Average-case optimization analysis for distributed consensus algorithms on regular graphs

TL;DR

This paper analyzes the average-case performance of distributed consensus algorithms on regular graphs, deriving optimal methods and comparing their efficiency to existing first-order algorithms.

Contribution

It introduces an average-case analysis framework for consensus algorithms on regular graphs, deriving an optimal method related to the Heavy Ball approach.

Findings

Derived the optimal consensus method for regular graphs.

Established the relation to the Heavy Ball method.

Compared convergence rates of various algorithms through numerical experiments.

Abstract

The consensus problem in distributed computing involves a network of agents aiming to compute the average of their initial vectors through local communication, represented by an undirected graph. This paper focuses on the studying of this problem using an average-case analysis approach, particularly over regular graphs. Traditional algorithms for solving the consensus problem often rely on worst-case performance evaluation scenarios, which may not reflect typical performance in real-world applications. Instead, we apply average-case analysis, focusing on the expected spectral distribution of eigenvalues to obtain a more realistic view of performance. Key contributions include deriving the optimal method for consensus on regular graphs, showing its relation to the Heavy Ball method, analyzing its asymptotic convergence rate, and comparing it to various first-order methods through numerical experiments.