Gossip of Statistical Observations using Orthogonal Polynomials

Consider a network of agents connected by communication links, where each agent holds a real value. The gossip problem consists in estimating the average of the values diffused in the network in a distributed manner. Current techniques for gossiping are designed to deal with worst-case scenarios, which is irrelevant in applications to distributed statistical learning and denoising in sensor networks. We design second-order gossip methods tailor-made for the case where the real values are i.i.d. samples from the same distribution. In some regular network structures, we are able to prove optimality of our methods, and simulations suggest that they are efficient in a wide range of random networks. Our approach of gossip stems from a new acceleration framework using the family of orthogonal polynomials with respect to the spectral measure of the network graph.