Distributed Random Reshuffling over Networks

TL;DR

This paper introduces a distributed random reshuffling algorithm for networked optimization, achieving fast convergence rates for both convex and nonconvex problems, matching centralized methods and outperforming distributed SGD.

Contribution

The paper proposes the D-RR algorithm, extending random reshuffling to distributed settings with proven convergence rates for convex and nonconvex functions.

Findings

D-RR achieves an $oldsymbol{O}(1/T^2)$ convergence rate for strongly convex functions.

D-RR attains an $oldsymbol{O}(1/T^{2/3})$ rate for nonconvex functions.

Numerical experiments confirm the efficiency of D-RR over distributed SGD.

Abstract

In this paper, we consider distributed optimization problems where $n$ agents, each possessing a local cost function, collaboratively minimize the average of the local cost functions over a connected network. To solve the problem, we propose a distributed random reshuffling (D-RR) algorithm that invokes the random reshuffling (RR) update in each agent. We show that D-RR inherits favorable characteristics of RR for both smooth strongly convex and smooth nonconvex objective functions. In particular, for smooth strongly convex objective functions, D-RR achieves $\mathcal{O}(1/T^2)$ rate of convergence (where $T$ counts epoch number) in terms of the squared distance between the iterate and the global minimizer. When the objective function is assumed to be smooth nonconvex, we show that D-RR drives the squared norm of gradient to $0$ at a rate of $\mathcal{O}(1/T^{2/3})$. These convergence results match those of centralized RR (up to constant factors) and outperform the distributed stochastic gradient descent (DSGD) algorithm if we run a relatively large number of epochs. Finally, we conduct a set of numerical experiments to illustrate the efficiency of the proposed D-RR method on both strongly convex and nonconvex distributed optimization problems.