Deep learning is recognized to be capable of discovering deep features for representation learning and pattern recognition without requiring elegant feature engineering techniques by taking advantage of human ingenuity and prior knowledge. Thus it has triggered enormous research activities in machine learning and pattern recognition. One of the most important challenge of deep learning is to figure out relations between a feature and the depth of deep neural networks (deep nets for short) to reflect the necessity of depth. Our purpose is to quantify this feature-depth correspondence in feature extraction and generalization. We present the adaptivity of features to depths and vice-verse via showing a depth-parameter trade-off in extracting both single feature and composite features. Based on these results, we prove that implementing the classical empirical risk minimization on deep nets can achieve the optimal generalization performance for numerous learning tasks. Our theoretical results are verified by a series of numerical experiments including toy simulations and a real application of earthquake seismic intensity prediction.
Stanford University（斯坦福大学）；City University of Hong Kong（香港城市大学）
The great success of deep learning poses urgent challenges for understanding its working mechanism and rationality. The depth, structure, and massive size of the data are recognized to be three key ingredients for deep learning. Most of the recent theoretical studies for deep learning focus on the necessity and advantages of depth and structures of neural networks. In this paper, we aim at rigorous verification of the importance of massive data in embodying the out-performance of deep learning. To approximate and learn spatially sparse and smooth functions, we establish a novel sampling theore…
Jiangxi Normal University（江西师范大学）；City University of Hong Kong（香港城市大学）；Hong Kong University of Science and Technology（香港科技大学）
In this paper, we develop an alternating direction method of multipliers (ADMM) for deep neural networks training with sigmoid-type activation functions (called \textit{sigmoid-ADMM pair}), mainly motivated by the gradient-free nature of ADMM in avoiding the saturation of sigmoid-type activations and the advantages of deep neural networks with sigmoid-type activations (called deep sigmoid nets) over their rectified linear unit (ReLU) counterparts (called deep ReLU nets) in terms of approximation. In particular, we prove that the approximation capability of deep sigmoid nets is not worse than t…
In spite of several notable efforts, explaining the generalization of deterministic non-smooth deep nets, e.g., ReLU-nets, has remained challenging. Existing approaches for deterministic non-smooth deep nets typically need to bound the Lipschitz constant of such deep nets but such bounds are quite large, may even increase with the training set size yielding vacuous generalization bounds. In this paper, we present a new family of de-randomized PAC-Bayes margin bounds for deterministic non-convex and non-smooth predictors, e.g., ReLU-nets. Unlike PAC-Bayes, which applies to Bayesian predictors, …
Yale University（耶鲁大学）；Rutgers University – New Brunswick（罗格斯大学新不伦瑞克分校）
For any ReLU network there is a representation in which the sum of the absolute values of the weights into each node is exactly $1$, and the input layer variables are multiplied by a value $V$ coinciding with the total variation of the path weights. Implications are given for Gaussian complexity, Rademacher complexity, statistical risk, and metric entropy, all of which are shown to be proportional to $V$. There is no dependence on the number of nodes per layer, except for the number of inputs $d$. For estimation with sub-Gaussian noise, the mean square generalization error bounds that can be o…
Stanford University（斯坦福大学）；Princeton University（普林斯顿大学）；University of Texas at Austin（得克萨斯大学奥斯汀分校）
Recent works have shown that on sufficiently over-parametrized neural nets, gradient descent with relatively large initialization optimizes a prediction function in the RKHS of the Neural Tangent Kernel (NTK). This analysis leads to global convergence results but does not work when there is a standard $\ell_2$ regularizer, which is useful to have in practice. We show that sample efficiency can indeed depend on the presence of the regularizer: we construct a simple distribution in d dimensions which the optimal regularized neural net learns with $O(d)$ samples but the NTK requires $Ω(d^2)$ samp…
University of Tübingen（蒂宾根大学）；MPI for Intelligent Systems, Tübingen（蒂宾根马克斯·普朗克智能系统研究所）
A Bayesian treatment can mitigate overconfidence in ReLU nets around the training data. But far away from them, ReLU Bayesian neural networks (BNNs) can still underestimate uncertainty and thus be asymptotically overconfident. This issue arises since the output variance of a BNN with finitely many features is quadratic in the distance from the data region. Meanwhile, Bayesian linear models with ReLU features converge, in the infinite-width limit, to a particular Gaussian process (GP) with a variance that grows cubically so that no asymptotic overconfidence can occur. While this may seem of mos…
Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）；Carnegie Mellon University（卡内基梅隆大学）
Recent works have cast some light on the mystery of why deep nets fit any data and generalize despite being very overparametrized. This paper analyzes training and generalization for a simple 2-layer ReLU net with random initialization, and provides the following improvements over recent works: (i) Using a tighter characterization of training speed than recent papers, an explanation for why training a neural net with random labels leads to slower training, as originally observed in [Zhang et al. ICLR'17]. (ii) Generalization bound independent of network size, using a data-dependent complexity …
In this paper, we study the generalization performance of global minima for implementing empirical risk minimization (ERM) on over-parameterized deep ReLU nets. Using a novel deepening scheme for deep ReLU nets, we rigorously prove that there exist perfect global minima achieving almost optimal generalization error bounds for numerous types of data under mild conditions. Since over-parameterization is crucial to guarantee that the global minima of ERM on deep ReLU nets can be realized by the widely used stochastic gradient descent (SGD) algorithm, our results indeed fill a gap between optimiza…
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…
This theoretical paper is devoted to developing a rigorous theory for demystifying the global convergence phenomenon in a challenging scenario: learning over-parameterized Rectified Linear Unit (ReLU) nets for very high dimensional dataset under very mild assumptions. A major ingredient of our analysis is a fine-grained analysis of random activation matrices. The essential virtue of dissecting activation matrices is that it bridges the dynamics of optimization and angular distribution in high-dimensional data space. This angle-based detailed analysis leads to asymptotic characterizations of gr…
Department of Mathematics, Texas A&M（德克萨斯农工大学数学系）；Trinity College, Cambridge（剑桥大学三一学院）
This article concerns the expressive power of depth in deep feed-forward neural nets with ReLU activations. Specifically, we answer the following question: for a fixed $d_{in}\geq 1,$ what is the minimal width $w$ so that neural nets with ReLU activations, input dimension $d_{in}$, hidden layer widths at most $w,$ and arbitrary depth can approximate any continuous, real-valued function of $d_{in}$ variables arbitrarily well? It turns out that this minimal width is exactly equal to $d_{in}+1.$ That is, if all the hidden layer widths are bounded by $d_{in}$, then even in the infinite depth limit…
University of Chicago（芝加哥大学）；Technion, Israel Institute of Technology（以色列理工学院）；Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
A key element of understanding the efficacy of overparameterized neural networks is characterizing how they represent functions as the number of weights in the network approaches infinity. In this paper, we characterize the norm required to realize a function $f:\mathbb{R}^d\rightarrow\mathbb{R}$ as a single hidden-layer ReLU network with an unbounded number of units (infinite width), but where the Euclidean norm of the weights is bounded, including precisely characterizing which functions can be realized with finite norm. This was settled for univariate univariate functions in Savarese et al.…
The ability to interpret machine learning models has become increasingly important now that machine learning is used to inform consequential decisions. We propose an approach called model extraction for interpreting complex, blackbox models. Our approach approximates the complex model using a much more interpretable model; as long as the approximation quality is good, then statistical properties of the complex model are reflected in the interpretable model. We show how model extraction can be used to understand and debug random forests and neural nets trained on several datasets from the UCI M…
"Deep Learning"/"Deep Neural Nets" is a technological marvel that is now increasingly deployed at the cutting-edge of artificial intelligence tasks. This dramatic success of deep learning in the last few years has been hinged on an enormous amount of heuristics and it has turned out to be a serious mathematical challenge to be able to rigorously explain them. In this thesis, submitted to the Department of Applied Mathematics and Statistics, Johns Hopkins University we take several steps towards building strong theoretical foundations for these new paradigms of deep-learning. In chapter 2 we sh…
Barcelona Supercomputing Center (BSC)（巴塞罗那超级计算中心（BSC））；Universitat Politècnica de Catalunya - BarcelonaTech (UPC)（加泰罗尼亚理工大学 - 巴塞罗那技术大学（UPC））；IBM T.J. Watson Research Center（IBM T.J. 沃森研究中心）
Deep neural networks are representation learning techniques. During training, a deep net is capable of generating a descriptive language of unprecedented size and detail in machine learning. Extracting the descriptive language coded within a trained CNN model (in the case of image data), and reusing it for other purposes is a field of interest, as it provides access to the visual descriptors previously learnt by the CNN after processing millions of images, without requiring an expensive training phase. Contributions to this field (commonly known as feature representation transfer or transfer l…
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…
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…
Michigan Technological University（密歇根理工大学）；Symbiosis Centre for Medical Image Analysis（共生医学图像分析中心）；Symbiosis Institute of Technology（共生技术学院）
This work introduces a novel activation unit that can be efficiently employed in deep neural nets (DNNs) and performs significantly better than the traditional Rectified Linear Units (ReLU). The function developed is a two parameter version of the specialized Richard's Curve and we call it Adaptive Richard's Curve weighted Activation (ARiA). This function is non-monotonous, analogous to the newly introduced Swish, however allows a precise control over its non-monotonous convexity by varying the hyper-parameters. We first demonstrate the mathematical significance of the two parameter ARiA follo…
MPI-MIS（马克斯·普朗克数学研究所）；Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）；UCLA（加州大学洛杉矶分校）
A recent line of work has analyzed the theoretical properties of deep neural networks via the Neural Tangent Kernel (NTK). In particular, the smallest eigenvalue of the NTK has been related to the memorization capacity, the global convergence of gradient descent algorithms and the generalization of deep nets. However, existing results either provide bounds in the two-layer setting or assume that the spectrum of the NTK matrices is bounded away from 0 for multi-layer networks. In this paper, we provide tight bounds on the smallest eigenvalue of NTK matrices for deep ReLU nets, both in the limit…
Despite the phenomenal success of deep learning in recent years, there remains a gap in understanding the fundamental mechanics of neural nets. More research is focussed on handcrafting complex and larger networks, and the design decisions are often ad-hoc and based on intuition. Some recent research has aimed to demystify the learning dynamics in neural nets by attempting to build a theory from first principles, such as characterising the non-linear dynamics of specialised \textit{linear} deep neural nets (such as orthogonal networks). In this work, we expand and derive properties of learning…
Motivated by the resurgence of neural networks in being able to solve complex learning tasks we undertake a study of high depth networks using ReLU gates which implement the function $x \mapsto \max\{0,x\}$. We try to understand the role of depth in such neural networks by showing size lowerbounds against such network architectures in parameter regimes hitherto unexplored. In particular we show the following two main results about neural nets computing Boolean functions of input dimension $n$, 1. We use the method of random restrictions to show almost linear, $Ω(ε^{2(1-δ)}n^{1-δ})$, lower boun…
Massachusetts Institute of Technology（麻省理工学院）；Georgia Institute of Technology（佐治亚理工学院）；Aalto University（阿尔托大学）
We propose semi-random features for nonlinear function approximation. The flexibility of semi-random feature lies between the fully adjustable units in deep learning and the random features used in kernel methods. For one hidden layer models with semi-random features, we prove with no unrealistic assumptions that the model classes contain an arbitrarily good function as the width increases (universality), and despite non-convexity, we can find such a good function (optimization theory) that generalizes to unseen new data (generalization bound). For deep models, with no unrealistic assumptions,…
The Hebrew University（希伯来大学）；Google Research（谷歌研究院）
We consider the natural problem of learning a ReLU network from queries, which was recently remotivated by model extraction attacks. In this work, we present a polynomial-time algorithm that can learn a depth-two ReLU network from queries under mild general position assumptions. We also present a polynomial-time algorithm that, under mild general position assumptions, can learn a rich class of depth-three ReLU networks from queries. For instance, it can learn most networks where the number of first layer neurons is smaller than the dimension and the number of second layer neurons. These two re…
Recent theoretical work has demonstrated that deep neural networks have superior performance over shallow networks, but their training is more difficult, e.g., they suffer from the vanishing gradient problem. This problem can be typically resolved by the rectified linear unit (ReLU) activation. However, here we show that even for such activation, deep and narrow neural networks (NNs) will converge to erroneous mean or median states of the target function depending on the loss with high probability. Deep and narrow NNs are encountered in solving partial differential equations with high-order de…
Aalto University（阿尔托大学）；University of Waterloo（滑铁卢大学）；F-Secure Corporation（F-Secure公司）
Recently, machine learning (ML) has introduced advanced solutions to many domains. Since ML models provide business advantage to model owners, protecting intellectual property of ML models has emerged as an important consideration. Confidentiality of ML models can be protected by exposing them to clients only via prediction APIs. However, model extraction attacks can steal the functionality of ML models using the information leaked to clients through the results returned via the API. In this work, we question whether model extraction is a serious threat to complex, real-life ML models. We eval…
ETH Zurich（苏黎世联邦理工学院）；INRIA Paris（法国国家信息与自动化研究所巴黎分部）
Randomly initialized neural networks are known to become harder to train with increasing depth, unless architectural enhancements like residual connections and batch normalization are used. We here investigate this phenomenon by revisiting the connection between random initialization in deep networks and spectral instabilities in products of random matrices. Given the rich literature on random matrices, it is not surprising to find that the rank of the intermediate representations in unnormalized networks collapses quickly with depth. In this work we highlight the fact that batch normalization…
Generative models for deep learning are promising both to improve understanding of the model, and yield training methods requiring fewer labeled samples. Recent works use generative model approaches to produce the deep net's input given the value of a hidden layer several levels above. However, there is no accompanying "proof of correctness" for the generative model, showing that the feedforward deep net is the correct inference method for recovering the hidden layer given the input. Furthermore, these models are complicated. The current paper takes a more theoretical tack. It presents a very …
Ecole Normale Supérieure（巴黎高等师范学院）；EPFL（洛桑联邦理工学院）；International School of Advanced Studies (SISSA)（国际高等研究学院）
Autoencoders are the simplest neural network for unsupervised learning, and thus an ideal framework for studying feature learning. While a detailed understanding of the dynamics of linear autoencoders has recently been obtained, the study of non-linear autoencoders has been hindered by the technical difficulty of handling training data with non-trivial correlations - a fundamental prerequisite for feature extraction. Here, we study the dynamics of feature learning in non-linear, shallow autoencoders. We derive a set of asymptotically exact equations that describe the generalisation dynamics of…
Deutsches Medizinrechenzentrum GmbH & Co KG（德国医学计算中心有限两合公司）
Baseline detection is still a challenging task for heterogeneous collections of historical documents. We present a novel approach to baseline extraction in such settings, turning out the winning entry to the ICDAR 2017 Competition on Baseline detection (cBAD). It utilizes deep convolutional nets (CNNs) for both, the actual extraction of baselines, as well as for a simple form of layout analysis in a pre-processing step. To the best of our knowledge it is the first CNN-based system for baseline extraction applying a U-net architecture and sliding window detection, profiting from a high local ac…
Arizona State University（亚利桑那州立大学）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
Modern audio source separation techniques rely on optimizing sequence model architectures such as, 1D-CNNs, on mixture recordings to generalize well to unseen mixtures. Specifically, recent focus is on time-domain based architectures such as Wave-U-Net which exploit temporal context by extracting multi-scale features. However, the optimality of the feature extraction process in these architectures has not been well investigated. In this paper, we examine and recommend critical architectural changes that forge an optimal multi-scale feature extraction process. To this end, we replace regular $1…
Center for Machine Perception（机器感知中心）；Czech Technical University in Prague（捷克布拉格技术大学）
Layer-sequential unit-variance (LSUV) initialization - a simple method for weight initialization for deep net learning - is proposed. The method consists of the two steps. First, pre-initialize weights of each convolution or inner-product layer with orthonormal matrices. Second, proceed from the first to the final layer, normalizing the variance of the output of each layer to be equal to one. Experiment with different activation functions (maxout, ReLU-family, tanh) show that the proposed initialization leads to learning of very deep nets that (i) produces networks with test accuracy better or…
Deep Reinforcement Learning (DRL) has been successfully applied in several research domains such as robot navigation and automated video game playing. However, these methods require excessive computation and interaction with the environment, so enhancements on sample efficiency are required. The main reason for this requirement is that sparse and delayed rewards do not provide an effective supervision for representation learning of deep neural networks. In this study, Proximal Policy Optimization (PPO) algorithm is augmented with Generative Adversarial Networks (GANs) to increase the sample ef…
The University of Iowa（爱荷华大学）；Snap Research（Snap研究院）
While going deeper has been witnessed to improve the performance of convolutional neural networks (CNN), going smaller for CNN has received increasing attention recently due to its attractiveness for mobile/embedded applications. It remains an active and important topic how to design a small network while retaining the performance of large and deep CNNs (e.g., Inception Nets, ResNets). Albeit there are already intensive studies on compressing the size of CNNs, the considerable drop of performance is still a key concern in many designs. This paper addresses this concern with several new contrib…
NRC "Kurchatov institute"（国家研究中心“库尔恰托夫研究所”）；MEPhI National Research Nuclear University（莫斯科工程物理学院国家核研究大学）；I.M. Sechenov First Moscow State Medical University (Sechenov University)（谢切诺夫第一莫斯科国立医科大学（谢切诺夫大学））
We present the full-size Russian complexly NER-labeled corpus of Internet user reviews, along with an evaluation of accuracy levels reached on this corpus by a set of advanced deep learning neural networks to extract the pharmacologically meaningful entities from Russian texts. The corpus annotation includes mentions of the following entities: Medication (33005 mentions), Adverse Drug Reaction (1778), Disease (17403), and Note (4490). Two of them - Medication and Disease - comprise a set of attributes. A part of the corpus has the coreference annotation with 1560 coreference chains in 300 docu…
NSF Center for Cloud and Autonomic Computing（NSF云与自主计算中心）；Department of Computer Science（计算机科学系）；The University of Arizona（亚利桑那大学）；The University of North Texas（北得克萨斯大学）
Anomaly detection is critically important for intelligent surveillance systems to detect in a timely manner any malicious activities. Many video anomaly detection approaches using deep learning methods focus on a single camera video stream with a fixed scenario. These deep learning methods use large-scale training data with large complexity. As a solution, in this paper, we show how to use pre-trained convolutional neural net models to perform feature extraction and context mining, and then use denoising autoencoder with relatively low model complexity to provide efficient and accurate surveil…
Knowledge extraction is used to convert neural networks into symbolic descriptions with the objective of producing more comprehensible learning models. The central challenge is to find an explanation which is more comprehensible than the original model while still representing that model faithfully. The distributed nature of deep networks has led many to believe that the hidden features of a neural network cannot be explained by logical descriptions simple enough to be comprehensible. In this paper, we propose a novel layerwise knowledge extraction method using M-of-N rules which seeks to obta…
Yale University（耶鲁大学）；University of Maryland（马里兰大学）
Deep learning models extract, before a final classification layer, features or patterns which are key for their unprecedented advantageous performance. However, the process of complex nonlinear feature extraction is not well understood, a major reason why interpretation, adversarial robustness, and generalization of deep neural nets are all open research problems. In this paper, we use wavelet transformation and spectral methods to analyze the contents of image classification datasets, extract specific patterns from the datasets and find the associations between patterns and classes. We show t…
