Controlling sparsity in deep networks springerlink. 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. The ones marked may be different from the article in the profile. In proceedings of the 28th conference on learning theory, pp.
Behnam neyshabur, ryota tomioka, nathan srebro submitted on 27 feb 2015 v1, last revised 14 apr 2015 this version, v2. Behnam neyshabur, srinadh bhojanapalli, david mcallester, and nati srebro. Capacity control of relu neural networks by basispath norm shuxin zheng 1. 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. Understanding the role of invariance in training neural. We investigate the capacity, convexity and characterization of a general family of. 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. Generalization error in deep learning springerlink. 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.
Fisherrao metric, geometry, and complexity of neural. Tomioka, srebro 2015 normbased capacity control in neural networks, colt. Pathnormalized optimization in deep neural networks, nips. Ors18sametoymak, benjaminrecht, andmahdisoltanolkotabi.
Computing nonvacuous generalization bounds for deep. 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. Sparse recovery, learning, and neural networks charles. Capacity control of relu neural networks by basispath. In advances in neural information processing systems, pages 5947.
We study how these measures can ensure generalization, highlighting the importance of scale normalization, and making a connection between sharpness and pacbayes theory. In particular, we show how perunit regularization is equivalent to a novel path based regularizer and how overall l2 regularization for twolayer networks is. On the other hand, norm based metrics can not distinguish good versusbad modelswhich, arguably is the point of needing quality metrics. 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. In particular, we show how perunit regularization is equivalent to a novel pathbased regularizer and how overall l2 regularization for twolayer networks is. This paper presents a general framework for norm based capacity control for lp,q weight normalized deep neural networks. This cited by count includes citations to the following articles in scholar. Normbased capacity control in neural networks authors. Their combined citations are counted only for the first article. Can we control the capacity of nns independent of num. We investigate the capacity, convexity and characterization of a. Sparsity is a potentially important property of neural networks, but is not explicitly controlled by dropoutbased regularization.
Norm based metrics correlate well with reported test accuracies for welltrained models across nearly all cv architecture series. In many applications, one works with deep neural network dnn models trained by someone else. Understanding weight normalized deep neural networks with. By behnam neyshabur, ryota tomioka and nathan srebro. Normbased capacity control in neural networks core. Normbased capacity control in neural networks proceedings of. 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. Understanding weight normalized deep neural networks with rectified linear units. In terms of capacity control, we show that perunit regularization allows sizeindependent capacitycontrol only with a perunit.
We discuss advantages and weaknesses of each of these complexity measures and examine their abilities to explain the observed generalization phenomena in deep. This paper presents a general framework for normbased capacity control for lp. Proceedings of the 32nd international conference on neural. 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. Fisherrao metric, geometry, and complexity of neural networks. Capacity control of relu neural networks by basispath norm. Advances in neural information processing systems, 2016. 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. Deep learning models have lately shown great performance in various fields such as computer vision, speech recognition, speech translation, and natural language processing. We establish the upper bound on the rademacher complexities of this family. Pdf generalization in deep learning semantic scholar. 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. 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.
This paper presents a general framework for normbased capacity control for lp,q weight normalized deep neural networks. These normbased bounds are the foundation of our current understanding of neural network. Structured pruning of recurrent neural networks through. 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. This capacity formula can be used to identify networks that achieve maximal capacity under various natural constraints. The statistical complexity, or capacity, of unregularized feedforward neural networks, as a function of the network size and depth, is fairly well understood. 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. Capacity control in terms of norm, when using a zeroone loss i. We investigate the capacity, convexity and characterization of a general family of normconstrained feedforward networks.
Normbased capacity control in neural networks pmlr. For inference time and memory usage measurements we have used torch7 collobert et al. Normbased capacity control in neural networks, colt. This raises the question why they do not easily overfit real data. Among different types of deep neural networks, relu networks i. We establish a generalization error bound based on this basis path norm, and show it. Exploring generalization in deep learning nips proceedings. Theoretical investigation of generalization bound for. Normbased capacity control in neural networks videolectures. For the regression problem, we analyze the rademacher complexity of the resnets family.
With a goal of understanding what drives generalization in deep networks, we consider several recently suggested explanations, including norm based control, sharpness and robustness. Deep stochastic neural networks with many more parameters than training data uai 2017 4. 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. With a goal of understanding what drives generalization in deep networks, we consider several recently suggested explanations, including normbased control, sharpness and robustness. We show that deep networks with finite weights or trained for finite number of. This paper presents a general framework for normbased capacity control for l p,q weight normalized deep neural networks. A pacbayesian approach to spectrallynormalized margin bounds for neural networks. Pdf exploring generalization in deep learning semantic. To answer this question, we study deep networks using fourier analysis. We study the relationship between geometry and capacity measures for deep neural networks from an invariance viewpoint. Capacity control of relu neural networks by basispath norm authors. Feb 27, 2015 normbased capacity control in neural networks. Proof sketch to show convexity, consider two functions f, g.
Norm based capacity control in neural networks authors. This paper presents a framework for norm based capacity control with respect to an lp,q norm in weightnormalized residual neural networks resnets. The 28th conference on learning theory colt, 2015 to appear. 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. This paper presents a general framework for norm based capacity control for lp. Understanding the role of invariances in training neural networks ryota tomioka. Understanding weight normalized deep neural networks. 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. Universal statistics of fisher information in deep neural. We theoretically find novel statistics of the fim, which are universal among a wide class of deep networks with any number of.
This paper presents a framework for normbased capacity control with respect to an lp,qnorm in weightnormalized residual neural networks resnets. Sparse recovery, learning, and neural networks charles delahunt. 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. 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. We first formulate the representation of each residual block. Normbased capacity control in neural networks journal of. 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.
Pdf normbased capacity control in neural networks semantic. More surprisingly, deep neural networks generalize well, even when the number of parameters is. Finitetime convergent complexvalued neural networks for. Enter your email into the cc field, and we will keep you updated with your requests status. Improved normbased bounds were obtained using rademacher and gaussian complexity by bartlett and mendelson bm02 and koltchinskii and panchenko kp02. This theoretical result is aligned with the designs used in the recent stateoftheart cnns, where. Behnam neyshabur, srinadh bhojanapalli, david mcallester, and nathan srebro. Shuxin zheng, qi meng, huishuai zhang, wei chen, nenghai yu, tieyan liu submitted on 19 sep 2018. Predicting trends in the quality of stateoftheart neural networks without access to training or testing data.
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. It is intractable to learn sparse parametric models by minimizing the l 0 norm based on gradient optimization. Behnam neyshabur, ryota tomioka, and nathan srebro. Understanding the role of invariance in training neural networks. Nts15behnam neyshabur, ryota tomioka, and nathan srebro. On the spectral bias of deep neural networks arxiv. Image inpainting via generative multicolumn convolutional. Learning with deep neural networks has enjoyed huge empirical success in recent years. Normbased capacity control in neural networks behnam neyshabur, ryota tomioka, nathan srebro toyota technological institute at chicago. Norm based capacity control in neural networks behnam neyshabur, ryota tomioka, nathan srebro toyota technological institute at chicago. Request pdf capacity control of relu neural networks by basispath norm.
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