Optimal Distributed Stochastic Mirror Descent for Strongly Convex Optimization

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.