Almost Sure Convergence and Non-asymptotic Concentration Bounds for Stochastic Mirror Descent Algorithm

TL;DR

This paper analyzes the convergence and concentration properties of the Stochastic Mirror Descent algorithm with biased stochastic subgradients, providing theoretical guarantees and bounds under various noise assumptions.

Contribution

It establishes almost sure convergence and derives new concentration bounds for SMD with biased and Sub-Gaussian stochastic subgradients.

Findings

Almost sure convergence under diminishing bias

Concentration bounds for function value discrepancies

Refined bounds assuming Sub-Gaussian noise

Abstract

This letter investigates the convergence and concentration properties of the Stochastic Mirror Descent (SMD) algorithm utilizing biased stochastic subgradients. We establish the almost sure convergence of the algorithm's iterates under the assumption of diminishing bias. Furthermore, we derive concentration bounds for the discrepancy between the iterates' function values and the optimal value, based on standard assumptions. Subsequently, leveraging the assumption of Sub-Gaussian noise in stochastic subgradients, we present refined concentration bounds for this discrepancy.