The Anisotropic Noise in Stochastic Gradient Descent: Its Behavior of Escaping from Sharp Minima and Regularization Effects

Understanding the behavior of stochastic gradient descent (SGD) in the context of deep neural networks has raised lots of concerns recently. Along this line, we study a general form of gradient based optimization dynamics with unbiased noise, which unifies SGD and standard Langevin dynamics. Through investigating this general optimization dynamics, we analyze the behavior of SGD on escaping from minima and its regularization effects. A novel indicator is derived to characterize the efficiency of escaping from minima through measuring the alignment of noise covariance and the curvature of loss function. Based on this indicator, two conditions are established to show which type of noise structure is superior to isotropic noise in term of escaping efficiency. We further show that the anisotropic noise in SGD satisfies the two conditions, and thus helps to escape from sharp and poor minima effectively, towards more stable and flat minima that typically generalize well. We systematically design various experiments to verify the benefits of the anisotropic noise, compared with full gradient descent plus isotropic diffusion (i.e. Langevin dynamics).

[1]  L. Bottou.  Stochastic Gradient Learning in Neural Networks , 1991 .

[2]  Michael I. Jordan,et al.  How to Escape Saddle Points Efficiently , 2017, ICML.

[3]  Shai Shalev-Shwartz,et al.  SGD Learns Over-parameterized Networks that Provably Generalize on Linearly Separable Data , 2017, ICLR.

[4]  Yann Dauphin,et al.  Empirical Analysis of the Hessian of Over-Parametrized Neural Networks , 2017, ICLR.

[5]  Yuchen Zhang,et al.  A Hitting Time Analysis of Stochastic Gradient Langevin Dynamics , 2017, COLT.

[6]  Kai Zheng,et al.  Generalization Bounds of SGLD for Non-convex Learning: Two Theoretical Viewpoints , 2017, COLT.

[7]  Zhanxing Zhu,et al.  Covariance-Controlled Adaptive Langevin Thermostat for Large-Scale Bayesian Sampling , 2015, NIPS.

[8]  Nathan Srebro,et al.  Exploring Generalization in Deep Learning , 2017, NIPS.

[9]  Zhanxing Zhu,et al.  Towards Understanding Generalization of Deep Learning: Perspective of Loss Landscapes , 2017, ArXiv.

[10]  Naftali Tishby,et al.  Opening the Black Box of Deep Neural Networks via Information , 2017, ArXiv.

[11]  Kaiming He,et al.  Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour , 2017, ArXiv.

[12]  Yuanzhi Li,et al.  Convergence Analysis of Two-layer Neural Networks with ReLU Activation , 2017, NIPS.

[13]  Jorge Nocedal,et al.  On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima , 2016, ICLR.

[14]  Quoc V. Le,et al.  A Bayesian Perspective on Generalization and Stochastic Gradient Descent , 2017, ICLR.

[15]  Jürgen Schmidhuber,et al.  Flat Minima , 1997, Neural Computation.

[16]  Stefano Soatto,et al.  Stochastic Gradient Descent Performs Variational Inference, Converges to Limit Cycles for Deep Networks , 2017, 2018 Information Theory and Applications Workshop (ITA).

[17]  Yoshua Bengio,et al.  Three Factors Influencing Minima in SGD , 2017, ArXiv.

[18]  E Weinan,et al.  Stochastic Modified Equations and Adaptive Stochastic Gradient Algorithms , 2015, ICML.

[19]  Thomas Hofmann,et al.  Escaping Saddles with Stochastic Gradients , 2018, ICML.

[20]  Elad Hoffer,et al.  Train longer, generalize better: closing the generalization gap in large batch training of neural networks , 2017, NIPS.

[21]  Ahn,et al.  Bayesian posterior sampling via stochastic gradient Fisher scoring Bayesian Posterior Sampling via Stochastic Gradient Fisher Scoring , 2012 .

[22]  Wenqing Hu,et al.  On the diffusion approximation of nonconvex stochastic gradient descent , 2017, Annals of Mathematical Sciences and Applications.

[23]  David M. Blei,et al.  Stochastic Gradient Descent as Approximate Bayesian Inference , 2017, J. Mach. Learn. Res..

[24]  B. Øksendal.  Stochastic Differential Equations , 1985 .

[25]  Y. Pawitan.  In all likelihood : statistical modelling and inference using likelihood , 2002 .

[26]  Tianqi Chen,et al.  Stochastic Gradient Hamiltonian Monte Carlo , 2014, ICML.