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

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

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 centralised control

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 1.How to model the phenomena? 2.How to value potential samples? 3.How to coordinate to gather samples of highest value?

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

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

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

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

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

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 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 Potential Sample Location

18. 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 Measure of uncertainty

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

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

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 Social welfare: Mobile Sensors

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

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

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

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

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

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

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

32. Unfortunately, the straightforward application of Max-Sum is too computationally expensive From variable i to 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 1.Try to prune the action spaces of individual sensors 2.Try to prune joint actions

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

36. 1. Neighbours send bounds ↑ [2, 2] ↓ [1, 1] ↑ [5, 6] ↓ [0, 1] ↑ [1, 2] ↓ [3, 4]

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

40. We developed two general pruning techniques that speed up Max-Sum Goal: Make as small as possible 1.Try to prune the action spaces of individual sensors 2.Try to prune joint actions

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

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

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

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

48. 91078 [7, 13][0, 4][2, 6] Sensor 1 Sensor 2 Sensor 3

49. The second pruning technique prunes the joint action space using Branch and Bound 91078 [7, 13][0, 4][2, 6] Sensor 1 Sensor 2 Sensor 3

50. This demonstration shows four sensors monitoring a spatial phenomenon

51. Sensors

52. This demonstration shows four sensors monitoring a spatial phenomenon Uncertainty Contours

53. This demonstration shows four sensors monitoring a spatial phenomenon

54. The two pruning techniques combined prune 95% of the action space with 6 neighbouring sensors Number of neighbouring sensors % of joint actions pruned

55. Avg. Root Mean Squared Error Our Algorithm reduces Root Mean Squared Error of predictions up to 50% compared to Greedy

56. In conclusion, the use Max-Sum leads to an effective coordination algorithm for mobile sensors 1. Decentralised

57. In conclusion, the use Max-Sum leads to an effective coordination algorithm for mobile sensors 1. Decentralised 2. Fast % Pruned

58. In conclusion, the use Max-Sum leads to an effective coordination algorithm for mobile sensors 1. Decentralised 2. Fast 3. Accurate predictions % Pruned Prediction Error

59. For future work, we wish to extend the algorithm to do non-myopic planning

60. In conclusion, the use Max-Sum leads to an effective coordination algorithm for mobile sensors 1. Decentralised 2. Fast 3. Accurate predictions % Pruned Prediction Error Questions?