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Ruben Stranders , Alessandro Farinelli , Alex Rogers, Nick Jennings

Decentralised Coordination of Continuously Valued Control Parameters using the Max-Sum Algorithm. Ruben Stranders , Alessandro Farinelli , Alex Rogers, Nick Jennings School of Electronics and Computer Science University of Southampton {rs06r, af2, acr , nrj }@ ecs.soton.ac.uk.

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Ruben Stranders , Alessandro Farinelli , Alex Rogers, Nick Jennings

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  1. Decentralised Coordination of Continuously Valued Control Parameters using the Max-Sum Algorithm Ruben Stranders, Alessandro Farinelli, Alex Rogers, Nick Jennings School of Electronics and Computer Science University of Southampton {rs06r, af2, acr, nrj}@ecs.soton.ac.uk

  2. This presentation focuses on the use of Max-Sum in coordination problems with continuous parameters Max-Sum for Decentralised Coordination From Discrete to Continuous Empirical Evaluation

  3. Max-Sum is a powerful algorithm for solving DCOPs Optimality Complete Algorithms DPOP OptAPO ADOPT Max-Sum Algorithm Iterative Algorithms Best Response (BR) Distributed Stochastic Algorithm (DSA) Fictitious Play (FP) Communication Cost

  4. Max-Sum solves the social welfare maximisation problem in a decentralised way Agents

  5. Max-Sum solves the social welfare maximisation problem in a decentralised way Control Parameters

  6. Max-Sum solves the social welfare maximisation problem in a decentralised way Utility Functions

  7. Max-Sum solves the social welfare maximisation problem in a decentralised way Localised Interaction

  8. Max-Sum solves the social welfare maximisation problem in a decentralised way Agents Social welfare:

  9. The input for the Max-Sum algorithm is a graphical representation of the problem: a Factor Graph Variable nodes Function nodes Agent 3 Agent 1 Agent 2

  10. Max-Sum solves the social welfare maximisation problem by message passing Variable nodes Function nodes Agent 3 Agent 1 Agent 2

  11. Max-Sum solves the social welfare maximisation problem by message passing From variableito function j • From function j to variable i

  12. Until now, Max-Sum was only defined for discretely valued variables Graph Colouring

  13. However, many problems are inherently continuous. Activation Time • Heading • and • Velocity Autonomous Ground Robot Unattended Ground Sensor PreferredRoom Temperature Thermostat

  14. So, we extended the Max-Sum algorithm to operate in continuous action spaces Discrete Continuous

  15. We focussed on utility functions that are Continuous Piecewise Linear Functions (CPLFs)

  16. We focussed on utility functions that are Continuous Piecewise Linear Functions (CPLFs) “Continuous” Graph Colouring

  17. A CPLF is defined by a domain partitioning followed by value assignment

  18. A CPLF is defined by a domain partitioning followed by value assignment

  19. A CPLF is defined by a domain partitioning followed by value assignment

  20. To make Max-Sum work on CPLFs, we need to define key two operations on them From variableito function j From function j to variable i

  21. To make Max-Sum work on CPLFs, we need to define key two operations on them From variableito function j Additionof two CPLFs From function j to variable i

  22. To make Max-Sum work on CPLFs, we need to define key two operations on them From variableito function j From function j to variable i 2. Marginal Maximisation to a single variable

  23. Addition of two CPLFs involves merging their domains, and then summing their values

  24. Addition of two CPLFs involves merging their domains, and then summing their values 1. Merge domains

  25. Addition of two CPLFs involves merging their domains, and then summing their values

  26. Addition of two CPLFs involves merging their domains, and then summing their values 2. Sum Values

  27. Marginal maximisation is the operation of finding the maximum value of a function, if we fix all but one variable From function j to variable i:

  28. Marginal maximisation involves finding the maximum value of a function, if we fix all but one variable

  29. Marginal maximisation involves finding the maximum value of a function, if we fix all but one variable Example: bivariatefunction:

  30. Marginal maximisation involves the projection of a CLPF on a 2-D plane, and upper envelope extraction Project onto axis

  31. Marginal maximisation involves the projection of a CLPF on a 2-D plane, and upper envelope extraction Project onto axis

  32. Marginal maximisation involves the projection of a CLPF on a 2-D plane, and upper envelope extraction Project onto axis Result of projection

  33. Marginal maximisation involves the projection of a CLPF on a 2-D plane, and upper envelope extraction Extract Upper Envelope

  34. Marginal maximisation involves the projection of a CLPF on a 2-D plane, and upper envelope extraction Extract Upper Envelope

  35. We empirically evaluated this algorithm in a wide-area surveillance scenario Unattended Ground Sensor Dense deployment of sensors to detect activity within an urban environment.

  36. Sensors adapt their duty cycles to maximise event detection by coordinating with overlapping sensors duty cycle time Discretised time • Discrete duty cycle time duty cycle time

  37. Sensors adapt their duty cycles to maximise event detection by coordinating with overlapping sensors Continuous duty cycle duty cycle time time • Discrete duty cycle duty cycle time time duty cycle duty cycle time time

  38. Continuous Max-Sum outperforms Discrete Max-Sum by up to 10% Average Solution Quality over 25 Iterations Solution Quality (as fraction of optimal) Discretisation

  39. Continuous Max-Sum leads to more effective use of communication resources than Discrete Max-Sum Total number of values exchanged between agents Total Message Size Discretisation

  40. In conclusion, we have shown that Continuous Max-Sum is more effective than Discrete Max-Sum time time • 1. No artificial • discretisation

  41. In conclusion, we have shown that Continuous Max-Sum is more effective than Discrete Max-Sum time Solution Quality time • 1. No artificial • discretisation 2. Better solutions

  42. In conclusion, we have shown that Continuous Max-Sum is more effective than Discrete Max-Sum time Solution Quality time • 1. No artificial • discretisation Message Size 2. Better solutions 3. Effective communication

  43. For future work, we wish to extend the algorithm to arbitrary continuous functions • For example, using Gaussian Processes

  44. In conclusion, we have shown that Continuous Max-Sum is more effective than Discrete Max-Sum time Solution Quality time • 1. No artificial • discretisation Message Size 2. Better solutions Questions? 3. Effective communication

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