How neural network behaves during the training over different choices of hyperparameters is an important question in the study of neural networks. In this work, inspired by the phase diagram in statistical mechanics, we draw the phase diagram for the two-layer ReLU neural network at the infinite-width limit for a complete characterization of its dynamical regimes and their dependence on hyperparameters related to initialization. Through both experimental and theoretical approaches, we identify three regimes in the phase diagram, i.e., linear regime, critical regime and condensed regime, based on the relative change of input weights as the width approaches infinity, which tends to $0$, $O(1)$ and $+\infty$, respectively. In the linear regime, NN training dynamics is approximately linear similar to a random feature model with an exponential loss decay. In the condensed regime, we demonstrate through experiments that active neurons are condensed at several discrete orientations. The critical regime serves as the boundary between above two regimes, which exhibits an intermediate nonlinear behavior with the mean-field model as a typical example. Overall, our phase diagram for the two-layer ReLU NN serves as a map for the future studies and is a first step towards a more systematical investigation of the training behavior and the implicit regularization of NNs of different structures.
Despite their prevalence in neural networks we still lack a thorough theoretical characterization of ReLU layers. This paper aims to further our understanding of ReLU layers by studying how the activation function ReLU interacts with the linear component of the layer and what role this interaction plays in the success of the neural network in achieving its intended task. To this end, we introduce two new tools: ReLU singular values of operators and the Gaussian mean width of operators. By presenting on the one hand theoretical justifications, results, and interpretations of these two concepts …
Minimum Block Width for Universal Approximation by Residual Neural Networks with Inner Width One
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology（湖北省工程建模与科学计算重点实验室）
National Institute of Advanced Industrial Science and Technology（产业技术综合研究所）
In this paper, we theoretically prove that the deep ReLU neural networks do not lie in spurious local minima in the loss landscape under the Neural Tangent Kernel (NTK) regime, that is, in the gradient descent training dynamics of the deep ReLU neural networks whose parameters are initialized by a normal distribution in the limit as the widths of the hidden layers tend to infinity.
University of Torino（都灵大学）；Collegio Carlo Alberto（卡洛·阿尔贝托学院）；Bocconi University（博科尼大学）；Cogent Labs（Cogent实验室）
There is a recent and growing literature on large-width asymptotic properties of Gaussian neural networks (NNs), namely NNs whose weights are initialized as Gaussian distributions. Two popular problems are: i) the study of the large-width distributions of NNs, which characterizes the infinitely wide limit of a rescaled NN in terms of a Gaussian stochastic process; ii) the study of the large-width training dynamics of NNs, which characterizes the infinitely wide dynamics in terms of a deterministic kernel, referred to as the neural tangent kernel (NTK), and shows that, for a sufficiently large …
The minimal width of universal $p$-adic ReLU neural networks
Mathematical Institute, E\"otv\"os Lor\'；University 1117 Budapest, P\'azm\'any P\'eter s\'et\'any 1/C, Hungary；Department of Algebra；Geometry, Institute of Mathemetics, Budapest University of Technology；4mm HUN-REN Alfr\'ed R\'enyi Institute of Mathematics, Re\'altanoda utca 13--15., H-1053 Budapest, Hungary
The universal approximation property of width-bounded networks has been studied as a dual of classical universal approximation results on depth-bounded networks. However, the critical width enabling the universal approximation has not been exactly characterized in terms of the input dimension $d_x$ and the output dimension $d_y$. In this work, we provide the first definitive result in this direction for networks using the ReLU activation functions: The minimum width required for the universal approximation of the $L^p$ functions is exactly $\max\{d_x+1,d_y\}$. We also prove that the same concl…
Before training a neural net, a classic rule of thumb is to randomly initialize the weights so the variance of activations is preserved across layers. This is traditionally interpreted using the total variance due to randomness in both weights \emph{and} samples. Alternatively, one can interpret the rule of thumb as preservation of the variance over samples for a fixed network. The two interpretations differ little for a shallow net, but the difference is shown to grow with depth for a deep ReLU net by decomposing the total variance into the network-averaged sum of the sample variance and squa…
Peking University（北京大学）；Key Laboratory of Machine Perception, MOE, School of EECS, Peking University（北京大学信息科学技术学院机器感知与智能教育部重点实验室）；Center for Data Science, Peking University（北京大学数据科学研究中心）；Beijing Institute of Big Data Research（北京大数据研究院）
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 depth-bounded (e.g. depth-$2$) networks with suitable activation functions are universal approximators. We show a universal approximation theorem for 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 funct…
We present a framework to derive upper bounds on the number of regions that feed-forward neural networks with ReLU activation functions are affine linear on. It is based on an inductive analysis that keeps track of the number of such regions per dimensionality of their images within the layers. More precisely, the information about the number regions per dimensionality is pushed through the layers starting with one region of the input dimension of the neural network and using a recursion based on an analysis of how many regions per output dimensionality a subsequent layer with a certain width …
The universal approximation property (UAP) of neural networks is fundamental for deep learning, and it is well known that wide neural networks are universal approximators of continuous functions within both the $L^p$ norm and the continuous/uniform norm. However, the exact minimum width, $w_{\min}$, for the UAP has not been studied thoroughly. Recently, using a decoder-memorizer-encoder scheme, \citet{Park2021Minimum} found that $w_{\min} = \max(d_x+1,d_y)$ for both the $L^p$-UAP of ReLU networks and the $C$-UAP of ReLU+STEP networks, where $d_x,d_y$ are the input and output dimensions, respec…
In this paper, we construct neural networks with ReLU, sine and $2^x$ as activation functions. For general continuous $f$ defined on $[0,1]^d$ with continuity modulus $ω_f(\cdot)$, we construct ReLU-sine-$2^x$ networks that enjoy an approximation rate $\mathcal{O}(ω_f(\sqrt{d})\cdot2^{-M}+ω_{f}\left(\frac{\sqrt{d}}{N}\right))$, where $M,N\in \mathbb{N}^{+}$ denote the hyperparameters related to widths of the networks. As a consequence, we can construct ReLU-sine-$2^x$ network with the depth $5$ and width $\max\left\{\left\lceil2d^{3/2}\left(\frac{3μ}ε\right)^{1/α}\right\rceil,2\left\lceil\log_…
National University of Singapore（新加坡国立大学）；Purdue University（普渡大学）
A new network with super approximation power is introduced. This network is built with Floor ($\lfloor x\rfloor$) or ReLU ($\max\{0,x\}$) activation function in each neuron and hence we call such networks Floor-ReLU networks. For any hyper-parameters $N\in\mathbb{N}^+$ and $L\in\mathbb{N}^+$, it is shown that Floor-ReLU networks with width $\max\{d,\, 5N+13\}$ and depth $64dL+3$ can uniformly approximate a Hölder function $f$ on $[0,1]^d$ with an approximation error $3λd^{α/2}N^{-α\sqrt{L}}$, where $α\in(0,1]$ and $λ$ are the Hölder order and constant, respectively. More generally for an arbit…
Cornell University（康奈尔大学）；University of Connecticut（康涅狄格大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
The standard Universal Approximation Theorem for operator neural networks (NNs) holds for arbitrary width and bounded depth. Here, we prove that operator NNs of bounded width and arbitrary depth are universal approximators for continuous nonlinear operators. In our main result, we prove that for non-polynomial activation functions that are continuously differentiable at a point with a nonzero derivative, one can construct an operator NN of width five, whose inputs are real numbers with finite decimal representations, that is arbitrarily close to any given continuous nonlinear operator. We deri…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；DeepMind（深度思维）
Training very deep neural networks is still an extremely challenging task. The common solution is to use shortcut connections and normalization layers, which are both crucial ingredients in the popular ResNet architecture. However, there is strong evidence to suggest that ResNets behave more like ensembles of shallower networks than truly deep ones. Recently, it was shown that deep vanilla networks (i.e. networks without normalization layers or shortcut connections) can be trained as fast as ResNets by applying certain transformations to their activation functions. However, this method (called…
A recurrent neural network (RNN) is a widely used deep-learning network for dealing with sequential data. Imitating a dynamical system, an infinite-width RNN can approximate any open dynamical system in a compact domain. In general, deep networks with bounded widths are more effective than wide networks in practice; however, the universal approximation theorem for deep narrow structures has yet to be extensively studied. In this study, we prove the universality of deep narrow RNNs and show that the upper bound of the minimum width for universality can be independent of the length of the data. …
We show that finite-width deep ReLU neural networks yield rate-distortion optimal approximation (Bölcskei et al., 2018) of polynomials, windowed sinusoidal functions, one-dimensional oscillatory textures, and the Weierstrass function, a fractal function which is continuous but nowhere differentiable. Together with their recently established universal approximation property of affine function systems (Bölcskei et al., 2018), this shows that deep neural networks approximate vastly different signal structures generated by the affine group, the Weyl-Heisenberg group, or through warping, and even c…
National University of Singapore（新加坡国立大学）；Purdue University（普渡大学）
This paper concentrates on the approximation power of deep feed-forward neural networks in terms of width and depth. It is proved by construction that ReLU networks with width $\mathcal{O}\big(\max\{d\lfloor N^{1/d}\rfloor,\, N+2\}\big)$ and depth $\mathcal{O}(L)$ can approximate a Hölder continuous function on $[0,1]^d$ with an approximation rate $\mathcal{O}\big(λ\sqrt{d} (N^2L^2\ln N)^{-α/d}\big)$, where $α\in (0,1]$ and $λ>0$ are Hölder order and constant, respectively. Such a rate is optimal up to a constant in terms of width and depth separately, while existing results are only nearly op…
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, …
Department of Mathematics, Texas A\&M, College Station, United States
This article concerns the expressive power of depth in neural nets with ReLU activations and bounded width. We are particularly interested in the following questions: what is the minimal width $w_{\text{min}}(d)$ so that ReLU nets of width $w_{\text{min}}(d)$ (and arbitrary depth) can approximate any continuous function on the unit cube $[0,1]^d$ aribitrarily well? For ReLU nets near this minimal width, what can one say about the depth necessary to approximate a given function? Our approach to this paper is based on the observation that, due to the convexity of the ReLU activation, ReLU nets a…
We focus on estimating \emph{a priori} generalization error of two-layer ReLU neural networks (NNs) trained by mean squared error, which only depends on initial parameters and the target function, through the following research line. We first estimate \emph{a priori} generalization error of finite-width two-layer ReLU NN with constraint of minimal norm solution, which is proved by \cite{zhang2019type} to be an equivalent solution of a linearized (w.r.t. parameter) finite-width two-layer NN. As the width goes to infinity, the linearized NN converges to the NN in Neural Tangent Kernel (NTK) regi…
We consider the approximation rates of shallow neural networks with respect to the variation norm. Upper bounds on these rates have been established for sigmoidal and ReLU activation functions, but it has remained an important open problem whether these rates are sharp. In this article, we provide a solution to this problem by proving sharp lower bounds on the approximation rates for shallow neural networks, which are obtained by lower bounding the $L^2$-metric entropy of the convex hull of the neural network basis functions. In addition, our methods also give sharp lower bounds on the Kolmogo…
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…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）
We explore the phase diagram of approximation rates for deep neural networks and prove several new theoretical results. In particular, we generalize the existing result on the existence of deep discontinuous phase in ReLU networks to functional classes of arbitrary positive smoothness, and identify the boundary between the feasible and infeasible rates. Moreover, we show that all networks with a piecewise polynomial activation function have the same phase diagram. Next, we demonstrate that standard fully-connected architectures with a fixed width independent of smoothness can adapt to smoothne…
Better NTK Conditioning: A Free Lunch from (ReLU) Nonlinear Activation in Wide Neural Networks
Elmore Family School of Electrical and Computer Engineering（埃尔莫尔电气与计算机工程学院）；Purdue University（普渡大学）；Department of Physics and Astronomy（物理与天文学系）；Halıcıoğlu Data Science Institute（哈利奇奥格卢数据科学研究所）；University of California San Diego（加州大学圣地亚哥分校）；Rosen Center For Advanced Computing（罗森先进计算中心）
We consider a deep ReLU / Leaky ReLU student network trained from the output of a fixed teacher network of the same depth, with Stochastic Gradient Descent (SGD). The student network is \emph{over-realized}: at each layer $l$, the number $n_l$ of student nodes is more than that ($m_l$) of teacher. Under mild conditions on dataset and teacher network, we prove that when the gradient is small at every data sample, each teacher node is \emph{specialized} by at least one student node \emph{at the lowest layer}. For two-layer network, such specialization can be achieved by training on any dataset o…
Department of Mathematics, Texas A&M（德克萨斯农工大学数学系）；Trinity College, Cambridge（剑桥大学三一学院）
This article concerns the expressive power of depth in deep feed-forward neural nets with ReLU activations. Specifically, we answer the following question: for a fixed $d_{in}\geq 1,$ what is the minimal width $w$ so that neural nets with ReLU activations, input dimension $d_{in}$, hidden layer widths at most $w,$ and arbitrary depth can approximate any continuous, real-valued function of $d_{in}$ variables arbitrarily well? It turns out that this minimal width is exactly equal to $d_{in}+1.$ That is, if all the hidden layer widths are bounded by $d_{in}$, then even in the infinite depth limit…
University of Chicago（芝加哥大学）；Technion, Israel Institute of Technology（以色列理工学院）；Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
A key element of understanding the efficacy of overparameterized neural networks is characterizing how they represent functions as the number of weights in the network approaches infinity. In this paper, we characterize the norm required to realize a function $f:\mathbb{R}^d\rightarrow\mathbb{R}$ as a single hidden-layer ReLU network with an unbounded number of units (infinite width), but where the Euclidean norm of the weights is bounded, including precisely characterizing which functions can be realized with finite norm. This was settled for univariate univariate functions in Savarese et al.…
We solve an open question from Lu et al. (2017), by showing that any target network with inputs in $\mathbb{R}^d$ can be approximated by a width $O(d)$ network (independent of the target network's architecture), whose number of parameters is essentially larger only by a linear factor. In light of previous depth separation theorems, which imply that a similar result cannot hold when the roles of width and depth are interchanged, it follows that depth plays a more significant role than width in the expressive power of neural networks. We extend our results to constructing networks with bounded w…
MPI-MIS（马克斯·普朗克数学研究所）；Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）；UCLA（加州大学洛杉矶分校）
A recent line of work has analyzed the theoretical properties of deep neural networks via the Neural Tangent Kernel (NTK). In particular, the smallest eigenvalue of the NTK has been related to the memorization capacity, the global convergence of gradient descent algorithms and the generalization of deep nets. However, existing results either provide bounds in the two-layer setting or assume that the spectrum of the NTK matrices is bounded away from 0 for multi-layer networks. In this paper, we provide tight bounds on the smallest eigenvalue of NTK matrices for deep ReLU nets, both in the limit…
Implicit deep learning has recently become popular in the machine learning community since these implicit models can achieve competitive performance with state-of-the-art deep networks while using significantly less memory and computational resources. However, our theoretical understanding of when and how first-order methods such as gradient descent (GD) converge on \textit{nonlinear} implicit networks is limited. Although this type of problem has been studied in standard feed-forward networks, the case of implicit models is still intriguing because implicit networks have \textit{infinitely} m…
UC Berkeley（加州大学伯克利分校）；Microsoft Research Redmond（微软研究院雷德蒙德分院）
We consider the phenomenon of adversarial examples in ReLU networks with independent gaussian parameters. For networks of constant depth and with a large range of widths (for instance, it suffices if the width of each layer is polynomial in that of any other layer), small perturbations of input vectors lead to large changes of outputs. This generalizes results of Daniely and Schacham (2020) for networks of rapidly decreasing width and of Bubeck et al (2021) for two-layer networks. The proof shows that adversarial examples arise in these networks because the functions that they compute are very…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；University of Guelph（圭尔夫大学）
Theoretical results show that neural networks can be approximated by Gaussian processes in the infinite-width limit. However, for fully connected networks, it has been previously shown that for any fixed network width, $n$, the Gaussian approximation gets worse as the network depth, $d$, increases. Given that modern networks are deep, this raises the question of how well modern architectures, like ResNets, are captured by the infinite-width limit. To provide a better approximation, we study ReLU ResNets in the infinite-depth-and-width limit, where both depth and width tend to infinity as their…
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…
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…
