Linear Convergence of Variance-Reduced Stochastic Gradient without Strong Convexity

Stochastic gradient algorithms estimate the gradient based on only one or a fewsamples and enjoy low computational cost per iteration. They have been widelyused in large-scale optimization problems. However, stochastic gradient algo-rithms are usually slow to converge and achieve sub-linear convergence rates,due to the inherent variance in the gradient computation. To accelerate the con-vergence, some variance-reduced stochastic gradient algorithms, e.g., proximalstochastic variance-reduced gradient (Prox-SVRG) algorithm, have recently beenproposed to solve strongly convex problems. Under the strongly convex condi-tion, these variance-reduced stochastic gradient algorithms achieve a linear con-vergence rate. However, many machine learning problems are convex but notstrongly convex. In this paper, we introduce Prox-SVRG and its projected variantcalled Variance-Reduced Projected Stochastic Gradient (VRPSG) to solve a classof non-strongly convex optimization problems widely used in machine learning.As the main technical contribution of this paper, we show that both VRPSG andProx-SVRG achieve a linear convergence rate without strong convexity. A keyingredient in our proof is a Semi-Strongly Convex (SSC) inequality which is thefirst to be rigorously proved for a class of non-strongly convex problems in bothconstrained and regularized settings. Moreover, the SSC inequality is independentof algorithms and may be applied to analyze other stochastic gradient algorithmsbesides VRPSG and Prox-SVRG, which may be of independent interest. To thebest of our knowledge, this is the first work that establishes the linear conver-gence rate for the variance-reduced stochastic gradient algorithms on solving bothconstrained and regularized problems without strong convexity.