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 (ADAM), and implicit ADAM (iADAM). 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, iADAM enjoy acceleration only with a mild multiplicative noise, while ADAM enjoys acceleration even with a large multiplicative noise. Furthermore, we propose a generic tail-averaged scheme that allows the accelerated rates of ADAM and iADAM to nearly attain the theoretical lower bound (up to a logarithmic factor in the variance term).
You-Lin Chen
PhD (2016-2021)
You-Lin Chen was a statistics PhD candidate at the University of Chicago advised by Mladen Kolar. He pursues his research interests in machine learning, stochastic and non-convex optimization, high-dimensional statistics.
Sen Na
PhD (2016-2021)
Sen Na was a PhD student in the Department of Statistics at The University of Chicago. Prior to graduate school, he obtained BS in mathematics at Nanjing University, China. His research interests lie in nonlinear and nonconvex optimization, dynamic programming, high-dimensional statistics and their interface.
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.