We formalize and interpret the geometric structure of $d$-dimensional fully connected ReLU layers in neural networks. The parameters of a ReLU layer induce a natural partition of the input domain, such that the ReLU layer can be significantly simplified in each sector of the partition. This leads to a geometric interpretation of a ReLU layer as a projection onto a polyhedral cone followed by an affine transformation, in line with the description in [doi:10.48550/arXiv.1905.08922] for convolutional networks with ReLU activations. Further, this structure facilitates simplified expressions for preimages of the intersection between partition sectors and hyperplanes, which is useful when describing decision boundaries in a classification setting. We investigate this in detail for a feed-forward network with one hidden ReLU-layer, where we provide results on the geometric complexity of the decision boundary generated by such networks, as well as proving that modulo an affine transformation, such a network can only generate $d$ different decision boundaries. Finally, the effect of adding more layers to the network is discussed.
Understanding theoretical properties of deep and locally connected nonlinear network, such as deep convolutional neural network (DCNN), is still a hard problem despite its empirical success. In this paper, we propose a novel theoretical framework for such networks with ReLU nonlinearity. The framework explicitly formulates data distribution, favors disentangled representations and is compatible with common regularization techniques such as Batch Norm. The framework is built upon teacher-student setting, by expanding the student forward/backward propagation onto the teacher's computational grap…
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…
Recent Progress has shown that exploitation of hidden layer neurons in convolution neural networks incorporating with a carefully designed activation function can yield better classification results in the field of computer vision. The paper firstly introduces a novel deep learning architecture aiming to mitigate the gradient-vanishing problem, in which the earlier hidden layer neurons could be directly connected with the last hidden layer and feed into the last layer for classification. We then design a generalized linear rectifier function as the activation function that can approximate arbi…
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…
Let F:R^n -> R be a feedforward ReLU neural network. It is well-known that for any choice of parameters, F is continuous and piecewise (affine) linear. We lay some foundations for a systematic investigation of how the architecture of F impacts the geometry and topology of its possible decision regions for binary classification tasks. Following the classical progression for smooth functions in differential topology, we first define the notion of a generic, transversal ReLU neural network and show that almost all ReLU networks are generic and transversal. We then define a partially-oriented line…
Faculty of Information Technology, Monash University（莫纳什大学信息技术学院）；Monash University（莫纳什大学）
Previous work has questioned the conditions under which the decision regions of a neural network are connected and further showed the implications of the corresponding theory to the problem of adversarial manipulation of classifiers. It has been proven that for a class of activation functions including leaky ReLU, neural networks having a pyramidal structure, that is no layer has more hidden units than the input dimension, produce necessarily connected decision regions. In this paper, we advance this important result by further developing the sufficient and necessary conditions under which the…
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…
Mathematical Institute（数学研究所）；University of Oxford（牛津大学）
This paper considers the growth in the length of one-dimensional trajectories as they are passed through deep ReLU neural networks, which, among other things, is one measure of the expressivity of deep networks. We generalise existing results, providing an alternative, simpler method for lower bounding expected trajectory growth through random networks, for a more general class of weights distributions, including sparsely connected networks. We illustrate this approach by deriving bounds for sparse-Gaussian, sparse-uniform, and sparse-discrete-valued random nets. We prove that trajectory growt…
State-of-the-art machine learning models frequently misclassify inputs that have been perturbed in an adversarial manner. Adversarial perturbations generated for a given input and a specific classifier often seem to be effective on other inputs and even different classifiers. In other words, adversarial perturbations seem to transfer between different inputs, models, and even different neural network architectures. In this work, we show that in the context of linear classifiers and two-layer ReLU networks, there provably exist directions that give rise to adversarial perturbations for many cla…
In this paper, we determine analytical bounds on the local Lipschitz constants of of affine functions composed with rectified linear units (ReLUs). Affine-ReLU functions represent a widely used layer in deep neural networks, due to the fact that convolution, fully-connected, and normalization functions are all affine, and are often followed by a ReLU activation function. Using an analytical approach, we mathematically determine upper bounds on the local Lipschitz constant of an affine-ReLU function, show how these bounds can be combined to determine a bound on an entire network, and discuss ho…
University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）
The NP-hard problem of optimizing a shallow ReLU network can be characterized as a combinatorial search over each training example's activation pattern followed by a constrained convex problem given a fixed set of activation patterns. We explore the implications of this combinatorial aspect of ReLU optimization in this work. We show that it can be naturally modeled via a geometric and combinatoric object known as a zonotope with its vertex set isomorphic to the set of feasible activation patterns. This assists in analysis and provides a foundation for further research. We demonstrate its usefu…
Institute for Adaptive and Neural Computation（自适应与神经计算研究所）
This informal technical report details the geometric illustration of decision boundaries for ReLU units in a three layer fully connected neural network. The network is designed and trained to predict pixel intensity from an (x, y) input location. The Geometric Illustration of Neural Networks (GINN) tool was built to visualise and track the points at which ReLU units switch from being active to off (or vice versa) as the network undergoes training. Several phenomenon were observed and are discussed herein. This technical report is a supporting document to the blog post with online demos and is …
SAILAB, University of Siena（锡耶纳大学SAILAB实验室）；University of Pavia（帕维亚大学）
Amongst others, the adoption of Rectified Linear Units (ReLUs) is regarded as one of the ingredients of the success of deep learning. ReLU activation has been shown to mitigate the vanishing gradient issue, to encourage sparsity in the learned parameters, and to allow for efficient backpropagation. In this paper, we recognize that the non-linear behavior of the ReLU function gives rise to a natural clustering when the pattern of active neurons is considered. This observation helps to deepen the learning mechanism of the network; in fact, we demonstrate that, within each cluster, the network ca…
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…
We propose a new notion of `non-linearity' of a network layer with respect to an input batch that is based on its proximity to a linear system, which is reflected in the non-negative rank of the activation matrix. We measure this non-linearity by applying non-negative factorization to the activation matrix. Considering batches of similar samples, we find that high non-linearity in deep layers is indicative of memorization. Furthermore, by applying our approach layer-by-layer, we find that the mechanism for memorization consists of distinct phases. We perform experiments on fully-connected and …
Neural networks have shown high successful performance in a wide range of tasks, but further studies are needed to improve its performance. We analyze the approximation error of the specific neural network architecture with a local connection and higher application than one with the full connection because the local-connected network can be used to explain diverse neural networks such as CNNs. Our error estimate depends on two parameters: one controlling the depth of the hidden layer, and the other, the width of the hidden layers.
The realization function of a shallow ReLU network is a continuous and piecewise affine function $f:\mathbb R^d\to \mathbb R$, where the domain $\mathbb R^{d}$ is partitioned by a set of $n$ hyperplanes into cells on which $f$ is affine. We show that the minimal representation for $f$ uses either $n$, $n+1$ or $n+2$ neurons and we characterize each of the three cases. In the particular case, where the input layer is one-dimensional, minimal representations always use at most $n+1$ neurons but in all higher dimensional settings there are functions for which $n+2$ neurons are needed. Then we sho…
We propose a system for calculating a "scaling constant" for layers and weights of neural networks. We relate this scaling constant to two important quantities that relate to the optimizability of neural networks, and argue that a network that is "preconditioned" via scaling, in the sense that all weights have the same scaling constant, will be easier to train. This scaling calculus results in a number of consequences, among them the fact that the geometric mean of the fan-in and fan-out, rather than the fan-in, fan-out, or arithmetic mean, should be used for the initialization of the variance…
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 …
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）
The implicit bias induced by the training of neural networks has become a topic of rigorous study. In the limit of gradient flow and gradient descent with appropriate step size, it has been shown that when one trains a deep linear network with logistic or exponential loss on linearly separable data, the weights converge to rank-1 matrices. In this paper, we extend this theoretical result to the last few linear layers of the much wider class of nonlinear ReLU-activated feedforward networks containing fully-connected layers and skip connections. Similar to the linear case, the proof relies on sp…
Numerical influence of ReLU'(0) on backpropagation
IRT Saint Exupéry（圣埃克苏佩里研究所）；ISAE-SUPAERO（法国国立高等航空航天学院）；ANITI（图卢兹人工智能跨学科研究所）；Toulouse School of Economics（图卢兹经济学院）；Université Toulouse 1 Capitole（图卢兹第一大学）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）；CNRS（法国国家科学研究中心）；IRIT, Université Paul Sabatier（图卢兹计算机科学研究所，保罗·萨巴捷大学）
Universiti Tun Hussein Onn Malaysia（马来西亚敦胡先翁大学）；Institute of Engineering and Management（工程与管理学院）
Activation functions are essential for deep learning methods to learn and perform complex tasks such as image classification. Rectified Linear Unit (ReLU) has been widely used and become the default activation function across the deep learning community since 2012. Although ReLU has been popular, however, the hard zero property of the ReLU has heavily hindered the negative values from propagating through the network. Consequently, the deep neural network has not been benefited from the negative representations. In this work, an activation function called Flatten-T Swish (FTS) that leverage the…
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…
Center for Data Science, New York University（纽约大学数据科学中心）；Inria - Département d’Informatique de l’École Normale Supérieure（法国国家信息与自动化研究所 - 巴黎高等师范学校计算机科学系）；PSL Research University（巴黎文理研究大学）
Deep networks are often considered to be more expressive than shallow ones in terms of approximation. Indeed, certain functions can be approximated by deep networks provably more efficiently than by shallow ones, however, no tractable algorithms are known for learning such deep models. Separately, a recent line of work has shown that deep networks trained with gradient descent may behave like (tractable) kernel methods in a certain over-parameterized regime, where the kernel is determined by the architecture and initialization, and this paper focuses on approximation for such kernels. We show …
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…
We apply neural nets with ReLU gates in online reinforcement learning. Our goal is to train these networks in an incremental manner, without the computationally expensive experience replay. By studying how individual neural nodes behave in online training, we recognize that the global nature of ReLU gates can cause undesirable learning interference in each node's learning behavior. We propose reducing such interferences with two efficient input transformation methods that are geometric in nature and match well the geometric property of ReLU gates. The first one is tile coding, a classic binary…
UC Berkeley（加州大学伯克利分校）；TTI Chicago（芝加哥丰田技术研究所）；Hebrew University（希伯来大学）；Google（谷歌）；University of Michigan（密歇根大学）
The implicit biases of gradient-based optimization algorithms are conjectured to be a major factor in the success of modern deep learning. In this work, we investigate the implicit bias of gradient flow and gradient descent in two-layer fully-connected neural networks with leaky ReLU activations when the training data are nearly-orthogonal, a common property of high-dimensional data. For gradient flow, we leverage recent work on the implicit bias for homogeneous neural networks to show that asymptotically, gradient flow produces a neural network with rank at most two. Moreover, this network is…
Institute for Computer Science and Control (SZTAKI)（计算机科学与控制研究所（SZTAKI））
Recent articles indicate that deep neural networks are efficient models for various learning problems. However they are often highly sensitive to various changes that cannot be detected by an independent observer. As our understanding of deep neural networks with traditional generalization bounds still remains incomplete, there are several measures which capture the behaviour of the model in case of small changes at a specific state. In this paper we consider adversarial stability in the tangent space and suggest tangent sensitivity in order to characterize stability. We focus on a particular …
Efficient Global Optimization of Two-Layer ReLU Networks: Quadratic-Time Algorithms and Adversarial Training
Department of Mechanical Engineering, University of California, Berkeley（机械工程系，加州大学伯克利分校）；Department of Electrical Engineering and Computer Science, University of California, Berkeley（电气工程与计算机科学系，加州大学伯克利分校）
