Network-Based Consensus Averaging With General Noisy Channels

This paper focuses on the consensus averaging problem on graphs under general imperfect communications. We study a particular class of distributed consensus algorithms based on damped updates, and using the ordinary differential equation method, we prove that the updates converge almost surely to the consensus average for various models of perturbation of data exchanged between nodes. The convergence is not asymptotic in the size of the network. Our analysis applies to various types of stochastic disturbances to the updates, including errors in update calculations, dithered quantization and imperfect data exchange among nodes. Under a suitable stability condition, we prove that the error between estimated and true averages is asymptotically Gaussian, and we show how the asymptotic covariance is specified by the graph Laplacian. For additive perturbations, we show how the scaling of the asymptotic MSE is controlled by the spectral gap of the Laplacian.

[1]  Carlos S. Kubrusly,et al.  Stochastic approximation algorithms and applications , 1973, CDC 1973.

[2]  M. Degroot.  Reaching a Consensus , 1974 .

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

[4]  John N. Tsitsiklis,et al.  Problems in decentralized decision making and computation , 1984 .

[5]  N. Alon.  Eigenvalues and expanders , 1986, Comb..

[6]  Pierre Priouret,et al.  Adaptive Algorithms and Stochastic Approximations , 1990, Applications of Mathematics.

[7]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[8]  Tamer Basar,et al.  Analysis of Recursive Stochastic Algorithms , 2001 .

[9]  Johannes Gehrke,et al.  Gossip-based computation of aggregate information , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[10]  Mathew D. Penrose,et al.  Random Geometric Graphs , 2003 .

[11]  Martin J. Wainwright,et al.  LP Decoding Corrects a Constant Fraction of Errors , 2004, IEEE Transactions on Information Theory.

[12]  Stephen P. Boyd,et al.  Fast linear iterations for distributed averaging , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[13]  Noga Alon,et al.  The Probabilistic Method, Second Edition , 2004 .

[14]  M. Mesbahi,et al.  Agreement in presence of noise: pseudogradients on random geometric networks , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

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

[16]  Sandro Zampieri,et al.  Average consensus with packet drop communication , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[17]  R. Srikant,et al.  Quantized Consensus , 2006, 2006 IEEE International Symposium on Information Theory.

[18]  R. Carli,et al.  Average consensus on networks with transmission noise or quantization , 2007, 2007 European Control Conference (ECC).

[19]  Stephen P. Boyd,et al.  Distributed average consensus with least-mean-square deviation , 2007, J. Parallel Distributed Comput..

[20]  A. Scaglione,et al.  Differential Nested Lattice Encoding for Consensus Problems , 2007, 2007 6th International Symposium on Information Processing in Sensor Networks.

[21]  M. Rabbat,et al.  Rates of Convergence for Distributed Average Consensus with Probabilistic Quantization , 2007 .

[22]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[23]  Alexandros G. Dimakis,et al.  Gossip along the way: order-optimal consensus through randomized path averaging , 2007 .

[24]  Soummya Kar,et al.  Distributed Consensus Algorithms in Sensor Networks: Link Failures and Channel Noise , 2007, ArXiv.

[25]  Giorgio Picci,et al.  Almost sure convergence of random gossip algorithms , 2007, 2007 46th IEEE Conference on Decision and Control.

[26]  Michael Rabbat,et al.  Distributed Average Consensus using Probabilistic Quantization , 2007, 2007 IEEE/SP 14th Workshop on Statistical Signal Processing.

[27]  T. C. Aysal,et al.  Distributed Average Consensus With Dithered Quantization , 2008, IEEE Transactions on Signal Processing.

[28]  Soummya Kar,et al.  Distributed average consensus in sensor networks with quantized inter-sensor communication , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.

[29]  Martin Vetterli,et al.  Which Distributed Averaging Algorithm Should I Choose for my Sensor Network? , 2008, IEEE INFOCOM 2008 - The 27th Conference on Computer Communications.

[30]  Mehmet E. Yildiz,et al.  Broadcast gossip algorithms , 2008, 2008 IEEE Information Theory Workshop.

[31]  Alexandros G. Dimakis,et al.  Geographic Gossip: Efficient Averaging for Sensor Networks , 2007, IEEE Transactions on Signal Processing.

[32]  Mehmet E. Yildiz,et al.  Coding With Side Information for Rate-Constrained Consensus , 2008, IEEE Transactions on Signal Processing.

[33]  Alejandro Ribeiro,et al.  Consensus in Ad Hoc WSNs With Noisy Links—Part I: Distributed Estimation of Deterministic Signals , 2008, IEEE Transactions on Signal Processing.

[34]  Soummya Kar,et al.  Distributed Consensus Algorithms in Sensor Networks With Imperfect Communication: Link Failures and Channel Noise , 2007, IEEE Transactions on Signal Processing.

[35]  Soummya Kar,et al.  A mixed time-scale algorithm for distributed parameter estimation : Nonlinear observation models and imperfect communication , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[36]  Soummya Kar,et al.  Distributed Consensus Algorithms in Sensor Networks: Quantized Data and Random Link Failures , 2007, IEEE Transactions on Signal Processing.

[37]  U. Feige,et al.  Spectral Graph Theory , 2015 .