Acceleration of gossip algorithms through the Euler–Poisson–Darboux Equation

Author:

Berthier Raphaël1,Li Mufan (Bill)2

Affiliation:

1. EPFL , Lausanne 1015 , Switzerland

2. University of Toronto and Vector Institute , Toronto, ON M5G 1Z5 , Canada

Abstract

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.

Funder

Ontario Graduate Scholarship

Vector Institute

Publisher

Oxford University Press (OUP)

Subject

Applied Mathematics

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3