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 BP 1 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? 1 [Wainwright et.al.’05, Kolmogorov ’04]

3. Goal Efficient computation of the MAP solution (x MAP ), using inference on simpler subgraphs OR Return an approximation (x *,gap) s.t. Score(x MAP ) – Score(x * ) < gap

4. 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 ++= GG T1T1 = ++ T2T2 T3T3 Score(x) Score 1 (x)Score 2 (x)Score 3 (x) =++ =>

5. Rank aggregation (contd.) [Step 2] Perform top-k MAP estimation on each constituent and compute upper bound (ub) S1S1 : SiSi : SLSL : x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 x7x7 x 20 x4x4 x1x1 x9x9 x 11 x8x8 x2x2 x 15 x2x2 x8x8 x 22 x3x3 x5x5 x4x4 x6x6 ub = score 1 (x 8 ) + score i (x 2 ) + score L (x 4 ) ties

6. Rank-Merge [Step 3] Merge the ranked lists using the aggregate function Score(x) =  i Score i (x) x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 x7x7 x 20 x4x4 x1x1 x9x9 x 11 x8x8 x2x2 x 15 x2x2 x8x8 x 22 x3x3 x5x5 x4x4 x6x6 x2x2 x1x1 x8x8 x * (MAP estimate) Computed directly from the model

7. Rank-Merge (contd.) If Score(x*) ≥ ub, then x MAP = x*  From the property of RankMerge algorithm If Score(x*) < ub, then Score(x MAP ) < 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 ub i incrementally.

8. Comparison with Tree Re-weighted BP TRW-BP Generates bounds only from K=1 Outputs x MAP 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 x MAP does not have to be in ALL the lists No agreement criteria  Comparison with ub sufficient  x MAP need not be the best in any list.

9. Reparameterization No guarantee that x MAP 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. =  i  T i =  i  ’ T i reparam GG ’G’G s.t. Score  G (x) ´ Score  ’ G (x)

10. Final Algorithm Construct potentials for simpler constituents Get Top-k MAP estimates for each constituent Rank Aggregate the sorted lists Reparameterize graph potentials found (X*, 0) (X*, gap) bored neither next iteration

11. 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 SuccessesFailures

12. 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.

13. Experiments : Tree Selection  Sensitivity to selection  Behavior also depends on reparameterization! Grid(M) FailuresClique(M) Failures Grid(M) Successes

14. Experiments : Gap Evolution Gap converges for RankAgg in both scenarios Cliques: Erratic gaps shown by TRW-S  Bounds are monotonic but MAP-estimates are not!! GridsCliques

15. 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