Exact Diffusion for Distributed Optimization and Learning --- Part II: Convergence Analysis

TL;DR

This paper analyzes the convergence and stability of the exact diffusion algorithm in distributed optimization, demonstrating its linear convergence, wider stability range, and faster convergence compared to previous methods through theoretical and numerical evidence.

Contribution

It provides a detailed convergence analysis of the exact diffusion algorithm, establishing its linear convergence and broader stability range over existing solutions.

Findings

Exact diffusion achieves linear convergence.

It has a wider stability range than the EXTRA method.

Numerical simulations confirm theoretical results.

Abstract

Part I of this work [2] developed the exact diffusion algorithm to remove the bias that is characteristic of distributed solutions for deterministic optimization problems. The algorithm was shown to be applicable to a larger set of combination policies than earlier approaches in the literature. In particular, the combination matrices are not required to be doubly stochastic, which impose stringent conditions on the graph topology and communications protocol. In this Part II, we examine the convergence and stability properties of exact diffusion in some detail and establish its linear convergence rate. We also show that it has a wider stability range than the EXTRA consensus solution, meaning that it is stable for a wider range of step-sizes and can, therefore, attain faster convergence rates. Analytical examples and numerical simulations illustrate the theoretical findings.