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 approximation rate for the class $\mathcal{G}_D$ achieved by depth $D^{\prime}$ networks is strictly worse than that achieved by depth $D$ networks.
This constitutes a fine-grained characterization of the representation power of feedforward networks of arbitrary depth $D$ and number of neurons $N$, in contrast to existing representation results which either require $D$ growing quickly with $N$ or assume that the function being represented is highly smooth. In the latter case similar rates can be obtained with a single nonlinear layer. Our results confirm the prevailing hypothesis that deeper networks are better at representing less smooth functions, and indeed, the main technical novelty is to fully exploit the fact that deep networks can produce highly oscillatory functions with few activation functions.
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…
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 …
We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in $L^2$. As a model class, we consider the set $\mathcal{E}^β(\mathbb R^d)$ of possibly discontinuous piecewise $C^β$ functions $f : [-1/2, 1/2]^d \to \mathbb R$, where the different smooth regions of $f$ are separated by $C^β$ hypersurfaces. For dimension $d \geq 2$, regularity $β> 0$, and accuracy $\varepsilon > 0$, we construct artificial neural networks with ReLU activation function that approximate functions from $\m…
We demonstrate that a very deep ResNet with stacked modules with one neuron per hidden layer and ReLU activation functions can uniformly approximate any Lebesgue integrable function in $d$ dimensions, i.e. $\ell_1(\mathbb{R}^d)$. Because of the identity mapping inherent to ResNets, our network has alternating layers of dimension one and $d$. This stands in sharp contrast to fully connected networks, which are not universal approximators if their width is the input dimension $d$ [Lu et al, 2017; Hanin and Sellke, 2017]. Hence, our result implies an increase in representational power for narrow …
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Microsoft Research（微软研究院）
It was empirically confirmed by Keskar et al.\cite{SharpMinima} that flatter minima generalize better. However, for the popular ReLU network, sharp minimum can also generalize well \cite{SharpMinimacan}. The conclusion demonstrates that the existing definitions of flatness fail to account for the complex geometry of ReLU neural networks because they can't cover the Positively Scale-Invariant (PSI) property of ReLU network. In this paper, we formalize the PSI causes problem of existing definitions of flatness and propose a new description of flatness - \emph{PSI-flatness}. PSI-flatness is defin…
Michigan Technological University（密歇根理工大学）；Symbiosis Centre for Medical Image Analysis（共生医学图像分析中心）；Symbiosis Institute of Technology（共生技术学院）
This work introduces a novel activation unit that can be efficiently employed in deep neural nets (DNNs) and performs significantly better than the traditional Rectified Linear Units (ReLU). The function developed is a two parameter version of the specialized Richard's Curve and we call it Adaptive Richard's Curve weighted Activation (ARiA). This function is non-monotonous, analogous to the newly introduced Swish, however allows a precise control over its non-monotonous convexity by varying the hyper-parameters. We first demonstrate the mathematical significance of the two parameter ARiA follo…
Sharp description of local minima in the loss landscape of high-dimensional two-layer ReLU neural networks
Physics, Chalmers University of Technology；University of Gothenburg；Ecole Normale Superieure, PSL \& CNRS；Engineering, Chalmers University of Technology；School of Computer Science；Applied Mathematics, University of the Witwatersrand
Deciphering Two Training Clocks in Grokking via Deep Linear Network Theory with Conditional ReLU Reduction
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China（数学科学国家重点实验室，数学与系统科学研究院，中国科学院，北京100190，中国）；School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China（中国科学院大学数学科学学院，北京100049，中国）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS), Shanghai, China（上海数学与交叉科学研究所（SIMIS），上海，中国）；Key Laboratory of Systems Health Science of Zhejiang Province, School of Life Science, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Hangzhou 310024, China（浙江省系统健康科学重点实验室，生命科学学院，杭州先进研究院，中国科学院大学，中国科学院，杭州310024，中国）
In a function approximation with a neural network, an input dataset is mapped to an output index by optimizing the parameters of each hidden-layer unit. For a unary function, we present constraints on the parameters and its second derivative by constructing a continuum version of a one-hidden-layer neural network with the rectified linear unit (ReLU) activation function. The network is accurately implemented because the constraints decrease the degrees of freedom of the parameters. We also explain the existence of a smooth binary function that cannot be precisely represented by any such neural…
ETH Zurich（苏黎世联邦理工学院）；UIUC（伊利诺伊大学厄巴纳-香槟分校）；VMware Research（威睿研究部）
Formal verification of neural networks is critical for their safe adoption in real-world applications. However, designing a precise and scalable verifier which can handle different activation functions, realistic network architectures and relevant specifications remains an open and difficult challenge. In this paper, we take a major step forward in addressing this challenge and present a new verification framework, called PRIMA. PRIMA is both (i) general: it handles any non-linear activation function, and (ii) precise: it computes precise convex abstractions involving multiple neurons via nove…
Rice University（莱斯大学）；University of Helsinki（赫尔辛基大学）；University of Basel（巴塞尔大学）
Injectivity plays an important role in generative models where it enables inference; in inverse problems and compressed sensing with generative priors it is a precursor to well posedness. We establish sharp characterizations of injectivity of fully-connected and convolutional ReLU layers and networks. First, through a layerwise analysis, we show that an expansivity factor of two is necessary and sufficient for injectivity by constructing appropriate weight matrices. We show that global injectivity with iid Gaussian matrices, a commonly used tractable model, requires larger expansivity between …
Is a sample rich enough to determine, at least locally, the parameters of a neural network? To answer this question, we introduce a new local parameterization of a given deep ReLU neural network by fixing the values of some of its weights. This allows us to define local lifting operators whose inverses are charts of a smooth manifold of a high dimensional space. The function implemented by the deep ReLU neural network composes the local lifting with a linear operator which depends on the sample. We derive from this convenient representation a geometrical necessary and sufficient condition of l…
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…
Integral Signatures of Activation Functions: A 9-Dimensional Taxonomy and Stability Theory for Deep Learning
Bellini College of AI, Cybersecurity and Computing（人工智能、网络安全与计算学院）；University of South Florida（佛罗里达州立大学）；College of Computing and Informatics（计算与信息学院）；Drexel University（德雷塞尔大学）；Departments of Physics（物理系）
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…
University of Vienna（维也纳大学）；Austrian Academy of Sciences（奥地利科学院）；Catholic University of Eichstätt-Ingolstadt（艾希施泰特-因戈尔施塔特天主教大学）
Statistical learning theory provides bounds on the necessary number of training samples needed to reach a prescribed accuracy in a learning problem formulated over a given target class. This accuracy is typically measured in terms of a generalization error, that is, an expected value of a given loss function. However, for several applications -- for example in a security-critical context or for problems in the computational sciences -- accuracy in this sense is not sufficient. In such cases, one would like to have guarantees for high accuracy on every input value, that is, with respect to the …
Yale-NUS College（耶鲁-新加坡国立大学学院）；Institute of Information Science, Academia Sinica（中央研究院资讯科学研究所）
We consider deep feedforward neural networks with rectified linear units from a signal processing perspective. In this view, such representations mark the transition from using a single (data-driven) linear representation to utilizing a large collection of affine linear representations tailored to particular regions of the signal space. This paper provides a precise description of the individual affine linear representations and corresponding domain regions that the (data-driven) neural network associates to each signal of the input space. In particular, we describe atomic decompositions of th…
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…
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.…
Purdue University（普渡大学）；New Jersey Institute of Technology（新泽西理工学院）
Classifiers built with neural networks handle large-scale high dimensional data, such as facial images from computer vision, extremely well while traditional statistical methods often fail miserably. In this paper, we attempt to understand this empirical success in high dimensional classification by deriving the convergence rates of excess risk. In particular, a teacher-student framework is proposed that assumes the Bayes classifier to be expressed as ReLU neural networks. In this setup, we obtain a sharp rate of convergence, i.e., $\tilde{O}_d(n^{-2/3})$, for classifiers trained using either …
Optimal Initialization in Depth: Lyapunov Initialization and Limit Theorems for Deep Leaky ReLU Networks
School of Mathematics, Institute for Advanced Study（数学系，高级研究院）；Mathematical Institute, University of Oxford（牛津大学数学学院）；Max Planck Institute for Multidisciplinary Sciences（多学科科学研究所）；Department of Mathematics, University College London（伦敦大学学院数学系）
Injectivity of ReLU networks: perspectives from statistical physics
ETH Zürich（苏黎世联邦理工学院）；University of Basel（巴塞尔大学）；University of Illinois at Urbana–Champaign（伊利诺伊大学厄巴纳-香槟分校）；Meiji University（明治大学）；UniDistance Suisse（瑞士远程大学）
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…
In this paper, we study the compression of a target two-layer neural network with N nodes into a compressed network with M<N nodes. More precisely, we consider the setting in which the weights of the target network are i.i.d. sub-Gaussian, and we minimize the population L_2 loss between the outputs of the target and of the compressed network, under the assumption of Gaussian inputs. By using tools from high-dimensional probability, we show that this non-convex problem can be simplified when the target network is sufficiently over-parameterized, and provide the error rate of this approximation …
We examine one-hidden-layer neural networks with random weights. It is well-known that in the limit of infinitely many neurons they simplify to Gaussian processes. For networks with a polynomial activation, we demonstrate that the rate of this convergence in 2-Wasserstein metric is $O(n^{-\frac{1}{2}})$, where $n$ is the number of hidden neurons. We suspect this rate is asymptotically sharp. We improve the known convergence rate for other activations, to power-law in $n$ for ReLU and inverse-square-root up to logarithmic factors for erf. We explore the interplay between spherical harmonics, St…
