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 how the bounds can be efficiently computed, even for larger layers and networks. We show several examples by applying our results to AlexNet, as well as several smaller networks based on the MNIST and CIFAR-10 datasets. The results show that our method produces tighter bounds than the standard conservative bound (i.e. the product of the spectral norms of the layers' linear matrices), especially for small perturbations.
In this paper, we determine analytical upper bounds on the local Lipschitz constants of feedforward neural networks with ReLU activation functions. We do so by deriving Lipschitz constants and bounds for ReLU, affine-ReLU, and max pooling functions, and combining the results to determine a network-wide bound. Our method uses several insights to obtain tight bounds, such as keeping track of the zero elements of each layer, and analyzing the composition of affine and ReLU functions. Furthermore, we employ a careful computational approach which allows us to apply our method to large networks such…
International Institute of Information Technology Bangalore（班加罗尔国际信息技术学院）
The Lipschitz constant of neural networks plays an important role in several contexts of deep learning ranging from robustness certification and regularization to stability analysis of systems with neural network controllers. Obtaining tight bounds of the Lipschitz constant is therefore important. We introduce LipBaB, a branch and bound framework to compute certified bounds of the local Lipschitz constant of deep neural networks with ReLU activation functions up to any desired precision. We achieve this by bounding the norm of the Jacobians, corresponding to different activation patterns of th…
Local Lipschitz Constant Computation of ReLU-FNNs: Upper Bound Computation with Exactness Verification
Graduate School of Information Science and Electrical Engineering, Kyushu University（九州大学信息科学与电气工程研究生院）；LAAS-CNRS（法国国家科学研究中心自动控制与系统分析实验室）；Université de Toulouse（图卢兹大学）；CNRS（法国国家科学研究中心）
We present a framework to derive upper bounds on the number of regions that feed-forward neural networks with ReLU activation functions are affine linear on. It is based on an inductive analysis that keeps track of the number of such regions per dimensionality of their images within the layers. More precisely, the information about the number regions per dimensionality is pushed through the layers starting with one region of the input dimension of the neural network and using a recursion based on an analysis of how many regions per output dimensionality a subsequent layer with a certain width …
Although for neural networks with locally Lipschitz continuous activation functions the classical derivative exists almost everywhere, the standard chain rule is in general not applicable. We will consider a way of introducing a derivative for neural networks that admits a chain rule, which is both rigorous and easy to work with. In addition we will present a method of converting approximation results on bounded domains to global (pointwise) estimates. This can be used to extend known neural network approximation theory to include the study of regularity properties. Of particular interest is t…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）
Training neural networks under a strict Lipschitz constraint is useful for provable adversarial robustness, generalization bounds, interpretable gradients, and Wasserstein distance estimation. By the composition property of Lipschitz functions, it suffices to ensure that each individual affine transformation or nonlinear activation is 1-Lipschitz. The challenge is to do this while maintaining the expressive power. We identify a necessary property for such an architecture: each of the layers must preserve the gradient norm during backpropagation. Based on this, we propose to combine a gradient …
California Institute of Technology（加州理工学院）；NVIDIA（英伟达）
Certified robustness is a desirable property for deep neural networks in safety-critical applications, and popular training algorithms can certify robustness of a neural network by computing a global bound on its Lipschitz constant. However, such a bound is often loose: it tends to over-regularize the neural network and degrade its natural accuracy. A tighter Lipschitz bound may provide a better tradeoff between natural and certified accuracy, but is generally hard to compute exactly due to non-convexity of the network. In this work, we propose an efficient and trainable \emph{local} Lipschitz…
Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense
Department of Mathematics & Informatics, University of Wuppertal（乌尔姆大学数学与信息学系）；School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen)（香港中文大学（深圳）数据科学学院及深圳大数据研究院）；Applied Mathematics: Institute for Analysis and Numerics, University of Münster（穆恩斯特大学应用数学系及分析与数值分析研究所）；Department of Mathematical Sciences, University of Arkansas（阿肯色大学数学科学系）；Center for Astrophysics, Space Physics, and Engineering Research, Baylor University（贝勒大学天体物理学、空间物理学与工程研究中心）
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.
The local Lipschitz constant of a neural network is a useful metric with applications in robustness, generalization, and fairness evaluation. We provide novel analytic results relating the local Lipschitz constant of nonsmooth vector-valued functions to a maximization over the norm of the generalized Jacobian. We present a sufficient condition for which backpropagation always returns an element of the generalized Jacobian, and reframe the problem over this broad class of functions. We show strong inapproximability results for estimating Lipschitz constants of ReLU networks, and then formulate …
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
Lipschitz-constrained neural networks have many applications in machine learning. Since designing and training expressive Lipschitz-constrained networks is very challenging, there is a need for improved methods and a better theoretical understanding. Unfortunately, it turns out that ReLU networks have provable disadvantages in this setting. Hence, we propose to use learnable spline activation functions with at least 3 linear regions instead. We prove that this choice is optimal among all component-wise $1$-Lipschitz activation functions in the sense that no other weight constrained architectur…
Basque Center for Applied Mathematics（巴斯克应用数学中心）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；University of Münster（明斯特大学）；University of Duisburg-Essen（杜伊斯堡-埃森大学）
The training of artificial neural networks (ANNs) is nowadays a highly relevant algorithmic procedure with many applications in science and industry. Roughly speaking, ANNs can be regarded as iterated compositions between affine linear functions and certain fixed nonlinear functions, which are usually multidimensional versions of a one-dimensional so-called activation function. The most popular choice of such a one-dimensional activation function is the rectified linear unit (ReLU) activation function which maps a real number to its positive part $ \mathbb{R} \ni x \mapsto \max\{ x, 0 \} \in \…
Nonlinearity is crucial to the performance of a deep (neural) network (DN). To date there has been little progress understanding the menagerie of available nonlinearities, but recently progress has been made on understanding the rôle played by piecewise affine and convex nonlinearities like the ReLU and absolute value activation functions and max-pooling. In particular, DN layers constructed from these operations can be interpreted as {\em max-affine spline operators} (MASOs) that have an elegant link to vector quantization (VQ) and $K$-means. While this is good theoretical progress, the entir…
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…
Several current bounds on the maximal number of affine regions of a ReLU feed-forward neural network are special cases of the framework [1] which relies on layer-wise activation histogram bounds. We analyze and partially solve a problem in algebraic topology the solution of which would fully exploit this framework. Our partial solution already induces slightly tighter bounds and suggests insight in how parameter initialization methods can affect the number of regions. Furthermore, we extend the framework to allow the composition of subnetwork instead of layer-wise activation histogram bounds t…
Ens de Lyon（里昂高等师范学院）；CNRS（法国国家科学研究中心）；Université Claude Bernard Lyon 1（里昂第一大学（克洛德·贝尔纳））；Inria（法国国家信息与自动化研究所）；Université de Toulouse（图卢兹大学）；IRIT（图卢兹计算机科学、集成与系统实验室）
Verification of neural networks relies on activation functions being piecewise affine (pwa) -- enabling an encoding of the verification problem for theorem provers. In this paper, we present the first formalization of pwa activation functions for an interactive theorem prover tailored to verifying neural networks within Coq using the library Coquelicot for real analysis. As a proof-of-concept, we construct the popular pwa activation function ReLU. We integrate our formalization into a Coq model of neural networks, and devise a verified transformation from a neural network N to a pwa function r…
An Accelerated Alternating Partial Bregman Algorithm for ReLU-based Matrix Decomposition
School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）
LAAS-CNRS（法国国家科学研究中心自动化、计算机工程、信号与机器人实验室）；Université de Toulouse（图卢兹大学）；IMT（图卢兹理工学院）；IRIT（图卢兹计算机研究所）
The Lipschitz constant of a network plays an important role in many applications of deep learning, such as robustness certification and Wasserstein Generative Adversarial Network. We introduce a semidefinite programming hierarchy to estimate the global and local Lipschitz constant of a multiple layer deep neural network. The novelty is to combine a polynomial lifting for ReLU functions derivatives with a weak generalization of Putinar's positivity certificate. This idea could also apply to other, nearly sparse, polynomial optimization problems in machine learning. We empirically demonstrate th…
University of Southern California（南加州大学）；Toyota Research Institute North America（丰田北美研究院）
Signal Temporal Logic (STL) has become a popular tool for expressing formal requirements of Cyber-Physical Systems (CPS). The problem of verifying STL properties of neural network-controlled CPS remains a largely unexplored problem. In this paper, we present a model for the verification of Neural Network (NN) controllers for general STL specifications using a custom neural architecture where we map an STL formula into a feed-forward neural network with ReLU activation. In the case where both our plant model and the controller are ReLU-activated neural networks, we reduce the STL verification p…
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…
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 …
On bounds for norms of reparameterized ReLU artificial neural network parameters: sums of fractional powers of the Lipschitz norm control the network parameter vector
School of Data Science and School of Artificial Intelligence, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen)（数据科学学院和人工智能学院，香港中文大学（深圳））；Applied Mathematics: Institute for Analysis and Numerics, Faculty of Mathematics and Computer Science, University of Münster（应用数学：分析与数值研究所，数学与计算机科学学院，穆恩斯特大学）
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…
Recent advances in adversarial attacks and Wasserstein GANs have advocated for use of neural networks with restricted Lipschitz constants. Motivated by these observations, we study the recently introduced GroupSort neural networks, with constraints on the weights, and make a theoretical step towards a better understanding of their expressive power. We show in particular how these networks can represent any Lipschitz continuous piecewise linear functions. We also prove that they are well-suited for approximating Lipschitz continuous functions and exhibit upper bounds on both the depth and size.…
In this note, we study how neural networks with a single hidden layer and ReLU activation interpolate data drawn from a radially symmetric distribution with target labels 1 at the origin and 0 outside the unit ball, if no labels are known inside the unit ball. With weight decay regularization and in the infinite neuron, infinite data limit, we prove that a unique radially symmetric minimizer exists, whose weight decay regularizer and Lipschitz constant grow as $d$ and $\sqrt{d}$ respectively. We furthermore show that the weight decay regularizer grows exponentially in $d$ if the label $1$ is i…
We present an explicit deep neural network construction that transforms uniformly distributed one-dimensional noise into an arbitrarily close approximation of any two-dimensional Lipschitz-continuous target distribution. The key ingredient of our design is a generalization of the "space-filling" property of sawtooth functions discovered in (Bailey & Telgarsky, 2018). We elicit the importance of depth - in our neural network construction - in driving the Wasserstein distance between the target distribution and the approximation realized by the network to zero. An extension to output distributio…
A ReLU neural network determines/is a continuous piecewise linear map from an input space to an output space. The weights in the neural network determine a decomposition of the input space into convex polytopes and on each of these polytopes the network can be described by a single affine mapping. The structure of the decomposition, together with the affine map attached to each polytope, can be analyzed to investigate the behavior of the associated neural network.
Exact ReLU realization of affine one-dimensional refinement iterates via residual memory and offset frames
National University of Mongolia（蒙古国国立大学）；McGill University（麦吉尔大学）；Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences（蒙古科学院数学与数字技术研究所）
Yale-NUS College（耶鲁-新加坡国立大学学院）；Institute of Information Science, Academia Sinica（中央研究院资讯科学研究所）
We consider deep feedforward neural networks with rectified linear units from a signal processing perspective. In this view, such representations mark the transition from using a single (data-driven) linear representation to utilizing a large collection of affine linear representations tailored to particular regions of the signal space. This paper provides a precise description of the individual affine linear representations and corresponding domain regions that the (data-driven) neural network associates to each signal of the input space. In particular, we describe atomic decompositions of th…
This paper considers the following question: how well can depth-two ReLU networks with randomly initialized bottom-level weights represent smooth functions? We give near-matching upper- and lower-bounds for $L_2$-approximation in terms of the Lipschitz constant, the desired accuracy, and the dimension of the problem, as well as similar results in terms of Sobolev norms. Our positive results employ tools from harmonic analysis and ridgelet representation theory, while our lower-bounds are based on (robust versions of) dimensionality arguments.
ETH Zurich（苏黎世联邦理工学院）；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 paper, we analyze the landscape of the true loss of neural networks with one hidden layer and ReLU, leaky ReLU, or quadratic activation. In all three cases, we provide a complete classification of the critical points in the case where the target function is affine and one-dimensional. In particular, we show that there exist no local maxima and clarify the structure of saddle points. Moreover, we prove that non-global local minima can only be caused by `dead' ReLU neurons. In particular, they do not appear in the case of leaky ReLU or quadratic activation. Our approach is of a combinato…
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…
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…
Exact ReLU realization of binary affine refinement iterates via reflection folding and cone switching
McGill University（麦吉尔大学）；National University of Mongolia（蒙古国立大学）；Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences（蒙古科学院数学与数字技术研究所）
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, …
