Convergence Analysis of Distributed Optimization: A Dissipativity Framework

TL;DR

This paper introduces a system-theoretic framework using dissipativity and contraction theory to analyze the convergence of distributed optimization algorithms modeled as interconnected dynamical systems.

Contribution

It provides a novel, systematic approach for convergence analysis applicable to any network structure, expressed through linear matrix inequalities.

Findings

The framework successfully analyzes distributed gradient descent convergence.

It offers a step-by-step analysis pipeline adaptable to various network topologies.

Numerical comparisons demonstrate advantages over traditional methods.

Abstract

We develop a system-theoretic framework for the structured analysis of distributed optimization algorithms with decomposable cost functions. We model such algorithms as a network of interacting dynamical systems and derive tests for convergence based on incremental dissipativity and contraction theory. This approach yields a step-by-step analysis pipeline suitable for any network structure, with conditions expressed as linear matrix inequalities. In addition, a numerical comparison with traditional analysis methods is presented, in the context of distributed gradient descent.