A proximal stochastic gradient method with progressive variance reduction

Lin Xiao, Tong Zhang

Research output: Contribution to journalArticlepeer-review

350 Scopus citations

Abstract

We consider the problem of minimizing the sum of two convex functions: one is the average of a large number of smooth component functions, and the other is a general convex function that admits a simple proximal mapping. We assume the whole objective function is strongly convex. Such problems often arise in machine learning, known as regularized empirical risk minimization. We propose and analyze a new proximal stochastic gradient method, which uses a multistage scheme to progressively reduce the variance of the stochastic gradient. While each iteration of this algorithm has similar cost as the classical stochastic gradient method (or incremental gradient method), we show that the expected objective value converges to the optimum at a geometric rate. The overall complexity of this method is much lower than both the proximal full gradient method and the standard proximal stochastic gradient method.

Original languageEnglish
Pages (from-to)2057-2075
Number of pages19
JournalSIAM Journal on Optimization
Volume24
Issue number4
DOIs
StatePublished - 2014
Externally publishedYes

Keywords

  • Proximal mapping
  • Stochastic gradient method
  • Variance reduction

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