Distributed Agreement Algorithms

Final MURI Review Meeting

John N. TsitsiklisDecember 2, 2005

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)

Motivation

Fusion of individual estimates(or of likelihood ratios)

Agreement on a pending decision Load balancing

Multiagent coordination and control Flocking, cooperative control….

The Agreement Algorithm [JNT et al. 1984-89]

Special cases: • Equal weight to yourself and

messages just received

• Pairwise averaging

Assumptions

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(update using outdated values of others)

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

Speed of Convergence

Fixed graphs:(tight for “bad graphs”)

Changing graphs:Have variation of the algorithm that guarantees:

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.