Distributed Averaging in the presence of a Sparse Cut

TL;DR

This paper introduces a decentralized algorithm using non-convex updates for distributed averaging on graphs with a sparse cut, achieving faster convergence than traditional convex methods.

Contribution

It proposes a novel non-convex averaging algorithm for graphs with a sparse cut, improving convergence speed over existing convex-based algorithms.

Findings

Non-convex updates can significantly reduce averaging time.

The algorithm outperforms known distributed averaging methods.

Stochastic dominance is used to prove the effectiveness of the approach.

Abstract

We consider the question of averaging on a graph that has one sparse cut separating two subgraphs that are internally well connected. While there has been a large body of work devoted to algorithms for distributed averaging, nearly all algorithms involve only {\it convex} updates. In this paper, we suggest that {\it non-convex} updates can lead to significant improvements. We do so by exhibiting a decentralized algorithm for graphs with one sparse cut that uses non-convex averages and has an averaging time that can be significantly smaller than the averaging time of known distributed algorithms, such as those of \cite{tsitsiklis, Boyd}. We use stochastic dominance to prove this result in a way that may be of independent interest.