Towards optimal convex combination rules for gossiping

• Published in: 2013 American Control Conference pp. 1261 - 1265
• Main Authors: Mangoubi, Oren, Shaoshuai Mou, Ji Liu, Morse, A. Stephen
• Format: Conference Proceeding
• Language: English
• Published: IEEE 01.06.2013
• Subjects: Convergence; Eigenvalues and eigenfunctions; Probabilistic logic; Protocols; Signal processing algorithms; Tree graphs; Vectors
• ISBN: 1479901776; 9781479901777
• ISSN: 0743-1619
• DOI: 10.1109/ACC.2013.6580009

By the distributed averaging problem is meant the problem of computing the average value y avg of a set of numbers possessed by the agents in a distributed network using only communication between neighboring agents. Gossiping is a well-known approach to the problem which seeks to iteratively arrive at a solution by allowing each agent to interchange information with at most one neighbor at each iterative step. In the most widely studied situation, gossiping agents i and j update their current estimates x i (t) and x j (t) of y avg by setting their new estimates x i (t+1) and x j (t+1) equal to the average of x i (t) and x j (t). A more general approach is for gossiping agents i and j to use the convex combination update rules x i (t+1) = wx i (t) + (1 - w)x j (t) and x j (t + 1) = wx j (t) + (1 - w)x i (t) respectively where w is a real number between 0 and 1. While for probabilistic gossiping protocols, a largest convergence rate is attained when w = 0.5, for deterministic gossiping protocols this is not the case. The aim of this paper is to demonstrate by computer experiments and analytically studied examples that for deterministic gossiping protocols which are periodic, the value of w which maximizes convergence rate is not necessarily w = 0.5 and moreover, convergence at the optimal value of w can be significantly faster than convergence at the value w = 0.5. Thus this paper's contribution is to provide clear justification for a deeper study of the optimum convergence rate question for gossiping algorithms using convex combination rules.