Linear Convergence of Variance-Reduced Stochastic Gradient without Strong Convexity

TL;DR

This paper proves that variance-reduced stochastic gradient algorithms can achieve linear convergence rates even for convex problems lacking strong convexity, by establishing a new semi-strong convexity inequality.

Contribution

It introduces the SSC inequality and demonstrates linear convergence of VRPSG and Prox-SVRG without requiring strong convexity, extending their applicability.

Findings

Prox-SVRG and VRPSG achieve linear convergence without strong convexity.

The SSC inequality is independent of algorithms and applicable broadly.

First proof of linear convergence for variance-reduced methods on non-strongly convex problems.

Abstract

Stochastic gradient algorithms estimate the gradient based on only one or a few samples and enjoy low computational cost per iteration. They have been widely used in large-scale optimization problems. However, stochastic gradient algorithms are usually slow to converge and achieve sub-linear convergence rates, due to the inherent variance in the gradient computation. To accelerate the convergence, some variance-reduced stochastic gradient algorithms, e.g., proximal stochastic variance-reduced gradient (Prox-SVRG) algorithm, have recently been proposed to solve strongly convex problems. Under the strongly convex condition, these variance-reduced stochastic gradient algorithms achieve a linear convergence rate. However, many machine learning problems are convex but not strongly convex. In this paper, we introduce Prox-SVRG and its projected variant called Variance-Reduced Projected Stochastic Gradient (VRPSG) to solve a class of non-strongly convex optimization problems widely used in machine learning. As the main technical contribution of this paper, we show that both VRPSG and Prox-SVRG achieve a linear convergence rate without strong convexity. A key ingredient in our proof is a Semi-Strongly Convex (SSC) inequality which is the first to be rigorously proved for a class of non-strongly convex problems in both constrained and regularized settings. Moreover, the SSC inequality is independent of algorithms and may be applied to analyze other stochastic gradient algorithms besides VRPSG and Prox-SVRG, which may be of independent interest. To the best of our knowledge, this is the first work that establishes the linear convergence rate for the variance-reduced stochastic gradient algorithms on solving both constrained and regularized problems without strong convexity.