Distributed Averaging via Lifted Markov Chains

TL;DR

This paper introduces a novel pseudo-lifting method to design non-reversible Markov chains that achieve the fastest convergence rates for distributed averaging on network graphs, optimizing for network topology constraints.

Contribution

The paper presents a new pseudo-lifting technique to construct non-reversible Markov chains with optimal convergence rates for distributed averaging algorithms.

Findings

Convergence time proportional to network diameter

Pseudo-lifting yields the fastest mixing Markov chain

Applicable to graphs with geometric or doubling dimension

Abstract

Motivated by applications of distributed linear estimation, distributed control and distributed optimization, we consider the question of designing linear iterative algorithms for computing the average of numbers in a network. Specifically, our interest is in designing such an algorithm with the fastest rate of convergence given the topological constraints of the network. As the main result of this paper, we design an algorithm with the fastest possible rate of convergence using a non-reversible Markov chain on the given network graph. We construct such a Markov chain by transforming the standard Markov chain, which is obtained using the Metropolis-Hastings method. We call this novel transformation pseudo-lifting. We apply our method to graphs with geometry, or graphs with doubling dimension. Specifically, the convergence time of our algorithm (equivalently, the mixing time of our Markov chain) is proportional to the diameter of the network graph and hence optimal. As a byproduct, our result provides the fastest mixing Markov chain given the network topological constraints, and should naturally find their applications in the context of distributed optimization, estimation and control.