Paving the Way for Consensus: Convergence of Block Gossip Algorithms

TL;DR

This paper establishes new convergence guarantees for various gossip protocols in distributed consensus, linking them to the block randomized Kaczmarz method, and extends theoretical results to more general settings with empirical validation.

Contribution

It introduces a novel theoretical framework connecting gossip algorithms with the block randomized Kaczmarz method, extending convergence results to broader scenarios.

Findings

Proves convergence guarantees for path, clique, and pairwise gossip protocols.

Extends Kaczmarz-based convergence analysis to rank-deficient and inconsistent systems.

Provides empirical validation across synthetic networks.

Abstract

Gossip protocols are popular methods for average consensus problems in distributed computing. We prove new convergence guarantees for a variety of such protocols, including path, clique, and synchronous pairwise gossip. These arise by exploiting the connection between these protocols and the block randomized Kaczmarz method for solving linear systems. Moreover, we extend existing convergence results for block randomized Kaczmarz to allow for a more general choice of blocks, rank-deficient systems, and provide a tighter convergence rate guarantee. We furthermore apply this analysis to inconsistent consensus models and obtain similar guarantees. An extensive empirical analysis of these methods is provided for a variety of synthetic networks.