are also strictly more expressive, as allowing to model complicated relationships beyond the functions of the dot-products of parameter-input vectors. We also introduce the neural network bundling process that applies SNNKs to compactify deep neural network architectures, resulting in additional compression gains. In its extreme version, it leads to the fully bundled network whose optimal parameters can be expressed via explicit formulae for several loss functions (e.g. mean squared error), opening a possibility to bypass backpropagation. As a by-product of our analysis, we introduce the mechanism of the universal random features (or URFs), applied to instantiate several SNNK variants, and interesting on its own in the context of scalable kernel methods. We provide rigorous theoretical analysis of all these concepts as well as an extensive empirical evaluation, ranging from point-wise kernel estimation to Transformers' fine-tuning with novel adapter layers inspired by SNNKs. Our mechanism provides up to 5x reduction in the number of trainable parameters, while maintaining competitive accuracy.

Transformers have become foundational architectures for both natural language and computer vision tasks. However, the high computational cost makes it quite challenging to deploy on resource-constraint devices. This paper investigates the computational bottleneck modules of efficient transformer, i.e., normalization layers and attention modules. LayerNorm is commonly used in transformer architectures but is not computational friendly due to statistic calculation during inference. However, replacing LayerNorm with more efficient BatchNorm in transformer often leads to inferior performance and collapse in training. To address this problem, we propose a novel method named PRepBN to progressively replace LayerNorm with re-parameterized BatchNorm in training. Moreover, we propose a simplified linear attention (SLA) module that is simple yet effective to achieve strong performance. Extensive experiments on image classification as well as object detection demonstrate the effectiveness of our proposed method. For example, our SLAB-Swin obtains 83.6% top-1 accuracy on ImageNet-1K with 16.2ms latency, which is 2.4ms less than that of Flatten-Swin with 0.1% higher accuracy. We also evaluated our method for language modeling task and obtain comparable performance and lower latency.Codes are publicly available at https://github.com/xinghaochen/SLAB and https://github.com/mindspore-lab/models/tree/master/research/huawei-noah/SLAB.

Large language models (LLMs) contain substantial factual knowledge which is commonly elicited by multiple-choice question-answering prompts. Internally, such models process the prompt through multiple transformer layers, building varying representations of the problem within its hidden states. Ultimately, however, only the hidden state corresponding to the final layer and token position are used to predict the answer label. In this work, we propose instead to learn a small separate neural network predictor module on a collection of training questions, that take the hidden states from all the layers at the last temporal position as input and outputs predictions. In effect, such a framework disentangles the representational abilities of LLMs from their predictive abilities. On a collection of hard benchmarks, our method achieves considerable improvements in performance, sometimes comparable to supervised fine-tuning procedures, but at a fraction of the computational cost.

A biological neural network in the cortex forms a neural field. Neurons in the field have their own receptive fields, and connection weights between two neurons are random but highly correlated when they are in close proximity in receptive fields. In this paper, we investigate such neural fields in a multilayer architecture to investigate the supervised learning of the fields. We empirically compare the performances of our field model with those of randomly connected deep networks. The behavior of a randomly connected network is investigated on the basis of the key idea of the neural tangent kernel regime, a recent development in the machine learning theory of over-parameterized networks; for most randomly connected neural networks, it is shown that global minima always exist in their small neighborhoods. We numerically show that this claim also holds for our neural fields. In more detail, our model has two structures: i) each neuron in a field has a continuously distributed receptive field, and ii) the initial connection weights are random but not independent, having correlations when the positions of neurons are close in each layer. We show that such a multilayer neural field is more robust than conventional models when input patterns are deformed by noise disturbances. Moreover, its generalization ability can be slightly superior to that of conventional models.

Machine learning techniques based on neural networks are achieving remarkable results in a wide variety of domains. Often, the training of models requires large, representative datasets, which may be crowdsourced and contain sensitive information. The models should not expose private information in these datasets. Addressing this goal, we develop new algorithmic techniques for learning and a refined analysis of privacy costs within the framework of differential privacy. Our implementation and experiments demonstrate that we can train deep neural networks with non-convex objectives, under a modest privacy budget, and at a manageable cost in software complexity, training efficiency, and model quality.

Machine unlearning methods seek to revise pretrained models such that effects of certain training samples can be removed. In addition to effective erasure, low computational cost and general utility retention are also highly desirable. Existing unlearning methods usually involve iterative updates over the model parameters, which incurs a high computational cost. In this work, we propose an efficient method that only requires a one-time gradient computation, with which we modify only a single layer of model parameters. Specifically, we first identify a small number of model layers that lie on the Pareto front of high forget importance and low retain influence as critical layers. Then we search for a suitable step size and take a step along the gradient direction of a single critical layer while keeping other layers frozen. This method is highly modular and can be used to unlearn multiple concepts simultaneously in a controllable manner. We demonstrate the effectiveness and efficiency of this method on various models including CLIP, stable diffusion, and VLMs, surpassing other state-of-the-art methods.

Deep learning yields great results across many fields, from speech recognition, image classification, to translation. But for each problem, getting a deep model to work well involves research into the architecture and a long period of tuning. We present a single model that yields good results on a number of problems spanning multiple domains. In particular, this single model is trained concurrently on ImageNet, multiple translation tasks, image captioning (COCO dataset), a speech recognition corpus, and an English parsing task. Our model architecture incorporates building blocks from multiple domains. It contains convolutional layers, an attention mechanism, and sparsely-gated layers. Each of these computational blocks is crucial for a subset of the tasks we train on. Interestingly, even if a block is not crucial for a task, we observe that adding it never hurts performance and in most cases improves it on all tasks. We also show that tasks with less data benefit largely from joint training with other tasks, while performance on large tasks degrades only slightly if at all.

Deep neural networks with millions of parameters are at the heart of many state of the art machine learning models today. However, recent works have shown that models with much smaller number of parameters can also perform just as well. In this work, we introduce the problem of architecture-learning, i.e; learning the architecture of a neural network along with weights. We introduce a new trainable parameter called tri-state ReLU, which helps in eliminating unnecessary neurons. We also propose a smooth regularizer which encourages the total number of neurons after elimination to be small. The resulting objective is differentiable and simple to optimize. We experimentally validate our method on both small and large networks, and show that it can learn models with a considerably small number of parameters without affecting prediction accuracy.

The Transformer is an extremely powerful and prominent deep learning architecture. In this work, we challenge the commonly held belief in deep learning that going deeper is better, and show an alternative design approach that is building wider attention Transformers. We demonstrate that wide single layer Transformer models can compete with or outperform deeper ones in a variety of Natural Language Processing (NLP) tasks when both are trained from scratch. The impact of changing the model aspect ratio on Transformers is then studied systematically. This ratio balances the number of layers and the number of attention heads per layer while keeping the total number of attention heads and all other hyperparameters constant. On average, across 4 NLP tasks and 10 attention types, single layer wide models perform 0.3% better than their deep counterparts. We show an in-depth evaluation and demonstrate how wide models require a far smaller memory footprint and can run faster on commodity hardware, in addition, these wider models are also more interpretable. For example, a single layer Transformer on the IMDb byte level text classification has 3.1x faster inference latency on a CPU than its equally accurate deeper counterpart, and is half the size. We therefore put forward wider and shallower models as a viable and desirable alternative for small models on NLP tasks, and as an important area of research for domains beyond this.

We apply supervised deep neural networks (DNNs) for pricing and calibration of both vanilla and exotic options under both diffusion and pure jump processes with and without stochastic volatility. We train our neural network models under different number of layers, neurons per layer, and various different activation functions in order to find which combinations work better empirically. For training, we consider various different loss functions and optimization routines. We demonstrate that deep neural networks exponentially expedite option pricing compared to commonly used option pricing methods which consequently make calibration and parameter estimation super fast.

Transformer layers, which use an alternating pattern of multi-head attention and multi-layer perceptron (MLP) layers, provide an effective tool for a variety of machine learning problems. As the transformer layers use residual connections to avoid the problem of vanishing gradients, they can be viewed as the numerical integration of a differential equation. In this extended abstract, we build upon this connection and propose a modification of the internal architecture of a transformer layer. The proposed model places the multi-head attention sublayer and the MLP sublayer parallel to each other. Our experiments show that this simple modification improves the performance of transformer networks in multiple tasks. Moreover, for the image classification task, we show that using neural ODE solvers with a sophisticated integration scheme further improves performance.

Graph Neural Networks (GNNs) are popular models for graph learning problems. GNNs show strong empirical performance in many practical tasks. However, the theoretical properties have not been completely elucidated. In this paper, we investigate whether GNNs can exploit the graph structure from the perspective of the expressive power of GNNs. In our analysis, we consider graph generation processes that are controlled by hidden (or latent) node features, which contain all information about the graph structure. A typical example of this framework is kNN graphs constructed from the hidden features. In our main results, we show that GNNs can recover the hidden node features from the input graph alone, even when all node features, including the hidden features themselves and any indirect hints, are unavailable. GNNs can further use the recovered node features for downstream tasks. These results show that GNNs can fully exploit the graph structure by themselves, and in effect, GNNs can use both the hidden and explicit node features for downstream tasks. In the experiments, we confirm the validity of our results by showing that GNNs can accurately recover the hidden features using a GNN architecture built based on our theoretical analysis.

We look critically at popular self-supervision techniques for learning deep convolutional neural networks without manual labels. We show that three different and representative methods, BiGAN, RotNet and DeepCluster, can learn the first few layers of a convolutional network from a single image as well as using millions of images and manual labels, provided that strong data augmentation is used. However, for deeper layers the gap with manual supervision cannot be closed even if millions of unlabelled images are used for training. We conclude that: (1) the weights of the early layers of deep networks contain limited information about the statistics of natural images, that (2) such low-level statistics can be learned through self-supervision just as well as through strong supervision, and that (3) the low-level statistics can be captured via synthetic transformations instead of using a large image dataset.

Designing machine learning architectures for processing neural networks in their raw weight matrix form is a newly introduced research direction. Unfortunately, the unique symmetry structure of deep weight spaces makes this design very challenging. If successful, such architectures would be capable of performing a wide range of intriguing tasks, from adapting a pre-trained network to a new domain to editing objects represented as functions (INRs or NeRFs). As a first step towards this goal, we present here a novel network architecture for learning in deep weight spaces. It takes as input a concatenation of weights and biases of a pre-trained MLP and processes it using a composition of layers that are equivariant to the natural permutation symmetry of the MLP's weights: Changing the order of neurons in intermediate layers of the MLP does not affect the function it represents. We provide a full characterization of all affine equivariant and invariant layers for these symmetries and show how these layers can be implemented using three basic operations: pooling, broadcasting, and fully connected layers applied to the input in an appropriate manner. We demonstrate the effectiveness of our architecture and its advantages over natural baselines in a variety of learning tasks.

As the size of deep learning models continues to grow, finding optimal models under memory and computation constraints becomes increasingly more important. Although usually the architecture and constituent building blocks of neural networks allow them to be used in a modular way, their training process is not aware of this modularity. Consequently, conventional neural network training lacks the flexibility to adapt the computational load of the model during inference. This paper proposes SortedNet, a generalized and scalable solution to harness the inherent modularity of deep neural networks across various dimensions for efficient dynamic inference. Our training considers a nested architecture for the sub-models with shared parameters and trains them together with the main model in a sorted and probabilistic manner. This sorted training of sub-networks enables us to scale the number of sub-networks to hundreds using a single round of training. We utilize a novel updating scheme during training that combines random sampling of sub-networks with gradient accumulation to improve training efficiency. Furthermore, the sorted nature of our training leads to a search-free sub-network selection at inference time; and the nested architecture of the resulting sub-networks leads to minimal storage requirement and efficient switching between sub-networks at inference. Our general dynamic training approach is demonstrated across various architectures and tasks, including large language models and pre-trained vision models. Experimental results show the efficacy of the proposed approach in achieving efficient sub-networks while outperforming state-of-the-art dynamic training approaches. Our findings demonstrate the feasibility of training up to 160 different sub-models simultaneously, showcasing the extensive scalability of our proposed method while maintaining 96% of the model performance.

The ability to learn tasks in a sequential fashion is crucial to the development of artificial intelligence. Neural networks are not, in general, capable of this and it has been widely thought that catastrophic forgetting is an inevitable feature of connectionist models. We show that it is possible to overcome this limitation and train networks that can maintain expertise on tasks which they have not experienced for a long time. Our approach remembers old tasks by selectively slowing down learning on the weights important for those tasks. We demonstrate our approach is scalable and effective by solving a set of classification tasks based on the MNIST hand written digit dataset and by learning several Atari 2600 games sequentially.

Despite its importance for federated learning, continuous learning and many other applications, on-device training remains an open problem for EdgeAI. The problem stems from the large number of operations (e.g., floating point multiplications and additions) and memory consumption required during training by the back-propagation algorithm. Consequently, in this paper, we propose a new gradient filtering approach which enables on-device CNN model training. More precisely, our approach creates a special structure with fewer unique elements in the gradient map, thus significantly reducing the computational complexity and memory consumption of back propagation during training. Extensive experiments on image classification and semantic segmentation with multiple CNN models (e.g., MobileNet, DeepLabV3, UPerNet) and devices (e.g., Raspberry Pi and Jetson Nano) demonstrate the effectiveness and wide applicability of our approach. For example, compared to SOTA, we achieve up to 19times speedup and 77.1% memory savings on ImageNet classification with only 0.1% accuracy loss. Finally, our method is easy to implement and deploy; over 20times speedup and 90% energy savings have been observed compared to highly optimized baselines in MKLDNN and CUDNN on NVIDIA Jetson Nano. Consequently, our approach opens up a new direction of research with a huge potential for on-device training.

Recent advances show that Neural Architectural Search (NAS) method is able to find state-of-the-art image classification deep architectures. In this paper, we consider the one-shot NAS problem for resource constrained applications. This problem is of great interest because it is critical to choose different architectures according to task complexity when the resource is constrained. Previous techniques are either too slow for one-shot learning or does not take the resource constraint into consideration. In this paper, we propose the resource constrained differentiable architecture search (RC-DARTS) method to learn architectures that are significantly smaller and faster while achieving comparable accuracy. Specifically, we propose to formulate the RC-DARTS task as a constrained optimization problem by adding the resource constraint. An iterative projection method is proposed to solve the given constrained optimization problem. We also propose a multi-level search strategy to enable layers at different depths to adaptively learn different types of neural architectures. Through extensive experiments on the Cifar10 and ImageNet datasets, we show that the RC-DARTS method learns lightweight neural architectures which have smaller model size and lower computational complexity while achieving comparable or better performances than the state-of-the-art methods.

Attention layers -- which map a sequence of inputs to a sequence of outputs -- are core building blocks of the Transformer architecture which has achieved significant breakthroughs in modern artificial intelligence. This paper presents a rigorous theoretical study on the learning and generalization of a single multi-head attention layer, with a sequence of key vectors and a separate query vector as input. We consider the random feature setting where the attention layer has a large number of heads, with randomly sampled frozen query and key matrices, and trainable value matrices. We show that such a random-feature attention layer can express a broad class of target functions that are permutation invariant to the key vectors. We further provide quantitative excess risk bounds for learning these target functions from finite samples, using random feature attention with finitely many heads. Our results feature several implications unique to the attention structure compared with existing random features theory for neural networks, such as (1) Advantages in the sample complexity over standard two-layer random-feature networks; (2) Concrete and natural classes of functions that can be learned efficiently by a random-feature attention layer; and (3) The effect of the sampling distribution of the query-key weight matrix (the product of the query and key matrix), where Gaussian random weights with a non-zero mean result in better sample complexities over the zero-mean counterpart for learning certain natural target functions. Experiments on simulated data corroborate our theoretical findings and further illustrate the interplay between the sample size and the complexity of the target function.

We investigate the expressive power of depth-2 bandlimited random neural networks. A random net is a neural network where the hidden layer parameters are frozen with random assignment, and only the output layer parameters are trained by loss minimization. Using random weights for a hidden layer is an effective method to avoid non-convex optimization in standard gradient descent learning. It has also been adopted in recent deep learning theories. Despite the well-known fact that a neural network is a universal approximator, in this study, we mathematically show that when hidden parameters are distributed in a bounded domain, the network may not achieve zero approximation error. In particular, we derive a new nontrivial approximation error lower bound. The proof utilizes the technique of ridgelet analysis, a harmonic analysis method designed for neural networks. This method is inspired by fundamental principles in classical signal processing, specifically the idea that signals with limited bandwidth may not always be able to perfectly recreate the original signal. We corroborate our theoretical results with various simulation studies, and generally, two main take-home messages are offered: (i) Not any distribution for selecting random weights is feasible to build a universal approximator; (ii) A suitable assignment of random weights exists but to some degree is associated with the complexity of the target function.

Neural networks have achieved tremendous success in a large variety of applications. However, their memory footprint and computational demand can render them impractical in application settings with limited hardware or energy resources. In this work, we propose a novel algorithm to find efficient low-rank subnetworks. Remarkably, these subnetworks are determined and adapted already during the training phase and the overall time and memory resources required by both training and evaluating them are significantly reduced. The main idea is to restrict the weight matrices to a low-rank manifold and to update the low-rank factors rather than the full matrix during training. To derive training updates that are restricted to the prescribed manifold, we employ techniques from dynamic model order reduction for matrix differential equations. This allows us to provide approximation, stability, and descent guarantees. Moreover, our method automatically and dynamically adapts the ranks during training to achieve the desired approximation accuracy. The efficiency of the proposed method is demonstrated through a variety of numerical experiments on fully-connected and convolutional networks.

Neural networks have enabled state-of-the-art approaches to achieve incredible results on computer vision tasks such as object detection. However, such success greatly relies on costly computation resources, which hinders people with cheap devices from appreciating the advanced technology. In this paper, we propose Cross Stage Partial Network (CSPNet) to mitigate the problem that previous works require heavy inference computations from the network architecture perspective. We attribute the problem to the duplicate gradient information within network optimization. The proposed networks respect the variability of the gradients by integrating feature maps from the beginning and the end of a network stage, which, in our experiments, reduces computations by 20% with equivalent or even superior accuracy on the ImageNet dataset, and significantly outperforms state-of-the-art approaches in terms of AP50 on the MS COCO object detection dataset. The CSPNet is easy to implement and general enough to cope with architectures based on ResNet, ResNeXt, and DenseNet. Source code is at https://github.com/WongKinYiu/CrossStagePartialNetworks.

Deep neural networks (DNNs) have been shown to outperform traditional machine learning algorithms in a broad variety of application domains due to their effectiveness in modeling complex problems and handling high-dimensional datasets. Many real-life datasets, however, are of increasingly high dimensionality, where a large number of features may be irrelevant for both supervised and unsupervised learning tasks. The inclusion of such features would not only introduce unwanted noise but also increase computational complexity. Furthermore, due to high non-linearity and dependency among a large number of features, DNN models tend to be unavoidably opaque and perceived as black-box methods because of their not well-understood internal functioning. Their algorithmic complexity is often simply beyond the capacities of humans to understand the interplay among myriads of hyperparameters. A well-interpretable model can identify statistically significant features and explain the way they affect the model's outcome. In this paper, we propose an efficient method to improve the interpretability of black-box models for classification tasks in the case of high-dimensional datasets. First, we train a black-box model on a high-dimensional dataset to learn the embeddings on which the classification is performed. To decompose the inner working principles of the black-box model and to identify top-k important features, we employ different probing and perturbing techniques. We then approximate the behavior of the black-box model by means of an interpretable surrogate model on the top-k feature space. Finally, we derive decision rules and local explanations from the surrogate model to explain individual decisions. Our approach outperforms state-of-the-art methods like TabNet and XGboost when tested on different datasets with varying dimensionality between 50 and 20,000 w.r.t metrics and explainability.

The exponential growth in numbers of parameters of neural networks over the past years has been accompanied by an increase in performance across several fields. However, due to their sheer size, the networks not only became difficult to interpret but also problematic to train and use in real-world applications, since hardware requirements increased accordingly. Tackling both issues, we present a novel approach that either drops a neural network's initial weights or inverts their respective sign. Put simply, a network is trained by weight selection and inversion without changing their absolute values. Our contribution extends previous work on masking by additionally sign-inverting the initial weights and follows the findings of the Lottery Ticket Hypothesis. Through this extension and adaptations of initialization methods, we achieve a pruning rate of up to 99%, while still matching or exceeding the performance of various baseline and previous models. Our approach has two main advantages. First, and most notable, signed Supermask models drastically simplify a model's structure, while still performing well on given tasks. Second, by reducing the neural network to its very foundation, we gain insights into which weights matter for performance. The code is available on GitHub.

We address the challenging problem of efficient inference across many devices and resource constraints, especially on edge devices. Conventional approaches either manually design or use neural architecture search (NAS) to find a specialized neural network and train it from scratch for each case, which is computationally prohibitive (causing CO_2 emission as much as 5 cars' lifetime) thus unscalable. In this work, we propose to train a once-for-all (OFA) network that supports diverse architectural settings by decoupling training and search, to reduce the cost. We can quickly get a specialized sub-network by selecting from the OFA network without additional training. To efficiently train OFA networks, we also propose a novel progressive shrinking algorithm, a generalized pruning method that reduces the model size across many more dimensions than pruning (depth, width, kernel size, and resolution). It can obtain a surprisingly large number of sub-networks (> 10^{19}) that can fit different hardware platforms and latency constraints while maintaining the same level of accuracy as training independently. On diverse edge devices, OFA consistently outperforms state-of-the-art (SOTA) NAS methods (up to 4.0% ImageNet top1 accuracy improvement over MobileNetV3, or same accuracy but 1.5x faster than MobileNetV3, 2.6x faster than EfficientNet w.r.t measured latency) while reducing many orders of magnitude GPU hours and CO_2 emission. In particular, OFA achieves a new SOTA 80.0% ImageNet top-1 accuracy under the mobile setting (<600M MACs). OFA is the winning solution for the 3rd Low Power Computer Vision Challenge (LPCVC), DSP classification track and the 4th LPCVC, both classification track and detection track. Code and 50 pre-trained models (for many devices & many latency constraints) are released at https://github.com/mit-han-lab/once-for-all.

Recent advances in Neural Architecture Search (NAS) such as one-shot NAS offer the ability to extract specialized hardware-aware sub-network configurations from a task-specific super-network. While considerable effort has been employed towards improving the first stage, namely, the training of the super-network, the search for derivative high-performing sub-networks is still under-explored. Popular methods decouple the super-network training from the sub-network search and use performance predictors to reduce the computational burden of searching on different hardware platforms. We propose a flexible search framework that automatically and efficiently finds optimal sub-networks that are optimized for different performance metrics and hardware configurations. Specifically, we show how evolutionary algorithms can be paired with lightly trained objective predictors in an iterative cycle to accelerate architecture search in a multi-objective setting for various modalities including machine translation and image classification.

Recent one-shot Neural Architecture Search algorithms rely on training a hardware-agnostic super-network tailored to a specific task and then extracting efficient sub-networks for different hardware platforms. Popular approaches separate the training of super-networks from the search for sub-networks, often employing predictors to alleviate the computational overhead associated with search. Additionally, certain methods also incorporate the quantization policy within the search space. However, while the quantization policy search for convolutional neural networks is well studied, the extension of these methods to transformers and especially foundation models remains under-explored. In this paper, we demonstrate that by using multi-objective search algorithms paired with lightly trained predictors, we can efficiently search for both the sub-network architecture and the corresponding quantization policy and outperform their respective baselines across different performance objectives such as accuracy, model size, and latency. Specifically, we demonstrate that our approach performs well across both uni-modal (ViT and BERT) and multi-modal (BEiT-3) transformer-based architectures as well as convolutional architectures (ResNet). For certain networks, we demonstrate an improvement of up to 4.80x and 3.44x for latency and model size respectively, without degradation in accuracy compared to the fully quantized INT8 baselines.

We introduce a modern Hopfield network with continuous states and a corresponding update rule. The new Hopfield network can store exponentially (with the dimension of the associative space) many patterns, retrieves the pattern with one update, and has exponentially small retrieval errors. It has three types of energy minima (fixed points of the update): (1) global fixed point averaging over all patterns, (2) metastable states averaging over a subset of patterns, and (3) fixed points which store a single pattern. The new update rule is equivalent to the attention mechanism used in transformers. This equivalence enables a characterization of the heads of transformer models. These heads perform in the first layers preferably global averaging and in higher layers partial averaging via metastable states. The new modern Hopfield network can be integrated into deep learning architectures as layers to allow the storage of and access to raw input data, intermediate results, or learned prototypes. These Hopfield layers enable new ways of deep learning, beyond fully-connected, convolutional, or recurrent networks, and provide pooling, memory, association, and attention mechanisms. We demonstrate the broad applicability of the Hopfield layers across various domains. Hopfield layers improved state-of-the-art on three out of four considered multiple instance learning problems as well as on immune repertoire classification with several hundreds of thousands of instances. On the UCI benchmark collections of small classification tasks, where deep learning methods typically struggle, Hopfield layers yielded a new state-of-the-art when compared to different machine learning methods. Finally, Hopfield layers achieved state-of-the-art on two drug design datasets. The implementation is available at: https://github.com/ml-jku/hopfield-layers

Recent advances in self-attention and pure multi-layer perceptrons (MLP) models for vision have shown great potential in achieving promising performance with fewer inductive biases. These models are generally based on learning interaction among spatial locations from raw data. The complexity of self-attention and MLP grows quadratically as the image size increases, which makes these models hard to scale up when high-resolution features are required. In this paper, we present the Global Filter Network (GFNet), a conceptually simple yet computationally efficient architecture, that learns long-term spatial dependencies in the frequency domain with log-linear complexity. Our architecture replaces the self-attention layer in vision transformers with three key operations: a 2D discrete Fourier transform, an element-wise multiplication between frequency-domain features and learnable global filters, and a 2D inverse Fourier transform. We exhibit favorable accuracy/complexity trade-offs of our models on both ImageNet and downstream tasks. Our results demonstrate that GFNet can be a very competitive alternative to transformer-style models and CNNs in efficiency, generalization ability and robustness. Code is available at https://github.com/raoyongming/GFNet

Designing an efficient model within the limited computational cost is challenging. We argue the accuracy of a lightweight model has been further limited by the design convention: a stage-wise configuration of the channel dimensions, which looks like a piecewise linear function of the network stage. In this paper, we study an effective channel dimension configuration towards better performance than the convention. To this end, we empirically study how to design a single layer properly by analyzing the rank of the output feature. We then investigate the channel configuration of a model by searching network architectures concerning the channel configuration under the computational cost restriction. Based on the investigation, we propose a simple yet effective channel configuration that can be parameterized by the layer index. As a result, our proposed model following the channel parameterization achieves remarkable performance on ImageNet classification and transfer learning tasks including COCO object detection, COCO instance segmentation, and fine-grained classifications. Code and ImageNet pretrained models are available at https://github.com/clovaai/rexnet.

Depth is the hallmark of deep neural networks. But more depth means more sequential computation and higher latency. This begs the question -- is it possible to build high-performing "non-deep" neural networks? We show that it is. To do so, we use parallel subnetworks instead of stacking one layer after another. This helps effectively reduce depth while maintaining high performance. By utilizing parallel substructures, we show, for the first time, that a network with a depth of just 12 can achieve top-1 accuracy over 80% on ImageNet, 96% on CIFAR10, and 81% on CIFAR100. We also show that a network with a low-depth (12) backbone can achieve an AP of 48% on MS-COCO. We analyze the scaling rules for our design and show how to increase performance without changing the network's depth. Finally, we provide a proof of concept for how non-deep networks could be used to build low-latency recognition systems. Code is available at https://github.com/imankgoyal/NonDeepNetworks.

This paper studies the problem of training a two-layer ReLU network for binary classification using gradient flow with small initialization. We consider a training dataset with well-separated input vectors: Any pair of input data with the same label are positively correlated, and any pair with different labels are negatively correlated. Our analysis shows that, during the early phase of training, neurons in the first layer try to align with either the positive data or the negative data, depending on its corresponding weight on the second layer. A careful analysis of the neurons' directional dynamics allows us to provide an O(log n{mu}) upper bound on the time it takes for all neurons to achieve good alignment with the input data, where n is the number of data points and mu measures how well the data are separated. After the early alignment phase, the loss converges to zero at a O(1{t}) rate, and the weight matrix on the first layer is approximately low-rank. Numerical experiments on the MNIST dataset illustrate our theoretical findings.