We study the effects of mild over-parameterization on the optimization landscape of a simple ReLU neural network of the form $\mathbf{x}\mapsto\sum_{i=1}^k\max\{0,\mathbf{w}_i^{\top}\mathbf{x}\}$, in a well-studied teacher-student setting where the target values are generated by the same architecture, and when directly optimizing over the population squared loss with respect to Gaussian inputs. We prove that while the objective is strongly convex around the global minima when the teacher and student networks possess the same number of neurons, it is not even \emph{locally convex} after any amount of over-parameterization. Moreover, related desirable properties (e.g., one-point strong convexity and the Polyak-Łojasiewicz condition) also do not hold even locally. On the other hand, we establish that the objective remains one-point strongly convex in \emph{most} directions (suitably defined), and show an optimization guarantee under this property. For the non-global minima, we prove that adding even just a single neuron will turn a non-global minimum into a saddle point. This holds under some technical conditions which we validate empirically. These results provide a possible explanation for why recovering a global minimum becomes significantly easier when we over-parameterize, even if the amount of over-parameterization is very moderate.
Center for Data Science, New York University（纽约大学数据科学中心）；Inria - Département d’Informatique de l’École Normale Supérieure（法国国家信息与自动化研究所 - 巴黎高等师范学校计算机科学系）；PSL Research University（巴黎文理研究大学）
Deep networks are often considered to be more expressive than shallow ones in terms of approximation. Indeed, certain functions can be approximated by deep networks provably more efficiently than by shallow ones, however, no tractable algorithms are known for learning such deep models. Separately, a recent line of work has shown that deep networks trained with gradient descent may behave like (tractable) kernel methods in a certain over-parameterized regime, where the kernel is determined by the architecture and initialization, and this paper focuses on approximation for such kernels. We show …
We study the dynamics and implicit bias of gradient flow (GF) on univariate ReLU neural networks with a single hidden layer in a binary classification setting. We show that when the labels are determined by the sign of a target network with $r$ neurons, with high probability over the initialization of the network and the sampling of the dataset, GF converges in direction (suitably defined) to a network achieving perfect training accuracy and having at most $\mathcal{O}(r)$ linear regions, implying a generalization bound. Unlike many other results in the literature, under an additional assumpti…
A recent line of research on deep learning focuses on the extremely over-parameterized setting, and shows that when the network width is larger than a high degree polynomial of the training sample size $n$ and the inverse of the target error $ε^{-1}$, deep neural networks learned by (stochastic) gradient descent enjoy nice optimization and generalization guarantees. Very recently, it is shown that under certain margin assumptions on the training data, a polylogarithmic width condition suffices for two-layer ReLU networks to converge and generalize (Ji and Telgarsky, 2019). However, whether dee…
Iowa State University（爱荷华州立大学）；The Ohio State University（俄亥俄州立大学）
Implicit deep learning has received increasing attention recently due to the fact that it generalizes the recursive prediction rules of many commonly used neural network architectures. Its prediction rule is provided implicitly based on the solution of an equilibrium equation. Although a line of recent empirical studies has demonstrated its superior performances, the theoretical understanding of implicit neural networks is limited. In general, the equilibrium equation may not be well-posed during the training. As a result, there is no guarantee that a vanilla (stochastic) gradient descent (SGD…
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Recent theoretical work has guaranteed that overparameterized networks trained by gradient descent achieve arbitrarily low training error, and sometimes even low test error. The required width, however, is always polynomial in at least one of the sample size $n$, the (inverse) target error $1/ε$, and the (inverse) failure probability $1/δ$. This work shows that $\widetildeΘ(1/ε)$ iterations of gradient descent with $\widetildeΩ(1/ε^2)$ training examples on two-layer ReLU networks of any width exceeding $\mathrm{polylog}(n,1/ε,1/δ)$ suffice to achieve a test misclassification error of $ε$. We a…
This theoretical paper is devoted to developing a rigorous theory for demystifying the global convergence phenomenon in a challenging scenario: learning over-parameterized Rectified Linear Unit (ReLU) nets for very high dimensional dataset under very mild assumptions. A major ingredient of our analysis is a fine-grained analysis of random activation matrices. The essential virtue of dissecting activation matrices is that it bridges the dynamics of optimization and angular distribution in high-dimensional data space. This angle-based detailed analysis leads to asymptotic characterizations of gr…
University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）
The NP-hard problem of optimizing a shallow ReLU network can be characterized as a combinatorial search over each training example's activation pattern followed by a constrained convex problem given a fixed set of activation patterns. We explore the implications of this combinatorial aspect of ReLU optimization in this work. We show that it can be naturally modeled via a geometric and combinatoric object known as a zonotope with its vertex set isomorphic to the set of feasible activation patterns. This assists in analysis and provides a foundation for further research. We demonstrate its usefu…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Empirical studies show that gradient-based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. Very recently, a line of work explains in theory that with over-parameterization and proper random initialization, gradient-based methods can find the global minima of the training loss for DNNs. However, existing generalization error bounds are unable to explain the good generalization performance of over-parameterized DNNs. The major limitation of most existing …
In this paper, we study the generalization performance of global minima for implementing empirical risk minimization (ERM) on over-parameterized deep ReLU nets. Using a novel deepening scheme for deep ReLU nets, we rigorously prove that there exist perfect global minima achieving almost optimal generalization error bounds for numerous types of data under mild conditions. Since over-parameterization is crucial to guarantee that the global minima of ERM on deep ReLU nets can be realized by the widely used stochastic gradient descent (SGD) algorithm, our results indeed fill a gap between optimiza…
We give a simple proof for the global convergence of gradient descent in training deep ReLU networks with the standard square loss, and show some of its improvements over the state-of-the-art. In particular, while prior works require all the hidden layers to be wide with width at least $Ω(N^8)$ ($N$ being the number of training samples), we require a single wide layer of linear, quadratic or cubic width depending on the type of initialization. Unlike many recent proofs based on the Neural Tangent Kernel (NTK), our proof need not track the evolution of the entire NTK matrix, or more generally, …
ReLU neural-networks have been in the focus of many recent theoretical works, trying to explain their empirical success. Nonetheless, there is still a gap between current theoretical results and empirical observations, even in the case of shallow (one hidden-layer) networks. For example, in the task of memorizing a random sample of size $m$ and dimension $d$, the best theoretical result requires the size of the network to be $\tildeΩ(\frac{m^2}{d})$, while empirically a network of size slightly larger than $\frac{m}{d}$ is sufficient. To bridge this gap, we turn to study a simplified model for…
We prove that the gradient descent training of a two-layer neural network on empirical or population risk may not decrease population risk at an order faster than $t^{-4/(d-2)}$ under mean field scaling. Thus gradient descent training for fitting reasonably smooth, but truly high-dimensional data may be subject to the curse of dimensionality. We present numerical evidence that gradient descent training with general Lipschitz target functions becomes slower and slower as the dimension increases, but converges at approximately the same rate in all dimensions when the target function lies in the …
While regression tasks aim at interpolating a relation on the entire input space, they often have to be solved with a limited amount of training data. Still, if the hypothesis functions can be sketched well with the data, one can hope for identifying a generalizing model. In this work, we introduce with the Neural Restricted Isometry Property (NeuRIP) a uniform concentration event, in which all shallow $\mathrm{ReLU}$ networks are sketched with the same quality. To derive the sample complexity for achieving NeuRIP, we bound the covering numbers of the networks in the Sub-Gaussian metric and ap…
The Hebrew University（希伯来大学）；Google Research（谷歌研究院）；Weizmann Institute of Science（魏茨曼科学研究所）
We prove hardness-of-learning results under a well-studied assumption on the existence of local pseudorandom generators. As we show, this assumption allows us to surpass the current state of the art, and prove hardness of various basic problems, with no hardness results to date. Our results include: hardness of learning shallow ReLU neural networks under the Gaussian distribution and other distributions; hardness of learning intersections of $ω(1)$ halfspaces, DNF formulas with $ω(1)$ terms, and ReLU networks with $ω(1)$ hidden neurons; hardness of weakly learning deterministic finite automata…
It has been noted in existing literature that over-parameterization in ReLU networks generally improves performance. While there could be several factors involved behind this, we prove some desirable theoretical properties at initialization which may be enjoyed by ReLU networks. Specifically, it is known that He initialization in deep ReLU networks asymptotically preserves variance of activations in the forward pass and variance of gradients in the backward pass for infinitely wide networks, thus preserving the flow of information in both directions. Our paper goes beyond these results and sho…
We present a theoretical and empirical study of the gradient dynamics of overparameterized shallow ReLU networks with one-dimensional input, solving least-squares interpolation. We show that the gradient dynamics of such networks are determined by the gradient flow in a non-redundant parameterization of the network function. We examine the principal qualitative features of this gradient flow. In particular, we determine conditions for two learning regimes:kernel and adaptive, which depend both on the relative magnitude of initialization of weights in different layers and the asymptotic behavio…
Microsoft Research AI（微软研究院AI）；Stanford University（斯坦福大学）；Princeton University（普林斯顿大学）；UT-Austin（德克萨斯大学奥斯汀分校）；University of Washington（华盛顿大学）；Harvard University（哈佛大学）
Deep neural networks (DNNs) have demonstrated dominating performance in many fields; since AlexNet, networks used in practice are going wider and deeper. On the theoretical side, a long line of works has been focusing on training neural networks with one hidden layer. The theory of multi-layer networks remains largely unsettled. In this work, we prove why stochastic gradient descent (SGD) can find $\textit{global minima}$ on the training objective of DNNs in $\textit{polynomial time}$. We only make two assumptions: the inputs are non-degenerate and the network is over-parameterized. The latter…
University of Chicago（芝加哥大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Steklov Mathematical Institute（斯捷克洛夫数学研究所）
We continue a long line of research aimed at proving convergence of depth 2 neural networks, trained via gradient descent, to a global minimum. Like in many previous works, our model has the following features: regression with quadratic loss function, fully connected feedforward architecture, RelU activations, Gaussian data instances and network initialization, adversarial labels. It is more general in the sense that we allow both layers to be trained simultaneously and at {\em different} rates. Our results improve on state-of-the-art [Oymak Soltanolkotabi 20] (training the first layer only) a…
In this note, we study how neural networks with a single hidden layer and ReLU activation interpolate data drawn from a radially symmetric distribution with target labels 1 at the origin and 0 outside the unit ball, if no labels are known inside the unit ball. With weight decay regularization and in the infinite neuron, infinite data limit, we prove that a unique radially symmetric minimizer exists, whose weight decay regularizer and Lipschitz constant grow as $d$ and $\sqrt{d}$ respectively. We furthermore show that the weight decay regularizer grows exponentially in $d$ if the label $1$ is i…
In this paper we focus on the problem of finding the optimal weights of the shallowest of neural networks consisting of a single Rectified Linear Unit (ReLU). These functions are of the form $\mathbf{x}\rightarrow \max(0,\langle\mathbf{w},\mathbf{x}\rangle)$ with $\mathbf{w}\in\mathbb{R}^d$ denoting the weight vector. We focus on a planted model where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to a planted weight vector. We first show that mini-batch stochastic gradient descent when suitably initialized, converges at a geometric rate to the…
We prove a precise geometric description of all one layer ReLU networks $z(x;θ)$ with a single linear unit and input/output dimensions equal to one that interpolate a given dataset $\mathcal D=\{(x_i,f(x_i))\}$ and, among all such interpolants, minimize the $\ell_2$-norm of the neuron weights. Such networks can intuitively be thought of as those that minimize the mean-squared error over $\mathcal D$ plus an infinitesimal weight decay penalty. We therefore refer to them as ridgeless ReLU interpolants. Our description proves that, to extrapolate values $z(x;θ)$ for inputs $x\in (x_i,x_{i+1})$ ly…
Institute of Informatics, University of Warsaw（华沙大学信息学研究所）
We study the overparametrization bounds required for the global convergence of stochastic gradient descent algorithm for a class of one hidden layer feed-forward neural networks, considering most of the activation functions used in practice, including ReLU. We improve the existing state-of-the-art results in terms of the required hidden layer width. We introduce a new proof technique combining nonlinear analysis with properties of random initializations of the network. First, we establish the global convergence of continuous solutions of the differential inclusion being a nonsmooth analogue of…
Incorporating a so-called "momentum" dynamic in gradient descent methods is widely used in neural net training as it has been broadly observed that, at least empirically, it often leads to significantly faster convergence. At the same time, there are very few theoretical guarantees in the literature to explain this apparent acceleration effect. Even for the classical strongly convex quadratic problems, several existing results only show Polyak's momentum has an accelerated linear rate asymptotically. In this paper, we first revisit the quadratic problems and show a non-asymptotic accelerated l…
UCLA（加州大学洛杉矶分校）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Center for Data Science, New York University（纽约大学数据科学中心）；Institute for Advanced Study（普林斯顿高等研究院）
The analysis of neural network training beyond their linearization regime remains an outstanding open question, even in the simplest setup of a single hidden-layer. The limit of infinitely wide networks provides an appealing route forward through the mean-field perspective, but a key challenge is to bring learning guarantees back to the finite-neuron setting, where practical algorithms operate. Towards closing this gap, and focusing on shallow neural networks, in this work we study the ability of different regularisation strategies to capture solutions requiring only a finite amount of neurons…
Baylor College of Medicine（贝勒医学院）；Rice University（莱斯大学）
Understanding the learning dynamics and inductive bias of neural networks (NNs) is hindered by the opacity of the relationship between NN parameters and the function represented. We propose reparametrizing ReLU NNs as continuous piecewise linear splines. Using this spline lens, we study learning dynamics in shallow univariate ReLU NNs, finding unexpected insights and explanations for several perplexing phenomena. We develop a surprisingly simple and transparent view of the structure of the loss surface, including its critical and fixed points, Hessian, and Hessian spectrum. We also show that s…
Texas A&M University（得克萨斯农工大学）；University of Toronto（多伦多大学）
We prove the precise scaling, at finite depth and width, for the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network. The standard deviation is exponential in the ratio of network depth to width. Thus, even in the limit of infinite overparameterization, the NTK is not deterministic if depth and width simultaneously tend to infinity. Moreover, we prove that for such deep and wide networks, the NTK has a non-trivial evolution during training by showing that the mean of its first SGD update is also exponential in the ratio of network depth to width. This is…
We consider the optimization problem associated with fitting two-layers ReLU networks with respect to the squared loss, where labels are generated by a target network. We leverage the rich symmetry structure to analytically characterize the Hessian at various families of spurious minima in the natural regime where the number of inputs $d$ and the number of hidden neurons $k$ is finite. In particular, we prove that for $d\ge k$ standard Gaussian inputs: (a) of the $dk$ eigenvalues of the Hessian, $dk - O(d)$ concentrate near zero, (b) $Ω(d)$ of the eigenvalues grow linearly with $k$. Although t…
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）；University of Münster（明斯特大学）
In this article we study fully-connected feedforward deep ReLU ANNs with an arbitrarily large number of hidden layers and we prove convergence of the risk of the GD optimization method with random initializations in the training of such ANNs under the assumption that the unnormalized probability density function of the probability distribution of the input data of the considered supervised learning problem is piecewise polynomial, under the assumption that the target function (describing the relationship between input data and the output data) is piecewise polynomial, and under the assumption …
PAG Investment Solutions（PAG投资解决方案公司）；Hunan University（湖南大学）；Xiaoxiang Research Institute of Big Data（潇湘大数据研究院）
In this paper, we prove that a shallow neural network with a monotone sigmoid, ReLU, ELU, Softplus, or LeakyReLU activation function can arbitrarily well approximate any L^p(p>=2) integrable functions defined on R*[0,1]^n. We also prove that a shallow neural network with a sigmoid, ReLU, ELU, Softplus, or LeakyReLU activation function expresses no nonzero integrable function defined on the Euclidean plane. Together with a recent result that the deep ReLU network can arbitrarily well approximate any integrable function on Euclidean spaces, we provide a new perspective on the advantage of multip…
We consider the optimization problem associated with training simple ReLU neural networks of the form $\mathbf{x}\mapsto \sum_{i=1}^{k}\max\{0,\mathbf{w}_i^\top \mathbf{x}\}$ with respect to the squared loss. We provide a computer-assisted proof that even if the input distribution is standard Gaussian, even if the dimension is arbitrarily large, and even if the target values are generated by such a network, with orthonormal parameter vectors, the problem can still have spurious local minima once $6\le k\le 20$. By a concentration of measure argument, this implies that in high input dimensions,…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）；Institute for Information Transmission Problems（信息传输问题研究所）
We study expressive power of shallow and deep neural networks with piece-wise linear activation functions. We establish new rigorous upper and lower bounds for the network complexity in the setting of approximations in Sobolev spaces. In particular, we prove that deep ReLU networks more efficiently approximate smooth functions than shallow networks. In the case of approximations of 1D Lipschitz functions we describe adaptive depth-6 network architectures more efficient than the standard shallow architecture.
Shallow ReLU$^s$ Networks in $L^p$-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization
School of Mathematical Sciences Fudan University（复旦大学数学学院）；School of Mathematical Sciences Institute of Natural Sciences and MOE-LSC Shanghai Jiao Tong University（上海交通大学数学学院、自然科学研究院和教育部低维量子体系科学重点实验室）；School of Mathematical Sciences Shanghai Key Laboratory for Contemporary Applied Mathematics Fudan University（复旦大学上海当代应用数学重点实验室数学学院）
