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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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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