Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm

1. Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm • Ruben Stranders, Alessandro Farinelli, Alex Rogers, Nick Jennings School of Electronics and Computer Science University of Southampton {rs06r2, af2, acr, nrj}@ecs.soton.ac.uk

2. This presentation focuses on the use of Max-Sum to coordinate mobile sensors Decentralised Control using Max-Sum Problem Formulation Model Value Coordinate Sensor Architecture

3. The key challenge is to monitor a spatial phenomenon with a team of autonomous sensors Sensors

4. The key challenge is to monitor a spatial phenomenon with a team of autonomous sensors • Limited • Communication

5. The key challenge is to monitor a spatial phenomenon with a team of autonomous sensors • No centralisedcontrol

6. Spatial phenomena are modelled as a spatial field over two spatial and one temporal dimensions

7. The aim of the sensors is to collectively minimise predictive uncertainty of the spatial phenomenon Predictive Uncertainty Contours

8. The main challenge is to coordinate the sensors in order to the state of these spatial phenomena • How to move • to minimise • uncertainty?

9. To solve this coordination problem, we had to address three challenges How to modelthe phenomena? How to value potential samples? How to coordinateto gather samples of highest value?

10. The three central challenges are clearly reflected in the architecture of our sensing agents Incoming negotiation messages Samples received from neighbouring agents Sensing Agent Value of potential samples Information processing Samples from own sensor Model Value Action Selection Model of Environment Coordinate Move Raw samples Outgoing negotiation messages Samples sent to neighbouring agents

11. These three challenges are clearly reflected in the architecture of our sensing agents Incoming negotiation messages Samples received from neighbouring agents Sensing Agent Value of potential samples Information processing Samples from own sensor Action Selection Model of Environment Move Raw samples Model Outgoing negotiation messages Samples sent to neighbouring agents

12. The sensors model the spatial phenomenon using the Gaussian Process Spatial Correlations Weak Strong

13. The sensors model the spatial phenomenon using the Gaussian Process Temporal Correlations Weak Strong

14. The value of a sample is determined how much it reduces uncertainty Incoming negotiation messages Samples received from neighbouring agents Sensing Agent Value of potential samples Information processing Samples from own sensor Action Selection Model of Environment Move Raw samples Value Outgoing negotiation messages Samples sent to neighbouring agents

15. The value of a sample is based on how much it reduces uncertainty • But how to determine uncertainty reduction before collecting a sample?

16. The value of a sample is based on how much it reduces uncertainty • But how to determine uncertainty reduction before collecting a sample? Prediction Confidence Interval Collected Sample Gaussian Process not only gives predictions, but also confidence intervals

17. The value of a sample is based on how much it reduces uncertainty Potential Sample Location • But how to determine uncertainty reduction before collecting a sample? Prediction Confidence Interval Collected Sample Gaussian Process not only gives predictions, but also confidence intervals

18. The value of a sample is based on how much it reduces uncertainty Measure of uncertainty • But how to determine uncertainty reduction before collecting a sample? Prediction Confidence Interval Collected Sample Gaussian Process not only gives predictions, but also confidence intervals

19. The value of a sample is based on how much it reduces uncertainty Specifically, we use Entropy , as information metric Prediction Confidence Interval Collected Sample

20. The sensor agents coordinate using the Max-Sum algorithm Incoming negotiation messages Samples received from neighbouring agents Sensing Agent Value of potential samples Information processing Samples from own sensor Action Selection Model of Environment Move Raw samples Coordinate Outgoing negotiation messages Samples sent to neighbouring agents

21. Using the Entropy criterion, the sum of the conditional values equals the team utility

22. The key problem is to maximise the social welfare of the team of sensors in a decentralised way Mobile Sensors Social welfare:

23. The key problem is to maximise the social welfare of the team of sensors in a decentralised way Variables Encode Movement

24. The key problem is to maximise the social welfare of the team of sensors in a decentralised way Utility Functions (These encode information value)

25. The key problem is to maximise the social welfare of the team of sensors in a decentralised way Localised Interaction

26. We can now use Max-Sum to solve the social welfare maximisation problem Optimality Complete Algorithms DPOP OptAPO ADOPT Max-Sum Algorithm Iterative Algorithms Best Response (BR) Distributed Stochastic Algorithm (DSA) Fictitious Play (FP) Communication Cost

27. 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

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

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

30. To use Max-Sum, we encode the mobile sensor coordination problem as a factor graph Sensor 1 Sensor 2 Sensor 3 Sensor 1 Sensor 2 Sensor 3

31. Unfortunately, the straightforward application of Max-Sum is too computationally expensive From variableito function j From function j to variable i

32. Unfortunately, the straightforward application of Max-Sum is too computationally expensive From variableito function j From function j to variable i Bottleneck!

33. Therefore, we developed two general pruning techniques that speed up Max-Sum Goal: Make as small as possible

34. Therefore, we developed two general pruning techniques that speed up Max-Sum Goal: Make as small as possible Try to prune the action spaces of individual sensors Try to prune joint actions

35. The first pruning technique prunes individual actions by identifying dominated actions

36. The first pruning technique prunes individual actions by identifying dominated actions ↑ [1, 2] ↓ [3, 4] 1. Neighbours send bounds ↑ [2, 2] ↓ [1, 1] ↑ [5, 6] ↓ [0, 1]

37. The first pruning technique prunes individual actions by identifying dominated actions ↑ [1, 2] ↓ [3, 4] 2. Bounds are summed ↑ [2, 2] ↓ [1, 1] ↑ [5, 6] ↓ [0, 1] ↑ [8, 10] • ↓ [4, 7]

38. The first pruning technique prunes individual actions by identifying dominated actions 2. Bounds are summed ↑ [8, 10] • ↓ [4, 7]

39. The first pruning technique prunes individual actions by identifying dominated actions 3. Dominated actions are pruned X ↑ [8, 10] [8, 10] • ↓ [4, 7] [4, 7]

40. We developed two general pruning techniques that speed up Max-Sum Goal: Make as small as possible ✔ Try to prune the action spaces of individual sensors Try to prune joint actions

41. The second pruning technique reduces the joint action space because exhaustive enumeration is too costly Sensor 1 Sensor 2 Sensor 3

42. The second pruning technique reduces the joint action space because exhaustive enumeration is too costly Sensor 1 Sensor 2 Sensor 3

43. The second pruning technique reduces the joint action space because exhaustive enumeration is too costly Sensor 1 Sensor 2 Sensor 3

44. The second pruning technique reduces the joint action space because exhaustive enumeration is too costly

45. The second pruning technique prunes the joint action space using Branch and Bound Sensor 1 Sensor 2 Sensor 3

46. The second pruning technique prunes the joint action space using Branch and Bound Sensor 1 [0, 4] [7, 13] [2, 6] Sensor 2 Sensor 3

47. The second pruning technique prunes the joint action space using Branch and Bound Sensor 1 X X [0, 4] [7, 13] [2, 6] Sensor 2 Sensor 3

48. The second pruning technique prunes the joint action space using Branch and Bound Sensor 1 X X [0, 4] [7, 13] [2, 6] Sensor 2 9 10 7 8 Sensor 3

49. The second pruning technique prunes the joint action space using Branch and Bound Sensor 1 X X [0, 4] [7, 13] [2, 6] Sensor 2 O X X X 9 10 7 8 Sensor 3

50. This demonstration shows four sensors monitoring a spatial phenomenon