A Stochastic Smoothing Algorithm for Semidefinite Programming

TL;DR

This paper introduces a stochastic smoothing technique for semidefinite programming that leverages Gaussian perturbations to efficiently approximate maximum eigenvalues, improving complexity over deterministic methods in specific regimes.

Contribution

The paper develops a novel stochastic smoothing algorithm for maximum eigenvalue problems, enhancing efficiency compared to existing deterministic approaches.

Findings

Reduced complexity in certain precision/dimension regimes

Effective stochastic approximation of maximum eigenvalue function

Improved efficiency over deterministic smoothing algorithms

Abstract

We use a rank one Gaussian perturbation to derive a smooth stochastic approximation of the maximum eigenvalue function. We then combine this smoothing result with an optimal smooth stochastic optimization algorithm to produce an efficient method for solving maximum eigenvalue minimization problems. We show that the complexity of this new method is lower than that of deterministic smoothing algorithms in certain precision/dimension regimes.