Popov Mirror-Prox Method for Variational Inequalities
Abhishek Chakraborty, Angelia Nedić
TL;DR
This work extends the Popov variant of the mirror-prox method to variational inequalities whose mappings satisfy a generalized polynomial-growth condition, introducing completely parameter-free step-size schemes usable in both stochastic and deterministic settings. It establishes optimal convergence rates in terms of the dual gap over bounded domains and, for Hölder-continuous deterministic mappings with $M_\nu=0$, improves the rate via the residual function under Minty solvability, while relaxing monotonicity and boundedness when a Minty solution exists. The analysis covers both constant and diminishing steps, with specialized averaging schemes that achieve robust $O(1/\sqrt{T})$ or $O(1/T^{(1+\nu)/2})$ rates depending on the setting, and demonstrates practical viability through experiments on matrix games, piecewise quadratic functions, and image-classification tasks with ResNet-18. Overall, the paper broadens the applicability of mirror-prox techniques to broader growth conditions, removes reliance on problem-parameter knowledge, and provides empirical validation across stochastic and deterministic VI scenarios. The results have potential impact on equilibrium computation, game-theoretic learning, and robust optimization in settings with non-Lipschitz or discontinuous dynamics.
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 Hölder 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 Hölder 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.