The document summarizes recent advances in convex relaxations for MAP inference in discrete models. It discusses the linear programming relaxation and how the primal formulation is useful for analysis. It then describes randomized rounding and move-making schemes for obtaining integer solutions from the fractional LP solution. Finally, it discusses going beyond the LP relaxation by incorporating cycle inequalities to handle cases where the LP is not tight.

1. MAP Inference in Discrete Models
Recent Advances in Convex
Relaxations
M. Pawan Kumar, Stanford University

2. Outline
• Revisiting the LP relaxation
• Rounding Schemes and Move Making
• Beyond the LP relaxation

3. 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

4. Linear Programming Relaxation
Primal formulation is useful
Easier to analyze
LP better than a large class of relaxations
- QP (Ravikumar, Lafferty 2006)
- SOCP (Muramatsu, Suzuki 2003)
Kumar, Kolmogorov and Torr, NIPS 2007

5. Linear Programming Relaxation
Primal fractional solution is useful
Multiplicative Bounds
Type of Problem Bound
Potts 2
Truncated Linear 2 + √2
Truncated Quadratic O(√M)
General Metric O(log |L|)

6. Outline
• Revisiting the LP relaxation
• Rounding Schemes and Move Making
• Beyond the LP relaxation

7. Randomized Rounding
0 y’a;0 y’a;i y’a;k y’a;h = 1
y’a;i = ya;0 + ya;1 + … + ya;i
Choose an interval of length L’

8. Randomized Rounding
r
0 y’a;0 y’a;i y’a;k y’a;h = 1
y’a;i = ya;0 + ya;1 + … + ya;i
Generate a random number r  (0,1]

9. Randomized Rounding
r
0 y’a;0 y’a;i y’a;k y’a;h = 1
y’a;i = ya;0 + ya;1 + … + ya;i
Assign label next to r (if within the interval)

10. Move Making
• Initialize the labeling
• Choose interval I of L’ labels
• Each variable can
• Retain old label
• Choose a label from I
• Choose best labeling
Va Vb Iterate over intervals
Truncated Convex Models

11. Two Problems
• Choose interval I of L’ labels
• Each variable can
• Retain old label
• Choose a label from I
• Choose best labeling
Large L’ => Non-submodular
Va Vb Non-submodular

12. First Problem
Va Vb Submodular problem
Ishikawa, 2003; Veksler, 2007

13. First Problem
Va Vb Non-submodular
Problem

14. First Problem
Va Vb Submodular problem
Veksler, 2007

15. First Problem
am+1 bm+1
am+2 bm+2
am+2 bm+2
an bn
Va Vb t

16. First Problem
am+1 bm+1
am+2 bm+2
am+2 bm+2
an bn
Va Vb t

17. First Problem
am+1 bm+1
am+2 bm+2
am+2 bm+2
an bn
Va Vb t

18. First Problem
am+1 bm+1
am+2 bm+2
am+2 bm+2
an bn
Va Vb t

19. First Problem
am+1 bm+1
am+2 bm+2
am+2 bm+2
an bn
Va Vb t
Model unary potentials exactly

20. First Problem
am+1 bm+1
am+2 bm+2
am+2 bm+2
an bn
Va Vb t
Similarly for Vb

21. First Problem
am+1 bm+1
am+2 bm+2
am+2 bm+2
an bn
Va Vb t
Model convex pairwise costs

22. First Problem
Wanted to model
ab;ik = wab min{ d(i-k), M }
For all li, lk  I
Have modelled
ab;ik = wab d(i-k)
For all li, lk  I
Va Vb
Overestimated pairwise potentials

23. Second Problem
• Choose interval I of L’ labels
• Each variable can
• Retain old label
• Choose a label from I
• Choose best labeling
Va Vb Non-submodular problem !!

24. Second Problem
am+1 bm+1
am+2 bm+2
an bn
Va Vb t
Previous labels may not lie in interval

25. Second Problem
s
ua ub
am+1 bm+1
am+2 bm+2
an bn
Va Vb t
ua and ub : unary potentials for previous labels

26. Second Problem
s
ua ub
Pab
M Mb
am+1 ab m+1
am+2 bm+2
an bn
Va Vb t
Pab : pairwise potential for previous labels

27. Second Problem
s
ua ub
Pab
M M
am+1 ab bm+1
am+2 bm+2
an bn
Va Vb t
wab d(i-k)

28. Second Problem
s
ua ub
Pab
M M
am+1 ab bm+1
am+2 bm+2
an bn
Va Vb t
wab ( d(i-m-1) + M )

29. Second Problem
s
ua ub
Pab
M M
am+1 ab bm+1
am+2 bm+2
an bn
Va Vb t
Pab

30. Graph Construction
Find st-MINCUT.
Retain old labeling
if energy increases.
am+1 bm+1
am+2 bm+2
an bn
Va Vb t
ITERATE

31. Move Making
LP Bounds In General?
Kumar and Torr, NIPS 08 Kumar and Koller, UAI 09
Type of Problem Bound
Potts 2
Truncated Linear 2 + √2
Truncated Quadratic O(√M)
General Metric O(log |L|)

32. Outline
• Revisiting the LP relaxation
• Rounding Schemes and Move Making
• Beyond the LP relaxation

33. LP over a Frustrated Cycle
0 1 0 0 1 0 0 1 0
l1
0 0 0 0 0 0
l0
0 1 0 0 1 0 0 1 0
Va Vb Vb Vc Vc Va
Optimal labeling has energy = 1
One takes label l0, two take label l1
One takes label l1, two take label l0

34. LP optimal solution
0.5 0 0.5 0.5 0 0.5 0.5 0 0.5
l1
0.5 0.5 0.5 0.5 0.5 0.5
l0
0.5 0 0.5 0.5 0 0.5 0.5 0 0.5
Va Vb Vb Vc Vc Va
Optimal fractional labeling has energy = 0
Need tighter relaxations

35. Cycle Inequalities
Vb
Va Vc
At least two variables take same label

36. Cycle Inequalities
Vb
Va Vc
Va and Vc take label 0, yac;00 = 1

37. Cycle Inequalities
Vb
Va Vc
Or Va and Vc take label 1, yac;11 = 1

38. Cycle Inequalities
Vb
Va Vc
∑ yab;00 + yab;11 ≥ 1

39. LP optimal solution
0.5 0 0.5 0.5 0 0.5 0.5 0 0.5
l1
0.5 0.5 0.5 0.5 0.5 0.5
l0
0.5 0 0.5 0.5 0 0.5 0.5 0 0.5
Va Vb Vb Vc Vc Va
Does not satisfy cycle inequality

40. Cycle Inequalities
Generalizes to cycles of arbitrary length
Barahona and Mahjoub, 1986
Generalizes to arbitrary label sets
Chopra and Rao, 1991
Sontag and Jaakkola, 2007
Modifies the primal
But weren’t we solving the dual?

41. Modifying the Dual
Do operations on trees and cycles
Which algorithm? Which cycles?
Kumar and Torr, 2008
TRW-S All cycles of length 3 and 4
Komodakis and Paragios, 2008
Dual Decomposition All frustrated cycles
Sontag et al, 2008
MPLP Iteratively add cycles
Maximum increase in the dual

42. Questions?