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Accelerating Approximate Consensus with Self-Organizing Overlays

This paper proposes a method called PLD-Consensus that accelerates the convergence of Laplacian-based consensus algorithms by creating self-organizing overlays. The approach biases the consensus process to trade precision for speed, making it suitable for spatial computing applications such as flocking, swarming, and collective decision making. The PLD-Consensus algorithm is shown to converge in O(diam) time, with robustness to mobility and parameter variations.

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Accelerating Approximate Consensus with Self-Organizing Overlays

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  1. Accelerating Approximate Consensus with Self-Organizing Overlays Jacob Beal 2013 Spatial Computing Workshop @ AAMAS May, 2013

  2. PLD-Consensus: With asymmetry from a self-organizing overlay… … we can cheaply trade precision for speed.

  3. Motivation: approximate consensus Many spatial computing applications: • Flocking & swarming • Localization • Collective decision • etc … Current Laplacian-based approach: • Sensor fusion • Modular robotics • DMIMO Many nice properties: exponential convergence, highly robust, etc. But there is a catch…

  4. Problem: Laplacian consensus too slow On spatial computers: • Converge O(diam2) • Stability constraint  very bad constant Root issue: symmetry [Elhage & Beal ‘10]

  5. Approach: shrink effective diameter • Self-organize an overlay network, linking all devices to a random regional “leaders” • Regions grow, until one covers whole network • Every device blends values with “leader” as well as neighbors In effect, randomly biasing consensus Cost: variation in converged value

  6. Self-organizing overlay Creating one overlay neighbor per device: • Random “dominance” from 1/f noise • Decrease dominance over space and time • Values flow down dominance gradient

  7. Power-Law-Driven Consensus Asymmetric overlay update Laplacian update

  8. Race: PLD-Consensus vs. Laplacian

  9. Analytic Sketch • O(1) time for 1/f noise to generate a dominating value at some device D [probably] • O(diam) for a dominance to cover network • Then O(1) to converge to D±δ, for any given α (blending converges exponentially) Thus: O(diam) convergence

  10. Simulation Results: Convergence 1000 devices randomly on square, 15 expected neighbors; α=0.01, β=0, ε=0.01] Much faster than Laplacian consensus, even without spatial correlations! O(diameter), but with a high constant.

  11. Simulation Results: Mobility No impact even from relatively high mobility (possible acceleration when very fast)

  12. Simulation Results: Blend Rates β irrelevant: Laplacian term simply cannot move information fast enough to affect result

  13. Current Weaknesses • Variability of consensus value: • OK for collective choice, not for error-rejection • Possible mitigations: • Feedback “up” dominance gradient • Hierarchical consensus • Fundamental robustness / speed / variance tradeoff? • Leader Locking: • OK for 1-shot consensus, not for long-term • Possible mitigations: • Assuming network dimension (elongated distributions?) • Upper bound from diameter estimation

  14. Contributions • PLD-Consensus converges to approximate consensus in O(diam) rather than O(diam2) • Convergence is robust to mobility, parameters • Future directions: • Formal proofs • Mitigating variability, leader locking • Putting fast consensus to use…

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