Achieving Geometric Convergence for Distributed Optimization over Time-Varying Graphs

TL;DR

This paper introduces the DIGing and Push-DIGing algorithms for distributed optimization over time-varying graphs, achieving geometric convergence rates with fixed step-sizes, applicable to both undirected and directed graphs.

Contribution

The paper proposes novel algorithms that ensure geometric convergence for distributed optimization over time-varying graphs, extending to directed graphs with push-sum protocol integration.

Findings

Algorithms converge at R-linear (geometric) rates under strong convexity.

DIGing scales polynomially with the number of agents.

Numerical experiments validate theoretical convergence and efficacy.

Abstract

This paper considers the problem of distributed optimization over time-varying graphs. For the case of undirected graphs, we introduce a distributed algorithm, referred to as DIGing, based on a combination of a distributed inexact gradient method and a gradient tracking technique. The DIGing algorithm uses doubly stochastic mixing matrices and employs fixed step-sizes and, yet, drives all the agents' iterates to a global and consensual minimizer. When the graphs are directed, in which case the implementation of doubly stochastic mixing matrices is unrealistic, we construct an algorithm that incorporates the push-sum protocol into the DIGing structure, thus obtaining Push-DIGing algorithm. The Push-DIGing uses column stochastic matrices and fixed step-sizes, but it still converges to a global and consensual minimizer. Under the strong convexity assumption, we prove that the algorithms converge at R-linear (geometric) rates as long as the step-sizes do not exceed some upper bounds. We establish explicit estimates for the convergence rates. When the graph is undirected it shows that DIGing scales polynomially in the number of agents. We also provide some numerical experiments to demonstrate the efficacy of the proposed algorithms and to validate our theoretical findings.