Low complexity convergence rate bounds for the synchronous gossip subclass of push-sum algorithms

TL;DR

This paper derives simple bounds on the exponential convergence rate of push-sum algorithms in synchronous gossip networks, relating the bounds to network spectrum and demonstrating their effectiveness through simulations.

Contribution

It introduces accessible bounds for convergence rates of push-sum algorithms in synchronous gossip scenarios, with analysis based on network spectral properties.

Findings

Bounds depend on network spectrum

More symmetric networks yield tighter bounds

Numerical simulations confirm the bounds' effectiveness

Abstract

We develop easily accessible quantities for bounding the almost sure exponential convergence rate of push-sum algorithms. We analyze the scenario of i.i.d. synchronous gossip, every agent communicating towards its single target at every step. Multiple bounding expressions are developed depending on the generality of the setup, all functions of the spectrum of the network. While the most general bound awaits further improvement, with more symmetries, close bounds can be established, as demonstrated by numerical simulations.