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Accelerating Approximate Consensus with Self-Organizing Overlays in Spatial Computing

This paper explores a novel approach to accelerate approximate consensus in spatial computing through self-organizing overlays. The proposed PLD-Consensus method outperforms traditional Laplacian consensus, achieving convergence in O(diameter) time instead of O(diameter^2). By linking devices through a random overlay, this technique allows for efficient value blending from regional leaders, significantly reducing convergence time while maintaining robustness against mobility. The research highlights the balance between speed and variability, offering insights for future work in dynamic consensus algorithms.

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Accelerating Approximate Consensus with Self-Organizing Overlays in Spatial Computing

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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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