On stochastic mirror descent with interacting particles: convergence properties and variance reduction
This addresses an open problem in optimization for researchers and practitioners dealing with noisy data, though it appears incremental as it builds on existing stochastic approximation methods.
The paper tackles the problem of computing an exact minimizer in optimization with noisy gradient information by comparing strategies like decreasing step-sizes, independent replicas, and interacting replicas. It finds that interaction in stochastic mirror descent improves convergence and reduces variance, supported by theoretical and numerical evidence.
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.