We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $\mathbb{R}^d$ with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/ε)^{O(k)}$, where $ε>0$ is the target accuracy. Prior work had given an algorithm for this problem with complexity $(dk/ε)^{h(k)}$, where the function $h(k)$ scales super-polynomially in $k$. Interestingly, the complexity of our algorithm is near-optimal within the class of Correlational Statistical Query algorithms. At a high-level, our algorithm uses tensor decomposition to identify a subspace such that all the $O(k)$-order moments are small in the orthogonal directions. Its analysis makes essential use of the theory of Schur polynomials to show that the higher-moment error tensors are small given that the lower-order ones are.
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加州大学圣地亚哥分校）
We study the problem of PAC learning one-hidden-layer ReLU networks with $k$ hidden units on $\mathbb{R}^d$ under Gaussian marginals in the presence of additive label noise. For the case of positive coefficients, we give the first polynomial-time algorithm for this learning problem for $k$ up to $\tilde{O}(\sqrt{\log d})$. Previously, no polynomial time algorithm was known, even for $k=3$. This answers an open question posed by~\cite{Kliv17}. Importantly, our algorithm does not require any assumptions about the rank of the weight matrix and its complexity is independent of its condition number…
We study the problem of learning one-hidden-layer neural networks with Rectified Linear Unit (ReLU) activation function, where the inputs are sampled from standard Gaussian distribution and the outputs are generated from a noisy teacher network. We analyze the performance of gradient descent for training such kind of neural networks based on empirical risk minimization, and provide algorithm-dependent guarantees. In particular, we prove that tensor initialization followed by gradient descent can converge to the ground-truth parameters at a linear rate up to some statistical error. To the best …
The existence of local minima for one-hidden-layer ReLU networks has been investigated theoretically in [8]. Based on the theory, in this paper, we first analyze how big the probability of existing local minima is for 1D Gaussian data and how it varies in the whole weight space. We show that this probability is very low in most regions. We then design and implement a linear programming based approach to judge the existence of genuine local minima, and use it to predict whether bad local minima exist for the MNIST and CIFAR-10 datasets, and find that there are no bad differentiable local minima…
We study finite sample expressivity, i.e., memorization power of ReLU networks. Recent results require $N$ hidden nodes to memorize/interpolate arbitrary $N$ data points. In contrast, by exploiting depth, we show that 3-layer ReLU networks with $Ω(\sqrt{N})$ hidden nodes can perfectly memorize most datasets with $N$ points. We also prove that width $Θ(\sqrt{N})$ is necessary and sufficient for memorizing $N$ data points, proving tight bounds on memorization capacity. The sufficiency result can be extended to deeper networks; we show that an $L$-layer network with $W$ parameters in the hidden l…
We prove that finding all globally optimal two-layer ReLU neural networks can be performed by solving a convex optimization program with cone constraints. Our analysis is novel, characterizes all optimal solutions, and does not leverage duality-based analysis which was recently used to lift neural network training into convex spaces. Given the set of solutions of our convex optimization program, we show how to construct exactly the entire set of optimal neural networks. We provide a detailed characterization of this optimal set and its invariant transformations. As additional consequences of o…
We give superpolynomial statistical query (SQ) lower bounds for learning two-hidden-layer ReLU networks with respect to Gaussian inputs in the standard (noise-free) model. No general SQ lower bounds were known for learning ReLU networks of any depth in this setting: previous SQ lower bounds held only for adversarial noise models (agnostic learning) or restricted models such as correlational SQ. Prior work hinted at the impossibility of our result: Vempala and Wilmes showed that general SQ lower bounds cannot apply to any real-valued family of functions that satisfies a simple non-degeneracy co…
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, …
In this paper, we consider regression problems with one-hidden-layer neural networks (1NNs). We distill some properties of activation functions that lead to $\mathit{local~strong~convexity}$ in the neighborhood of the ground-truth parameters for the 1NN squared-loss objective. Most popular nonlinear activation functions satisfy the distilled properties, including rectified linear units (ReLUs), leaky ReLUs, squared ReLUs and sigmoids. For activation functions that are also smooth, we show $\mathit{local~linear~convergence}$ guarantees of gradient descent under a resampling rule. For homogeneou…
University of Münster（明斯特大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
In this article we study the stochastic gradient descent (SGD) optimization method in the training of fully-connected feedforward artificial neural networks with ReLU activation. The main result of this work proves that the risk of the SGD process converges to zero if the target function under consideration is constant. In the established convergence result the considered artificial neural networks consist of one input layer, one hidden layer, and one output layer (with $d \in \mathbb{N}$ neurons on the input layer, $H \in \mathbb{N}$ neurons on the hidden layer, and one neuron on the output l…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We study the sample complexity of learning one-hidden-layer convolutional neural networks (CNNs) with non-overlapping filters. We propose a novel algorithm called approximate gradient descent for training CNNs, and show that, with high probability, the proposed algorithm with random initialization grants a linear convergence to the ground-truth parameters up to statistical precision. Compared with existing work, our result applies to general non-trivial, monotonic and Lipschitz continuous activation functions including ReLU, Leaky ReLU, Sigmod and Softplus etc. Moreover, our sample complexity …
We consider the problem of learning an unknown ReLU network with respect to Gaussian inputs and obtain the first nontrivial results for networks of depth more than two. We give an algorithm whose running time is a fixed polynomial in the ambient dimension and some (exponentially large) function of only the network's parameters. Our bounds depend on the number of hidden units, depth, spectral norm of the weight matrices, and Lipschitz constant of the overall network (we show that some dependence on the Lipschitz constant is necessary). We also give a bound that is doubly exponential in the size…
University of Münster（明斯特大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
Gradient descent (GD) type optimization schemes are the standard methods to train artificial neural networks (ANNs) with rectified linear unit (ReLU) activation. Such schemes can be considered as discretizations of gradient flows (GFs) associated to the training of ANNs with ReLU activation and most of the key difficulties in the mathematical convergence analysis of GD type optimization schemes in the training of ANNs with ReLU activation seem to be already present in the dynamics of the corresponding GF differential equations. It is the key subject of this work to analyze such GF differential…
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…
Understanding the implicit regularization (or implicit bias) of gradient descent has recently been a very active research area. However, the implicit regularization in nonlinear neural networks is still poorly understood, especially for regression losses such as the square loss. Perhaps surprisingly, we prove that even for a single ReLU neuron, it is impossible to characterize the implicit regularization with the square loss by any explicit function of the model parameters (although on the positive side, we show it can be characterized approximately). For one hidden-layer networks, we prove a …
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…
Understanding the fundamental mechanism behind the success of deep neural networks is one of the key challenges in the modern machine learning literature. Despite numerous attempts, a solid theoretical analysis is yet to be developed. In this paper, we develop a novel unified framework to reveal a hidden regularization mechanism through the lens of convex optimization. We first show that the training of multiple three-layer ReLU sub-networks with weight decay regularization can be equivalently cast as a convex optimization problem in a higher dimensional space, where sparsity is enforced via a…
University of Minnesota（明尼苏达大学）；Beijing Institute of Technology（北京理工大学）；Tongji University（同济大学）
Neural networks with REctified Linear Unit (ReLU) activation functions (a.k.a. ReLU networks) have achieved great empirical success in various domains. Nonetheless, existing results for learning ReLU networks either pose assumptions on the underlying data distribution being e.g. Gaussian, or require the network size and/or training size to be sufficiently large. In this context, the problem of learning a two-layer ReLU network is approached in a binary classification setting, where the data are linearly separable and a hinge loss criterion is adopted. Leveraging the power of random noise pertu…
We provide a theoretical algorithm for checking local optimality and escaping saddles at nondifferentiable points of empirical risks of two-layer ReLU networks. Our algorithm receives any parameter value and returns: local minimum, second-order stationary point, or a strict descent direction. The presence of $M$ data points on the nondifferentiability of the ReLU divides the parameter space into at most $2^M$ regions, which makes analysis difficult. By exploiting polyhedral geometry, we reduce the total computation down to one convex quadratic program (QP) for each hidden node, $O(M)$ (in)equa…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Technion - Israel Institute of Technology（以色列理工学院）
We study the implicit bias of ReLU neural networks trained by a variant of SGD where at each step, the label is changed with probability $p$ to a random label (label smoothing being a close variant of this procedure). Our experiments demonstrate that label noise propels the network to a sparse solution in the following sense: for a typical input, a small fraction of neurons are active, and the firing pattern of the hidden layers is sparser. In fact, for some instances, an appropriate amount of label noise does not only sparsify the network but further reduces the test error. We then turn to th…
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 develop exact representations of training two-layer neural networks with rectified linear units (ReLUs) in terms of a single convex program with number of variables polynomial in the number of training samples and the number of hidden neurons. Our theory utilizes semi-infinite duality and minimum norm regularization. We show that ReLU networks trained with standard weight decay are equivalent to block $\ell_1$ penalized convex models. Moreover, we show that certain standard convolutional linear networks are equivalent semi-definite programs which can be simplified to $\ell_1$ regularized li…
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…
For one-hidden-layer ReLU networks, we prove that all differentiable local minima are global inside differentiable regions. We give the locations and losses of differentiable local minima, and show that these local minima can be isolated points or continuous hyperplanes, depending on an interplay between data, activation pattern of hidden neurons and network size. Furthermore, we give necessary and sufficient conditions for the existence of saddle points as well as non-differentiable local minima, and their locations if they exist.
Recent work has shown that the training of a one-hidden-layer, scalar-output fully-connected ReLU neural network can be reformulated as a finite-dimensional convex program. Unfortunately, the scale of such a convex program grows exponentially in data size. In this work, we prove that a stochastic procedure with a linear complexity well approximates the exact formulation. Moreover, we derive a convex optimization approach to efficiently solve the "adversarial training" problem, which trains neural networks that are robust to adversarial input perturbations. Our method can be applied to binary c…
Jiangxi Normal University（江西师范大学）；City University of Hong Kong（香港城市大学）；Hong Kong University of Science and Technology（香港科技大学）
In this paper, we develop an alternating direction method of multipliers (ADMM) for deep neural networks training with sigmoid-type activation functions (called \textit{sigmoid-ADMM pair}), mainly motivated by the gradient-free nature of ADMM in avoiding the saturation of sigmoid-type activations and the advantages of deep neural networks with sigmoid-type activations (called deep sigmoid nets) over their rectified linear unit (ReLU) counterparts (called deep ReLU nets) in terms of approximation. In particular, we prove that the approximation capability of deep sigmoid nets is not worse than t…
Deep learning models are often successfully trained using gradient descent, despite the worst case hardness of the underlying non-convex optimization problem. The key question is then under what conditions can one prove that optimization will succeed. Here we provide a strong result of this kind. We consider a neural net with one hidden layer and a convolutional structure with no overlap and a ReLU activation function. For this architecture we show that learning is NP-complete in the general case, but that when the input distribution is Gaussian, gradient descent converges to the global optimu…
Large neural network models have been successful in learning functions of importance in many branches of science, including physics, chemistry and biology. Recent theoretical work has shown explicit learning bounds for wide networks and kernel methods on some simple classes of functions, but not on more complex functions which arise in practice. We extend these techniques to provide learning bounds for analytic functions on the sphere for any kernel method or equivalent infinitely-wide network with the corresponding activation function trained with SGD. We show that a wide, one-hidden layer Re…
The success of neural networks over the past decade has established them as effective models for many relevant data generating processes. Statistical theory on neural networks indicates graceful scaling of sample complexity. For example, Joen & Van Roy (arXiv:2203.00246) demonstrate that, when data is generated by a ReLU teacher network with $W$ parameters, an optimal learner needs only $\tilde{O}(W/ε)$ samples to attain expected error $ε$. However, existing computational theory suggests that, even for single-hidden-layer teacher networks, to attain small error for all such teacher networks, t…
Neural Tangent Kernels and Fisher Information Matrices for Simple ReLU Networks with Random Hidden Weights
Faculty of Information Science and Electrical Engineering, Kyushu University（九州大学信息科学与电气工程学部）；Faculty of Science and Engineering, Waseda University（早稻田大学理工学部）；Faculty of Information Sciences, Hiroshima City University（广岛城大学信息科学学部）；Joint Graduate School of Math for Innovation, Kyushu University（九州大学创新数学研究生院）；The University of Electro-Communications（电子通信大学）
Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence（伯努利数学、计算机科学与人工智能研究所）；University of Groningen（格罗宁根大学）
We study layered neural networks of rectified linear units (ReLU) in a modelling framework for stochastic training processes. The comparison with sigmoidal activation functions is in the center of interest. We compute typical learning curves for shallow networks with K hidden units in matching student teacher scenarios. The systems exhibit sudden changes of the generalization performance via the process of hidden unit specialization at critical sizes of the training set. Surprisingly, our results show that the training behavior of ReLU networks is qualitatively different from that of networks …
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 …
Model extraction attacks have renewed interest in the classic problem of learning neural networks from queries. In this work we give the first polynomial-time algorithm for learning arbitrary one hidden layer neural networks activations provided black-box access to the network. Formally, we show that if $F$ is an arbitrary one hidden layer neural network with ReLU activations, there is an algorithm with query complexity and running time that is polynomial in all parameters that outputs a network $F'$ achieving low square loss relative to $F$ with respect to the Gaussian measure. While a number…
University of Münster（明斯特大学）；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）
Gradient descent (GD) type optimization methods are the standard instrument to train artificial neural networks (ANNs) with rectified linear unit (ReLU) activation. Despite the great success of GD type optimization methods in numerical simulations for the training of ANNs with ReLU activation, it remains - even in the simplest situation of the plain vanilla GD optimization method with random initializations and ANNs with one hidden layer - an open problem to prove (or disprove) the conjecture that the risk of the GD optimization method converges in the training of such ANNs to zero as the widt…
We propose and analyze a new family of algorithms for training neural networks with ReLU activations. Our algorithms are based on the technique of alternating minimization: estimating the activation patterns of each ReLU for all given samples, interleaved with weight updates via a least-squares step. The main focus of our paper are 1-hidden layer networks with $k$ hidden neurons and ReLU activation. We show that under standard distributional assumptions on the $d-$dimensional input data, our algorithm provably recovers the true `ground truth' parameters in a linearly convergent fashion. This h…
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…
Ant Group（蚂蚁集团）；University at Buffalo, the State University of New York（纽约州立大学布法罗分校）
The first provably efficient algorithm for learning graph neural networks (GNNs) with one hidden layer for node information convolution is provided in this paper. Two types of GNNs are investigated, depending on whether labels are attached to nodes or graphs. A comprehensive framework for designing and analyzing convergence of GNN training algorithms is developed. The algorithm proposed is applicable to a wide range of activation functions including ReLU, Leaky ReLU, Sigmod, Softplus and Swish. It is shown that the proposed algorithm guarantees a linear convergence rate to the underlying true …
Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation
University of Duisburg-Essen（杜伊斯堡-埃森大学）；University of Münster（明斯特大学）；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）
Carnegie Mellon University（卡内基梅隆大学）；University of Southern California（南加州大学）；Facebook Artificial Intelligence Research（Facebook人工智能研究院）
We consider the problem of learning a one-hidden-layer neural network with non-overlapping convolutional layer and ReLU activation, i.e., $f(\mathbf{Z}, \mathbf{w}, \mathbf{a}) = \sum_j a_jσ(\mathbf{w}^T\mathbf{Z}_j)$, in which both the convolutional weights $\mathbf{w}$ and the output weights $\mathbf{a}$ are parameters to be learned. When the labels are the outputs from a teacher network of the same architecture with fixed weights $(\mathbf{w}^*, \mathbf{a}^*)$, we prove that with Gaussian input $\mathbf{Z}$, there is a spurious local minimizer. Surprisingly, in the presence of the spurious …
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…
Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院工业与系统工程学院）
In this paper, we explore some basic questions on the complexity of training neural networks with ReLU activation function. We show that it is NP-hard to train a two-hidden layer feedforward ReLU neural network. If dimension of the input data and the network topology is fixed, then we show that there exists a polynomial time algorithm for the same training problem. We also show that if sufficient over-parameterization is provided in the first hidden layer of ReLU neural network, then there is a polynomial time algorithm which finds weights such that output of the over-parameterized ReLU neural…
We argue the Fisher information matrix (FIM) of one hidden layer networks with the ReLU activation function. For a network, let $W$ denote the $d \times p$ weight matrix from the $d$-dimensional input to the hidden layer consisting of $p$ neurons, and $v$ the $p$-dimensional weight vector from the hidden layer to the scalar output. We focus on the FIM of $v$, which we denote as $I$. Under certain conditions, we characterize the first three clusters of eigenvalues and eigenvectors of the FIM. Specifically, we show that 1) Since $I$ is non-negative owing to the ReLU, the first eigenvalue is the …
