Optimal stochastic extragradient schemes for pseudomonotone stochastic variational inequality problems and their variants

TL;DR

This paper develops and analyzes extragradient-based stochastic approximation algorithms for solving stochastic variational inequality problems with non-monotone maps, achieving optimal convergence rates under pseudomonotonicity.

Contribution

It introduces extragradient and mirror-prox schemes for pseudomonotone stochastic variational inequalities, providing convergence proofs and optimal rates, extending beyond monotone cases.

Findings

Convergence of iterates under pseudomonotonicity and acute angle conditions.

Achieved optimal O(1/k) convergence rate under strong pseudomonotonicity.

Empirical results show the importance of initial step length for performance.

Abstract

We consider the stochastic variational inequality problem in which the map is expectation-valued in a component-wise sense. Much of the available convergence theory and rate statements for stochastic approximation schemes are limited to monotone maps. However, non-monotone stochastic variational inequality problems are not uncommon and are seen to arise from product pricing, fractional optimization problems, and subclasses of economic equilibrium problems. Motivated by the need to address a broader class of maps, we make the following contributions: (i) We present an extragradient-based stochastic approximation scheme and prove that the iterates converge to a solution of the original problem under either pseudomonotonicity requirements or a suitably defined acute angle condition. Such statements are shown to be generalizable to the stochastic mirror-prox framework; (ii) Under strong pseudomonotonicity, we show that the mean-squared error in the solution iterates produced by the extragradient SA scheme converges at the optimal rate of O(1/k) statements that were hitherto unavailable K in this regime. Notably, we optimize the initial steplength by obtaining an {\epsilon}-infimum of a discontinuous nonconvex function. Similar statements are derived for mirror-prox generalizations and can accommodate monotone SVIs under a weak-sharpness requirement. Finally, both the asymptotics and the empirical rates of the schemes are studied on a set of variational problems where it is seen that the theoretically specified initial steplength leads to significant performance benefits.