Popov Mirror-Prox Method for Variational Inequalities

TL;DR

This paper introduces a parameter-free Popov mirror-prox algorithm that achieves optimal convergence for stochastic and deterministic variational inequalities, including non-monotone cases, under polynomial growth conditions.

Contribution

It extends mirror-prox methods to handle polynomial growth mappings without prior parameter knowledge, addressing both monotone and certain non-monotone variational inequalities.

Findings

Achieves optimal convergence rates for stochastic and deterministic VIs.

Handles non-monotone VIs with residual convergence without boundedness.

Validated effectiveness on matrix games, quadratic functions, and image classification.

Abstract

This paper establishes the convergence properties of the Popov mirror-prox algorithm for solving stochastic and deterministic variational inequalities (VIs) under a polynomial growth condition on the mapping variation. Unlike existing methods that require prior knowledge of problem-specific parameters, we propose step-size schemes that are entirely parameter-free in both constant and diminishing forms. For stochastic and deterministic monotone VIs, we establish optimal convergence rates in terms of the dual gap function over a bounded constraint set. Additionally, for deterministic VIs with H\"older continuous mapping, we prove convergence in terms of the residual function without requiring a bounded set or a monotone mapping, provided a Minty solution exists. This allows our method to address certain classes of non-monotone VIs. However, knowledge of the H\"older exponent is necessary to achieve the best convergence rates in this case. By extending mirror-prox techniques to mappings with arbitrary polynomial growth, our work bridges an existing gap in the literature. We validate our theoretical findings with empirical results on matrix games, piecewise quadratic functions, and image classification tasks using ResNet-18.