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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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**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**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**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**Max-Sum solves the social welfare maximisation problem in a**decentralised way Agents**Max-Sum solves the social welfare maximisation problem in a**decentralised way Control Parameters**Max-Sum solves the social welfare maximisation problem in a**decentralised way Utility Functions**Max-Sum solves the social welfare maximisation problem in a**decentralised way Localised Interaction**Max-Sum solves the social welfare maximisation problem in a**decentralised way Agents Social welfare:**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**Max-Sum solves the social welfare maximisation problem by**message passing Variable nodes Function nodes Agent 3 Agent 1 Agent 2**Max-Sum solves the social welfare maximisation problem by**message passing From variableito function j • From function j to variable i**Until now, Max-Sum was only defined for discretely valued**variables Graph Colouring**However, many problems are inherently continuous.**Activation Time • Heading • and • Velocity Autonomous Ground Robot Unattended Ground Sensor PreferredRoom Temperature Thermostat**So, we extended the Max-Sum algorithm to operate in**continuous action spaces Discrete Continuous**We focussed on utility functions that are Continuous**Piecewise Linear Functions (CPLFs)**We focussed on utility functions that are Continuous**Piecewise Linear Functions (CPLFs) “Continuous” Graph Colouring**A CPLF is defined by a domain partitioning followed by value**assignment**A CPLF is defined by a domain partitioning followed by value**assignment**A CPLF is defined by a domain partitioning followed by value**assignment**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**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**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**Addition of two CPLFs involves merging their domains, and**then summing their values**Addition of two CPLFs involves merging their domains, and**then summing their values 1. Merge domains**Addition of two CPLFs involves merging their domains, and**then summing their values**Addition of two CPLFs involves merging their domains, and**then summing their values 2. Sum Values**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:**Marginal maximisation involves finding the maximum value of**a function, if we fix all but one variable**Marginal maximisation involves finding the maximum value of**a function, if we fix all but one variable Example: bivariatefunction:**Marginal maximisation involves the projection of a CLPF on a**2-D plane, and upper envelope extraction Project onto axis**Marginal maximisation involves the projection of a CLPF on a**2-D plane, and upper envelope extraction Project onto axis**Marginal maximisation involves the projection of a CLPF on a**2-D plane, and upper envelope extraction Project onto axis Result of projection**Marginal maximisation involves the projection of a CLPF on a**2-D plane, and upper envelope extraction Extract Upper Envelope**Marginal maximisation involves the projection of a CLPF on a**2-D plane, and upper envelope extraction Extract Upper Envelope**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.**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**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**Continuous Max-Sum outperforms Discrete Max-Sum by up to 10%**Average Solution Quality over 25 Iterations Solution Quality (as fraction of optimal) Discretisation**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**In conclusion, we have shown that Continuous Max-Sum is more**effective than Discrete Max-Sum time time • 1. No artificial • discretisation**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**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**For future work, we wish to extend the algorithm to**arbitrary continuous functions • For example, using Gaussian Processes**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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