In the present study, we investigate a universality of neural networks, which concerns a density of the set of two-layer neural networks in a function spaces. There are many works that handle the convergence over compact sets. In the present paper, we consider a global convergence by introducing a norm suitably, so that our results will be uniform over any compact set.
Old Dominion University（老道明大学）；Sun Yat-sen University（中山大学）
We explore convergence of deep neural networks with the popular ReLU activation function, as the depth of the networks tends to infinity. To this end, we introduce the notion of activation domains and activation matrices of a ReLU network. By replacing applications of the ReLU activation function by multiplications with activation matrices on activation domains, we obtain an explicit expression of the ReLU network. We then identify the convergence of the ReLU networks as convergence of a class of infinite products of matrices. Sufficient and necessary conditions for convergence of these infini…
Deep neural networks, as a powerful system to represent high dimensional complex functions, play a key role in deep learning. Convergence of deep neural networks is a fundamental issue in building the mathematical foundation for deep learning. We investigated the convergence of deep ReLU networks and deep convolutional neural networks in two recent researches (arXiv:2107.12530, 2109.13542). Only the Rectified Linear Unit (ReLU) activation was studied therein, and the important pooling strategy was not considered. In this current work, we study the convergence of deep neural networks as the dep…
In recent years novel activation functions have been proposed to improve the performance of neural networks, and they show superior performance compared to the ReLU counterpart. However, there are environments, where the availability of complex activations is limited, and usually only the ReLU is supported. In this paper we propose methods that can be used to improve the performance of ReLU networks by using these efficient novel activations during model training. More specifically, we propose ensemble activations that are composed of the ReLU and one of these novel activations. Furthermore, t…
Brownian ReLU(Br-ReLU): A New Activation Function for a Long-Short Term Memory (LSTM) Network
Department of Statistics and Actuarial Science, Kwame Nkrumah University of Science and Technology（统计与精算科学系，库马西大学科学技术学院）；Department of Mathematics, Toronto Metropolitan University（数学系，多伦多 metropolitan 大学）
Toyota Technological Institute at Chicago（芝加哥丰田技术研究院）；University of Toronto（多伦多大学）；Carnegie Mellon University（卡内基梅隆大学）
We investigate the parameter-space geometry of recurrent neural networks (RNNs), and develop an adaptation of path-SGD optimization method, attuned to this geometry, that can learn plain RNNs with ReLU activations. On several datasets that require capturing long-term dependency structure, we show that path-SGD can significantly improve trainability of ReLU RNNs compared to RNNs trained with SGD, even with various recently suggested initialization schemes.
Department of Mathematics, Texas A\&M, College Station, United States
This article concerns the expressive power of depth in neural nets with ReLU activations and bounded width. We are particularly interested in the following questions: what is the minimal width $w_{\text{min}}(d)$ so that ReLU nets of width $w_{\text{min}}(d)$ (and arbitrary depth) can approximate any continuous function on the unit cube $[0,1]^d$ aribitrarily well? For ReLU nets near this minimal width, what can one say about the depth necessary to approximate a given function? Our approach to this paper is based on the observation that, due to the convexity of the ReLU activation, ReLU nets a…
Overwhelming theoretical and empirical evidence shows that mildly overparametrized neural networks -- those with more connections than the size of the training data -- are often able to memorize the training data with $100\%$ accuracy. This was rigorously proved for networks with sigmoid activation functions and, very recently, for ReLU activations. Addressing a 1988 open question of Baum, we prove that this phenomenon holds for general multilayered perceptrons, i.e. neural networks with threshold activation functions, or with any mix of threshold and ReLU activations. Our construction is prob…
Indian Institute Of Technology–Roorkee（印度理工学院鲁尔基分校）
Activation functions have come up as one of the essential components of neural networks. The choice of adequate activation function can impact the accuracy of these methods. In this study, we experiment for finding an optimal activation function by defining it as a weighted sum of existing activation functions and then further optimizing these weights while training the network. The study uses three activation functions, ReLU, tanh, and sin, over three popular image datasets, MNIST, FashionMNIST, and KMNIST. We observe that the ReLU activation function can easily overlook other activation func…
London South Bank University（伦敦南岸大学）；IST College（IST学院）
Despite the unresolved 'dying ReLU problem', the classical ReLU activation function (AF) has been extensively applied in Deep Neural Networks (DNN), in particular Convolutional Neural Networks (CNN), for image classification. The common gradient issues of ReLU pose challenges in applications on academy and industry sectors. Recent approaches for improvements are in a similar direction by just proposing variations of the AF, such as Leaky ReLU (LReLU), while maintaining the solution within the same unresolved gradient problems. In this paper, the Absolute Leaky ReLU (ALReLU) AF, a variation of …
Oak Ridge National Laboratory（橡树岭国家实验室）；University of Tennessee at Knoxville（田纳西大学诺克斯维尔分校）
In this effort, we derive a formula for the integral representation of a shallow neural network with the ReLU activation function. We assume that the outer weighs admit a finite $L_1$-norm with respect to Lebesgue measure on the sphere. For univariate target functions we further provide a closed-form formula for all possible representations. Additionally, in this case our formula allows one to explicitly solve the least $L_1$-norm neural network representation for a given function.
ReLU$^2$ Wins: Discovering Efficient Activation Functions for Sparse LLMs
Tsinghua University（清华大学）；Shanghai Jiao Tong University（上海交通大学）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；Renmin University of China（中国人民大学）
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…
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…
Google Research（谷歌研究院）；Northwestern University（西北大学）
We present polynomial time and sample efficient algorithms for learning an unknown depth-2 feedforward neural network with general ReLU activations, under mild non-degeneracy assumptions. In particular, we consider learning an unknown network of the form $f(x) = {a}^{\mathsf{T}}σ({W}^\mathsf{T}x+b)$, where $x$ is drawn from the Gaussian distribution, and $σ(t) := \max(t,0)$ is the ReLU activation. Prior works for learning networks with ReLU activations assume that the bias $b$ is zero. In order to deal with the presence of the bias terms, our proposed algorithm consists of robustly decomposing…
SwishReLU: A Unified Approach to Activation Functions for Enhanced Deep Neural Networks Performance
Jiangsu University（江苏大学）；Abdus Salam School of Mathematical Sciences, Govt College University（政府学院大学阿卜杜斯·萨拉姆数学科学学院）；Metaverse Research Institute, School of Computer Science and Cyber Engineering, Guangzhou University（广州大学计算机科学与网络工程学院元宇宙研究院）
Texas Southern University（德克萨斯南方大学）；Prairie View A & M University（Prairie View A & M 大学）
The nonlinearity of activation functions used in deep learning models are crucial for the success of predictive models. There are several commonly used simple nonlinear functions, including Rectified Linear Unit (ReLU) and Leaky-ReLU (L-ReLU). In practice, these functions remarkably enhance the model accuracy. However, there is limited insight into the functionality of these nonlinear activation functions in terms of why certain models perform better than others. Here, we investigate the model performance when using ReLU or L-ReLU as activation functions in different model architectures and da…
Well-known activation functions like ReLU or Leaky ReLU are non-differentiable at the origin. Over the years, many smooth approximations of ReLU have been proposed using various smoothing techniques. We propose new smooth approximations of a non-differentiable activation function by convolving it with approximate identities. In particular, we present smooth approximations of Leaky ReLU and show that they outperform several well-known activation functions in various datasets and models. We call this function Smooth Activation Unit (SAU). Replacing ReLU by SAU, we get 5.12% improvement with Shuf…
Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore（数学科学系，物理与数学科学学院，南洋理工大学，新加坡）；Faculty of Mathematics, Bielefeld University, Germany（数学学院，比勒菲尔德大学，德国）
York University（约克大学）；Department of Electrical Engineering and Computer Science（电气工程与计算机科学系）
In this paper, we have extended the well-established universal approximator theory to neural networks that use the unbounded ReLU activation function and a nonlinear softmax output layer. We have proved that a sufficiently large neural network using the ReLU activation function can approximate any function in $L^1$ up to any arbitrary precision. Moreover, our theoretical results have shown that a large enough neural network using a nonlinear softmax output layer can also approximate any indicator function in $L^1$, which is equivalent to mutually-exclusive class labels in any realistic multipl…
Technische Universität Berlin（柏林工业大学）；University of Oxford（牛津大学）
We analyze approximation rates of deep ReLU neural networks for Sobolev-regular functions with respect to weaker Sobolev norms. First, we construct, based on a calculus of ReLU networks, artificial neural networks with ReLU activation functions that achieve certain approximation rates. Second, we establish lower bounds for the approximation by ReLU neural networks for classes of Sobolev-regular functions. Our results extend recent advances in the approximation theory of ReLU networks to the regime that is most relevant for applications in the numerical analysis of partial differential equation…
University of California, Berkeley（加州大学伯克利分校）；Google（谷歌）
We establish conditions under which gradient descent applied to fixed-width deep networks drives the logistic loss to zero, and prove bounds on the rate of convergence. Our analysis applies for smoothed approximations to the ReLU, such as Swish and the Huberized ReLU, proposed in previous applied work. We provide two sufficient conditions for convergence. The first is simply a bound on the loss at initialization. The second is a data separation condition used in prior analyses.
National Digital Switching System Engineering and Technological R&D Center（国家数字交换系统工程技术研究中心）
Activation functions play a key role in providing remarkable performance in deep neural networks, and the rectified linear unit (ReLU) is one of the most widely used activation functions. Various new activation functions and improvements on ReLU have been proposed, but each carry performance drawbacks. In this paper, we propose an improved activation function, which we name the natural-logarithm-rectified linear unit (NLReLU). This activation function uses the parametric natural logarithmic transform to improve ReLU and is simply defined as. NLReLU not only retains the sparse activation charac…
The choice of activation functions in deep networks has a significant effect on the training dynamics and task performance. Currently, the most successful and widely-used activation function is the Rectified Linear Unit (ReLU). Although various hand-designed alternatives to ReLU have been proposed, none have managed to replace it due to inconsistent gains. In this work, we propose to leverage automatic search techniques to discover new activation functions. Using a combination of exhaustive and reinforcement learning-based search, we discover multiple novel activation functions. We verify the …
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
In the era of Deep Neural Network based solutions for a variety of real-life tasks, having a compact and energy-efficient deployable model has become fairly important. Most of the existing deep architectures use Rectifier Linear Unit (ReLU) activation. In this paper, we propose a novel idea of rotating the ReLU activation to give one more degree of freedom to the architecture. We show that this activation wherein the rotation is learned via training results in the elimination of those parameters/filters in the network which are not important for the task. In other words, rotated ReLU seems to …
MINES Paris（巴黎矿业学院）；PSL University（巴黎文理研究大学）；Centre for mathematical morphology (CMM)（数学形态学中心（CMM））
This paper analyses both nonlinear activation functions and spatial max-pooling for Deep Convolutional Neural Networks (DCNNs) by means of the algebraic basis of mathematical morphology. Additionally, a general family of activation functions is proposed by considering both max-pooling and nonlinear operators in the context of morphological representations. Experimental section validates the goodness of our approach on classical benchmarks for supervised learning by DCNN.
Neural networks have been widely used to solve complex real-world problems. Due to the complicate, nonlinear, non-convex nature of neural networks, formal safety guarantees for the output behaviors of neural networks will be crucial for their applications in safety-critical systems.In this paper, the output reachable set computation and safety verification problems for a class of neural networks consisting of Rectified Linear Unit (ReLU) activation functions are addressed. A layer-by-layer approach is developed to compute output reachable set. The computation is formulated in the form of a set…
Deep learning researchers have a keen interest in proposing two new novel activation functions which can boost network performance. A good choice of activation function can have significant consequences in improving network performance. A handcrafted activation is the most common choice in neural network models. ReLU is the most common choice in the deep learning community due to its simplicity though ReLU has some serious drawbacks. In this paper, we have proposed a new novel activation function based on approximation of known activation functions like Leaky ReLU, and we call this function Sm…
We study the approximation properties of random ReLU features through their reproducing kernel Hilbert space (RKHS). We first prove a universality theorem for the RKHS induced by random features whose feature maps are of the form of nodes in neural networks. The universality result implies that the random ReLU features method is a universally consistent learning algorithm. We prove that despite the universality of the RKHS induced by the random ReLU features, composition of functions in it generates substantially more complicated functions that are harder to approximate than those functions si…
Activation function is a key component in deep learning that performs non-linear mappings between the inputs and outputs. Rectified Linear Unit (ReLU) has been the most popular activation function across the deep learning community. However, ReLU contains several shortcomings that can result in inefficient training of the deep neural networks, these are: 1) the negative cancellation property of ReLU tends to treat negative inputs as unimportant information for the learning, resulting in a performance degradation; 2) the inherent predefined nature of ReLU is unlikely to promote additional flexi…
Facebook AI Research（Facebook人工智能研究院）；Texas A&M University（得克萨斯农工大学）；University of Pennsylvania（宾夕法尼亚大学）
The success of deep networks has been attributed in part to their expressivity: per parameter, deep networks can approximate a richer class of functions than shallow networks. In ReLU networks, the number of activation patterns is one measure of expressivity; and the maximum number of patterns grows exponentially with the depth. However, recent work has showed that the practical expressivity of deep networks - the functions they can learn rather than express - is often far from the theoretical maximum. In this paper, we show that the average number of activation patterns for ReLU networks at i…
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 …
We propose a novel Shapley value approach to help address neural networks' interpretability and "vanishing gradient" problems. Our method is based on an accurate analytical approximation to the Shapley value of a neuron with ReLU activation. This analytical approximation admits a linear propagation of relevance across neural network layers, resulting in a simple, fast and sensible interpretation of neural networks' decision making process. We then derived a globally continuous and non-vanishing Shapley gradient, which can replace the conventional gradient in training neural network layers with…
Deep networks are gradually penetrating almost every domain in our lives due to their amazing success. However, with substantive performance accuracy improvements comes the price of \emph{irreproducibility}. Two identical models, trained on the exact same training dataset may exhibit large differences in predictions on individual examples even when average accuracy is similar, especially when trained on highly distributed parallel systems. The popular Rectified Linear Unit (ReLU) activation has been key to recent success of deep networks. We demonstrate, however, that ReLU is also a catalyzer …
Topological Signatures of ReLU Neural Network Activation Patterns
Applied Math and Computational Sciences University of Pennsylvania（应用数学与计算科学大学公园大学）；Department of Mathematics Colorado State University（数学系科罗拉多州立大学）；Departments of Mechanical Engineering & Computational Mathematics, Sciences and Engineering (CMSE) Michigan State University（机械工程与计算数学、科学与工程系密歇根州立大学）；Department of Mathematics University of Washington（数学系华盛顿大学）；Georgia Tech Research Institute（佐治亚理工研究学院）
Deep neural networks paved the way for significant improvements in image visual categorization during the last years. However, even though the tasks are highly varying, differing in complexity and difficulty, existing solutions mostly build on the same architectural decisions. This also applies to the selection of activation functions (AFs), where most approaches build on Rectified Linear Units (ReLUs). In this paper, however, we show that the choice of a proper AF has a significant impact on the classification accuracy, in particular, if fine, subtle details are of relevance. Therefore, we pr…
