On the mean square error of randomized averaging algorithms

This paper considers randomized discrete-time consensus systems that preserve the average ''on average''. As a main result, we provide an upper bound on the mean square deviation of the consensus value from the initial average. Then, we apply our result to systems in which few or weakly correlated interactions take place: these assumptions cover several algorithms proposed in the literature. For such systems we show that, when the network size grows, the deviation tends to zero, and that the speed of this decay is not slower than the inverse of the size. Our results are based on a new approach, which is unrelated to the convergence properties of the system.

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

[2]  Fabio Fagnani,et al.  The asymptotical error of broadcast gossip averaging algorithms , 2010, ArXiv.

[3]  Anand D. Sarwate,et al.  Broadcast Gossip Algorithms for Consensus , 2009, IEEE Transactions on Signal Processing.

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

[5]  John S. Baras,et al.  Convergence results for the agreement problem on Markovian random topologies , 2011 .

[6]  F. Fagnani,et al.  Asymmetric Randomized Gossip Algorithms for Consensus , 2008 .

[7]  B. Touri.  Product of Random Stochastic Matrices and Distributed Averaging , 2012 .

[8]  Fabio Fagnani,et al.  Broadcast Gossip Averaging: Interference and Unbiasedness in Large Abelian Cayley Networks , 2011, IEEE Journal of Selected Topics in Signal Processing.

[9]  John N. Tsitsiklis,et al.  Introduction to the Issue on Gossiping Algorithms Design and Applications , 2011, IEEE J. Sel. Top. Signal Process..

[10]  Soummya Kar,et al.  Gossip Algorithms for Distributed Signal Processing , 2010, Proceedings of the IEEE.

[11]  Sandro Zampieri,et al.  Randomized consensus algorithms over large scale networks , 2007, 2007 Information Theory and Applications Workshop.

[12]  Jean-Charles Delvenne,et al.  Optimal strategies in the average consensus problem , 2007, 2007 46th IEEE Conference on Decision and Control.

[13]  Julien M. Hendrickx,et al.  Large network consensus is robust to packet losses and interferences , 2013, 2013 European Control Conference (ECC).

[14]  Alexandros G. Dimakis,et al.  Reaching consensus in wireless networks with probabilistic broadcast , 2009, 2009 47th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[15]  Jean-Charles Delvenne,et al.  Democracy in Markov chains and its preservation under local perturbations , 2010, 49th IEEE Conference on Decision and Control (CDC).