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 weights, and to constructing networks with width at most $d+2$, which is close to the minimal possible width due to previous lower bounds. Both of these constructions cause an extra polynomial factor in the number of parameters over the target network. We also show an exact representation of wide and shallow networks using deep and narrow networks which, in certain cases, does not increase the number of parameters over the target network.
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…
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…
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 …
Freie Universität Berlin（柏林自由大学）；London School of Economics and Political Science（伦敦政治经济学院）；University of Twente（特文特大学）
We prove that the set of functions representable by ReLU neural networks with integer weights strictly increases with the network depth while allowing arbitrary width. More precisely, we show that $\lceil\log_2(n)\rceil$ hidden layers are indeed necessary to compute the maximum of $n$ numbers, matching known upper bounds. Our results are based on the known duality between neural networks and Newton polytopes via tropical geometry. The integrality assumption implies that these Newton polytopes are lattice polytopes. Then, our depth lower bounds follow from a parity argument on the normalized vo…
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, …
On the Depth of Monotone ReLU Neural Networks and ICNNs
Department of Computer Science, the University of Copenhagen（计算机科学系，哥本哈根大学）；University of Technology Nuremberg（纽伦堡技术大学）；Worcester Polytechnic Institute（沃思堡理工学院）；Department of Mathematics, Technion-IIT（数学系，技术学院-以色列理工学院）
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…
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…
Google Research（谷歌研究院）；Northwestern University（西北大学）
We present polynomial time and sample efficient algorithms for learning an unknown depth-2 feedforward neural network with general ReLU activations, under mild non-degeneracy assumptions. In particular, we consider learning an unknown network of the form $f(x) = {a}^{\mathsf{T}}σ({W}^\mathsf{T}x+b)$, where $x$ is drawn from the Gaussian distribution, and $σ(t) := \max(t,0)$ is the ReLU activation. Prior works for learning networks with ReLU activations assume that the bias $b$ is zero. In order to deal with the presence of the bias terms, our proposed algorithm consists of robustly decomposing…
For any positive integer $k$, there exist neural networks with $Θ(k^3)$ layers, $Θ(1)$ nodes per layer, and $Θ(1)$ distinct parameters which can not be approximated by networks with $\mathcal{O}(k)$ layers unless they are exponentially large --- they must possess $Ω(2^k)$ 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 maxi…
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 …
We prove sharp dimension-free representation results for neural networks with $D$ ReLU layers under square loss for a class of functions $\mathcal{G}_D$ defined in the paper. These results capture the precise benefits of depth in the following sense: 1. The rates for representing the class of functions $\mathcal{G}_D$ via $D$ ReLU layers is sharp up to constants, as shown by matching lower bounds. 2. For each $D$, $\mathcal{G}_{D} \subseteq \mathcal{G}_{D+1}$ and as $D$ grows the class of functions $\mathcal{G}_{D}$ contains progressively less smooth functions. 3. If $D^{\prime} < D$, then the…
Stellenbosch University（Stellenbosch University（斯泰伦博斯大学））；University of the Witwatersrand（University of the Witwatersrand（金山大学））；Council for Scientific and Industrial Research（Council for Scientific and Industrial Research（科学与工业研究理事会））
Recent work in signal propagation theory has shown that dropout limits the depth to which information can propagate through a neural network. In this paper, we investigate the effect of initialisation on training speed and generalisation for ReLU networks within this depth limit. We ask the following research question: given that critical initialisation is crucial for training at large depth, if dropout limits the depth at which networks are trainable, does initialising critically still matter? We conduct a large-scale controlled experiment, and perform a statistical analysis of over $12000$ t…
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…
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…
State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences（中国科学院沈阳自动化研究所机器人学国家重点实验室）
Deep learning is recognized to be capable of discovering deep features for representation learning and pattern recognition without requiring elegant feature engineering techniques by taking advantage of human ingenuity and prior knowledge. Thus it has triggered enormous research activities in machine learning and pattern recognition. One of the most important challenge of deep learning is to figure out relations between a feature and the depth of deep neural networks (deep nets for short) to reflect the necessity of depth. Our purpose is to quantify this feature-depth correspondence in feature…
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 …
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）
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…
Motivated by the resurgence of neural networks in being able to solve complex learning tasks we undertake a study of high depth networks using ReLU gates which implement the function $x \mapsto \max\{0,x\}$. We try to understand the role of depth in such neural networks by showing size lowerbounds against such network architectures in parameter regimes hitherto unexplored. In particular we show the following two main results about neural nets computing Boolean functions of input dimension $n$, 1. We use the method of random restrictions to show almost linear, $Ω(ε^{2(1-δ)}n^{1-δ})$, lower boun…
Microsoft Research NYC（微软研究院纽约）；UT Austin（得克萨斯大学奥斯汀分校）；Google Research（谷歌研究院）；UC Berkeley（加州大学伯克利分校）；WPI（伍斯特理工学院）
We prove several hardness results for training depth-2 neural networks with the ReLU activation function; these networks are simply weighted sums (that may include negative coefficients) of ReLUs. Our goal is to output a depth-2 neural network that minimizes the square loss with respect to a given training set. We prove that this problem is NP-hard already for a network with a single ReLU. We also prove NP-hardness for outputting a weighted sum of $k$ ReLUs minimizing the squared error (for $k>1$) even in the realizable setting (i.e., when the labels are consistent with an unknown depth-2 ReLU…
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
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 study the approximation properties of random ReLU features through their reproducing kernel Hilbert space (RKHS). We first prove a universality theorem for the RKHS induced by random features whose feature maps are of the form of nodes in neural networks. The universality result implies that the random ReLU features method is a universally consistent learning algorithm. We prove that despite the universality of the RKHS induced by the random ReLU features, composition of functions in it generates substantially more complicated functions that are harder to approximate than those functions si…
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…
Towards Lower Bounds on the Depth of ReLU Neural Networks
London School of Economics and Political Science（伦敦政治经济学院）；Johns Hopkins University（约翰斯·霍普金斯大学）；Università degli Studi di Padova（帕多瓦大学）；Technische Universität Berlin（柏林工业大学）
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…
The Hebrew University（希伯来大学）；Google Research（谷歌研究院）
We consider the natural problem of learning a ReLU network from queries, which was recently remotivated by model extraction attacks. In this work, we present a polynomial-time algorithm that can learn a depth-two ReLU network from queries under mild general position assumptions. We also present a polynomial-time algorithm that, under mild general position assumptions, can learn a rich class of depth-three ReLU networks from queries. For instance, it can learn most networks where the number of first layer neurons is smaller than the dimension and the number of second layer neurons. These two re…
IBM Research AI（IBM研究院人工智能）；New York University（纽约大学）
A well-known line of work (Barron, 1993; Breiman, 1993; Klusowski & Barron, 2018) provides bounds on the width $n$ of a ReLU two-layer neural network needed to approximate a function $f$ over the ball $\mathcal{B}_R(\mathbb{R}^d)$ up to error $ε$, when the Fourier based quantity $C_f = \frac{1}{(2π)^{d/2}} \int_{\mathbb{R}^d} \|ξ\|^2 |\hat{f}(ξ)| \ dξ$ is finite. More recently Ongie et al. (2019) used the Radon transform as a tool for analysis of infinite-width ReLU two-layer networks. In particular, they introduce the concept of Radon-based $\mathcal{R}$-norms and show that a function defined…
Bar Ilan University（巴伊兰大学）；Allen Institute for Artificial Intelligence（艾伦人工智能研究所）
We present a simple proof for the benefit of depth in multi-layer feedforward network with rectified activation ("depth separation"). Specifically we present a sequence of classification problems indexed by $m$ such that (a) for any fixed depth rectified network there exist an $m$ above which classifying problem $m$ correctly requires exponential number of parameters (in $m$); and (b) for any problem in the sequence, we present a concrete neural network with linear depth (in $m$) and small constant width ($\leq 4$) that classifies the problem with zero error. The constructive proof is based on…
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…
This paper considers the following question: how well can depth-two ReLU networks with randomly initialized bottom-level weights represent smooth functions? We give near-matching upper- and lower-bounds for $L_2$-approximation in terms of the Lipschitz constant, the desired accuracy, and the dimension of the problem, as well as similar results in terms of Sobolev norms. Our positive results employ tools from harmonic analysis and ridgelet representation theory, while our lower-bounds are based on (robust versions of) dimensionality arguments.
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.
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…
Princeton University（普林斯顿大学）；University of Pennsylvania（宾夕法尼亚大学）；McGill University（麦吉尔大学）；Mila（米拉人工智能研究所）
Assessing the complexity of functions computed by a neural network helps us understand how the network will learn and generalize. One natural measure of complexity is how the network distorts length - if the network takes a unit-length curve as input, what is the length of the resulting curve of outputs? It has been widely believed that this length grows exponentially in network depth. We prove that in fact this is not the case: the expected length distortion does not grow with depth, and indeed shrinks slightly, for ReLU networks with standard random initialization. We also generalize this re…
Singapore University of Technology and Design（新加坡科技设计大学）；Stanford University（斯坦福大学）；SUTD（新加坡科技设计大学）
Understanding the representational power of Deep Neural Networks (DNNs) and how their structural properties (e.g., depth, width, type of activation unit) affect the functions they can compute, has been an important yet challenging question in deep learning and approximation theory. In a seminal paper, Telgarsky highlighted the benefits of depth by presenting a family of functions (based on simple triangular waves) for which DNNs achieve zero classification error, whereas shallow networks with fewer than exponentially many nodes incur constant error. Even though Telgarsky's work reveals the lim…
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…
Katholische Universität Eichstätt–Ingolstadt（艾希施泰特-因戈尔施塔特天主教大学）；University of Oxford（牛津大学）
We discuss the expressive power of neural networks which use the non-smooth ReLU activation function $\varrho(x) = \max\{0,x\}$ by analyzing the approximation theoretic properties of such networks. The existing results mainly fall into two categories: approximation using ReLU networks with a fixed depth, or using ReLU networks whose depth increases with the approximation accuracy. After reviewing these findings, we show that the results concerning networks with fixed depth--- which up to now only consider approximation in $L^p(λ)$ for the Lebesgue measure $λ$--- can be generalized to approxima…
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…
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 study the memorization power of feedforward ReLU neural networks. We show that such networks can memorize any $N$ points that satisfy a mild separability assumption using $\tilde{O}\left(\sqrt{N}\right)$ parameters. Known VC-dimension upper bounds imply that memorizing $N$ samples requires $Ω(\sqrt{N})$ parameters, and hence our construction is optimal up to logarithmic factors. We also give a generalized construction for networks with depth bounded by $1 \leq L \leq \sqrt{N}$, for memorizing $N$ samples using $\tilde{O}(N/L)$ parameters. This bound is also optimal up to logarithmic factors…
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…
National University of Singapore（新加坡国立大学）；Purdue University（普渡大学）
This paper establishes the (nearly) optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms of both width and depth simultaneously. To that end, we first prove that multivariate polynomials can be approximated by deep ReLU networks of width $\mathcal{O}(N)$ and depth $\mathcal{O}(L)$ with an approximation error $\mathcal{O}(N^{-L})$. Through local Taylor expansions and their deep ReLU network approximations, we show that deep ReLU networks of width $\mathcal{O}(N\ln N)$ and depth $\mathcal{O}(L\ln L)$ can approximate $f\in C^s([0…
