On Centralized and Distributed Mirror Descent: Convergence Analysis Using Quadratic Constraints

TL;DR

This paper introduces a semi-definite programming framework using quadratic constraints to analyze and certify convergence rates of centralized and distributed mirror descent algorithms, including new results for distributed settings.

Contribution

It develops a novel SDP-based analysis framework for MD, providing explicit convergence rates and extending understanding to distributed algorithms with new rate characterizations.

Findings

Certifies exponential convergence for centralized MD under strong convexity.

Provides $O(1/k)$ convergence rate for convex problems in both centralized and distributed MD.

Shows superior convergence rates for distributed MD compared to existing methods.

Abstract

Mirror descent (MD) is a powerful first-order optimization technique that subsumes several optimization algorithms including gradient descent (GD). In this work, we develop a semi-definite programming (SDP) framework to analyze the convergence rate of MD in centralized and distributed settings under both strongly convex and non-strongly convex assumptions. We view MD with a dynamical system lens and leverage quadratic constraints (QCs) to provide explicit convergence rates based on Lyapunov stability. For centralized MD under strongly convex assumption, we develop a SDP that certifies exponential convergence rates. We prove that the SDP always has a feasible solution that recovers the optimal GD rate as a special case. We complement our analysis by providing the $O(1/k)$ convergence rate for convex problems. Next, we analyze the convergence of distributed MD and characterize the rate using SDP. To the best of our knowledge, the numerical rate of distributed MD has not been previously reported in the literature. We further prove an $O(1/k)$ convergence rate for distributed MD in the convex setting. Our numerical experiments on strongly convex problems indicate that our framework certifies superior convergence rates compared to the existing rates for distributed GD.