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Deep learning has received considerable empirical successes in recent years. However, while many ad hoc tricks have been discovered by practitioners, until recently, there has been a lack of theoretical understanding for tricks invented in the deep learning literature. Known by practitioners that overparameterized neural networks are easy to learn, in the past few years there have been important theoretical developments in the analysis of overparameterized neural networks. In particular, it was shown that such systems behave like convex systems under various restricted settings, such as for two-layer NNs, and when learning is restricted locally in the so-called neural tangent kernel space around specialized initializations. This paper discusses some of these recent progresses leading to significant better understanding of neural networks. We will focus on the analysis of two-layer neural networks, and explain the key mathematical models, with their algorithmic implications. We will then discuss challenges in understanding deep neural networks and some current research directions.

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Significance
Modern deep neural networks for image classification have achieved superhuman performance. Yet, the complex details of trained networks have forced most practitioners and researchers to regard them as black boxes with little that could be understood. This paper considers in detail a now-standard training methodology: driving the cross-entropy loss to zero, continuing long after the classification error is already zero. Applying this methodology to an authoritative collection of standard deepnets and datasets, we observe the emergence of a simple and highly symmetric geometry of the deepnet features and of the deepnet classifier, and we document important benefits that the geometry conveys—thereby helping us understand an important component of the modern deep learning training paradigm.

We study the problem of training deep fully connected neural networks with Rectified Linear Unit (ReLU) activation function and cross entropy loss function for binary classification using gradient descent. We show that with proper random weight initialization, gradient descent can find the global minima of the training loss for an over-parameterized deep ReLU network, under certain assumption on the training data. The key idea of our proof is that Gaussian random initialization followed by gradient descent produces a sequence of iterates that stay inside a small perturbation region centered at the initial weights, in which the training loss function of the deep ReLU networks enjoys nice local curvature properties that ensure the global convergence of gradient descent. At the core of our proof technique is (1) a milder assumption on the training data; (2) a sharp analysis of the trajectory length for gradient descent; and (3) a finer characterization of the size of the perturbation region. Compared with the concurrent work (Allen-Zhu et al. in A convergence theory for deep learning via over-parameterization, 2018a; Du et al. in Gradient descent finds global minima of deep neural networks, 2018a) along this line, our result relies on milder over-parameterization condition on the neural network width, and enjoys faster global convergence rate of gradient descent for training deep neural networks.

Significance
Multilayer neural networks have proven extremely successful in a variety of tasks, from image classification to robotics. However, the reasons for this practical success and its precise domain of applicability are unknown. Learning a neural network from data requires solving a complex optimization problem with millions of variables. This is done by stochastic gradient descent (SGD) algorithms. We study the case of two-layer networks and derive a compact description of the SGD dynamics in terms of a limiting partial differential equation. Among other consequences, this shows that SGD dynamics does not become more complex when the network size increases.

The expressive power of neural networks is important for understanding deep learning. Most existing works consider this problem from the view of the depth of a network. In this paper, we study how width affects the expressiveness of neural networks. Classical results state that \emph{depth-bounded} (e.g. depth-$2$) networks with suitable activation functions are universal approximators. We show a universal approximation theorem for \emph{width-bounded} ReLU networks: width-$(n+4)$ ReLU networks, where $n$ is the input dimension, are universal approximators. Moreover, except for a measure zero set, all functions cannot be approximated by width-$n$ ReLU networks, which exhibits a phase transition. Several recent works demonstrate the benefits of depth by proving the depth-efficiency of neural networks. That is, there are classes of deep networks which cannot be realized by any shallow network whose size is no more than an \emph{exponential} bound. Here we pose the dual question on the width-efficiency of ReLU networks: Are there wide networks that cannot be realized by narrow networks whose size is not substantially larger? We show that there exist classes of wide networks which cannot be realized by any narrow network whose depth is no more than a \emph{polynomial} bound. On the other hand, we demonstrate by extensive experiments that narrow networks whose size exceed the polynomial bound by a constant factor can approximate wide and shallow network with high accuracy. Our results provide more comprehensive evidence that depth is more effective than width for the expressiveness of ReLU networks.

This article conerns the expressive power of depth in neural nets with ReLU activations. We prove that ReLU nets with width $2d+2$ can approximate any continuous scalar function on the $d$-dimensional cube $[0,1]^d$ arbitrarily well. We obtain quantitative depth estimates for such approximations. Our approach is based on the observation that ReLU nets are particularly well-suited for representing convex functions. Indeed, we give a constructive proof that ReLU nets with width $d+1$ can approximate any continuous convex function of $d$ arbitrarily well. Moreover, when approximating convex, piecewise affine functions by width $d+1$ ReLU nets, we obtain matching upper and lower bounds on the required depth, proving that our construction is essentially optimal.

While the optimization problem behind deep neural networks is highly non-convex, it is frequently observed in practice that training deep networks seems possible without getting stuck in suboptimal points. It has been argued that this is the case as all local minima are close to being globally optimal. We show that this is (almost) true, in fact almost all local minima are globally optimal, for a fully connected network with squared loss and analytic activation function given that the number of hidden units of one layer of the network is larger than the number of training points and the network structure from this layer on is pyramidal.

A class of measure-valued processes which model multilevel populations
undergoing mutation, selection, genetic drift and spatial migration is
considered. We investigate the qualitative behaviour of models with multilevel
selection and the interaction between the different levels of selection. The
basic tools in our analysis include the martingale problem formulation for
measure-valued processes and a generalization of the set-valued dual
representation introduced in Dawson-Greven (2014). In particular we describe
the ergodic behaviour of these processes and review some situations under which
deme level selection determines the long-time behaviour.

Deep architectures have demonstrated state-of-the-art results in a variety of settings, especially with vision datasets. Beyond the model definitions and the quantitative analyses, there is a need for qualitative comparisons of the solutions learned by various deep architectures. The goal of this paper is to find good qualita-tive interpretations of high level features represented by such models. To this end, we contrast and compare several techniques applied on Stacked Denoising Auto-encoders and Deep Belief Networks, trained on several vision datasets. We show that, perhaps counter-intuitively, such interpretation is possible at the unit level, that it is simple to accomplish and that the results are consistent across various techniques. We hope that such techniques will allow researchers in deep architec-tures to understand more of how and why deep architectures work.

Fleming and Viot have established the existence of a continuous-state-space version of the Ohta-Kimura ladder or stepwise-mutation model of population genetics for describing allelic frequencies within a selectively neutral population undergoing mutation and random genetic drift. Their model is given by a probability-measure-valued Markov diffusion process. In this paper, we investigate the qualitative behavior of such measure-valued processes. It is demonstrated that the random measure is supported on a bounded generalized Cantor set and that this set performs a "wandering" but "coherent" motion that, if appropriately rescaled, approaches a Brownian motion. The method used involves the construction of an interacting infinite particle system determined by the moment measures of the process and an analysis of the function-valued process that is "dual" to the measure-valued process of Fleming and Viot.

Approximation properties of a class of artificial neural networks
are established. It is shown that feedforward networks with one layer of
sigmoidal nonlinearities achieve integrated squared error of order O
(1/ n ), where n is the number of nodes. The
approximated function is assumed to have a bound on the first moment of
the magnitude distribution of the Fourier transform. The nonlinear
parameters associated with the sigmoidal nodes, as well as the
parameters of linear combination, are adjusted in the approximation. In
contrast, it is shown that for series expansions with n terms,
in which only the parameters of linear combination are adjusted, the
integrated squared approximation error cannot be made smaller than order
1/ n <sup>2</sup>d/ uniformly for functions satisfying the same
smoothness assumption, where d is the dimension of the input to
the function. For the class of functions examined, the approximation
rate and the parsimony of the parameterization of the networks are shown
to be advantageous in high-dimensional settings

Multilayer neural networks trained with the back-propagation
algorithm constitute the best example of a successful gradient based
learning technique. Given an appropriate network architecture,
gradient-based learning algorithms can be used to synthesize a complex
decision surface that can classify high-dimensional patterns, such as
handwritten characters, with minimal preprocessing. This paper reviews
various methods applied to handwritten character recognition and
compares them on a standard handwritten digit recognition task.
Convolutional neural networks, which are specifically designed to deal
with the variability of 2D shapes, are shown to outperform all other
techniques. Real-life document recognition systems are composed of
multiple modules including field extraction, segmentation recognition,
and language modeling. A new learning paradigm, called graph transformer
networks (GTN), allows such multimodule systems to be trained globally
using gradient-based methods so as to minimize an overall performance
measure. Two systems for online handwriting recognition are described.
Experiments demonstrate the advantage of global training, and the
flexibility of graph transformer networks. A graph transformer network
for reading a bank cheque is also described. It uses convolutional
neural network character recognizers combined with global training
techniques to provide record accuracy on business and personal cheques.
It is deployed commercially and reads several million cheques per day

We lower bound the complexity of finding ϵ-stationary points (with gradient norm at most ϵ) using stochastic first-order methods. In a well-studied model where algorithms access smooth, potentially non-convex functions through queries to an unbiased stochastic gradient oracle with bounded variance, we prove that (in the worst case) any algorithm requires at least ϵ-4 queries to find an ϵ-stationary point. The lower bound is tight, and establishes that stochastic gradient descent is minimax optimal in this model. In a more restrictive model where the noisy gradient estimates satisfy a mean-squared smoothness property, we prove a lower bound of ϵ-3 queries, establishing the optimality of recently proposed variance reduction techniques.

For a certain scaling of the initialization of stochastic gradient descent (SGD), wide neural networks (NN) have been shown to be well approximated by reproducing kernel Hilbert space (RKHS) methods. Recent empirical work showed that, for some classification tasks, RKHS methods can replace NNs without a large loss in performance. On the other hand, two-layers NNs are known to encode richer smoothness classes than RKHS and we know of special examples for which SGD-trained NN provably outperform RKHS. This is true even in the wide network limit, for a different scaling of the initialization. How can we reconcile the above claims? For which tasks do NNs outperform RKHS? If covariates are nearly isotropic, RKHS methods suffer from the curse of dimensionality, while NNs can overcome it by learning the best low-dimensional representation. Here we show that this curse of dimensionality becomes milder if the covariates display the same low-dimensional structure as the target function, and we precisely characterize this tradeoff. Building on these results, we present the spiked covariates model that can capture in a unified framework both behaviors observed in earlier work. We hypothesize that such a latent low-dimensional structure is present in image classification. We test numerically this hypothesis by showing that specific perturbations of the training distribution degrade the performances of RKHS methods much more significantly than NNs.

We analyze multilayer neural networks in the asymptotic regime of simultaneously (a) large network sizes and (b) large numbers of stochastic gradient descent training iterations. We rigorously establish the limiting behavior of the multilayer neural network output. The limit procedure is valid for any number of hidden layers, and it naturally also describes the limiting behavior of the training loss. The ideas that we explore are to (a) take the limits of each hidden layer sequentially and (b) characterize the evolution of parameters in terms of their initialization. The limit satisfies a system of deterministic integro-differential equations. The proof uses methods from weak convergence and stochastic analysis. We show that, under suitable assumptions on the activation functions and the behavior for large times, the limit neural network recovers a global minimum (with zero loss for the objective function).

A longstanding goal in deep learning research has been to precisely characterize training and generalization. However, the often complex loss landscapes of neural networks (NNs) have made a theory of learning dynamics elusive. In this work, we show that for wide NNs the learning dynamics simplify considerably and that, in the infinite width limit, they are governed by a linear model obtained from the first-order Taylor expansion of the network around its initial parameters. Furthermore, mirroring the correspondence between wide Bayesian NNs and Gaussian processes (GPs), gradient-based training of wide NNs with a squared loss produces test set predictions drawn from a GP with a particular compositional kernel. While these theoretical results are only exact in the infinite width limit, we nevertheless find excellent empirical agreement between the predictions of the original network and those of the linearized version even for finite practically-sized networks. This agreement is robust across different architectures, optimization methods, and loss functions.

One of the major concerns for neural network training is that the nonconvexity of the associated loss functions may cause a bad landscape. The recent success of neural networks suggests that their loss landscape is not too bad, but what specific results do we know about the landscape? In this article, we review recent findings and results on the global landscape of neural networks.

Learning is one of the things that humans do naturally, and it has always been a challenge for us to understand the process. Nowadays this challenge has another dimension as we try to build machines that are able to learn and to undertake tasks such as datamining, image processing and pattern recognition. We can formulate a simple framework, artificial neural networks, in which learning from examples may be described and understood. The contribution to this subject made over the last decade by researchers applying the techniques of statistical mechanics is the subject of this book. The authors provide a coherent account of various important concepts and techniques that are currently only found scattered in papers, supplement this with background material in mathematics and physics and include many examples and exercises to make a book that can be used with courses, or for self-teaching, or as a handy reference.

Consider the problem: given the data pair (x,y) drawn from a population with f*(x)=E[y|x=x], specify a neural network model and run gradient flow on the weights over time until reaching any stationarity. How does ft, the function computed by the neural network at time t, relate to f*, in terms of approximation and representation? What are the provable benefits of the adaptive representation by neural networks compared to the pre-specified fixed basis representation in the classical nonparametric literature? We answer the above questions via a dynamic reproducing kernel Hilbert space (RKHS) approach indexed by the training process of neural networks. Firstly, we show that when reaching any local stationarity, gradient flow learns an adaptive RKHS representation and performs the global least-squares projection onto the adaptive RKHS, simultaneously. Secondly, we prove that as the RKHS is data-adaptive and task-specific, the residual for f* lies in a subspace that is potentially much smaller than the orthogonal complement of the RKHS. The result formalizes the representation and approximation benefits of neural networks. Lastly, we show that the neural network function computed by gradient flow converges to the kernel ridgeless regression with an adaptive kernel, in the limit of vanishing regularization. The adaptive kernel viewpoint provides new angles of studying the approximation, representation, generalization, and optimization advantages of neural networks.

We rigorously prove a central limit theorem for neural network models with a single hidden layer. The central limit theorem is proven in the asymptotic regime of simultaneously (A) large numbers of hidden units and (B) large numbers of stochastic gradient descent training iterations. Our result describes the neural network’s fluctuations around its mean-field limit. The fluctuations have a Gaussian distribution and satisfy a stochastic partial differential equation. The proof relies upon weak convergence methods from stochastic analysis. In particular, we prove relative compactness for the sequence of processes and uniqueness of the limiting process in a suitable Sobolev space.

Neural network training relies on our ability to find "good" minimizers of highly non-convex loss functions. It is well known that certain network architecture designs (e.g., skip connections) produce loss functions that train easier, and well-chosen training parameters (batch size, learning rate, optimizer) produce minimizers that generalize better. However, the reasons for these differences, and their effect on the underlying loss landscape, is not well understood. In this paper, we explore the structure of neural loss functions, and the effect of loss landscapes on generalization, using a range of visualization methods. First, we introduce a simple "filter normalization" method that helps us visualize loss function curvature, and make meaningful side-by-side comparisons between loss functions. Then, using a variety of visualizations, we explore how network architecture affects the loss landscape, and how training parameters affect the shape of minimizers.

Rectified activation units (rectifiers) are essential for state-of-the-art neural networks. In this work, we study rectifier neural networks for image classification from two aspects. First, we propose a Parametric Rectified Linear Unit (PReLU) that generalizes the traditional rectified unit. PReLU improves model fitting with nearly zero extra computational cost and little overfitting risk. Second, we derive a robust initialization method that particularly considers the rectifier nonlinearities. This method enables us to train extremely deep rectified models directly from scratch and to investigate deeper or wider network architectures. Based on our PReLU networks (PReLU-nets), we achieve 4.94% top-5 test error on the ImageNet 2012 classification dataset. This is a 26% relative improvement over the ILSVRC 2014 winner (GoogLeNet, 6.66%). To our knowledge, our result is the first to surpass human-level performance (5.1%, Russakovsky et al.) on this visual recognition challenge.

How can we perform efficient inference and learning in directed probabilistic models, in the presence of continuous latent variables with intractable posterior distributions, and large datasets? We introduce a stochastic variational inference and learning algorithm that scales to large datasets and, under some mild differentiability conditions, even works in the intractable case. Our contributions is two-fold. First, we show that a reparameterization of the variational lower bound yields a lower bound estimator that can be straightforwardly optimized using standard stochastic gradient methods. Second, we show that for i.i.d. datasets with continuous latent variables per datapoint, posterior inference can be made especially efficient by fitting an approximate inference model (also called a recognition model) to the intractable posterior using the proposed lower bound estimator. Theoretical advantages are reflected in experimental results.

We trained a large, deep convolutional neural network to classify the 1.2 million high-resolution images in the ImageNet LSVRC-2010 contest into the 1000 dif- ferent classes. On the test data, we achieved top-1 and top-5 error rates of 37.5% and 17.0% which is considerably better than the previous state-of-the-art. The neural network, which has 60 million parameters and 650,000 neurons, consists of five convolutional layers, some of which are followed by max-pooling layers, and three fully-connected layers with a final 1000-way softmax. To make training faster, we used non-saturating neurons and a very efficient GPU implemen- tation of the convolution operation. To reduce overfitting in the fully-connected layers we employed a recently-developed regularization method called dropout that proved to be very effective. We also entered a variant of this model in the ILSVRC-2012 competition and achieved a winning top-5 test error rate of 15.3%, compared to 26.2% achieved by the second-best entry

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 fam- ily, or to the regularization techniques used during training. Through extensive systematic experiments, we show how these traditional ap- proaches 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 ran- dom labeling of the training data. This phenomenon is qualitatively unaffected by explicit regularization, and occurs even if we replace the true images by com- pletely unstructured random noise. We corroborate these experimental findings with a theoretical construction showing that simple depth two neural networks al- ready 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.

For any positive integer $k$, there exist neural networks with $backslashTheta(k^3)$ layers, $backslashTheta(1)$ nodes per layer, and $backslashTheta(1)$ distinct parameters which can not be approximated by networks with $backslashmathcalO(k)$ layers unless they are exponentially large --- they must possess $backslashOmega(2^k)$ nodes. This result is proved here for a class of nodes termed "semi-algebraic gates" which includes the common choices of ReLU, maximum, indicator, and piecewise polynomial functions, therefore establishing benefits of depth against not just standard networks with ReLU gates, but also convolutional networks with ReLU and maximization gates, sum-product networks, and boosted decision trees (in this last case with a stronger separation: $backslashOmega(2^{k^}3)$ total tree nodes are required).

We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a theoretical analysis of the number of required samples for a given approximation error, leading to both upper and lower bounds that are based solely on the eigenvalues of the associated integral operator and match up to logarithmic terms. In particular, we show that the upper bound may be obtained from independent and identically distributed samples from a specific non-uniform distribution, while the lower bound if valid for any set of points. Applying our results to kernel-based quadrature, while our results are fairly general, we recover known upper and lower bounds for the special cases of Sobolev spaces. Moreover, our results extend to the more general problem of full function approximations (beyond simply computing an integral), with results in L2- And L1-norm that match known results for special cases. Applying our results to random features, we show an improvement of the number of random features needed to preserve the generalization guarantees for learning with Lipshitz-continuous losses.

We show that the standard stochastic gradient decent (SGD) algorithm is guaranteed to learn, in polynomial time, a function that is competitive with the best function in the conjugate kernel space, as defined in Daniely, Frostig and Singer (2016). The result holds for log-depth networks from a rich family of architectures. To the best of our knowledge, it is the first polynomial-time guarantee for the standard neural network learning algorithm for networks of depth $\ge 3$.

In this paper, we prove a conjecture published in 1989 and also partially address an open problem announced at the Conference on Learning Theory (COLT) 2015. For an expected loss function of a deep nonlinear neural network, we prove the following statements under the independence assumption adopted from recent work: 1) the function is non-convex and non-concave, 2) every local minimum is a global minimum, 3) every critical point that is not a global minimum is a saddle point, and 4) the property of saddle points differs for shallow networks (with three layers) and deeper networks (with more than three layers). Moreover, we prove that the same four statements hold for deep linear neural networks with any depth, any widths and no unrealistic assumptions. As a result, we present an instance, for which we can answer to the following question: how difficult to directly train a deep model in theory? It is more difficult than the classical machine learning models (because of the non-convexity), but not too difficult (because of the nonexistence of poor local minima and the property of the saddle points). We note that even though we have advanced the theoretical foundations of deep learning, there is still a gap between theory and practice.

We develop a general duality between neural networks and compositional kernels, striving towards a better understanding of deep learning. We show that initial representations generated by common random initializations are sufficiently rich to express all functions in the dual kernel space. Hence, though the training objective is hard to optimize in the worst case, the initial weights form a good starting point for optimization. Our dual view also reveals a pragmatic and aesthetic perspective of neural networks and underscores their expressive power.

Deep neural nets with a large number of parameters are very powerful machine learning systems. However, overfitting is a serious problem in such networks. Large networks are also slow to use, making it difficult to deal with overfitting by combining the predictions of many different large neural nets at test time. Dropout is a technique for addressing this problem. The key idea is to randomly drop units (along with their connections) from the neural network during training. This prevents units from co-adapting too much. During training, dropout samples from an exponential number of different "thinned" networks. At test time, it is easy to approximate the effect of averaging the predictions of all these thinned networks by simply using a single unthinned network that has smaller weights. This significantly reduces overfitting and gives major improvements over other regularization methods. We show that dropout improves the performance of neural networks on supervised learning tasks in vision, speech recognition, document classification and computational biology, obtaining state-of-the-art results on many benchmark data sets. © 2014 Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever and Ruslan Salakhutdinov.

Two features distinguish the Bayesian approach to learning models from data. First, beliefs derived from background knowledge are used to select a prior probability distribution for the model parameters. Second, predictions of future observations are made by integrating the model's predictions with respect to the posterior parameter distribution obtained by updating this prior to take account of the data. For neural network models, both these aspects present diiculties | the prior over network parameters has no obvious relation to our prior knowledge, and integration over the posterior is computationally very demanding. I address the problem by deening classes of prior distributions for network param-eters that reach sensible limits as the size of the network goes to innnity. In this limit, the properties of these priors can be elucidated. Some priors converge to Gaussian processes, in which functions computed by the network may be smooth, Brownian, or fractionally Brownian. Other priors converge to non-Gaussian stable processes. Interesting eeects are obtained by combining priors of both sorts in networks with more than one hidden layer.

Most current speech recognition systems use hidden Markov models (HMMs) to deal with the temporal variability of speech and Gaussian mixture models (GMMs) to determine how well each state of each HMM fits a frame or a short window of frames of coefficients that represents the acoustic input. An alternative way to evaluate the fit is to use a feed-forward neural network that takes several frames of coefficients as input and produces posterior probabilities over HMM states as output. Deep neural networks (DNNs) that have many hidden layers and are trained using new methods have been shown to outperform GMMs on a variety of speech recognition benchmarks, sometimes by a large margin. This article provides an overview of this progress and represents the shared views of four research groups that have had recent successes in using DNNs for acoustic modeling in speech recognition.

Random networks of nonlinear functions have a long history of empirical success in function fitting but few theoretical guarantees. In this paper, using techniques from probability on Banach Spaces, we analyze a specific architecture of random nonlinearities, provide L<sub>infin</sub> and L<sub>2</sub> error bounds for approximating functions in Reproducing Kernel Hilbert Spaces, and discuss scenarios when these expansions are dense in the continuous functions. We discuss connections between these random nonlinear networks and popular machine learning algorithms and show experimentally that these networks provide competitive performance at far lower computational cost on large-scale pattern recognition tasks.

Several researchers characterized the activation function under which multilayer feedforward networks can act as universal approximators. We show that most of all the characterizations that were reported thus far in the literature are special cases of the following general result: A standard multilayer feedforward network with a locally bounded piecewise continuous activation function can approximate any continuous function to any degree of accuracy if and only if the network's activation function is not a polynomial. We also emphasize the important role of the threshold, asserting that without it the last theorem does not hold.

For neural networks with a wide class of weight-priors, it can be shown that in the limit of an infinite number of hidden units the prior over functions tends to a Gaussian process. In this paper analytic forms are derived for the covariance function of the Gaussian processes corresponding to networks with sigmoidal and Gaussian hidden units. This allows predictions to be made efficiently using networks with an infinite number of hidden units, and shows that, somewhat paradoxically, it may be easier to compute with infinite networks than finite ones. 1 Introduction To someone training a neural network by maximizing the likelihood of a finite amount of data it makes no sense to use a network with an infinite number of hidden units; the network will "overfit" the data and so will be expected to generalize poorly. However, the idea of selecting the network size depending on the amount of training data makes little sense to a Bayesian; a model should be chosen that reflects the u...

Neural machine translation by jointly learning to align and translate

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Gradient descent finds global minima of deep neural networks

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What can resnet learn efficiently, going beyond kernels

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