1. Distributed cooperation and coordination using the Max-Sum algorithm
Farinelli, A., Rogers, A., Petcu, A. and Jennings, N. R. (2008) Decentralised Coordination of Low-Power Embedded Devices Using the Max-Sum Algorithm. In: Seventh International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS-08), May 2008, Estoril, Portugal.
Waldock, A., Nicholson, D. and Rogers, A. (2008) Cooperative Control using the Max-Sum Algorithm. In: Second International Workshop on Agent Technology for Sensor Networks, Estoril, Portugal.

2. What is Max-Sum?
Max-sum is a derivative of the sum-product algorithm, formed by transforming the max-product algorithm into log-space.

3. What is Sum-Product? General form of the forward-backward algorithm
Iterative algorithm to compute the marginals of functions
a.k.a Belief Propagation (when applied to probabilistic graphical models)
Information Theory
Machine Learning
Vision

4. An abstract problem
Consider a function, F, that is dependent on N variables, x = {x xN}, and is defined as a product of M factors such that:
where each of the factors fm(xm) is a function of a subset xm of the variables that make up x.

5. An abstract problem
Find the marginal function zn(xn) that describes the dependency of F(x) on variable xn:
The marginal zn(a) is defined to be the sum of F(x) over all configurations of variables where
xn = a.

6. Factor graph representation
Bipartite graph consisting of two types of nodes
Variable nodes
Function nodes

7. Factor graph representation
Example: F = f1(x1,x2)f2(x2,x3)

8. What is Sum-Product? Given a factor graph
Apply a local message passing procedure
Compute the marginals of functions

9. Two local message types
From variable to function:
From function to variable:

10. Canonical update procedure
Graph is cycle-free (a tree)
Leaf nodes send identity messages to parents
When a node receives a message on some edge, it computes and transmits the outgoing message for all other edges
After variable node has received a message from every neighbor, it can compute its exact marginal value:

11. Message Passing

12. Message Passing

13. Message Passing

14. “Loopy” Belief Propagation
Sum-product algorithm is only guaranteed to converge on graphs containing up to one loop
Could oscillate, or fail to converge on cyclic graphs
Empirically, it often converges pretty well on many types of graphs

15. “Loopy” Belief Propagation
All variable messages initialized to 1
All messages updated at every iteration
Normalize variable to function message:
Choose αnm such that

16. What is Max-Product?
Instead of computing the marginal functions, we can calculate arg maxx F(x)
Replace function-to-variable message:

17. What is Max-Product? Now, when we take the product of the messages:
For arg maxx F(x), each component is given by:
A.k.a: Viterbi algorithm

18. What is Max-Sum?
Suppose we have an F(x) that is the sum of function nodes, e.g.
Use max-product in logarithm space

19. Two new local message types
From variable to function:
From function to variable:

20. What is Max-Sum?
Now, when we take the sum of the messages:

21. Advantages Existing theoretical results
Simple, scalable local algorithm
Exponential only in # of neighbors in graph
Already decentralized and asynchronous
Partition nodes as desired
Only exchange local messages
Can estimate variables before convergence

22. Caveats No strong theoretical results for loopy BP
Convergence not guaranteed
Must exchange local messages between related factor and variable nodes
Not necessarily easy to evaluate functions and compute messages

23. How to use Max-Sum to solve your distributed problem:
Construct utility function

24. How to use Max-Sum to solve your distributed problem:
Turn it into factor graph
Construct utility function

25. Application #1: Decentralized Coordination
Farinelli, A., Rogers, A., Petcu, A. and Jennings, N. R. (2008) Decentralised Coordination of Low-Power Embedded Devices Using the Max-Sum Algorithm. In: Seventh International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS-08), May 2008, Estoril, Portugal.

26. Problem Solve graph coloring on sensor network
Utility can be formulated for each agent:
Penalty for same color:
A priori color preference:

27. Factor-graph Formulation
Utility function is only dependent on neighbors
Let each sensor maintain its state and utility nodes:
Exchange local messages as per Max-Sum

28. Comparison Algorithms
BR - Best Response
Each agent chooses the “best” state for its variable (i.e. the one that minimizes the conflicts) according to the current states of its neighbors
When a state change occurs, the agent sends the updated state to all of its neighbors
Represents a lower bound on the performance of any algorithm since it uses the minimum computation and communication possible
DSA - Distributed Stochastic Algorithm
Same as best response, except that an agent only actually performs the state change according to a predefined probability (called the activation probability)
Whenever an agent’s preferred state actually changes, it sends a message to its neighbors informing them of this fact
Set the activation probability to 0.6 (additional experiments indicate little sensitivity to the exact value of this parameter in our setting)
DPOP - Dynamic Programming Optimization Protocol
A complete algorithm that maintains optimality by preprocessing the constraint graph to produce a pseudo-tree, and then performs local message passing on this tree

29. Test Conditions Known colorable ADOPT graph repository Lattice graphs
3-color random graphs
10, 20, 30, 40, 50 nodes
Link density of 3
50 instances
ADOPT graph repository
Random graphs
10, 14, 18, 25 nodes
Not generally colorable with 3-colors
Lattice graphs
4-color colorable
36, 49, 64, 81, 100 nodes
Nodes connected to eight neighbors

30. Conflicts over time

31. Conflicts over time

32. Conflicts over time

33. What happened here? Combination of small and large irregular loops
Cycling causes poor convergence
Modify utility function:

34. Summary of conflict results

35. Message Size

36. Message Loss

37. Hardware Implementation

38. Application #2: Cooperative Control
Waldock, A., Nicholson, D. and Rogers, A. (2008) Cooperative Control using the Max-Sum Algorithm. In: Second International Workshop on Agent Technology for Sensor Networks, Estoril, Portugal.

39. Problem Sensor network doing distributed tracking
Sensors share information using decentralized data fusion (DDF)

40. Problem Select control parameters for each sensor
Maximize sum of global info for all targets

41. Problem
Define U as a summation of target factors Uj over only sensors that can observe target, e.g.
where Ub is defined as:

42. Problem
Unfortunately, we can’t break into smaller factors: ≠

43. Factor-graph Formulation
Each sensor maintains variable node for its own control parameter
Define a function node for each target
Maintained by the sensor that first saw target
Use DDF to determine edges of graph
Add/remove edges as target moves in and out of sensor range
Exchange local messages as per Max-Sum

44. Factor-graph Formulation
Sensor 1 θ1 UA
Sensor 2 θ2 UB
Sensor 3 θ2
Sensor 4 θ4
Sensor 5 θ5 UC

45. Test conditions
Sensors initialized with weak priors and target responsibilities
Performance evaluated using global information from one sensor

46. Global Information

47. Adjusting negotiation time

48. How to use Max-Sum to solve your distributed problem:
Turn into factor graph
Construct utility function