Variance-Based Bregman Extragradient Algorithm with Line Search for Solving Stochastic Variational Inequalities

TL;DR

This paper introduces a variance-based Bregman extragradient algorithm with line search for stochastic variational inequalities, achieving robust convergence without knowing the Lipschitz constant and providing optimal convergence rates.

Contribution

It proposes a novel variance-based extragradient method with line search that ensures convergence under unknown Lipschitz constants and establishes optimal convergence and complexity rates.

Findings

Almost sure convergence proved using a new method.

Achieves $ ext{O}(1/k)$ convergence rate for bounded sets.

Demonstrates superiority through numerical experiments.

Abstract

The main purpose of this paper is to propose a variance-based Bregman extragradient algorithm with line search for solving stochastic variational inequalities, which is robust with respect an unknown Lipschitz constant. We prove the almost sure convergence of the algorithm by a more concise and effective method instead of using the supermartingale convergence theorem. Furthermore, we obtain not only the convergence rate $\mathcal{O}(1/k)$ with the gap function when $X$ is bounded, but also the same convergence rate in terms of the natural residual function when $X$ is unbounded. Under the Minty variational inequality condition, we derive the iteration complexity $\mathcal{O}(1/\varepsilon)$ and the oracle complexity $\mathcal{O}(1/\varepsilon^2)$ in both cases. Finally, some numerical results demonstrate the superiority of the proposed algorithm.