Universal Extra-Gradient Method for Stochastic Variational Inequalities

We study the stochastic Extra-Gradient Method (EGM) for solving a stochastic variational inequality (VI) over a compact convex set. The main assumption is $(\delta ,L)$-continuity of the VI mapping, which includes H\"older mappings and their linear combinations. We develop new adaptive, line-search-free step-size rules for stochastic EGM. These rules can be interpreted as enhanced AdaGrad-type updates and require no prior knowledge of either the continuity modulus of the VI mapping or the variance of the stochastic oracle. The resulting methods are universal and attain the best complexity bound among the admissible continuity regimes. We prove convergence guarantees in terms of an expected residual function that upper bounds the relevant accuracy measures for several important problem classes, including the dual gap for monotone VIs, and the primal-dual gap for saddle-point and optimization problems. For H\"older-continuous mappings, our bounds recover the optimal dimension-independent complexity of first-order methods.