MAP estimation in MRFs via rank aggregation

1
MAP estimation in MRFs via rank aggregation

• Rahul Gupta Sunita Sarawagi
• (IBM India Research Lab) (IIT Bombay)

2
Background

• Approximate MAP estimation a must for complex
  models used in collective inference tasks
• Min-cuts, belief propagation, mean field,
  sampling
• Family of Tree Re-weighted BP1 algorithms
• Decompose graph into trees
• Perform inference on trees and combine results
• Can we generalize and do better?
• Can we provide better upper bounds on the MAP
  score?

• 1Wainwright et.al.05, Kolmogorov 04

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Goal

• Efficient computation of the MAP solution (xMAP),
  using inference on simpler subgraphs
• OR
• Return an approximation (x,gap) s.t.
• Score(xMAP) Score(x) lt gap

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MAP via rank-aggregation

• Step 1 Decompose graph potentials into a convex
  combination of simpler potentials
• E.g. Set of spanning trees that cover all edges

?T1


?T2
?T3
?G

Score(x)
Score1(x)
Score2(x)
Score3(x)



gt
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Rank aggregation (contd.)

• Step 2 Perform top-k MAP estimation on each
  constituent and compute upper bound (ub)

?S1

x1
x2
x3
x4
x5
x6
x7
x8
ub score1(x8)
?Si

x7
x20
x4
x1
x9
x11
x8
x2
scorei(x2)
?SL

x15
x2
x8
x22
x3
x5
x4
x6
scoreL(x4)
ties
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Rank-Merge

• Step 3 Merge the ranked lists using the
  aggregate function
• Score(x) ?i Scorei(x)

Computed directly from the model
x1
x2
x3
x4
x5
x6
x7
x8
x7
x20
x4
x1
x9
x11
x8
x2
x2
x1
x8
x (MAP estimate)
x15
x2
x8
x22
x3
x5
x4
x6
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Rank-Merge (contd.)

• If Score(x) ub, then xMAP x
• From the property of RankMerge algorithm
• If Score(x) lt ub, then Score(xMAP) lt ub
• From convexity of max and decomposition of ?G
• Tighter bounds can be obtained by increasing k
• Can do even better
• Generate top-K and upper bounds ubi
  incrementally.

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Comparison with Tree Re-weighted BP

• TRW-BP
• Generates bounds only from K1
• Outputs xMAP only when all trees agree on a
  common MAP
• May require enumerating all MAPs if MAP is not
  unique

• Rank-Aggregation
• Tighter bounds obtainable by increasing K
• xMAP does not have to be in ALL the lists
• No agreement criteria
• Comparison with ub sufficient
• xMAP need not be the best in any list.

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Reparameterization

• No guarantee that xMAP will be in the top-K list
  of ANY tree
• Need to align tree potentials with the
  max-marginals of the graph
• Can use existing reparam algorithms TRW-T,
  TRW-E, TRW-S
• TRW-S most expensive but gives monotonically
  decreasing bounds and converges fastest.
• Rank-aggregation gives significant improvements
  with all the reparameterization algorithms.

?G
?G
reparam
s.t. Score?G(x) Score?G(x)
?i?Ti
?i?Ti
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Final Algorithm

• Construct potentials for simpler constituents
• Get Top-k MAP estimates for each constituent
• Rank Aggregate the sorted lists
• Reparameterize graph potentials

next iteration
found
(X, 0)
bored
neither
(X, gap)
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Experiments (synthetic data)

• Improves even upon TRW-S
• Success in fewer iterations
• Smaller gap values on failure
• Effect much more significant for TRW-Tree and
  TRW-Edge

Successes
Failures
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Experiments (real data)

• Bibliographic data
• 24 labels (AUTHOR, TITLE, .), avg. record size
  11.
• Uniqueness constraints for some labels ? Clique
  models
• Substantially better MAP estimates and Gap
  values, less number of iterations.

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Experiments Tree Selection

• Sensitivity to selection
• Behavior also depends on reparameterization!

Grid(M) Failures
Clique(M) Failures
Grid(M) Successes
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Experiments Gap Evolution
Grids
Cliques

• Gap converges for RankAgg in both scenarios
• Cliques Erratic gaps shown by TRW-S
• Bounds are monotonic but MAP-estimates are not!!

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Summary and future work

• Rank aggregation significantly better in
  MAP-estimation
• Fewer iterations, Much tighter bounds, Low k.
• No dependence on the tree-agreement criteria
• Can generalize to non-tree constituents as long
  as top-k is supported.
• Future work
• Collective inference on constrained models
• Intelligent constraint generation
• Decrease sensitivity to tree selection