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Distributed Agreement Algorithms

Distributed Agreement Algorithms. Final MURI Review Meeting John N. Tsitsiklis December 2, 2005. The Problem. Each sensor has a number x i They wish to reach agreement on a common value: Some value in the range [Min x i , Max x i ], or The average of their values

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Distributed Agreement Algorithms

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  1. Distributed Agreement Algorithms Final MURI Review Meeting John N. Tsitsiklis December 2, 2005

  2. The Problem • Each sensor has a number xi • They wish to reach agreement on a common value: • Some value in the range [Min xi , Max xi], or • The average of their values • Using a distributed algorithm • Without assuming synchronization • Without preexisting infrastructure (such as a spanning tree)

  3. Motivation • Fusion of individual estimates (or of likelihood ratios) • Agreement on a pending decision • Load balancing • Multiagent coordination and control • Flocking, cooperative control….

  4. The Agreement Algorithm [JNT et al. 1984-89] • Special cases: • Equal weight to yourself and messages just received • Pairwise averaging

  5. Assumptions

  6. Convergence Theory • Under Bounded Asynchronism • Convergence to a common value • Average preserving variants: convergence to the average of initial values • Convergence happens at a geometric rate • Even in the presence of communication delays (update using outdated values of others)

  7. Impact of Initial Values • Non-average-preserving cases • Equal weights • Starting values in [0,1] • Fixed graph • Limit as high as 1(1/n) • Time-Varying graph • Limit as high as

  8. Speed of Convergence Fixed graphs: (tight for “bad graphs”) Changing graphs: Have variation of the algorithm that guarantees:

  9. References • V. D. Blondel, J. M. Hendrickx, A. Olshevsky, and J. N. Tsitsiklis, “Convergence in Multiagent Coordination, Consensus, and Flocking,” in Proceedings of the Joint 44th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC'05)} Seville, Spain, December 2005. • A. Olshevsky, MS thesis, EECS, MIT, in preparation. • J. N. Tsitsiklis, ``Problems in Decentralized Decision Making and Computation", Ph.D. Thesis, Department of EECS, MIT, November 1984. • D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice Hall, 1989. • J. N. Tsitsiklis, D. P. Bertsekas and M. Athans, “Distributed Asynchronous Deterministic and Stochastic Gradient Optimization Algorithms", IEEE Transactions on Automatic Control}, Vol. 31, No. 9, 1986, pp. 803-812.

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