Optimal Distributed Stochastic Mirror Descent for Strongly Convex Optimization

TL;DR

This paper introduces two non-Euclidean stochastic subgradient algorithms for distributed strongly convex optimization over time-varying networks, achieving optimal convergence rates with theoretical guarantees and simulation validation.

Contribution

It proposes novel Bregman divergence-based algorithms that improve convergence rates for distributed stochastic strongly convex optimization.

Findings

First algorithm achieves O(ln(T)/T) rate

Second epoch algorithm attains optimal O(1/T) rate

Algorithms outperform Euclidean-based methods in simulations

Abstract

In this paper we consider convergence rate problems for stochastic strongly-convex optimization in the non-Euclidean sense with a constraint set over a time-varying multi-agent network. We propose two efficient non-Euclidean stochastic subgradient descent algorithms based on the Bregman divergence as distance-measuring function rather than the Euclidean distances that were employed by the standard distributed stochastic projected subgradient algorithms. For distributed optimization of nonsmooth and strongly convex functions whose only stochastic subgradients are available, the first algorithm recovers the best previous known rate of O(ln(T)/T) (where T is the total number of iterations). The second algorithm is an epoch variant of the first algorithm that attains the optimal convergence rate of O(1/T), matching that of the best previously known centralized stochastic subgradient algorithm. Finally, we report some simulation results to illustrate the proposed algorithms.