Convergence Analysis of Accelerated Stochastic Gradient Descent under the Growth Condition

Abstract

We study the convergence of accelerated stochastic gradient descent for strongly convex objectives under the growth condition, which states that the variance of stochastic gradient is bounded by a multiplicative part that grows with the full gradient, and a constant additive part. Through the lens of the growth condition, we investigate four widely used accelerated methods: Nesterov’s accelerated method (NAM), robust momentum method (RMM), accelerated dual averaging method (DAM+), and implicit DAM+ (iDAM+). While these methods are known to improve the convergence rate of SGD under the condition that the stochastic gradient has bounded variance, it is not well understood how their convergence rates are affected by the multiplicative noise. In this paper, we show that these methods all converge to a neighborhood of the optimum with accelerated convergence rates (compared to SGD) even under the growth condition. In particular, NAM, RMM, iDAM+ enjoy acceleration only with a mild multiplicative noise, while DAM+ enjoys acceleration even with a large multiplicative noise. Furthermore, we propose a generic tail-averaged scheme that allows the accelerated rates of DAM+ and iDAM+ to nearly attain the theoretical lower bound (up to a logarithmic factor in the variance term). We conduct numerical experiments to support our theoretical conclusions.

You-Lin Chen

PhD in Statistics (2016-2021)

You-Lin Chen was a statistics PhD candidate at the University of Chicago. He pursues his research interests in machine learning, stochastic and non-convex optimization, and high-dimensional statistics.

Sen Na

Postdoc in Statistics and ICSI

Sen Na is a postdoctoral scholar in the Statistics department and ICSI at UC Berkeley. His research interests broadly lie in the mathematical foundations of data science, with topics including high-dimensional statistics, graphical models, semiparametric models, optimal control, and large-scale and stochastic nonlinear optimization. He is also interested in applying machine learning methods in biology, neuroscience, and engineering.

Mladen Kolar

Associate Professor of Econometrics and Statistics

Mladen Kolar is an Associate Professor of Econometrics and Statistics at the University of Chicago Booth School of Business. His research is focused on high-dimensional statistical methods, graphical models, varying-coefficient models and data mining, driven by the need to uncover interesting and scientifically meaningful structures from observational data.