Nonasymptotic convergence rates for cooperative learning over time-varying directed graphs

We study the problem of cooperative learning with a network of agents where some agents repeatedly access information about a random variable with unknown distribution. The group objective is to globally agree on a joint hypothesis (distribution) that best describes the observed data at all nodes. The agents interact with their neighbors in an unknown sequence of time-varying directed graphs. Following the pioneering work of Jadbabaie, Molavi, Sandroni, and Tahbaz-Salehi and others, we propose local learning dynamics which combine Bayesian updates at each node with a local aggregation rule of private agent signals. We show that these learning dynamics drive all agents to the set of hypotheses which best explain the data collected at all nodes as long as the sequence of interconnection graphs is uniformly strongly connected. Our main result establishes a non-asymptotic, explicit, geometric convergence rate for the learning dynamic.

[1]  Kamiar Rahnama Rad,et al.  Distributed parameter estimation in networks , 2010, 49th IEEE Conference on Decision and Control (CDC).

[2]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[3]  Venkatesh Saligrama,et al.  Distributed Tracking in Multihop Sensor Networks With Communication Delays , 2007, IEEE Transactions on Signal Processing.

[4]  Ali Jadbabaie,et al.  Non-Bayesian Social Learning , 2011, Games Econ. Behav..

[5]  Angelia Nedic,et al.  Distributed optimization over time-varying directed graphs , 2013, 52nd IEEE Conference on Decision and Control.

[6]  A. Zellner.  Optimal Information Processing and Bayes's Theorem , 1988 .

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

[8]  Christian Genest,et al.  Combining Probability Distributions: A Critique and an Annotated Bibliography , 1986 .

[9]  Gustavo L. Gilardoni,et al.  On Reaching a Consensus Using Degroot's Iterative Pooling , 1993 .

[10]  Polly S Nichols,et al.  Agreeing to disagree. , 2005, General dentistry.

[11]  M. Alanyali,et al.  Distributed Detection in Sensor Networks With Packet Losses and Finite Capacity Links , 2006, IEEE Transactions on Signal Processing.

[12]  Michael Athans,et al.  Convergence and asymptotic agreement in distributed decision problems , 1982, 1982 21st IEEE Conference on Decision and Control.

[13]  S. Venkatesh,et al.  Distributed Bayesian hypothesis testing in sensor networks , 2004, Proceedings of the 2004 American Control Conference.

[14]  Qipeng Liu,et al.  Non-Bayesian learning in social networks with time-varying weights , 2011, Proceedings of the 30th Chinese Control Conference.

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

[16]  Manuel Mueller-Frank,et al.  A general framework for rational learning in social networks , 2011 .

[17]  T. S. Jayram,et al.  Generalized Opinion Pooling , 2004, ISAIM.

[18]  Pooya Molavi,et al.  Information Heterogeneity and the Speed of Learning in Social Networks , 2013 .

[19]  Shahin Shahrampour,et al.  Exponentially fast parameter estimation in networks using distributed dual averaging , 2013, 52nd IEEE Conference on Decision and Control.

[20]  A. Madansky.  EXTERNALLY BAYESIAN GROUPS , 1964 .

[21]  Angelia Nedic,et al.  Distributed Optimization Over Time-Varying Directed Graphs , 2015, IEEE Trans. Autom. Control..

[22]  Colin McDiarmid,et al.  Surveys in Combinatorics, 1989: On the method of bounded differences , 1989 .

[23]  Lin Wang,et al.  Social learning with time-varying weights , 2014, Journal of Systems Science and Complexity.

[24]  Subhashis Ghosal,et al.  A Review of Consistency and Convergence of Posterior Distribution , 2022 .

[25]  Anand D. Sarwate,et al.  Distributed Learning of Distributions via Social Sampling , 2013, IEEE Transactions on Automatic Control.

[26]  Alvaro Sandroni,et al.  Non-Bayesian Learning , 2010 .

[27]  Douglas Gale,et al.  Bayesian learning in social networks , 2003, Games Econ. Behav..

[28]  Stephen G. Walker,et al.  Bayesian inference via a minimization rule , 2006 .

[29]  Ilan Lobel,et al.  BAYESIAN LEARNING IN SOCIAL NETWORKS , 2008 .

[30]  Qipeng Liu,et al.  Social Learning in Networks with Time‐Varying Topologies , 2014 .

[31]  Saptarshi Bandyopadhyay,et al.  Distributed estimation using Bayesian consensus filtering , 2014, 2014 American Control Conference.

[32]  Roger M. Cooke,et al.  Statistics in Expert Resolution: A Theory of Weights for Combining Expert Opinion , 1990 .

[33]  T. Javidi,et al.  Social learning and distributed hypothesis testing , 2014, 2014 IEEE International Symposium on Information Theory.

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

[35]  Manuel Mueller-Frank,et al.  A general framework for rational learning in social networks: Framework for rational learning , 2013 .

[36]  John N. Tsitsiklis,et al.  On distributed averaging algorithms and quantization effects , 2007, 2008 47th IEEE Conference on Decision and Control.

[37]  Qipeng Liu,et al.  Social learning in networks with time-varying topologies , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[38]  Shahin Shahrampour,et al.  Distributed Detection: Finite-Time Analysis and Impact of Network Topology , 2014, IEEE Transactions on Automatic Control.

[39]  Marco Dall'Aglio,et al.  Bayesian Posteriors Without Bayes' Theorem , 2012, ArXiv.

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