A Proximal Stochastic Gradient Method with Progressive Variance Reduction

TL;DR

This paper introduces a new proximal stochastic gradient method with progressive variance reduction, achieving faster convergence rates for strongly convex composite optimization problems common in machine learning.

Contribution

It presents a multi-stage stochastic gradient algorithm that progressively reduces variance, leading to geometric convergence and lower complexity than existing methods.

Findings

Expected objective value converges geometrically to the optimum.

Method has similar per-iteration cost as classical stochastic gradient methods.

Overall complexity is significantly lower than proximal full gradient and standard stochastic methods.

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 multi-stage 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.