A randomized gossip consensus algorithm on convex metric spaces

Abstract

ISR develops, applies and teaches advanced methodologies of design and analysis to solve complex, hierarchical, heterogeneous and dynamic problems of engineering technology and systems for industry and government.ISR is a permanent institute of the University of Maryland, within the A. James Clark School of Engineering. It is a graduated National Science Foundation Engineering Research Center. AbstractA 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. We study the convergence properties of the algorithm using stochastic differential equations theory. We show that the dynamics of the distances between the states of the agents can be upper bounded by the dynamics of a stochastic differential equation driven by Poisson counters. In addition, we introduce instances of the generalized consensus algorithm for several examples of convex metric spaces together with numerical simulations.I. Introduction Distributed algorithms are found in applications related to sensor, peer-to-peer and ad-hoc networks. A particular distributed algorithm is the consensus (or agreement) algorithm, where a group of dynamic agents seek to agree upon certain quantities of interest by exchanging information among them, according to a set of rules. This problem can model many phenomena involving information exchange between agents such as cooperative control of vehicles, formation control, flocking, synchronization, parallel computing, etc. Distributed computation over networks has a long history in control theory starting with the work of Borkar [21]. In addition to network variability, nodes in sensor networks operate under limited computational, communication, and energy resources. These constraints have motivated the design of gossip algorithms, in which a node communicates with a randomly chosen neighbor. Studies of randomized gossip consensus algorithms can be found in [2], [22].In this paper we introduce and analyze a generalized randomized gossip algorithm for achieving consensus. The algorithm acts on convex metric spaces, that are metric spaces endowed with a convex structure. We show that under the given algorithm, the agents' states converge to consensus with probability one and in the r th mean sense. The convergence study is based on analyzing the dynamics of a set of stochastic differential equations driven by poisson counters. Additionally, for a particular network topology we investigate in more depth the rate of convergence of the first and second moment of the distances between the agents' states. We present instances of the generalized gossip algorithm for three convex

A Generalized Gossip Algorithm on Convex Metric Spaces

“…Compared with the aforementioned work, the stochastic framework assumed in the current paper requires a completely different approach for studying the convergence properties of the algorithm. A preliminary short version of this paper can be found in [14], where due to space limitations most of the results are introduced without proof. Here, we refine and improve the results initially introduced in [14], we include all necessary proofs together and some new examples of convex metric spaces and their corresponding agreement algorithms.…”