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Probabilistic Inference Lecture 5: Final Exam Prep

Prepare for the final exam of Probabilistic Inference with topics like Belief Propagation, Relaxations, and Integer Programming Formulation. Study the slides and practice solving reparameterization constants and minimum cut problems. Understand the intricacies of Linear Programming Relaxation and Dual decomposition for efficient solutions.

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Probabilistic Inference Lecture 5: Final Exam Prep

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  1. Probabilistic InferenceLecture 5 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online http://cvc.centrale-ponts.fr/personnel/pawan/

  2. What to Expect in the Final Exam • Open Book • Textbooks • Research Papers • Course Slides • No Electronic Devices • Easy Questions – 10 points • Hard Questions – 10 points

  3. Easy Question – BP Compute the reparameterization constants for (a,b) and (c,b) such that the unary potentials of b are equal to its min-marginals. -2 2 6 -6 12 -3 -2 -1 -4 5 -3 -5 9 5 Vb Vc Va

  4. Hard Question – BP Provide an O(h) algorithm to compute the reparameterization constants of BP for an edge whose pairwise potentials are specified by a truncated linear model.

  5. Easy Question – Minimum Cut Provide the graph corresponding to the MAP estimation problem in the following MRF. -2 2 6 -6 12 -3 -2 -1 -4 5 -3 -5 9 5 Vb Vc Va

  6. Hard Question – Minimum Cut Show that the expansion algorithm provides a bound of 2M for the truncated linear metric, where M is the value of the truncation.

  7. Easy Question – Relaxations Using an example, show that the LP-S relaxation is not tight for a frustrated cycle (cycle with an odd number of supermodular pairwise potentials).

  8. Hard Question – Relaxations Prove or disprove that the LP-S and SOCP-MS relaxations are invariant to reparameterization.

  9. Recap

  10. Integer Programming Formulation min ∑a ∑i a;i ya;i + ∑(a,b) ∑ik ab;ik yab;ik ya;i  {0,1} ∑i ya;i = 1 yab;ik =ya;i yb;k

  11. Integer Programming Formulation min Ty ya;i  {0,1} ∑i ya;i = 1 yab;ik =ya;i yb;k  = [ … a;i …. ; … ab;ik ….] y = [ … ya;i …. ; … yab;ik ….]

  12. Linear Programming Relaxation min Ty ya;i  {0,1} ∑i ya;i = 1 yab;ik =ya;i yb;k Two reasons why we can’t solve this

  13. Linear Programming Relaxation min Ty ya;i  [0,1] ∑i ya;i = 1 yab;ik =ya;i yb;k One reason why we can’t solve this

  14. Linear Programming Relaxation min Ty ya;i  [0,1] ∑i ya;i = 1 ∑k yab;ik =∑kya;i yb;k One reason why we can’t solve this

  15. Linear Programming Relaxation min Ty ya;i  [0,1] ∑i ya;i = 1 ∑k yab;ik =ya;i∑k yb;k = 1 One reason why we can’t solve this

  16. Linear Programming Relaxation min Ty ya;i  [0,1] ∑i ya;i = 1 ∑k yab;ik =ya;i One reason why we can’t solve this

  17. Linear Programming Relaxation min Ty ya;i  [0,1] ∑i ya;i = 1 ∑k yab;ik =ya;i No reason why we can’t solve this * *memory requirements, time complexity

  18. Dual of the LP Relaxation Wainwright et al., 2001 1 Va Vb Vc Va Vb Vc 2 Vd Ve Vf Vd Ve Vf 3 Vg Vh Vi Vg Vh Vi 4 5 6  Va Vb Vc Vd Ve Vf Vg Vh Vi  i = 

  19. Dual of the LP Relaxation Wainwright et al., 2001 q*(1) Va Vb Vc Va Vb Vc Vd Ve Vf q*(2) Vd Ve Vf Vg Vh Vi q*(3) Vg Vh Vi q*(4) q*(5) q*(6)  Va Vb Vc Dual of LP Vd Ve Vf max  q*(i) Vg Vh Vi  i = 

  20. Dual of the LP Relaxation Wainwright et al., 2001 q*(1) Va Vb Vc Va Vb Vc Vd Ve Vf q*(2) Vd Ve Vf Vg Vh Vi q*(3) Vg Vh Vi q*(4) q*(5) q*(6)  Va Vb Vc Dual of LP Vd Ve Vf max  q*(i) Vg Vh Vi  i  

  21. Dual of the LP Relaxation Wainwright et al., 2001 max  q*(i)  i   I can easily compute q*(i) I can easily maintain reparam constraint So can I easily solve the dual?

  22. Outline • TRW Message Passing • Dual Decomposition

  23. Things to Remember • BP is exact for trees • Every iteration provides a reparameterization • Forward-pass computes min-marginals of root

  24. TRW Message Passing Kolmogorov, 2006 4 5 6 1 Va Vb Vc Vb Vc Va 2 Vd Ve Vf Ve Vf Vd 3 Vg Vh Vi Vh Vi Vg Va Pick a variable  q*(i)  i  

  25. TRW Message Passing Kolmogorov, 2006 1c;1 1b;1 1a;1 4a;1 4d;1 4g;1 1c;0 1b;0 1a;0 4a;0 4d;0 4g;0 Vc Vb Va Va Vd Vg  q*(i)  i  

  26. TRW Message Passing Kolmogorov, 2006 1c;1 1b;1 1a;1 4a;1 4d;1 4g;1 1c;0 1b;0 1a;0 4a;0 4d;0 4g;0 Vc Vb Va Va Vd Vg Reparameterize to obtain min-marginals of Va q*(1) + q*(4) + K 1 +4 + rest 

  27. TRW Message Passing Kolmogorov, 2006 ’1c;1 ’1b;1 ’1a;1 ’4a;1 ’4d;1 ’4g;1 ’1c;0 ’1b;0 ’1a;0 ’4a;0 ’4d;0 ’4g;0 Vc Vb Va Va Vd Vg One pass of Belief Propagation q*(’1) + q*(’4) + K ’1 +’4 + rest

  28. TRW Message Passing Kolmogorov, 2006 ’1c;1 ’1b;1 ’1a;1 ’4a;1 ’4d;1 ’4g;1 ’1c;0 ’1b;0 ’1a;0 ’4a;0 ’4d;0 ’4g;0 Vc Vb Va Va Vd Vg Remain the same q*(’1) + q*(’4) + K ’1 +’4 + rest 

  29. TRW Message Passing Kolmogorov, 2006 ’1c;1 ’1b;1 ’1a;1 ’4a;1 ’4d;1 ’4g;1 ’1c;0 ’1b;0 ’1a;0 ’4a;0 ’4d;0 ’4g;0 Vc Vb Va Va Vd Vg min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K ’1 +’4 + rest 

  30. TRW Message Passing Kolmogorov, 2006 ’1c;1 ’1b;1 ’1a;1 ’4a;1 ’4d;1 ’4g;1 ’1c;0 ’1b;0 ’1a;0 ’4a;0 ’4d;0 ’4g;0 Vc Vb Va Va Vd Vg Compute average of min-marginals of Va min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K ’1 +’4 + rest 

  31. TRW Message Passing Kolmogorov, 2006 ’1c;1 ’1b;1 ’1a;1 ’4a;1 ’4d;1 ’4g;1 ’1c;0 ’1b;0 ’1a;0 ’4a;0 ’4d;0 ’4g;0 Vc Vb Va Va Vd Vg ’’a;0 = ’1a;0+ ’4a;0 ’’a;1 = ’1a;1+ ’4a;1 2 2 min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K ’1 +’4 + rest 

  32. TRW Message Passing Kolmogorov, 2006 ’1c;1 ’1b;1 ’’a;1 ’’a;1 ’4d;1 ’4g;1 ’1c;0 ’1b;0 ’’a;0 ’’a;0 ’4d;0 ’4g;0 Vc Vb Va Va Vd Vg ’’a;0 = ’1a;0+ ’4a;0 ’’a;1 = ’1a;1+ ’4a;1 2 2 min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K ’’1 +’’4 + rest

  33. TRW Message Passing Kolmogorov, 2006 ’1c;1 ’1b;1 ’’a;1 ’’a;1 ’4d;1 ’4g;1 ’1c;0 ’1b;0 ’’a;0 ’’a;0 ’4d;0 ’4g;0 Vc Vb Va Va Vd Vg ’’a;0 = ’1a;0+ ’4a;0 ’’a;1 = ’1a;1+ ’4a;1 2 2 min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K ’’1 +’’4 + rest  

  34. TRW Message Passing Kolmogorov, 2006 ’1c;1 ’1b;1 ’’a;1 ’’a;1 ’4d;1 ’4g;1 ’1c;0 ’1b;0 ’’a;0 ’’a;0 ’4d;0 ’4g;0 Vc Vb Va Va Vd Vg ’’a;0 = ’1a;0+ ’4a;0 ’’a;1 = ’1a;1+ ’4a;1 2 2 2 min{’’a;0, ’’a;1} + K ’’1 +’’4 + rest  

  35. TRW Message Passing Kolmogorov, 2006 ’1c;1 ’1b;1 ’’a;1 ’’a;1 ’4d;1 ’4g;1 ’1c;0 ’1b;0 ’’a;0 ’’a;0 ’4d;0 ’4g;0 Vc Vb Va Va Vd Vg ≥ min {p1+p2, q1+q2} min {p1, q1} + min {p2, q2} 2 min{’’a;0, ’’a;1} + K ’’1 +’’4 + rest  

  36. TRW Message Passing Kolmogorov, 2006 ’1c;1 ’1b;1 ’’a;1 ’’a;1 ’4d;1 ’4g;1 ’1c;0 ’1b;0 ’’a;0 ’’a;0 ’4d;0 ’4g;0 Vc Vb Va Va Vd Vg Objective function increases or remains constant 2 min{’’a;0, ’’a;1} + K ’’1 +’’4 + rest  

  37. TRW Message Passing Initialize i. Take care of reparam constraint Choose random variable Va Compute min-marginals of Va for all trees Node-average the min-marginals Can also do edge-averaging REPEAT Kolmogorov, 2006

  38. Example 1 2 0 4 4 0 6 6 1 6 l1 1 2 4 1 3 1 l0 5 0 2 1 3 0 2 3 4 Vb Vc Va Va Vb Vc 5 6 7 Pick variable Va. Reparameterize.

  39. Example 1 5 -3 4 4 0 6 6 -3 10 l1 2 1 -1 3 -3 -2 l0 7 -2 2 1 3 -3 2 3 7 Vb Vc Va Va Vb Vc 5 6 7 Average the min-marginals of Va

  40. Example 1 7.5 -3 4 4 0 6 6 -3 7.5 l1 2 1 -1 3 -3 -2 l0 7 -2 2 1 3 -3 2 3 7 Vb Vc Va Va Vb Vc 7 6 7 Pick variable Vb. Reparameterize.

  41. Example 1 7.5 -7.5 8.5 9 -5 6 6 -3 7.5 l1 1 -5.5 -3 -1 -3 -7 l0 7 -7 6 -3 3 -3 7 3 7 Vb Vc Va Va Vb Vc 7 6 7 Average the min-marginals of Vb

  42. Example 1 7.5 -7.5 8.75 8.75 -5 6 6 -3 7.5 l1 1 -5.5 -3 -1 -3 -7 l0 7 -7 6.5 -3 3 -3 6.5 3 7 Vb Vc Va Va Vb Vc 6.5 6.5 7 Value of dual does not increase

  43. Example 1 7.5 -7.5 8.75 8.75 -5 6 6 -3 7.5 l1 1 -5.5 -3 -1 -3 -7 l0 7 -7 6.5 -3 3 -3 6.5 3 7 Vb Vc Va Va Vb Vc 6.5 6.5 7 Maybe it will increase for Vc NO

  44. Example 1 7.5 -7.5 8.75 8.75 -5 6 6 -3 7.5 l1 1 -5.5 -3 -1 -3 -7 l0 7 -7 6.5 -3 3 -3 6.5 3 7 Vb Vc Va Va Vb Vc f1(a) = 0 f1(b) = 0 f2(b) = 0 f2(c) = 0 f3(c) = 0 f3(a) = 0 Strong Tree Agreement Exact MAP Estimate

  45. Example 2 2 0 2 0 1 0 0 0 4 l1 1 0 1 1 0 1 l0 5 0 0 1 3 0 2 0 8 Vb Vc Va Va Vb Vc 4 0 4 Pick variable Va. Reparameterize.

  46. Example 2 4 -2 2 0 1 0 0 0 4 l1 0 -1 0 1 -1 0 l0 7 -2 0 1 3 -1 2 0 9 Vb Vc Va Va Vb Vc 4 0 4 Average the min-marginals of Va

  47. Example 2 4 -2 2 0 1 0 0 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 3 -1 2 0 8 Vb Vc Va Va Vb Vc 4 0 4 Value of dual does not increase

  48. Example 2 4 -2 2 0 1 0 0 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 3 -1 2 0 8 Vb Vc Va Va Vb Vc 4 0 4 Maybe it will decrease for Vb or Vc NO

  49. Example 2 4 -2 2 0 1 0 0 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 3 -1 2 0 8 Vb Vc Va Va Vb Vc f1(a) = 1 f1(b) = 1 f2(b) = 1 f2(c) = 0 f3(c) = 1 f3(a) = 1 f2(b) = 0 f2(c) = 1 Weak Tree Agreement Not Exact MAP Estimate

  50. Example 2 4 -2 2 0 1 0 0 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 3 -1 2 0 8 Vb Vc Va Va Vb Vc f1(a) = 1 f1(b) = 1 f2(b) = 1 f2(c) = 0 f3(c) = 1 f3(a) = 1 f2(b) = 0 f2(c) = 1 Weak Tree Agreement Convergence point of TRW

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