Linear Convergence of Distributed Mirror Descent with Integral Feedback for Strongly Convex Problems

TL;DR

This paper proves that a decentralized mirror descent algorithm with integral feedback achieves exponential convergence for strongly convex problems, enhancing distributed optimization efficiency.

Contribution

It establishes the exponential convergence of a continuous-time decentralized mirror descent algorithm with integral feedback for strongly convex functions, using control theory tools.

Findings

Proves local exponential convergence of the algorithm.

Validates convergence speed with a real data-set experiment.

Extends previous asymptotic convergence results to exponential rates.

Abstract

Distributed optimization often requires finding the minimum of a global objective function written as a sum of local functions. A group of agents work collectively to minimize the global function. We study a continuous-time decentralized mirror descent algorithm that uses purely local gradient information to converge to the global optimal solution. The algorithm enforces consensus among agents using the idea of integral feedback. Recently, Sun and Shahrampour (2020) studied the asymptotic convergence of this algorithm for when the global function is strongly convex but local functions are convex. Using control theory tools, in this work, we prove that the algorithm indeed achieves (local) exponential convergence. We also provide a numerical experiment on a real data-set as a validation of the convergence speed of our algorithm.