On stochastic mirror descent with interacting particles: convergence properties and variance reduction

TL;DR

This paper investigates stochastic mirror descent with interacting particles, analyzing how interaction influences convergence and variance reduction, and compares it to traditional methods using decreasing or fixed step-sizes.

Contribution

It introduces an analysis of interacting particles in stochastic mirror descent, highlighting their benefits in convergence speed and variance reduction over standard approaches.

Findings

Interaction improves convergence rates.

Interaction reduces variance of estimates.

Tradeoffs exist between communication and variance reduction.

Abstract

An open problem in optimization with noisy information is the computation of an exact minimizer that is independent of the amount of noise. A standard practice in stochastic approximation algorithms is to use a decreasing step-size. This however leads to a slower convergence. A second alternative is to use a fixed step-size and run independent replicas of the algorithm and average these. A third option is to run replicas of the algorithm and allow them to interact. It is unclear which of these options works best. To address this question, we reduce the problem of the computation of an exact minimizer with noisy gradient information to the study of stochastic mirror descent with interacting particles. We study the convergence of stochastic mirror descent and make explicit the tradeoffs between communication and variance reduction. We provide theoretical and numerical evidence to suggest that interaction helps to improve convergence and reduce the variance of the estimate.