我们研究深度ReLU神经网络在膨胀平移不变空间中逼近函数的表达能力，这类空间广泛应用于信号处理、图像处理、通信等领域。逼近误差界根据神经网络的宽度和深度进行估计。网络构造基于深度神经网络的比特提取和数据拟合能力。作为我们主要结果的应用，获得了经典函数空间（如Sobolev空间和Besov空间）的逼近速率。我们还给出了Sobolev空间在$L^p (1\le p \le \infty)$范数下逼近误差的下界，这表明我们的神经网络构造在相差一个对数因子的意义下是渐近最优的。
英文摘要：
We study the expressive power of deep ReLU neural networks for approximating functions in dilated shift-invariant spaces, which are widely used in signal processing, image processing, communications and so on. Approximation error bounds are estimated with respect to the width and depth of neural networks. The network construction is based on the bit extraction and data-fitting capacity of deep neural networks. As applications of our main results, the approximation rates of classical function spaces such as Sobolev spaces and Besov spaces are obtained. We also give lower bounds of the $L^p (1\le p \le \infty)$ approximation error for Sobolev spaces, which show that our construction of neural network is asymptotically optimal up to a logarithmic factor.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Microsoft Research（微软研究院）
It was empirically confirmed by Keskar et al.\cite{SharpMinima} that flatter minima generalize better. However, for the popular ReLU network, sharp minimum can also generalize well \cite{SharpMinimacan}. The conclusion demonstrates that the existing definitions of flatness fail to account for the complex geometry of ReLU neural networks because they can't cover the Positively Scale-Invariant (PSI) property of ReLU network. In this paper, we formalize the PSI causes problem of existing definitions of flatness and propose a new description of flatness - \emph{PSI-flatness}. PSI-flatness is defin…
Microsoft Research Asia（微软亚洲研究院）；University of Chinese Academy of Sciences（中国科学院大学）；University of Science and Technology of China（中国科学技术大学）
It is well known that neural networks with rectified linear units (ReLU) activation functions are positively scale-invariant. Conventional algorithms like stochastic gradient descent optimize the neural networks in the vector space of weights, which is, however, not positively scale-invariant. This mismatch may lead to problems during the optimization process. Then, a natural question is: \emph{can we construct a new vector space that is positively scale-invariant and sufficient to represent ReLU neural networks so as to better facilitate the optimization process }? In this paper, we provide o…
We give a complete characterisation of families of probability distributions that are invariant under the action of ReLU neural network layers. The need for such families arises during the training of Bayesian networks or the analysis of trained neural networks, e.g., in the context of uncertainty quantification (UQ) or explainable artificial intelligence (XAI). We prove that no invariant parametrised family of distributions can exist unless at least one of the following three restrictions holds: First, the network layers have a width of one, which is unreasonable for practical neural networks…
Central Institute of Mental Health, Mannheim, Germany（德国曼海姆中央心理健康研究所）；Heidelberg University, Heidelberg, Germany（德国海德堡大学）；Interdisciplinary Center for Scientific Computing, Heidelberg University, Germany（德国海德堡大学跨学科科学计算中心）；Faculty of Mathematics and Computer Science, Heidelberg University, Heidelberg, Germany（德国海德堡大学数学与计算机科学学院）
Institute of Bioinformatics（生物信息学研究所）；Johannes Kepler University（约翰内斯·开普勒大学）
We introduce the "exponential linear unit" (ELU) which speeds up learning in deep neural networks and leads to higher classification accuracies. Like rectified linear units (ReLUs), leaky ReLUs (LReLUs) and parametrized ReLUs (PReLUs), ELUs alleviate the vanishing gradient problem via the identity for positive values. However, ELUs have improved learning characteristics compared to the units with other activation functions. In contrast to ReLUs, ELUs have negative values which allows them to push mean unit activations closer to zero like batch normalization but with lower computational complex…
University of Science and Technology of China（中国科学技术大学）；Microsoft Research Asia（微软亚洲研究院）
Recently, path norm was proposed as a new capacity measure for neural networks with Rectified Linear Unit (ReLU) activation function, which takes the rescaling-invariant property of ReLU into account. It has been shown that the generalization error bound in terms of the path norm explains the empirical generalization behaviors of the ReLU neural networks better than that of other capacity measures. Moreover, optimization algorithms which take path norm as the regularization term to the loss function, like Path-SGD, have been shown to achieve better generalization performance. However, the path…
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…
Neural networks with the Rectified Linear Unit (ReLU) nonlinearity are described by a vector of parameters $θ$, and realized as a piecewise linear continuous function $R_θ: x \in \mathbb R^{d} \mapsto R_θ(x) \in \mathbb R^{k}$. Natural scalings and permutations operations on the parameters $θ$ leave the realization unchanged, leading to equivalence classes of parameters that yield the same realization. These considerations in turn lead to the notion of identifiability -- the ability to recover (the equivalence class of) $θ$ from the sole knowledge of its realization $R_θ$. The overall objectiv…
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…
Offline reinforcement learning (RL) leverages previously collected data for policy optimization without any further active exploration. Despite the recent interest in this problem, its theoretical results in neural network function approximation settings remain elusive. In this paper, we study the statistical theory of offline RL with deep ReLU network function approximation. In particular, we establish the sample complexity of $n = \tilde{\mathcal{O}}( H^{4 + 4 \frac{d}α} κ_μ^{1 + \frac{d}α} ε^{-2 - 2\frac{d}α} )$ for offline RL with deep ReLU networks, where $κ_μ$ is a measure of distributio…
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…
Quantitative Approximation Rates for Group Equivariant Learning
Department of Mathematics（数学系）；Texas A&M University（德克萨斯A&M大学）；Faculty of Mathematics（数学系）；Technion- Israel Institute of Technology（技术学院-以色列理工学院）；Computer Science and Artificial Intelligence Laboratory（计算机科学与人工智能实验室）；Massachusetts Institute of Technology（麻省理工学院）
We propose a simple extension to the ReLU-family of activation functions that allows them to shift the mean activation across a layer towards zero. Combined with proper weight initialization, this alleviates the need for normalization layers. We explore the training of deep vanilla recurrent neural networks (RNNs) with up to 144 layers, and show that bipolar activation functions help learning in this setting. On the Penn Treebank and Text8 language modeling tasks we obtain competitive results, improving on the best reported results for non-gated networks. In experiments with convolutional neur…
Approximating Positive Homogeneous Functions with Scale Invariant Neural Networks
Technical University of Munich（慕尼黑工业大学）；Holsten Systems GmbH（霍尔斯特恩系统有限公司）；Munich Center for Machine Learning（慕尼黑机器学习中心）；Munich Data Science Institute（慕尼黑数据科学研究所）
Successfully training deep neural networks often requires either batch normalization, appropriate weight initialization, both of which come with their own challenges. We propose an alternative, geometrically motivated method for training. Using elementary results from linear programming, we introduce Farkas layers: a method that ensures at least one neuron is active at a given layer. Focusing on residual networks with ReLU activation, we empirically demonstrate a significant improvement in training capacity in the absence of batch normalization or methods of initialization across a broad range…
Adversarial attacks add perturbations to the input features with the intent of changing the classification produced by a machine learning system. Small perturbations can yield adversarial examples which are misclassified despite being virtually indistinguishable from the unperturbed input. Classifiers trained with standard neural network techniques are highly susceptible to adversarial examples, allowing an adversary to create misclassifications of their choice. We introduce a new type of network unit, called MWD (max of weighed distance) units that have a built-in resistant to adversarial att…
We describe new results and algorithms for two different, but related, problems which deal with circulant matrices: learning shift-invariant components from training data and calculating the shift (or alignment) between two given signals. In the first instance, we deal with the shift-invariant dictionary learning problem while the latter bears the name of (compressive) shift retrieval. We formulate these problems using circulant and convolutional matrices (including unions of such matrices), define optimization problems that describe our goals and propose efficient ways to solve them. Based on…
The universal approximation property of various machine learning models is currently only understood on a case-by-case basis, limiting the rapid development of new theoretically justified neural network architectures and blurring our understanding of our current models' potential. This paper works towards overcoming these challenges by presenting a characterization, a representation, a construction method, and an existence result, each of which applies to any universal approximator on most function spaces of practical interest. Our characterization result is used to describe which activation f…
Using weight decay to penalize the L2 norms of weights in neural networks has been a standard training practice to regularize the complexity of networks. In this paper, we show that a family of regularizers, including weight decay, is ineffective at penalizing the intrinsic norms of weights for networks with positively homogeneous activation functions, such as linear, ReLU and max-pooling functions. As a result of homogeneity, functions specified by the networks are invariant to the shifting of weight scales between layers. The ineffective regularizers are sensitive to such shifting and thus p…
University of Queensland（昆士兰大学）；International Computer Science Institute（国际计算机科学研究所）
An interesting approach to analyzing neural networks that has received renewed attention is to examine the equivalent kernel of the neural network. This is based on the fact that a fully connected feedforward network with one hidden layer, a certain weight distribution, an activation function, and an infinite number of neurons can be viewed as a mapping into a Hilbert space. We derive the equivalent kernels of MLPs with ReLU or Leaky ReLU activations for all rotationally-invariant weight distributions, generalizing a previous result that required Gaussian weight distributions. Additionally, th…
Homomorphic Encryption (HE) is one of the most promising security solutions to emerging Machine Learning as a Service (MLaaS). Leveled-HE (LHE)-enabled Convolutional Neural Networks (LHECNNs) are proposed to implement MLaaS to avoid large bootstrapping overhead. However, prior LHECNNs have to pay significant computing overhead but achieve only low inference accuracy, due to their polynomial approximation activations and poolings. Stacking many polynomial approximation activation layers in a network greatly reduces inference accuracy, since the polynomial approximation activation errors lead to…
In adversarial attacks to machine-learning classifiers, small perturbations are added to input that is correctly classified. The perturbations yield adversarial examples, which are virtually indistinguishable from the unperturbed input, and yet are misclassified. In standard neural networks used for deep learning, attackers can craft adversarial examples from most input to cause a misclassification of their choice. We introduce a new type of network units, called RBFI units, whose non-linear structure makes them inherently resistant to adversarial attacks. On permutation-invariant MNIST, in ab…
When machine learning models are deployed on a test distribution different from the training distribution, they can perform poorly, but overestimate their performance. In this work, we aim to better estimate a model's performance under distribution shift, without supervision. To do so, we use a set of domain-invariant predictors as a proxy for the unknown, true target labels. Since the error of the resulting risk estimate depends on the target risk of the proxy model, we study generalization of domain-invariant representations and show that the complexity of the latent representation has a sig…
University of South Australia（南澳大学）；University of California, San Diego（加州大学圣地亚哥分校）；Western Sydney University（西悉尼大学）
Batch-normalization (BN) layers are thought to be an integrally important layer type in today's state-of-the-art deep convolutional neural networks for computer vision tasks such as classification and detection. However, BN layers introduce complexity and computational overheads that are highly undesirable for training and/or inference on low-power custom hardware implementations of real-time embedded vision systems such as UAVs, robots and Internet of Things (IoT) devices. They are also problematic when batch sizes need to be very small during training, and innovations such as residual connec…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Basel（巴塞尔大学）
Thanks to the use of convolution and pooling layers, convolutional neural networks were for a long time thought to be shift-invariant. However, recent works have shown that the output of a CNN can change significantly with small shifts in input: a problem caused by the presence of downsampling (stride) layers. The existing solutions rely either on data augmentation or on anti-aliasing, both of which have limitations and neither of which enables perfect shift invariance. Additionally, the gains obtained from these methods do not extend to image patterns not seen during training. To address thes…
A shift-invariant variational autoencoder (shift-VAE) is developed as an unsupervised method for the analysis of spectral data in the presence of shifts along the parameter axis, disentangling the physically-relevant shifts from other latent variables. Using synthetic data sets, we show that the shift-VAE latent variables closely match the ground truth parameters. The shift VAE is extended towards the analysis of band-excitation piezoresponse force microscopy (BE-PFM) data, disentangling the resonance frequency shifts from the peak shape parameters in a model-free unsupervised manner. The exte…
Deep neural networks are known to be vulnerable to adversarially perturbed inputs. A commonly used defense is adversarial training, whose performance is influenced by model capacity. While previous works have studied the impact of varying model width and depth on robustness, the impact of increasing capacity by using learnable parametric activation functions (PAFs) has not been studied. We study how using learnable PAFs can improve robustness in conjunction with adversarial training. We first ask the question: how should we incorporate parameters into activation functions to improve robustness…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Purdue University（普渡大学）
We propose learnable polyphase sampling (LPS), a pair of learnable down/upsampling layers that enable truly shift-invariant and equivariant convolutional networks. LPS can be trained end-to-end from data and generalizes existing handcrafted downsampling layers. It is widely applicable as it can be integrated into any convolutional network by replacing down/upsampling layers. We evaluate LPS on image classification and semantic segmentation. Experiments show that LPS is on-par with or outperforms existing methods in both performance and shift consistency. For the first time, we achieve true shi…
