Noise and Fluctuation of Finite Learning Rate Stochastic Gradient Descent

• URL: http://arxiv.org/abs/2012.03636v3
• Date: Fri, 12 Feb 2021 08:43:27 GMT
• Title: Noise and Fluctuation of Finite Learning Rate Stochastic Gradient Descent
• Authors: Kangqiao Liu, Liu Ziyin, Masahito Ueda
• Abstract summary: gradient descent (SGD) is relatively well understood in the vanishing learning rate regime. We propose to study the basic properties of SGD and its variants in the non-vanishing learning rate regime.
• Score: 3.0079490585515343
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In the vanishing learning rate regime, stochastic gradient descent (SGD) is now relatively well understood. In this work, we propose to study the basic properties of SGD and its variants in the non-vanishing learning rate regime. The focus is on deriving exactly solvable results and discussing their implications. The main contributions of this work are to derive the stationary distribution for discrete-time SGD in a quadratic loss function with and without momentum; in particular, one implication of our result is that the fluctuation caused by discrete-time dynamics takes a distorted shape and is dramatically larger than a continuous-time theory could predict. Examples of applications of the proposed theory considered in this work include the approximation error of variants of SGD, the effect of minibatch noise, the optimal Bayesian inference, the escape rate from a sharp minimum, and the stationary distribution of a few second-order methods including damped Newton's method and natural gradient descent.

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