Loopy Belief Propagation for Bipartite Maximum Weight b-matching

Transcription

1 Loopy Belief Propagation for Bipartite Maximum Weight b-matching Bert Huang and Tony Jebara Computer Science Department Columbia University New York, NY 10027

2 Outline 1. Bipartite Weighted b-matching 2. Edge Weights As a Distribution 3. Efficient Max-Product 4. Convergence Proof Sketch 5. Experiments 6. Discussion

3 Outline 1. Bipartite Weighted b-matching 2. Edge Weights As a Distribution 3. Efficient Max-Product 4. Convergence Proof Sketch 5. Experiments 6. Discussion

4 Bipartite Weighted b-matching

5 Bipartite Weighted b-matching On bipartite graph, G = (U, V, E) {u 1,..., u n } U {v 1,..., v n } V E = (u i, v j ), i j u 1 u 2 u 3 u 4 v 1 v 2 v 3 v 4

6 Bipartite Weighted b-matching On bipartite graph, {u 1,..., u n } U {v 1,..., v n } V E = (u i, v j ), i j G = (U, V, E) u 1 u 2 u 3 u 4 A = weight matrix s.t. weight of edge v 1 v 2 v 3 v 4 (u i, v j ) = A ij

7 Bipartite Weighted b-matching Task: Find the maximum weight subset of E such that each vertex has exactly b neighbors.

8 Bipartite Weighted b-matching Task: Find the maximum weight subset of E such that each vertex has exactly b neighbors. Example: u 1 u 2 u 3 u 4 n = 4 b = 2 v 1 v 2 v 3 v 4

9 Bipartite Weighted b-matching Classical Application: Resource Allocation - Manual labor Workers u 1 u 2 u 3 u 4 - n workers - n tasks Tasks - Team of b workers needed per task. - skill of worker at performing task. A ij v 1 v 2 v 3 v 4

10 Bipartite Weighted b-matching Alternate uses of b-matching: - Balanced k-nearest-neighbors - Each node can only be picked k times. - Robust to translations of test data. - When test data is collected under different conditions (e.g. time, location, instrument calibration).

11 Bipartite Weighted b-matching Classical algorithms solve Max-Weighted b- Matching in O(bn 3 ) running time, such as: - Blossom Algorithm (Edmonds 1965) - Balanced Network Flow (Fremuth-Paeger, Jungnickel 1999)

12 Outline 1. Bipartite Weighted b-matching 2. Edge Weights As a Distribution 3. Efficient Max-Product 4. Convergence Proof Sketch 5. Experiments 6. Discussion

13 Edge Weights As a Distribution - Bayati, Shah, and Sharma (2005) formulated the 1-matching problem as a probability distribution. - This work generalizes to arbitrary b.

14 Edge Weights As a Distribution Variables: Each vertex chooses b neighbors.

15 Example: u i - u 1 u 1 u 1 v 1 v 2 v 3 v 4 v 1 v 2 v 3 v 4 v 1 v 2 v 3 v 4 u 1 u 1 u 1 v 1 v 2 v 3 v 4 v 1 v 2 v 3 v 4 v 1 v 2 v 3 v 4

16 Edge Weights As a Distribution Variables: Each vertex chooses b neighbors.

17 Edge Weights As a Distribution Variables: Each vertex chooses b neighbors. For vertex, u i X i V, X i = b Similarly, for v j have variable Y j Note: variables have ( ) n b possible settings.

18 Edge Weights As a Distribution Weights as probabilities: Since we sum weights but multiply probabilities, exponentiate. φ(x i ) = exp( 1 2 φ(y j ) = exp( 1 2 v j X i A ij ) u i Y j A ij ) ( ) n b

19 Edge Weights As a Distribution Enforce b-matching: Neighbor choices must agree

20 Example: Invalid settings u 1 u 1 u 2 u 3 u 4 v 1 v 2 v 3 v 4 v 1

21 Edge Weights As a Distribution Enforce b-matching: Neighbor choices must agree

22 Edge Weights As a Distribution Enforce b-matching: Neighbor choices must agree Pairwise compatibility function: ψ(x i, Y j ) = = (v j X i u i Y j ). ( ) n b ( ) n b

23 Edge Weights As a Distribution P (X, Y ) = 1 Z n i,j=1 ψ(x i, Y j ) n k=1 φ(x k )φ(y j ) φ(x i ) = exp( 1 A ij ) φ(y j ) = exp( v j X i ψ(x i, Y j ) = (v j X i u i Y j ). A ij ) u i Y j

24 Edge Weights As a Distribution P (X, Y ) = 1 Z n i,j=1 ψ(x i, Y j ) n k=1 φ(x k )φ(y j ) φ(x i ) = exp( 1 A ij ) φ(y j ) = exp( v j X i ψ(x i, Y j ) = (v j X i u i Y j ). A ij ) u i Y j Ignore the Z normalization, P(X,Y) is exactly the exponentiated weight of the b-matching.

25 Edge Weights As a Distribution P (X, Y ) = 1 Z n i,j=1 ψ(x i, Y j ) n k=1 φ(x k )φ(y j ) φ(x i ) = exp( 1 A ij ) φ(y j ) = exp( v j X i ψ(x i, Y j ) = (v j X i u i Y j ). A ij ) u i Y j Also, since we re maximizing, ignore the 1/2 (makes the math more readable).

26 Outline 1. Bipartite Weighted b-matching 2. Edge Weights As a Distribution 3. Efficient Max-Product 4. Convergence Proof Sketch 5. Experiments 6. Discussion

27 Standard Max-Product Send messages between variables: m Xi (Y j ) = 1 Z max X i φ(x i )ψ(x i, Y j ) m Yk (X i ) k j Fuse messages to obtain beliefs (or estimate of max-marginals): b(x i ) = 1 Z φ(x i) k m Yk (X i )

28 Standard Max-Product Converges to true maximum on any tree structured graph (Pearl 1986). We show that it converges to the correct maximum on our graph.

29 Standard Max-Product Converges to true maximum on any tree structured graph (Pearl 1986). We show that it converges to the correct maximum on our graph. But what about the belief vectors? ( ) n b -length message and

30 Efficient Max-Product ( ) n - Use algebraic tricks to reduce b -length message vectors to scalars. - Derive new update rule for scalar messages. - Use similar trick to maximize belief vectors efficiently. Let s speed through the math.

31 Efficient Max-Product Take advantage of binary ψ(x i, y j ) function: Message vectors consist of only two values m xi (y j ) m xi (y j ) max v j x i φ(x i ) k j max v j / x i φ(x i ) k j m yk (x i ), if u i y j m yk (x i ), if u i / y j

32 Efficient Max-Product Take advantage of binary ψ(x i, y j ) function: Message vectors consist of only two values m xi (y j ) m xi (y j ) max v j x i φ(x i ) k j max v j / x i φ(x i ) k j m yk (x i ), if u i y j m yk (x i ), if u i / y j If we rename these two values we can break up the product.

33 Efficient Max-Product Take advantage of binary function: Message vectors consist of only two values µ xi y j max v j x i φ(x i ) ν xi y j max v j / x i φ(x i ) ψ(x i, y j ) u k x i \v j µ ki u k x i \v j µ ki u k / x i \v j ν ki u k / x i \v j ν ki. If we rename these two values we can break up the product.

34 Efficient Max-Product Normalize messages by dividing whole vector by ν xi y j ˆµ xi y j = µ x i y j ν xi y j and ˆν xi y j = 1

35 Efficient Max-Product Normalize messages by dividing whole vector by ν xi y j ˆµ xi y j = µ x i y j ν xi y j and ˆν xi y j = 1 ( ) n b -length vector scalar

36 Efficient Max-Product Derive update rule: ˆµ xi y j = max j x i φ(x i ) k x i \j ˆµ ki max j / xi φ(x i ) k x i \j ˆµ ki = max j x i k x i exp(a ik ) k x i \j ˆµ ki max j / xi k x i exp(a ik ) k x i \j ˆµ ki = exp(a ij) max j xi k x i \j exp(a ik)ˆµ ki max j / xi k x i exp(a ij )ˆµ ki

37 Efficient Max-Product Derive update rule: ˆµ xi y j = max j x i φ(x i ) k x i \j ˆµ ki max j / xi φ(x i ) k x i \j ˆµ ki = max j x i k x i exp(a ik ) k x i \j ˆµ ki max j / xi k x i exp(a ik ) k x i \j ˆµ ki φ(x = exp(a i ) exp(a ij) max j xi k x i \j exp(a ik ) k x ik)ˆµ ki i max j / xi k x i exp(a ij )ˆµ ki

38 Efficient Max-Product Derive update rule: ˆµ xi y j = max j x i φ(x i ) k x i \j ˆµ ki max j / xi φ(x i ) k x i \j ˆµ ki = max j x i k x i exp(a ik ) k x i \j ˆµ ki max j / xi k x i exp(a ik ) k x i \j ˆµ ki = exp(a ij) max j xi k x i \j exp(a ik)ˆµ ki max j / xi k x i exp(a ij )ˆµ ki

39 Efficient Max-Product Derive update rule: ˆµ xi y j = max j x i φ(x i ) k x i \j ˆµ ki max j / xi φ(x i ) k x i \j ˆµ ki = max j x i k x i exp(a ik ) k x i \j ˆµ ki max j / xi k x i exp(a ik ) k x i \j ˆµ ki = exp(a ij) max j xi k x i \j exp(a ik)ˆµ ki max j / xi k x i exp(a ij )ˆµ ki

40 Efficient Max-Product After canceling terms message update simplifies to ˆµ xi y j = exp(a ij) exp(a il )ˆµ yl x i. l = and we maximize beliefs with max b(x i ) max φ(x i ) ˆµ yk x i x i x i k x i max exp(a ik )ˆµ yk x i x i k x i bth greatest setting of k for the term exp(a ik )m yk (x i ), s. t. k j Both these updates take O(bn) time per vertex.

41 Outline 1. Bipartite Weighted b-matching 2. Edge Weights As a Distribution 3. Efficient Max-Product 4. Convergence Proof Sketch 5. Experiments 6. Discussion

42 Convergence Proof Sketch

43 Convergence Proof Sketch Assumptions: - Optimal b-matching is unique. - ɛ = difference between weight of best and 2nd best b-matching is constant. - Weights treated as constants.

44 Convergence Proof Sketch Basic mechanism: Unwrapped Graph, T 1. Pick root node 2. Copy all neighbors 3. Continue but don t backtrack 4. Continue to depth d u 1 u 2 u 3 u 4 v 1 v 2 v 3 v 4

45 the same asymptotic rate of O(bn 3 ). Furthe more, the algorithm shows promising perfo mance in machine learning applications. Convergence Proof Sketch 1 INTRODUCTION Basic mechanism: Unwrapped Graph, T 1. Pick root node 2. Copy all neighbors 3. Continue but don t backtrack 4. Continue to depth d u 1 u 2 u 3 u 4 v 1 v 2 v 3 v 4 u 1 Figure 1: Example b-matching M G on a bipartite gr v Dashed 1 v lines represent 2 v possible edges, solid 3 v lines re 4 b-matched edges. In this case b = 2. u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 2 EXPERIMENTS v 2 v 3 v 4 v 2 v 3 v 4 v 2 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3

46 Convergence Proof Sketch Basic mechanism: Unwrapped Graph, T 1. Pick root node 2. Copy all neighbors 3. Continue but don t backtrack 4. Continue to depth d u 1 u 2 u 3 u 4 v 1 v 2 v 3 v 4 u 1 v 1 v 2 v 3 v 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 v 2 v 3 v 4 v 2 v 3 v 4 v 2 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3

47 Convergence Proof Sketch Basic mechanism: Unwrapped Graph, T 1. Pick root node 2. Copy all neighbors 3. Continue but don t backtrack 4. Continue to depth d u 1 u 2 u 3 u 4 v 1 v 2 v 3 v 4 u 1 v 1 v 2 v 3 v 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 v 2 v 3 v 4 v 2 v 3 v 4 v 2 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3

48 Convergence Proof Sketch Basic mechanism: Unwrapped Graph, T 1. Pick root node 2. Copy all neighbors 3. Continue but don t backtrack 4. Continue to depth d u 1 u 2 u 3 u 4 Too complex to color... v 1 v 2 v 3 v 4 u 1 v 1 v 2 v 3 v 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 v 2 v 3 v 4 v 2 v 3 v 4 v 2 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3

49 Convergence Proof Sketch Basic mechanism: Unwrapped Graph, T - Construction follows loopy BP messages in reverse. - True max-marginals of root node are exactly belief at iteration d. - Max of unwrapped graph distribution is the maximum weight b-matching on tree.

50 Convergence Proof Sketch Proof by contradiction: What happens if optimal b-matching on T differs from optimal b-matching on G at root?

51 Convergence Proof Sketch Best b-matching on T u 1 v 1 v 2 v 3 v 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 v 2 v 3 v 4 v 2 v 3 v 4 v 2 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 Best b-matching on G copied onto T ple unwrapped graph T of G at 3 iterations. u 1 The matching M T is highlighted based on af nodes cannot have perfect b-matchings, but all inner nodes and the root do. One po ich is discussed in Lemma??. v 1 v 2 v 3 v 4 u 2 u First 3 cycle u 4 u 2 u 3 u 4 u 2 v 1 u 3 u 4 u 4 u 2 u 3 u 4 u 3 v 1 u 4 v 3 u 1 v 1 u 4 v 2 u 2... u 1 v 3 u 3 v 1 u 4 v 2 u 2 v 2 v 3 v 4 v 2 v 3 v 4 v 2 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3

52 Convergence Proof Sketch There exists at least one path on T that alternates between edges that are b-matched in each b-matching. u 1 v 1 v 2 v 3 v 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 v 2 v 3 v 4 v 2 v 3 v 4 v 2 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3

53 Convergence Proof Sketch There exists at least one path on T that alternates between edges that are b-matched in each b-matching. u 1 v 1 v 2 v 3 v 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 v 2 v 3 v 4 v 2 v 3 v 4 v 2 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3

54 Convergence Proof Sketch Claim: If depth d is great enough, if we replace the blue edges of this path with the red edges in the optimal b-matching on T, we get a new b-matching with greater weight. u 1 v 1 v 2 v 3 v 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 v 2 v 3 v 4 v 2 v 3 v 4 v 2 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3

55 Convergence Proof Sketch Claim: If depth d is great enough, if we replace the blue edges of this path with the red edges in the optimal b-matching on T, we get a new b-matching with greater weight. u 1 v 1 v 2 v 3 v 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 v 2 v 3 v 4 v 2 v 3 v 4 v 2 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3

56 Convergence Proof Sketch Claim: If depth d is great enough, if we replace the blue edges of this path with the red edges in the optimal b-matching on T, we get a new b-matching with greater weight. u 1 v 1 v 2 v 3 v 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 v 2 v 3 v 4 v 2 v 3 v 4 v 2 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3

57 Convergence Proof Sketch Modifying optimal b-matching produced better b-matching Original contradiction impossible. We can analyze the change in weight by looking only at edges on path. u 1 v 1 v 2 v 3 v 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 u 2 u 3 u 4 v 2 v 3 v 4 v 2 v 3 v 4 v 2 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 3 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 4 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3

58 Convergence Proof Sketch Loopy BP converges to true maximum weight b-matching in d iterations d n ɛ max i,j A ij = O(n) ɛ = difference between weight of best and 2nd best b-matching.

59 Convergence Proof Sketch Loopy BP converges to true maximum weight b-matching in d iterations d n ɛ max i,j A ij = O(n) ɛ = difference between weight of best and 2nd best b-matching. Running time of full algorithm: O(bn 3 )

60 Outline 1. Bipartite Weighted b-matching 2. Edge Weights As a Distribution 3. Efficient Max-Product 4. Convergence Proof Sketch 5. Experiments 6. Discussion

61 Experiments Running time comparison against GOBLIN graph optimization library. - Random weights. - Varied graph size n from 3 to Varied b from 1 to n/2

62 Experiments 3 Running time comparison against GOBLIN GOBLIN BP graph optimization library. 2 t BP median running time n b GOBLIN median running time t 1/3 1 0 Median Running time when B= n Median Running time when B=! n/2 " 4 GOBLIN 3 BP t 100 t 1/ n b n

63 Experiments: Translated Test Data $ % On toy data, translation cripples KNN but b-matching makes no classification errors. '()*+,*-./01*1 1 Accuracy & "!& Accuracy !%!$!! "! #" #! 231)451*,6/7,1*83, 0.2 b!matching k!nearest!neighbor k

64 Experiments MNIST Digits with pseudo-translation - Image data with background changes is like translation. - Train on MNIST digits 3, 5, and 8. - Test on new examples with various bluescreen textures.

65 Experiments Average accuracy Background texture BM KNN 5 6 Average accuracy BM KNN k

66 Outline 1. Bipartite Weighted b-matching 2. Edge Weights As a Distribution 3. Efficient Max-Product 4. Convergence Proof Sketch 5. Experiments 6. Discussion

67 Discussion Provably convergent belief propagation for a new type of graph (b-matchings). + Empirically faster than previous algorithms. + Parallelizeable - Only bipartite case. - Requires unique maximum. Interesting theoretical results coming out of sum-product for approximating marginals.