Understanding the loss surface of a neural network is fundamentally important to the understanding of deep learning. This paper presents how piecewise linear activation functions substantially shape the loss surfaces of neural networks. We first prove that {\it the loss surfaces of many neural networks have infinite spurious local minima} which are defined as the local minima with higher empirical risks than the global minima. Our result demonstrates that the networks with piecewise linear activations possess substantial differences to the well-studied linear neural networks. This result holds for any neural network with arbitrary depth and arbitrary piecewise linear activation functions (excluding linear functions) under most loss functions in practice. Essentially, the underlying assumptions are consistent with most practical circumstances where the output layer is narrower than any hidden layer. In addition, the loss surface of a neural network with piecewise linear activations is partitioned into multiple smooth and multilinear cells by nondifferentiable boundaries. The constructed spurious local minima are concentrated in one cell as a valley: they are connected with each other by a continuous path, on which empirical risk is invariant. Further for one-hidden-layer networks, we prove that all local minima in a cell constitute an equivalence class; they are concentrated in a valley; and they are all global minima in the cell.
College of Computer Science, Faculty of Information Technology（信息技术学院计算机学院）；Beijing University of Technology（北京工业大学）
In this paper, it is shown theoretically that spurious local minima are common for deep fully-connected networks and convolutional neural networks (CNNs) with piecewise linear activation functions and datasets that cannot be fitted by linear models. A motivating example is given to explain the reason for the existence of spurious local minima: each output neuron of deep fully-connected networks and CNNs with piecewise linear activations produces a continuous piecewise linear (CPWL) output, and different pieces of CPWL output can fit disjoint groups of data samples when minimizing the empirical…
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…
Simon Fraser University（西蒙弗雷泽大学）；Microsoft Research（微软研究院）
It is fundamental to measure model complexity of deep neural networks. The existing literature on model complexity mainly focuses on neural networks with piecewise linear activation functions. Model complexity of neural networks with general curve activation functions remains an open problem. To tackle the challenge, in this paper, we first propose the linear approximation neural network (LANN for short), a piecewise linear framework to approximate a given deep model with curve activation function. LANN constructs individual piecewise linear approximation for the activation function of each ne…
ARC Centre of Excellence for Robotic Vision（ARC机器人视觉卓越中心）；University of Adelaide（阿德莱德大学）
Deep feedforward neural networks with piecewise linear activations are currently producing the state-of-the-art results in several public datasets. The combination of deep learning models and piecewise linear activation functions allows for the estimation of exponentially complex functions with the use of a large number of subnetworks specialized in the classification of similar input examples. During the training process, these subnetworks avoid overfitting with an implicit regularization scheme based on the fact that they must share their parameters with other subnetworks. Using this framewo…
We present a technique for neural network verification using mixed-integer programming (MIP) formulations. We derive a \emph{strong formulation} for each neuron in a network using piecewise linear activation functions. Additionally, as in general, these formulations may require an exponential number of inequalities, we also derive a separation procedure that runs in super-linear time in the input dimension. We first introduce and develop our technique on the class of \emph{staircase} functions, which generalizes the ReLU, binarized, and quantized activation functions. We then use results for s…
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…
Department of Electrical Engineering, Tsinghua University（清华大学电子工程系）
The developments of deep neural networks (DNN) in recent years have ushered a brand new era of artificial intelligence. DNNs are proved to be excellent in solving very complex problems, e.g., visual recognition and text understanding, to the extent of competing with or even surpassing people. Despite inspiring and encouraging success of DNNs, thorough theoretical analyses still lack to unravel the mystery of their magics. The design of DNN structure is dominated by empirical results in terms of network depth, number of neurons and activations. A few of remarkable works published recently in an…
Norwegian University of Science and Technology（挪威科技大学）；SINTEF Digital（SINTEF数字）；Oklahoma State University（俄克拉荷马州立大学）
In exchange for large quantities of data and processing power, deep neural networks have yielded models that provide state of the art predication capabilities in many fields. However, a lack of strong guarantees on their behaviour have raised concerns over their use in safety-critical applications. A first step to understanding these networks is to develop alternate representations that allow for further analysis. It has been shown that neural networks with piecewise affine activation functions are themselves piecewise affine, with their domains consisting of a vast number of linear regions. S…
We prove new upper and lower bounds on the VC-dimension of deep neural networks with the ReLU activation function. These bounds are tight for almost the entire range of parameters. Letting $W$ be the number of weights and $L$ be the number of layers, we prove that the VC-dimension is $O(W L \log(W))$, and provide examples with VC-dimension $Ω( W L \log(W/L) )$. This improves both the previously known upper bounds and lower bounds. In terms of the number $U$ of non-linear units, we prove a tight bound $Θ(W U)$ on the VC-dimension. All of these bounds generalize to arbitrary piecewise linear act…
The activation function is at the heart of a deep neural networks nonlinearity; the choice of the function has great impact on the success of training. Currently, many practitioners prefer the Rectified Linear Unit (ReLU) due to its simplicity and reliability, despite its few drawbacks. While most previous functions proposed to supplant ReLU have been hand-designed, recent work on learning the function during training has shown promising results. In this paper we propose an adaptive piecewise linear activation function, the Piecewise Linear Unit (PiLU), which can be learned independently for e…
University of California - Irvine（加州大学尔湾分校）；Adobe Research（奥多比研究院）
Artificial neural networks typically have a fixed, non-linear activation function at each neuron. We have designed a novel form of piecewise linear activation function that is learned independently for each neuron using gradient descent. With this adaptive activation function, we are able to improve upon deep neural network architectures composed of static rectified linear units, achieving state-of-the-art performance on CIFAR-10 (7.51%), CIFAR-100 (30.83%), and a benchmark from high-energy physics involving Higgs boson decay modes.
Samsung AI Centre Toronto（多伦多三星人工智能中心）；University of Toronto（多伦多大学）；Vector Institute for AI（矢量人工智能研究所）
Modern generative adversarial networks (GANs) predominantly use piecewise linear activation functions in discriminators (or critics), including ReLU and LeakyReLU. Such models learn piecewise linear mappings, where each piece handles a subset of the input space, and the gradients per subset are piecewise constant. Under such a class of discriminator (or critic) functions, we present Gradient Normalization (GraN), a novel input-dependent normalization method, which guarantees a piecewise K-Lipschitz constraint in the input space. In contrast to spectral normalization, GraN does not constrain pr…
Today, it is more important than ever before for users to have trust in the models they use. As Machine Learning models fall under increased regulatory scrutiny and begin to see more applications in high-stakes situations, it becomes critical to explain our models. Piecewise Linear Neural Networks (PLNN) with the ReLU activation function have quickly become extremely popular models due to many appealing properties; however, they still present many challenges in the areas of robustness and interpretation. To this end, we introduce novel methodology toward simplification and increased interpreta…
Certified defenses against small-norm adversarial examples have received growing attention in recent years; though certified accuracies of state-of-the-art methods remain far below their non-robust counterparts, despite the fact that benchmark datasets have been shown to be well-separated at far larger radii than the literature generally attempts to certify. In this work, we offer insights that identify potential factors in this performance gap. Specifically, our analysis reveals that piecewise linearity imposes fundamental limitations on the tightness of leading certification techniques. Thes…
This paper shows that every sublevel set of the loss function of a class of deep over-parameterized neural nets with piecewise linear activation functions is connected and unbounded. This implies that the loss has no bad local valleys and all of its global minima are connected within a unique and potentially very large global valley.
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…
The recently introduced dropout training criterion for neural networks has been the subject of much attention due to its simplicity and remarkable effectiveness as a regularizer, as well as its interpretation as a training procedure for an exponentially large ensemble of networks that share parameters. In this work we empirically investigate several questions related to the efficacy of dropout, specifically as it concerns networks employing the popular rectified linear activation function. We investigate the quality of the test time weight-scaling inference procedure by evaluating the geometri…
Pennsylvania State University（宾夕法尼亚州立大学）；University of Southern California（南加州大学）
Neural networks are universal function approximators which are known to generalize well despite being dramatically overparameterized. We study this phenomenon from the point of view of the spectral bias of neural networks. Our contributions are two-fold. First, we provide a theoretical explanation for the spectral bias of ReLU neural networks by leveraging connections with the theory of finite element methods. Second, based upon this theory we predict that switching the activation function to a piecewise linear B-spline, namely the Hat function, will remove this spectral bias, which we verify …
Cornell University（康奈尔大学）；National Technical University of Athens（雅典国立技术大学）
We present a new, unifying approach following some recent developments on the complexity of neural networks with piecewise linear activations. We treat neural network layers with piecewise linear activations as tropical polynomials, which generalize polynomials in the so-called $(\max, +)$ or tropical algebra, with possibly real-valued exponents. Motivated by the discussion in (arXiv:1402.1869), this approach enables us to refine their upper bounds on linear regions of layers with ReLU or leaky ReLU activations to $\min\left\{ 2^m, \sum_{j=0}^n \binom{m}{j} \right\}$, where $n, m$ are the numb…
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）；Université de Montréal（蒙特利尔大学）
We study the complexity of functions computable by deep feedforward neural networks with piecewise linear activations in terms of the symmetries and the number of linear regions that they have. Deep networks are able to sequentially map portions of each layer's input-space to the same output. In this way, deep models compute functions that react equally to complicated patterns of different inputs. The compositional structure of these functions enables them to re-use pieces of computation exponentially often in terms of the network's depth. This paper investigates the complexity of such composi…
The approximation power of general feedforward neural networks with piecewise linear activation functions is investigated. First, lower bounds on the size of a network are established in terms of the approximation error and network depth and width. These bounds improve upon state-of-the-art bounds for certain classes of functions, such as strongly convex functions. Second, an upper bound is established on the difference of two neural networks with identical weights but different activation functions.
Deep neural networks (DNNs) generate much richer function spaces than shallow networks. Since the function spaces induced by shallow networks have several approximation theoretic drawbacks, this explains, however, not necessarily the success of deep networks. In this article we take another route by comparing the expressive power of DNNs with ReLU activation function to piecewise linear spline methods. We show that MARS (multivariate adaptive regression splines) is improper learnable by DNNs in the sense that for any given function that can be expressed as a function in MARS with $M$ parameter…
AffineLens: Capturing the Continuous Piecewise Affine Functions of Neural Networks
State Key Laboratory of Novel Software Technology School of Intelligence Science and Technology Nanjing University, Jiangsu, China（软件新技术国家重点实验室 智能科学与技术学院 南京大学 江苏 中国）；AI for Good Istituto Italiano di Tecnologia, Genoa, Italy DITEN University of Genoa, Genoa, Italy（AI for Good 意大利技术研究院，热那亚，意大利 DITEN 热那亚大学，热那亚，意大利）；State Key Laboratory of Novel Software Technology School of Artificial Intelligence Nanjing University, Jiangsu, China（软件新技术国家重点实验室 人工智能学院 南京大学 江苏 中国）；AI for Good Istituto Italiano di Tecnologia, Genoa, Italy Department of Computer Science University of Verona, Verona, Italy（AI for Good 意大利技术研究院，热那亚，意大利 计算机科学系 热那亚大学，热那亚，意大利）
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
We propose to optimize the activation functions of a deep neural network by adding a corresponding functional regularization to the cost function. We justify the use of a second-order total-variation criterion. This allows us to derive a general representer theorem for deep neural networks that makes a direct connection with splines and sparsity. Specifically, we show that the optimal network configuration can be achieved with activation functions that are nonuniform linear splines with adaptive knots. The bottom line is that the action of each neuron is encoded by a spline whose parameters (i…
University of Milan-Bicocca（米兰比可卡大学）；Consiglio Nazionale delle Ricerche (IBFM-CNR)（意大利国家研究委员会（分子生物成像与生理学研究所））
The current understanding of deep neural networks can only partially explain how input structure, network parameters and optimization algorithms jointly contribute to achieve the strong generalization power that is typically observed in many real-world applications. In order to improve the comprehension and interpretability of deep neural networks, we here introduce a novel theoretical framework based on the compositional structure of piecewise linear activation functions. By defining a direct acyclic graph representing the composition of activation patterns through the network layers, it is p…
We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in $L^2$. As a model class, we consider the set $\mathcal{E}^β(\mathbb R^d)$ of possibly discontinuous piecewise $C^β$ functions $f : [-1/2, 1/2]^d \to \mathbb R$, where the different smooth regions of $f$ are separated by $C^β$ hypersurfaces. For dimension $d \geq 2$, regularity $β> 0$, and accuracy $\varepsilon > 0$, we construct artificial neural networks with ReLU activation function that approximate functions from $\m…
Memory footprint is one of the main limiting factors for large neural network training. In backpropagation, one needs to store the input to each operation in the computational graph. Every modern neural network model has quite a few pointwise nonlinearities in its architecture, and such operation induces additional memory costs which -- as we show -- can be significantly reduced by quantization of the gradients. We propose a systematic approach to compute optimal quantization of the retained gradients of the pointwise nonlinear functions with only a few bits per each element. We show that such…
Although artificial neural networks have shown great promise in applications including computer vision and speech recognition, there remains considerable practical and theoretical difficulty in optimizing their parameters. The seemingly unreasonable success of gradient descent methods in minimizing these non-convex functions remains poorly understood. In this work we offer some theoretical guarantees for networks with piecewise affine activation functions, which have in recent years become the norm. We prove three main results. Firstly, that the network is piecewise convex as a function of the…
Indiana University（印第安纳大学）；Georgia Institute of Technology（佐治亚理工学院）
The number of linear regions is one of the distinct properties of the neural networks using piecewise linear activation functions such as ReLU, comparing with those conventional ones using other activation functions. Previous studies showed this property reflected the expressivity of a neural network family ([14]); as a result, it can be used to characterize how the structural complexity of a neural network model affects the function it aims to compute. Nonetheless, it is challenging to directly compute the number of linear regions; therefore, many researchers focus on estimating the bounds (i…
Rational Neural Networks have Expressivity Advantages
Center for Applied Mathematics, Cornell University, United States（应用数学中心，康奈尔大学，美国）；Department of Mathematics, Cornell University, United States（数学系，康奈尔大学，美国）
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）
We explore the phase diagram of approximation rates for deep neural networks and prove several new theoretical results. In particular, we generalize the existing result on the existence of deep discontinuous phase in ReLU networks to functional classes of arbitrary positive smoothness, and identify the boundary between the feasible and infeasible rates. Moreover, we show that all networks with a piecewise polynomial activation function have the same phase diagram. Next, we demonstrate that standard fully-connected architectures with a fixed width independent of smoothness can adapt to smoothne…
This paper presents a new reachability analysis approach to compute interval over-approximations of the output set of feedforward neural networks with input uncertainty. We adapt to neural networks an existing mixed-monotonicity method for the reachability analysis of dynamical systems and apply it to each partial network within the main network. This ensures that the intersection of the obtained results is the tightest interval over-approximation of the output of each layer that can be obtained using mixed-monotonicity on any partial network decomposition. Unlike other tools in the literature…
