Distributed Averaging using non-convex updates

Motivated by applications in distributed sensing, a significant amount of effort has been directed towards developing energy efficient algorithms for information exchange on graphs. The problem of distributed averaging has been studied intensively because it appears in several applications such as estimation on ad hoc wireless and sensor networks. A Gossip Algorithm is an averaging algorithm that, after a certain number of information exchanges and updates, leaves each node with a value close to the average of all the originally held values. 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 and present a case (the dumbbell graph) where they lead to an exponential speed-up. We do so by exhibiting a decentralized algorithm for averaging on a dumbbell graph with $2n$ nodes, that uses non-convex averages and has an averaging time that is $O(\ln n)$. On the other hand, we show that any decentralized algorithm using only convex updates on this graph must have an averaging time that is $\Omega(n)$. We use stochastic dominance to prove the last result in a way that may be of independent interest.