Acceleration of Gossip Algorithms through the Euler-Poisson-Darboux Equation

TL;DR

This paper studies the scaling limits of gossip algorithms, revealing that accelerated versions converge to the Euler-Poisson-Darboux PDE, which exhibits optimal spreading properties compared to the heat equation, supported by simulations.

Contribution

It introduces a novel PDE-based analysis of gossip algorithms in the scaling limit, connecting accelerated gossip to the Euler-Poisson-Darboux equation and demonstrating optimal spreading behavior.

Findings

Accelerated gossip converges to the Euler-Poisson-Darboux PDE.

Fundamental solution spreads at a linear rate, faster than the heat equation.

Simulations confirm the PDE approximation of gossip algorithms.

Abstract

Gossip algorithms and their accelerated versions have been studied exclusively in discrete time on graphs. In this work, we take a different approach, and consider the scaling limit of gossip algorithms in both large graphs and large number of iterations. These limits lead to well-known partial differential equations (PDEs) with insightful properties. On lattices, we prove that the non-accelerated gossip algorithm of Boyd et al. [2006] converges to the heat equation, and the accelerated Jacobi polynomial iteration of Berthier et al. [2020] converges to the Euler-Poisson-Darboux (EPD) equation - a damped wave equation. Remarkably, with appropriate parameters, the fundamental solution of the EPD equation has the ideal gossip behaviour: a uniform density over an ellipsoid, whose radius increases at a rate proportional to t - the fastest possible rate for locally communicating gossip algorithms. This is in contrast with the heat equation where the density spreads on a typical scale of $\sqrt{t}$. Additionally, we provide simulations demonstrating that the gossip algorithms are accurately approximated by their limiting PDEs.