Distributed Agreement Algorithms

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Distributed Agreement Algorithms

• Final MURI Review Meeting
• John N. Tsitsiklis
• December 2, 2005

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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)

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Motivation

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

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The Agreement Algorithm JNT et al. 1984-89

• Special cases
• Equal weight to yourself and messages just
• received
• Pairwise averaging

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Assumptions
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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• (update using outdated values of others)

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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

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