We study norm-based uniform convergence bounds for neural networks, aiming at a tight understanding of how these are affected by the architecture and type of norm constraint, for the simple class of scalar-valued one-hidden-layer networks, and inputs bounded in Euclidean norm. We begin by proving that in general, controlling the spectral norm of the hidden layer weight matrix is insufficient to get uniform convergence guarantees (independent of the network width), while a stronger Frobenius norm control is sufficient, extending and improving on previous work. Motivated by the proof constructions, we identify and analyze two important settings where (perhaps surprisingly) a mere spectral norm control turns out to be sufficient: First, when the network's activation functions are sufficiently smooth (with the result extending to deeper networks); and second, for certain types of convolutional networks. In the latter setting, we study how the sample complexity is additionally affected by parameters such as the amount of overlap between patches and the overall number of patches.
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 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 Ohio State University（俄亥俄州立大学）；Carnegie Mellon University（卡内基梅隆大学）
We study model recovery for data classification, where the training labels are generated from a one-hidden-layer neural network with sigmoid activations, also known as a single-layer feedforward network, and the goal is to recover the weights of the neural network. We consider two network models, the fully-connected network (FCN) and the non-overlapping convolutional neural network (CNN). We prove that with Gaussian inputs, the empirical risk based on cross entropy exhibits strong convexity and smoothness {\em uniformly} in a local neighborhood of the ground truth, as soon as the sample comple…
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…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；American Family Insurance（美国家庭保险集团）
With the rise of big data analytics, multi-layer neural networks have surfaced as one of the most powerful machine learning methods. However, their theoretical mathematical properties are still not fully understood. Training a neural network requires optimizing a non-convex objective function, typically done using stochastic gradient descent (SGD). In this paper, we seek to extend the mean field results of Mei et al. (2018) from two-layer neural networks with one hidden layer to three-layer neural networks with two hidden layers. We will show that the SGD dynamics is captured by a set of non-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…
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…
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…
Although deep learning has shown its powerful performance in many applications, the mathematical principles behind neural networks are still mysterious. In this paper, we consider the problem of learning a one-hidden-layer neural network with quadratic activations. We focus on the under-parameterized regime where the number of hidden units is smaller than the dimension of the inputs. We shall propose to solve the problem via a provable gradient-based method with random initialization. For the non-convex neural networks training problem we reveal that the gradient descent iterates are able to e…
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…
This paper analyzes the convergence and generalization of training a one-hidden-layer neural network when the input features follow the Gaussian mixture model consisting of a finite number of Gaussian distributions. Assuming the labels are generated from a teacher model with an unknown ground truth weight, the learning problem is to estimate the underlying teacher model by minimizing a non-convex risk function over a student neural network. With a finite number of training samples, referred to the sample complexity, the iterations are proved to converge linearly to a critical point with guaran…
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 consider the problem of learning a one-hidden-layer neural network: we assume the input $x\in \mathbb{R}^d$ is from Gaussian distribution and the label $y = a^\top σ(Bx) + ξ$, where $a$ is a nonnegative vector in $\mathbb{R}^m$ with $m\le d$, $B\in \mathbb{R}^{m\times d}$ is a full-rank weight matrix, and $ξ$ is a noise vector. We first give an analytic formula for the population risk of the standard squared loss and demonstrate that it implicitly attempts to decompose a sequence of low-rank tensors simultaneously. Inspired by the formula, we design a non-convex objective function $G(\cdot)…
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 …
Rutgers, The State University of New Jersey（罗格斯大学新泽西州立大学）；Michigan State University（密歇根州立大学）
We provide high-probability sample complexity guarantees for exact structure recovery and accurate predictive learning using noise-corrupted samples from an acyclic (tree-shaped) graphical model. The hidden variables follow a tree-structured Ising model distribution, whereas the observable variables are generated by a binary symmetric channel taking the hidden variables as its input (flipping each bit independently with some constant probability $q\in [0,1/2)$). In the absence of noise, predictive learning on Ising models was recently studied by Bresler and Karzand (2020); this paper quantifie…
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.
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…
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 …
We study the problem of policy optimization for infinite-horizon discounted Markov Decision Processes with softmax policy and nonlinear function approximation trained with policy gradient algorithms. We concentrate on the training dynamics in the mean-field regime, modeling e.g., the behavior of wide single hidden layer neural networks, when exploration is encouraged through entropy regularization. The dynamics of these models is established as a Wasserstein gradient flow of distributions in parameter space. We further prove global optimality of the fixed points of this dynamics under mild con…
Peking University（北京大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Sparse neural networks have received increasing interest due to their small size compared to dense networks. Nevertheless, most existing works on neural network theory have focused on dense neural networks, and the understanding of sparse networks is very limited. In this paper, we study the loss landscape of one-hidden-layer sparse networks. First, we consider sparse networks with a dense final layer. We show that linear networks can have no spurious valleys under special sparse structures, and non-linear networks could also admit no spurious valleys under a wide final layer. Second, we disco…
George Cybenko's landmark 1989 paper showed that there exists a feedforward neural network, with exactly one hidden layer (and a finite number of neurons), that can arbitrarily approximate a given continuous function $f$ on the unit hypercube. The paper did not address how to find the weight/parameters of such a network, or if finding them would be computationally feasible. This paper outlines an algorithm for a neural network with exactly one hidden layer to reconstruct any continuous scalar or vector valued continuous function.
In this note we consider setups in which variational objectives for Bayesian neural networks can be computed in closed form. In particular we focus on single-layer networks in which the activation function is piecewise polynomial (e.g. ReLU). In this case we show that for a Normal likelihood and structured Normal variational distributions one can compute a variational lower bound in closed form. In addition we compute the predictive mean and variance in closed form. Finally, we also show how to compute approximate lower bounds for other likelihoods (e.g. softmax classification). In experiments…
It is well known that Artificial Neural Networks are universal approximators. The classical result proves that, given a continuous function on a compact set on an n-dimensional space, then there exists a one-hidden-layer feedforward network which approximates the function. Such result proves the existence, but it does not provide a method for finding it. In this paper, a constructive approach to the proof of this property is given for the case of two-hidden-layer feedforward networks. This approach is based on an approximation of continuous functions by simplicial maps. Once a triangulation of…
Indian Institute of Technology, Roorkee（印度理工学院鲁尔基分校）
This paper proposes an architecture for deep neural networks with hidden layer branches that learn targets of lower hierarchy than final layer targets. The branches provide a channel for enforcing useful information in hidden layer which helps in attaining better accuracy, both for the final layer and hidden layers. The shared layers modify their weights using the gradients of all cost functions higher than the branching layer. This model provides a flexible inference system with many levels of targets which is modular and can be used efficiently in situations requiring different levels of res…
Igor Sikorsky Kyiv Polytechnic Institute（伊戈尔·西科尔斯基基辅理工学院）
Here is presented an analysis of an autoencoder with binary activations $\{0, 1\}$ and binary $\{0, 1\}$ random weights. Such set up puts this model at the intersection of different fields: neuroscience, information theory, sparse coding, and machine learning. It is shown that the sparse activation of the hidden layer arises naturally in order to preserve information between layers. Furthermore, with a large enough hidden layer, it is possible to get zero reconstruction error for any input just by varying the thresholds of neurons. The model preserves the similarity of inputs at the hidden lay…
Mila, University of Montreal（蒙特利尔大学Mila）；University of California, Berkeley（加州大学伯克利分校）；CentraleSupelec, University of Paris-Saclay（巴黎-萨克雷大学中央理工-高等电力学院）
Shallow supervised 1-hidden layer neural networks have a number of favorable properties that make them easier to interpret, analyze, and optimize than their deep counterparts, but lack their representational power. Here we use 1-hidden layer learning problems to sequentially build deep networks layer by layer, which can inherit properties from shallow networks. Contrary to previous approaches using shallow networks, we focus on problems where deep learning is reported as critical for success. We thus study CNNs on image classification tasks using the large-scale ImageNet dataset and the CIFAR-…
Friedrich-Alexander University Erlangen (FAU)（埃尔朗根-纽伦堡弗里德里希-亚历山大大学（FAU））；University Hospital Erlangen（埃尔朗根大学医院）
Deep neural networks typically outperform more traditional machine learning models in their ability to classify complex data, and yet is not clear how the individual hidden layers of a deep network contribute to the overall classification performance. We thus introduce a Generalized Discrimination Value (GDV) that measures, in a non-invasive manner, how well different data classes separate in each given network layer. The GDV can be used for the automatic tuning of hyper-parameters, such as the width profile and the total depth of a network. Moreover, the layer-dependent GDV(L) provides new in…
Purdue University（普渡大学）；The Ohio State University（俄亥俄州立大学）
In this paper, we study the generalization performance of overparameterized 3-layer NTK models. We show that, for a specific set of ground-truth functions (which we refer to as the "learnable set"), the test error of the overfitted 3-layer NTK is upper bounded by an expression that decreases with the number of neurons of the two hidden layers. Different from 2-layer NTK where there exists only one hidden-layer, the 3-layer NTK involves interactions between two hidden-layers. Our upper bound reveals that, between the two hidden-layers, the test error descends faster with respect to the number o…
The University of Western Australia（西澳大利亚大学）；The University of New South Wales（新南威尔士大学）
In this paper, we introduce transformations of deep rectifier networks, enabling the conversion of deep rectifier networks into shallow rectifier networks. We subsequently prove that any rectifier net of any depth can be represented by a maximum of a number of functions that can be realized by a shallow network with a single hidden layer. The transformations of both deep rectifier nets and deep residual nets are conducted to demonstrate the advantages of the residual nets over the conventional neural nets and the advantages of the deep neural nets over the shallow neural nets. In summary, for …
Despite their increasing popularity and success in a variety of supervised learning problems, deep neural networks are extremely hard to interpret and debug: Given and already trained Deep Neural Net, and a set of test inputs, how can we gain insight into how those inputs interact with different layers of the neural network? Furthermore, can we characterize a given deep neural network based on it's observed behavior on different inputs? In this paper we propose a novel factorization based approach on understanding how different deep neural networks operate. In our preliminary results, we ident…
In this paper, we study the problem of optimizing a two-layer artificial neural network that best fits a training dataset. We look at this problem in the setting where the number of parameters is greater than the number of sampled points. We show that for a wide class of differentiable activation functions (this class involves "almost" all functions which are not piecewise linear), we have that first-order optimal solutions satisfy global optimality provided the hidden layer is non-singular. Our results are easily extended to hidden layers given by a flat matrix from that of a square matrix. R…
We derive global convergence bounds for the Frank Wolfe algorithm when training one hidden layer neural networks. When using the ReLU activation function, and under tractable preconditioning assumptions on the sample data set, the linear minimization oracle used to incrementally form the solution can be solved explicitly as a second order cone program. The classical Frank Wolfe algorithm then converges with rate $O(1/T)$ where $T$ is both the number of neurons and the number of calls to the oracle.
In this paper, we propose a defence strategy to improve adversarial robustness by incorporating hidden layer representation. The key of this defence strategy aims to compress or filter input information including adversarial perturbation. And this defence strategy can be regarded as an activation function which can be applied to any kind of neural network. We also prove theoretically the effectiveness of this defense strategy under certain conditions. Besides, incorporating hidden layer representation we propose three types of adversarial attacks to generate three types of adversarial examples…
Purdue University（普渡大学）；Center for Computational and Applied Mathematics（计算与应用数学中心）；Department of Mathematics（数学系）
The extreme learning machine (ELM) method can yield highly accurate solutions to linear/nonlinear partial differential equations (PDEs), but requires the last hidden layer of the neural network to be wide to achieve a high accuracy. If the last hidden layer is narrow, the accuracy of the existing ELM method will be poor, irrespective of the rest of the network configuration. In this paper we present a modified ELM method, termed HLConcELM (hidden-layer concatenated ELM), to overcome the above drawback of the conventional ELM method. The HLConcELM method can produce highly accurate solutions to…
Machine Learning Department, Carnegie Mellon University（卡内基梅隆大学机器学习系）；Department of Computer Science, University of Texas at Austin（德克萨斯大学奥斯汀分校计算机科学系）
We propose a new algorithm to learn a one-hidden-layer convolutional neural network where both the convolutional weights and the outputs weights are parameters to be learned. Our algorithm works for a general class of (potentially overlapping) patches, including commonly used structures for computer vision tasks. Our algorithm draws ideas from (1) isotonic regression for learning neural networks and (2) landscape analysis of non-convex matrix factorization problems. We believe these findings may inspire further development in designing provable algorithms for learning neural networks and other…
McGill University（麦吉尔大学）；Mila - Quebec AI Institute（米拉 - 魁北克人工智能研究所）
Group invariant and equivariant Multilayer Perceptrons (MLP), also known as Equivariant Networks, have achieved remarkable success in learning on a variety of data structures, such as sequences, images, sets, and graphs. Using tools from group theory, this paper proves the universality of a broad class of equivariant MLPs with a single hidden layer. In particular, it is shown that having a hidden layer on which the group acts regularly is sufficient for universal equivariance (invariance). A corollary is unconditional universality of equivariant MLPs for Abelian groups, such as CNNs with a sin…
Although Bidirectional Encoder Representations from Transformers (BERT) have achieved tremendous success in many natural language processing (NLP) tasks, it remains a black box. A variety of previous works have tried to lift the veil of BERT and understand each layer's functionality. In this paper, we found that surprisingly the output layer of BERT can reconstruct the input sentence by directly taking each layer of BERT as input, even though the output layer has never seen the input other than the final hidden layer. This fact remains true across a wide variety of BERT-based models, even when…
Hebei University of Technology（河北工业大学）；Southwest Petroleum University（西南石油大学）
The three-way decisions strategy has been employed to construct network topology in a single hidden layer feedforward neural network (SFNN). However, this model has a general performance, and does not consider the process costs, since it has fixed threshold parameters. Inspired by the sequential three-way decisions (STWD), this paper proposes STWD with an SFNN (STWD-SFNN) to enhance the performance of networks on structured datasets. STWD-SFNN adopts multi-granularity levels to dynamically learn the number of hidden layer nodes from coarse to fine, and set the sequential threshold parameters. …
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 …
Imperial College London（伦敦帝国学院）；CentraleSupélec-CNRS-Université Paris-Sud（中央理工-高等电力学院-法国国家科学研究中心-巴黎南大学）
The muti-layer information bottleneck (IB) problem, where information is propagated (or successively refined) from layer to layer, is considered. Based on information forwarded by the preceding layer, each stage of the network is required to preserve a certain level of relevance with regards to a specific hidden variable, quantified by the mutual information. The hidden variables and the source can be arbitrarily correlated. The optimal trade-off between rates of relevance and compression (or complexity) is obtained through a single-letter characterization, referred to as the rate-relevance re…
International University of Business Agriculture and Technology（国际商业农业与技术大学）；Mississippi State University（密西西比州立大学）；Rajshahi University of Engineering and Technology（拉杰沙希工程技术大学）
Nowadays, deep learning can be employed to a wide ranges of fields including medicine, engineering, etc. In deep learning, Convolutional Neural Network (CNN) is extensively used in the pattern and sequence recognition, video analysis, natural language processing, spam detection, topic categorization, regression analysis, speech recognition, image classification, object detection, segmentation, face recognition, robotics, and control. The benefits associated with its near human level accuracies in large applications lead to the growing acceptance of CNN in recent years. The primary contribution…
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…
Recent results in nonparametric regression show that deep learning, i.e., neural network estimates with many hidden layers, are able to circumvent the so-called curse of dimensionality in case that suitable restrictions on the structure of the regression function hold. One key feature of the neural networks used in these results is that their network architecture has a further constraint, namely the network sparsity. In this paper we show that we can get similar results also for least squares estimates based on simple fully connected neural networks with ReLU activation functions. Here either …
