## A Generalized Gossip Algorithm on Convex Metric Spaces

**Title**: A Generalized Gossip Algorithm on Convex Metric Spaces

**Authors**:

Somasundaram, Kiran K

Matei, Ion

**Conference**: IEEE Transactions on Automatic Control Vol. 60, No. 5, pp. 1175 - 1187

**Date**: May 01 - May 01, 2015

A consensus problem consists of a group of dynamic agents who seek to agree upon certain quantities of interest. This problem can be generalized in the context of convex metric spaces that extend the standard notion of convexity. In this paper, we introduce and analyze a randomized gossip algorithm for solving the generalized consensus problem on convex metric spaces, where the communication between agents is based on a set of Poisson counters. We study the convergence properties of the algorithm using stochastic differential equations theory. In particular, we show that the distances between the states of the agents converge to zero with probability one and in the r^{th} mean sense. In the special case of complete connectivity and uniform Poisson counters, we give upper bounds on the dynamics of the first and second moments of the distances between the states of the agents. In addition, we introduce instances of the generalized consensus algorithm for several examples of convex metric spaces together with numerical simulations.^{ th} mean sense. In the special case of complete connectivity and uniform Poisson counters, we give upper bounds on the dynamics of the first and second moments of the distances between the states of the agents. In addition, we introduce instances of the generalized consensus algorithm for several examples of convex metric spaces together with numerical simulations.