We consider the fundamental problem of ReLU regression, where the goal is to output the best fitting ReLU with respect to square loss given access to draws from some unknown distribution. We give the first efficient, constant-factor approximation algorithm for this problem assuming the underlying distribution satisfies some weak concentration and anti-concentration conditions (and includes, for example, all log-concave distributions). This solves the main open problem of Goel et al., who proved hardness results for any exact algorithm for ReLU regression (up to an additive $ε$). Using more sophisticated techniques, we can improve our results and obtain a polynomial-time approximation scheme for any subgaussian distribution. Given the aforementioned hardness results, these guarantees can not be substantially improved.
Our main insight is a new characterization of surrogate losses for nonconvex activations. While prior work had established the existence of convex surrogates for monotone activations, we show that properties of the underlying distribution actually induce strong convexity for the loss, allowing us to relate the global minimum to the activation's Chow parameters.
We study the fundamental problem of ReLU regression, where the goal is to fit Rectified Linear Units (ReLUs) to data. This supervised learning task is efficiently solvable in the realizable setting, but is known to be computationally hard with adversarial label noise. In this work, we focus on ReLU regression in the Massart noise model, a natural and well-studied semi-random noise model. In this model, the label of every point is generated according to a function in the class, but an adversary is allowed to change this value arbitrarily with some probability, which is {\em at most} $η< 1/2$. W…
Real world data often exhibit low-dimensional geometric structures, and can be viewed as samples near a low-dimensional manifold. This paper studies nonparametric regression of Hölder functions on low-dimensional manifolds using deep ReLU networks. Suppose $n$ training data are sampled from a Hölder function in $\mathcal{H}^{s,α}$ supported on a $d$-dimensional Riemannian manifold isometrically embedded in $\mathbb{R}^D$, with sub-gaussian noise. A deep ReLU network architecture is designed to estimate the underlying function from the training data. The mean squared error of the empirical esti…
Institute of Environmental and Agricultural Biology (X-BIO), University of Tyumen（秋明大学生态与农业生物学研究所（X-BIO））；University of Mississippi（密西西比大学）
We consider regression estimation with modified ReLU neural networks in which network weight matrices are first modified by a function $α$ before being multiplied by input vectors. We give an example of continuous, piecewise linear function $α$ for which the empirical risk minimizers over the classes of modified ReLU networks with $l_1$ and squared $l_2$ penalties attain, up to a logarithmic factor, the minimax rate of prediction of unknown $β$-smooth function.
We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples $(\mathbf{x},y)$ from an unknown distribution on $\mathbb{R}^n \times \{ \pm 1\}$, whose marginal distribution on $\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\mathrm{OPT}+ε$, where $\mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors…
Consider the multivariate nonparametric regression model. It is shown that estimators based on sparsely connected deep neural networks with ReLU activation function and properly chosen network architecture achieve the minimax rates of convergence (up to $\log n$-factors) under a general composition assumption on the regression function. The framework includes many well-studied structural constraints such as (generalized) additive models. While there is a lot of flexibility in the network architecture, the tuning parameter is the sparsity of the network. Specifically, we consider large networks…
Faculty of Information Technology, Beijing University of Technology（北京工业大学信息学部）
We consider in this paper the optimal approximations of convex univariate functions with feed-forward Relu neural networks. We are interested in the following question: what is the minimal approximation error given the number of approximating linear pieces? We establish the necessary and sufficient conditions and uniqueness of optimal approximations, and give lower and upper bounds of the optimal approximation errors. Relu neural network architectures are then presented to generate these optimal approximations. Finally, we propose an algorithm to find the optimal approximations, as well as pro…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Peking University（北京大学）；The Pennsylvania State University（宾夕法尼亚州立大学）
This paper focuses on establishing $L^2$ approximation properties for deep ReLU convolutional neural networks (CNNs) in two-dimensional space. The analysis is based on a decomposition theorem for convolutional kernels with a large spatial size and multi-channels. Given the decomposition result, the property of the ReLU activation function, and a specific structure for channels, a universal approximation theorem of deep ReLU CNNs with classic structure is obtained by showing its connection with one-hidden-layer ReLU neural networks (NNs). Furthermore, approximation properties are obtained for o…
ETH Z\"urich, D-Math, R\"amistrasse 101, CH-8092 Z\"urich, Switzerland
Randomized neural networks (randomized NNs), where only the terminal layer's weights are optimized constitute a powerful model class to reduce computational time in training the neural network model. At the same time, these models generalize surprisingly well in various regression and classification tasks. In this paper, we give an exact macroscopic characterization (i.e., a characterization in function space) of the generalization behavior of randomized, shallow NNs with ReLU activation (RSNs). We show that RSNs correspond to a generalized additive model (GAM)-typed regression in which infini…
Finite-Sample Analysis of Learning High-Dimensional Single ReLU Neuron
Johns Hopkins University（约翰斯·霍普金斯大学）；The University of Hong Kong（香港大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Rice University（莱斯大学）；Harvard University（哈佛大学）
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…
University of Science and Technology of China（中国科学技术大学）；State University of New York at Buffalo（纽约州立大学布法罗分校）；Guangzhou University（广州大学）；Zhejiang University（浙江大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
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…
Transfer Learning in Nonparametric Regression with Deep ReLU Networks
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Washington University in St. Louis（圣路易斯华盛顿大学）；Hong Kong University of Science and Technology（香港科技大学）
Technische Universität Berlin（柏林工业大学）；University of Oxford（牛津大学）
We analyze approximation rates of deep ReLU neural networks for Sobolev-regular functions with respect to weaker Sobolev norms. First, we construct, based on a calculus of ReLU networks, artificial neural networks with ReLU activation functions that achieve certain approximation rates. Second, we establish lower bounds for the approximation by ReLU neural networks for classes of Sobolev-regular functions. Our results extend recent advances in the approximation theory of ReLU networks to the regime that is most relevant for applications in the numerical analysis of partial differential equation…
Whereas recovery of the manifold from data is a well-studied topic, approximation rates for functions defined on manifolds are less known. In this work, we study a regression problem with inputs on a $d^*$-dimensional manifold that is embedded into a space with potentially much larger ambient dimension. It is shown that sparsely connected deep ReLU networks can approximate a Hölder function with smoothness index $β$ up to error $ε$ using of the order of $ε^{-d^*/β}\log(1/ε)$ many non-zero network parameters. As an application, we derive statistical convergence rates for the estimator minimizin…
University of Tübingen（图宾根大学）；MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）
The point estimates of ReLU classification networks---arguably the most widely used neural network architecture---have been shown to yield arbitrarily high confidence far away from the training data. This architecture, in conjunction with a maximum a posteriori estimation scheme, is thus not calibrated nor robust. Approximate Bayesian inference has been empirically demonstrated to improve predictive uncertainty in neural networks, although the theoretical analysis of such Bayesian approximations is limited. We theoretically analyze approximate Gaussian distributions on the weights of ReLU netw…
National Institute of Standards and Technology (NIST)（美国国家标准与技术研究院）；École de Technologie Supérieure (ÉTS)（高等技术学院）
Among the several paradigms of artificial intelligence (AI) or machine learning (ML), a remarkably successful paradigm is deep learning. Deep learning's phenomenal success has been hoped to be interpreted via fundamental research on the theory of deep learning. Accordingly, applied research on deep learning has spurred the theory of deep learning-oriented depth and breadth of developments. Inspired by such developments, we pose these fundamental questions: can we accurately approximate an arbitrary matrix-vector product using deep rectified linear unit (ReLU) feedforward neural networks (FNNs)…
Center for Foundation Models and Generative AI, Northwestern University（基础模型与生成式人工智能中心，西北大学）；Department of Computer Science, Northwestern University（计算机科学系，西北大学）；Department of Physics, National Taiwan University（物理系，国立台湾大学）；Department of Statistics and Data Science, Northwestern University（统计与数据科学系，西北大学）
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…
Two-Dimensional Deep ReLU CNN Approximation for Korobov Functions: A Constructive Approach
Information and Engineering College, Dalian University（大连大学信息与工程学院）；School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（复旦大学数学学院及上海当代应用数学重点实验室）；School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学学院）；School of Mathematics and Statistics, University of Sydney（悉尼大学数学与统计学学院）
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…
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…
Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore（数学科学系，物理与数学科学学院，南洋理工大学，新加坡）；Faculty of Mathematics, Bielefeld University, Germany（数学学院，比勒菲尔德大学，德国）
York University（约克大学）；Department of Electrical Engineering and Computer Science（电气工程与计算机科学系）
In this paper, we have extended the well-established universal approximator theory to neural networks that use the unbounded ReLU activation function and a nonlinear softmax output layer. We have proved that a sufficiently large neural network using the ReLU activation function can approximate any function in $L^1$ up to any arbitrary precision. Moreover, our theoretical results have shown that a large enough neural network using a nonlinear softmax output layer can also approximate any indicator function in $L^1$, which is equivalent to mutually-exclusive class labels in any realistic multipl…
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…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Peking University（北京大学）；The Pennsylvania State University（宾夕法尼亚州立大学）
We study ReLU deep neural networks (DNNs) by investigating their connections with the hierarchical basis method in finite element methods. First, we show that the approximation schemes of ReLU DNNs for $x^2$ and $xy$ are composition versions of the hierarchical basis approximation for these two functions. Based on this fact, we obtain a geometric interpretation and systematic proof for the approximation result of ReLU DNNs for polynomials, which plays an important role in a series of recent exponential approximation results of ReLU DNNs. Through our investigation of connections between ReLU DN…
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.
We consider the problem of computing the best-fitting ReLU with respect to square-loss on a training set when the examples have been drawn according to a spherical Gaussian distribution (the labels can be arbitrary). Let $\mathsf{opt} < 1$ be the population loss of the best-fitting ReLU. We prove: 1. Finding a ReLU with square-loss $\mathsf{opt} + ε$ is as hard as the problem of learning sparse parities with noise, widely thought to be computationally intractable. This is the first hardness result for learning a ReLU with respect to Gaussian marginals, and our results imply -{\emph uncondition…
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 develop Banach spaces for ReLU neural networks of finite depth $L$ and infinite width. The spaces contain all finite fully connected $L$-layer networks and their $L^2$-limiting objects under bounds on the natural path-norm. Under this norm, the unit ball in the space for $L$-layer networks has low Rademacher complexity and thus favorable generalization properties. Functions in these spaces can be approximated by multi-layer neural networks with dimension-independent convergence rates. The key to this work is a new way of representing functions in some form of expectations, motivated by mult…
