This study analyzes the fisher information matrix fim by applying meanfield theory to deep neural networks with random weights. Computing nonvacuous generalization bounds for deep. Understanding weight normalized deep neural networks with rectified linear units. Normbased capacity control in neural networks pmlr. We study how these measures can ensure generalization, highlighting the importance of scale normalization, and making a connection between sharpness and pacbayes theory.

We first formulate the representation of each residual block. Edu toyota technological institute at chicago, chicago, il 60637, usa abstract we investigate the capacity, convexity and characterization of a general family of norm constrained feedforward networks. Normbased capacity control in neural networks authors. On the spectral bias of deep neural networks arxiv. Norm based metrics correlate well with reported test accuracies for welltrained models across nearly all cv architecture series. Enter your email into the cc field, and we will keep you updated with your requests status. We theoretically find novel statistics of the fim, which are universal among a wide class of deep networks with any number of layers and various activation functions. On the other hand, norm based metrics can not distinguish good versusbad modelswhich, arguably is the point of needing quality metrics. In particular, we show how perunit regularization is equivalent to a novel path based regularizer and how overall l2 regularization for twolayer networks is. The ones marked may be different from the article in the profile. Finitetime convergent complexvalued neural networks for. Edu toyota technological institute at chicago, chicago, il 60637, usa abstract we investigate the capacity, convexity and characterization of a general family of normconstrained feedforward networks. Deep learning models have lately shown great performance in various fields such as computer vision, speech recognition, speech translation, and natural language processing.

In this way, w i j controls the strength of the link from the input neuron i to the hidden neuron j, while z i and s j control the presence of neurons. Computing nonvacuous generalization bounds for deep stochastic neural networks with many more parameters than training data gintare karolina dziugaite department of engineering university of cambridge daniel m. Tomioka, srebro 2015 normbased capacity control in neural networks, colt. For the purposes of the pacbayes bound, it is the kl divergence klq jjp that upper bounds the performance of the stochastic neural network q. Normbased capacity control provides a possible answer and is being actively studied for deep networks krogh and hertz,1992,neyshabur et al. For inference time and memory usage measurements we have used torch7 collobert et al. Sparsity is a potentially important property of neural networks, but is not explicitly controlled by dropoutbased regularization. Predicting trends in the quality of stateoftheart neural networks without access to training or testing data. While training neural networks is known to be intractable in general, simple local search heuristics are often surprisingly e ective. Research predicting trends in the quality of stateofthe. This paper presents a general framework for norm based capacity control for lp,q weight normalized deep neural networks. Sparse recovery, learning, and neural networks charles. Behnam neyshabur, ryota tomioka, nathan srebro submitted on 27 feb 2015 v1, last revised 14 apr 2015 this version, v2.

Pdf exploring generalization in deep learning semantic. To answer this question, we study deep networks using fourier analysis. We show that deep networks with finite weights or trained for finite number of. This capacity formula can be used to identify networks that achieve maximal capacity under various natural constraints. Behnam neyshabur, srinadh bhojanapalli, david mcallester, and nathan srebro. Deep stochastic neural networks with many more parameters than training data uai 2017 4. We investigate the capacity, convexity and characterization of a. This paper presents a general framework for normbased capacity control for l p,q weight normalized deep neural networks. In many applications, one works with deep neural network dnn models trained by someone else. This paper presents a framework for norm based capacity control with respect to an lp,q norm in weightnormalized residual neural networks resnets. In particular, by viewing a tlayer neural network as a discretetime dynamical system with time horizon t, the minmax robust optimization problem can be seen as a. Understanding weight normalized deep neural networks with.

Shuxin zheng, qi meng, huishuai zhang, wei chen, nenghai yu, tieyan liu submitted on 19 sep 2018. Universal statistics of fisher information in deep neural. Jun 26, 2019 norm based measures do not explicitly depend on the amount of parameters in the model and therefore have a better potential to represent its capacity 14. Sparsity is a potentially important property of neural networks, but is not explicitly controlled by dropout based regularization. Image inpainting via generative multicolumn convolutional. Their combined citations are counted only for the first article. This paper presents a framework for normbased capacity control with respect to an lp,qnorm in weightnormalized residual neural networks resnets. Understanding the role of invariances in training neural networks ryota tomioka. This paper presents a general framework for norm based capacity control for lp. Capacity control of relu neural networks by basispath norm shuxin zheng 1. This paper provides nonvacuous and numericallytight generalization guarantees for deep learning, as well as theoretical insights into why and how deep learning can generalize well, despite its large capacity, complexity, possible algorithmic instability, nonrobustness, and sharp minima, responding to an open question in the literature. We establish a generalization error bound based on this basis path norm, and show it. Capacity control of relu neural networks by basispath norm shuxin zheng 1, qi meng 2, huishuai zhang, wei chen 2, nenghai yu 1, and tieyan liu 2 1 university of science and technology of china. We establish the upper bound on the rademacher complexities of this family.

Theoretical investigation of generalization bound for. Proof sketch to show convexity, consider two functions f, g. Among different types of deep neural networks, relu networks i. With a goal of understanding what drives generalization in deep networks, we consider several recently suggested explanations, including normbased control, sharpness and robustness. In conference on learning theory, pages 761401, 2015. Controlling sparsity in deep networks springerlink. Exploring generalization in deep learning nips proceedings. Generalization and capacity in order to understand the effect of the norm on the sample complexity, we bound the rademacher complexity of the classes nd. These normbased bounds are the foundation of our current understanding of neural network. Understanding the role of invariance in training neural networks. The statistical complexity, or capacity, of unregularized feedforward neural networks, as a function of the network size and depth, is fairly well understood. Norm based capacity control in neural networks behnam neyshabur, ryota tomioka, nathan srebro toyota technological institute at chicago. Improved normbased bounds were obtained using rademacher and gaussian complexity by bartlett and mendelson bm02 and koltchinskii and panchenko kp02.

Learning with deep neural networks has enjoyed huge empirical success in recent years. Normbased capacity control in neural networks proceedings of. This paper presents a general framework for normbased capacity control for lp. This raises the question why they do not easily overfit real data. Generalization error in deep learning springerlink. Sparse recovery, learning, and neural networks charles delahunt. Pdf generalization in deep learning semantic scholar. Fisherrao metric, geometry, and complexity of neural. Can we control the capacity of nns independent of num. For the regression problem, we analyze the rademacher complexity of the resnets family. Structured pruning of recurrent neural networks through. Normbased capacity control in neural networks behnam neyshabur, ryota tomioka, nathan srebro toyota technological institute at chicago.

Capacity control in terms of norm, when using a zeroone loss i. Pathnormalized optimization in deep neural networks, nips. Normbased capacity control in neural networks core. In particular, we show how perunit regularization is equivalent to a novel pathbased regularizer and how overall l2 regularization for twolayer networks is. Normbased capacity control in neural networks journal of. Normbased capacity control in neural networks videolectures.

We find an formula that approximately determines this number for any fullyconnected, feedforward network with any number of layers, virtually any sizes of layers, and with the threshold activation function. Proceedings of the 32nd international conference on neural. In terms of capacity control, we show that perunit regularization allows sizeindependent capacitycontrol only with a perunit. In this work, we propose sparseout a simple and efficient variant of dropout that can be used to control the sparsity of the activations in a neural network. Request pdf capacity control of relu neural networks by basispath norm. We theoretically find novel statistics of the fim, which are universal among a wide class of deep networks with any number of. Then p and qinduce distributions on h, which we will denote by p and q, respectively. With a goal of understanding what drives generalization in deep networks, we consider several recently suggested explanations, including norm based control, sharpness and robustness. We investigate the capacity, convexity and characterization of a general family of normconstrained feedforward networks. Nts15behnam neyshabur, ryota tomioka, and nathan srebro.

Capacity control of relu neural networks by basispath. Recently, path norm was proposed as a new capacity measure for neural networks with rectified linear unit relu activation function, which takes the rescalinginvariant property of relu into account. We study the relationship between geometry and capacity measures for deep neural networks from an invariance viewpoint. In advances in neural information processing systems, pages 5947. A major challenge is that training neural networks correspond to extremely highdimensional and nonconvex optimization problems and it is not clear how to provably solve them to global optimality. Norm based capacity control provides a possible answer and is being actively studied for deep networks krogh and hertz,1992,neyshabur et al.

Ors18sametoymak, benjaminrecht, andmahdisoltanolkotabi. This theoretical result is aligned with the designs used in the recent stateoftheart cnns, where. It is intractable to learn sparse parametric models by minimizing the l 0 norm based on gradient optimization. More surprisingly, deep neural networks generalize well, even when the number of parameters is. Capacity control of relu neural networks by basispath norm authors. Fisherrao metric, geometry, and complexity of neural networks. A pacbayesian approach to spectrallynormalized margin bounds for neural networks. This cited by count includes citations to the following articles in scholar. We investigate the capacity, convexity and characterization of a general family of. Normbased capacity control in neural networks, colt. Norm based capacity control in neural networks authors. Report a problem or upload files if you have found a problem with this lecture or would like to send us extra material, articles, exercises, etc.

Advances in neural information processing systems, 2016. We discuss advantages and weaknesses of each of these complexity measures and examine their abilities to explain the observed generalization phenomena in deep. Behnam neyshabur, srinadh bhojanapalli, david mcallester, and nati srebro. Norm based measures do not explicitly depend on the amount of parameters in the model and therefore have a better potential to represent its capacity 14. Research predicting trends in the quality of stateof. In proceedings of the 28th conference on learning theory, pp. Understanding the role of invariance in training neural. Capacity control of relu neural networks by basispath norm. The 28th conference on learning theory colt, 2015 to appear. This paper presents a general framework for normbased capacity control for lp,q weight normalized deep neural networks. Deep neural networks have pushed the frontiers of a wide variety of ai tasks in recent years such as speech recognition xiong et al.

Capacity control of relu neural networks by basispath norm shuxin zheng1, qi meng 2, huishuai zhang, wei chen2, nenghai yu1, and tieyan liu2 1university of science and technology of china. It is well known that overparametrized deep neural networks dnns are an overly expressive class of functions that can memorize even random data with 100% training accuracy. Behnam neyshabur, ryota tomioka, and nathan srebro. By behnam neyshabur, ryota tomioka and nathan srebro. Feb 27, 2015 normbased capacity control in neural networks. Finitetime convergent complexvalued neural networks for the timevarying complex linear matrix equations xuezhong wang, lu liang and maolin che, abstractin this paper, we propose two complexvalued neural networks for solving a timevarying complex linear matrix equation by constructing two new types of nonlinear activation functions.

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