A randomized Mirror-Prox method for solving structured large-scale matrix saddle-point problems

TL;DR

This paper introduces a randomized Mirror-Prox algorithm that efficiently solves large-scale structured matrix saddle-point problems by using stochastic matrix exponential approximations, reducing computational complexity.

Contribution

The paper presents a novel randomized Mirror-Prox method that leverages stochastic approximations to handle large-scale matrix saddle-point problems more efficiently than existing deterministic methods.

Findings

Significant reduction in computational complexity.

Effective stochastic approximation of matrix exponentials.

Numerical results confirm theoretical efficiency gains.

Abstract

In this paper, we derive a randomized version of the Mirror-Prox method for solving some structured matrix saddle-point problems, such as the maximal eigenvalue minimization problem. Deterministic first-order schemes, such as Nesterov's Smoothing Techniques or standard Mirror-Prox methods, require the exact computation of a matrix exponential at every iteration, limiting the size of the problems they can solve. Our method allows us to use stochastic approximations of matrix exponentials. We prove that our randomized scheme decreases significantly the complexity of its deterministic counterpart for large-scale matrix saddle-point problems. Numerical experiments illustrate and confirm our theoretical results.