A randomized gossip consensus algorithm on convex metric spaces

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. 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.

[1]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[2]  Devavrat Shah,et al.  Gossip Algorithms , 2009, Found. Trends Netw..

[3]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[4]  John S. Baras,et al.  The geometry of a probabilistic consensus of opinion algorithm , 2011, Proceedings of the 2011 American Control Conference.

[5]  G. Grimmett,et al.  Probability and random processes , 2002 .

[6]  Asuman E. Ozdaglar,et al.  Convergence rate for consensus with delays , 2010, J. Glob. Optim..

[7]  Ιωαννησ Τσιτσικλησ,et al.  PROBLEMS IN DECENTRALIZED DECISION MAKING AND COMPUTATION , 1984 .

[8]  John S. Baras,et al.  Consensus problems with directed Markovian communication patterns , 2009, 2009 American Control Conference.

[9]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[10]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[11]  Stephen P. Boyd,et al.  Randomized gossip algorithms , 2006, IEEE Transactions on Information Theory.

[12]  R. Murray,et al.  Consensus protocols for networks of dynamic agents , 2003, Proceedings of the 2003 American Control Conference, 2003..

[13]  Randal W. Beard,et al.  Consensus seeking in multiagent systems under dynamically changing interaction topologies , 2005, IEEE Transactions on Automatic Control.

[14]  Asuman E. Ozdaglar,et al.  Constrained Consensus and Optimization in Multi-Agent Networks , 2008, IEEE Transactions on Automatic Control.

[15]  Alba Pagès-Zamora.  Consensus in correlated random wireless sensor networks , 2011, ACWR '11.

[16]  Mehran Mesbahi,et al.  Agreement over random networks , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[17]  John S. Baras,et al.  Almost sure convergence to consensus in Markovian random graphs , 2008, 2008 47th IEEE Conference on Decision and Control.

[18]  Alireza Tahbaz-Salehi,et al.  Consensus Over Ergodic Stationary Graph Processes , 2010, IEEE Transactions on Automatic Control.

[19]  Maurizio Porfiri,et al.  Consensus Seeking Over Random Weighted Directed Graphs , 2007, IEEE Transactions on Automatic Control.

[20]  Alireza Tahbaz-Salehi,et al.  A Necessary and Sufficient Condition for Consensus Over Random Networks , 2008, IEEE Transactions on Automatic Control.

[21]  Luc Moreau,et al.  Stability of multiagent systems with time-dependent communication links , 2005, IEEE Transactions on Automatic Control.

[22]  Wataru Takahashi,et al.  A convexity in metric space and nonexpansive mappings, I , 1970 .

[23]  John S. Baras,et al.  The asymptotic consensus problem on convex metric spaces , 2010 .

[24]  V. Borkar,et al.  Asymptotic agreement in distributed estimation , 1982 .

[25]  John N. Tsitsiklis,et al.  Distributed Asynchronous Deterministic and Stochastic Gradient Optimization Algorithms , 1984, 1984 American Control Conference.

[26]  F. Smithies.  Linear Operators , 2019, Nature.