Convergence of Extragradient SVRG for Variational Inequalities: Error Bounds and Increasing Iterate Averaging

Abstract

We study the last-iterate convergence of variance reduction methods for extragradient (EG) algorithms for a class of variational inequalities satisfying error-bound conditions. Previously, last-iterate linear convergence was only known under strong monotonicity. We show that EG algorithms with SVRG-style variance reduction, denoted SVRG-EG, attain last-iterate linear convergence under a general error-bound condition much weaker than strong monotonicity. This condition captures a broad class of non-strongly monotone problems, such as bilinear saddle-point problems commonly encountered in two-player zero-sum Nash equilibrium computation. Next, we establish linear last-iterate convergence of SVRG-EG with an improved guarantee under the weak sharpness assumption. Furthermore, motivated by the empirical efficiency of increasing iterate averaging techniques in solving saddle-point problems, we also establish new convergence results for SVRG-EG with such techniques.