Stochastic variance reduced extragradient methods for solving hierarchical variational inequalities

TL;DR

This paper introduces stochastic variance reduced extragradient methods for hierarchical variational inequalities, providing the first convergence rates and complexity analysis for such algorithms in Euclidean and Bregman frameworks.

Contribution

It is the first to establish convergence rates and complexity bounds for variance-reduced stochastic algorithms solving hierarchical variational inequalities.

Findings

Proved convergence rates for hierarchical VI algorithms.

Established complexity bounds in Euclidean and Bregman setups.

Extended analysis to broad classes of variational inequality problems.

Abstract

We are concerned with optimization in a broad sense through the lens of solving variational inequalities (VIs) -- a class of problems that are so general that they cover as particular cases minimization of functions, saddle-point (minimax) problems, Nash equilibrium problems, and many others. The key challenges in our problem formulation are the two-level hierarchical structure and finite-sum representation of the smooth operators in each level. For this setting, we are the first to prove convergence rates and complexity statements for variance-reduced stochastic algorithms approaching the solution of hierarchical VIs in Euclidean and Bregman setups.