Convergence Rate Bounds for the Mirror Descent Method: IQCs and the Bregman Divergence

TL;DR

This paper develops a new IQC-based framework for analyzing the convergence rates of the mirror descent method in convex optimization, providing tighter bounds and insights into the role of Bregman divergence.

Contribution

It introduces an IQC-based analysis for mirror descent, connecting Bregman divergence with Popov criterion to derive less conservative convergence bounds.

Findings

IQC framework effectively analyzes convergence rates

Bregman divergence is a special Lyapunov function case

Derived bounds are shown to be tight in examples

Abstract

This paper is concerned with convergence analysis for the mirror descent (MD) method, a well-known algorithm in convex optimization. An analysis framework via integral quadratic constraints (IQCs) is constructed to analyze the convergence rate of the MD method with strongly convex objective functions in both continuous-time and discrete-time. We formulate the problem of finding convergence rates of the MD algorithms into feasibility problems of linear matrix inequalities (LMIs) in both schemes. In particular, in continuous-time, we show that the Bregman divergence function, which is commonly used as a Lyapunov function for this algorithm, is a special case of the class of Lyapunov functions associated with the Popov criterion, when the latter is applied to an appropriate reformulation of the problem. Thus, applying the Popov criterion and its combination with other IQCs, can lead to convergence rate bounds with reduced conservatism. We also illustrate via examples that the convergence rate bounds derived can be tight.