# New Version of Mirror Prox for Variational Inequalities with Adaptation to Inexactness

## TL;DR

This paper introduces an adaptive Mirror Prox method for variational inequalities that adjusts to both Lipschitz constant and oracle error, achieving near-optimal complexity and demonstrating effectiveness through experiments.

## Contribution

It presents a novel adaptive Mirror Prox algorithm that accounts for inexact oracle information and adapts to problem parameters, improving convergence guarantees.

## Key findings

- Achieves near $O(1/\varepsilon \log(1/\varepsilon))$ complexity for certain monotone operators.
- Demonstrates superior performance in experiments compared to existing methods.
- Shows effectiveness in matrix games with non-Euclidean setups.

## Abstract

Some adaptive analogue of the Mirror Prox method for variational inequalities is proposed. In this work we consider the adaptation not only to the value of the Lipschitz constant, but also to the magnitude of the oracle error. This approach, in particular, allows us to prove a complexity near $O\left(\frac{1}{\varepsilon}\log_2\frac{1}{\varepsilon}\right)$ for variational inequalities for a special class of monotone bounded operators. This estimate is optimal for variational inequalities with monotone Lipschitz-continuous operators. However, there exists some error, which may be insignificant. The results of experiments on the comparison of the proposed approach with some known analogues are presented. Also, we discuss the results of the experiments for matrix games in the case of using non-Euclidean proximal setup.