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Discrete Optimization Lecture 3 – Part 2: Dual Decomposition

This lecture discusses the concept of dual decomposition in discrete optimization, including its formulation and applications. The lecture also compares dual decomposition with tree-reweighted message passing methods.

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Discrete Optimization Lecture 3 – Part 2: Dual Decomposition

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  1. Discrete OptimizationLecture 3 – Part 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online http://cvn.ecp.fr/personnel/pawan/

  2. Outline • Dual Decomposition

  3. Dual Decomposition minx ∑igi(x) s.t.x  C

  4. Dual Decomposition minx,xi ∑igi(xi) s.t.xi  C xi = x

  5. Dual Decomposition minx,xi ∑igi(xi) s.t.xi  C

  6. Dual Decomposition maxλi minx,xi ∑igi(xi) + ∑iλiT(xi-x) s.t.xi  C KKT Condition: ∑iλi= 0

  7. Dual Decomposition maxλi minx,xi ∑igi(xi) + ∑iλiTxi s.t.xi  C

  8. Dual Decomposition maxλi minxi ∑i (gi(xi) + λiTxi) s.t.xi  C Projected Supergradient Ascent Supergradients of h(z) at z0 h(z) - h(z0) ≤ sT(z-z0), for all z in the feasible region

  9. Dual Decomposition maxλi minxi ∑i (gi(xi) + λiTxi) s.t.xi  C Initialize λi0= 0

  10. Dual Decomposition maxλi minxi ∑i (gi(xi) + λiTxi) s.t.xi  C Compute supergradients si= argminxi ∑i (gi(xi) + (λit)Txi)

  11. Dual Decomposition maxλi minxi ∑i (gi(xi) + λiTxi) s.t.xi  C Project supergradients pi = si - ∑jsj/m where ‘m’ = number of subproblems (slaves)

  12. Dual Decomposition maxλi minxi ∑i (gi(xi) + λiTxi) s.t.xi  C Update dual variables λit+1= λit + ηtpi where ηt = learning rate = 1/(t+1) for example

  13. Dual Decomposition Initialize λi0= 0 Compute projected supergradients si= argminxi ∑i (gi(xi) + (λit)Txi) pi = si - ∑jsj/m REPEAT Update dual variables λit+1= λit + ηtpi

  14. Outline • Dual Decomposition • TRW vs. DD • DD for Energy Minimization • Extensions of DD

  15. TRW 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

  16. TRW 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

  17. DD 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 ya;0 ya;1 yb;0 yb;1 yc;0 yc;1 1 0 1 0 - - Optimal LP solution Values of yab;ik not shown. But we know yab;ik = ya;iyb;k

  18. Supergradients 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 sa;0 sa;1 sb;0 sb;1 sc;0 sc;1 1 0 1 0 - - - - 1 0 1 0 1 0 - - 1 0

  19. Projected Supergradients 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 pa;0 pa;1 pb;0 pb;1 pc;0 pc;1 0 0 0 0 - - - - 0 0 0 0 0 0 - - 0 0

  20. Objective 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 No further increase in dual objective

  21. DD 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 No further increase in dual objective Strong Tree Agreement implies DD stops

  22. TRW 4 -2 2 0 1 0 0 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 -1 2 0 8 -0.2 Vb Vc Va Va Vb Vc 4 0 4

  23. TRW 4 -2 2 0 1 0 0 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 -1 2 0 8 -0.2 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

  24. DD 4 -2 2 0 1 0 0 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 -1 2 0 8 -0.2 Vb Vc Va Va Vb Vc ya;0 ya;1 yb;0 yb;1 yc;0 yc;1 0 1 0 1 - - Optimal LP solution Values of yab;ik not shown. But we know yab;ik = ya;iyb;k

  25. Supergradients 4 -2 2 0 1 0 0 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 -1 2 0 8 -0.2 Vb Vc Va Va Vb Vc sa;0 sa;1 sb;0 sb;1 sc;0 sc;1 0 1 0 1 - - - - 0 1 1 0 0 1 - - 0 1

  26. Projected Supergradients 4 -2 2 0 1 0 0 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 -1 2 0 8 -0.2 Vb Vc Va Va Vb Vc pa;0 pa;1 pb;0 pb;1 pc;0 pc;1 0 0 0 0 - - - - 0 0 0.5 -0.5 0 0 - - -0.5 0.5

  27. Update with Learning Rate ηt = 1 4 -2 2 0 1 0 0 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 -1 2 0 8 -0.2 Vb Vc Va Va Vb Vc pa;0 pa;1 pb;0 pb;1 pc;0 pc;1 0 0 0 0 - - - - 0 0 0.5 -0.5 0 0 - - -0.5 0.5

  28. Objective 4 -2 2 0 1 -0.5 0.5 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 -1 2 0.5 8 -0.7 Vb Vc Va Va Vb Vc -0.5 4 4.3 Decrease in dual objective

  29. Supergradients 4 -2 2 0 1 -0.5 0.5 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 -1 2 0.5 8 -0.7 Vb Vc Va Va Vb Vc sa;0 sa;1 sb;0 sb;1 sc;0 sc;1 0 1 0 1 - - - - 1 0 0 1 0 1 - - 1 0

  30. Projected Supergradients 4 -2 2 0 1 -0.5 0.5 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 -1 2 0.5 8 -0.7 Vb Vc Va Va Vb Vc pa;0 pa;1 pb;0 pb;1 pc;0 pc;1 0 0 -0.5 0.5 - - - - 0.5 -0.5 -0.5 0.5 0 0 - - 0.5 -0.5

  31. Update with Learning Rate ηt = 1/2 4 -2 2 0 1 -0.5 0.5 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 -1 2 0.5 8 -0.7 Vb Vc Va Va Vb Vc pa;0 pa;1 pb;0 pb;1 pc;0 pc;1 0 0 -0.5 0.5 - - - - 0.5 -0.5 -0.5 0.5 0 0 - - 0.5 -0.5

  32. Updated Subproblems 4 -2 2.25 -0.25 1 -0.25 0.25 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0.25 1 -1 1.75 0.25 8 -0.45 Vb Vc Va Va Vb Vc

  33. Objective 4 -2 2.25 -0.25 1 -0.25 0.25 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0.25 1 -1 1.75 0.25 8 -0.45 Vb Vc Va Va Vb Vc 0 4.25 4.25 Increase in dual objective DD goes beyond TRW

  34. DD 4 -2 2.25 -0.25 1 -0.25 0.25 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0.25 1 -1 1.75 0.25 8 -0.45 Vb Vc Va Va Vb Vc 0 4.25 4.25 Increase in dual objective DD provides the optimal dual objective

  35. Outline • Dual Decomposition • TRW vs. DD • DD for Energy Minimization • Extensions of DD

  36. Dual Decomposition Initialize λi0= 0 Compute projected supergradients si= argminxi ∑i (gi(xi) + (λit)Txi) pi = si - ∑jsj/m REPEAT Update dual variables λit+1= λit + ηtpi

  37. Dual Decomposition Komodakis et al., 2007 4 5 6 1 Va Vb Vc Vb Vc Va 2 Vd Ve Vf Ve Vf Vd 3 Vg Vh Vi Vh Vi Vg 1 0 s1a = Slaves agree on label for Va 1 0 s4a =

  38. Dual Decomposition Komodakis et al., 2007 4 5 6 1 Va Vb Vc Vb Vc Va 2 Vd Ve Vf Ve Vf Vd 3 Vg Vh Vi Vh Vi Vg 1 0 0 0 s1a = p1a = 1 0 0 0 s4a = p4a =

  39. Dual Decomposition Komodakis et al., 2007 4 5 6 1 Va Vb Vc Vb Vc Va 2 Vd Ve Vf Ve Vf Vd 3 Vg Vh Vi Vh Vi Vg 1 0 s1a = Slaves disagree on label for Va 0 1 s4a =

  40. Dual Decomposition Komodakis et al., 2007 4 5 6 1 Va Vb Vc Vb Vc Va 2 Vd Ve Vf Ve Vf Vd 3 Vg Vh Vi Vh Vi Vg 1 0 0.5 -0.5 s1a = p1a = Unary cost increases 0 1 -0.5 0.5 s4a = p4a =

  41. Dual Decomposition Komodakis et al., 2007 4 5 6 1 Va Vb Vc Vb Vc Va 2 Vd Ve Vf Ve Vf Vd 3 Vg Vh Vi Vh Vi Vg 1 0 0.5 -0.5 s1a = p1a = Unary cost decreases 0 1 -0.5 0.5 s4a = p4a =

  42. Dual Decomposition Komodakis et al., 2007 4 5 6 1 Va Vb Vc Vb Vc Va 2 Vd Ve Vf Ve Vf Vd 3 Vg Vh Vi Vh Vi Vg 1 0 0.5 -0.5 s1a = p1a = Push the slaves towards agreement 0 1 -0.5 0.5 s4a = p4a =

  43. Outline • Dual Decomposition • TRW vs. DD • DD for Energy Minimization • Extensions of DD

  44. Comparison TRW DD Fast Slow Local Maximum Global Maximum Requires MAP Estimate Requires Min-Marginals Also possible in the TRW framework Other forms of subproblems Tighter relaxations Sparse high-order potentials Easier in the DD framework

  45. Subproblems Va Vb Vc Va Vb Vc Vd Ve Vf Vd Ve Vf Vg Vh Vi Vg Vh Vi Binary labeling problem Va Vb Vc Black edges submodular Vd Ve Vf Red edges supermodular Vg Vh Vi

  46. Subproblems Va Vb Va Vb Vc Vd Ve Vf Vh Vi Vg Vh Vi Binary labeling problem Va Vb Vc Black edges submodular Vd Ve Vf Red edges supermodular Vg Vh Vi Remains submodular over iterations

  47. Tighter Relaxations Va Vb Vb Vc Va Vb Vc Vd Ve Ve Vf Vd Ve Vf Vg Vh Vi Vd Ve Ve Vf Vg Vh Vh Vi Relaxation that is tight for the above 4-cycles LP-S + Cycle inequalities

  48. High-Order Potentials Vb Vc Va Vb Vc Va Vb Ve Vf Vd Ve Vf Vd Ve Vg Vh Vi Vg Vh Vi Va Vd Ve Vf Vg Vh Vi

  49. High-Order Potentials Vb Vc Ve Vf Value of Potential θc;y Labeling y for Clique O(h|C|)!! Subproblem: minyθc;y + λTy

  50. Sparse High-Order Potentials Vb Vc Ve Vf Value of Potential θc;y Labeling y for Clique Σaya;0 = 0 Σaya;0 > 0 O(h|C|)!! Subproblem: minyθc;y + λTy

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