Variational algorithms for marginal MAP
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1 Variational algorithms for marginal MAP Alexander Ihler UC Irvine CIOG Workshop November 2011
2 Variational algorithms for marginal MAP Alexander Ihler UC Irvine CIOG Workshop November 2011 Work with Qiang Liu
3 Graphical models where (partition function) A C A C A C B Bayesian Network B Factor Graph B Markov Random Field
4 Types of queries Maximum a posterior (MAP) query wcsps, minimum energy configurations 4
5 Types of queries Maximum a posterior (MAP) query wcsps, minimum energy configurations Summation queries Partition function, #CSP 5
6 Types of queries Maximum a posterior (MAP) query wcsps, minimum energy configurations Summation queries Partition function, #CSP Mixed queries Combine more than one elimination operator Marginal-MAP max (B) sum (A) and many others where 6
7 Three type of queries Max-Inference Sum-Inference Harder Mixed-Inference NP-hard: exponentially many terms We will focus on approximation algorithms 7
8 Variable Elimination (exact sum) 8
9 Variable Elimination (exact sum) 9
10 Variable Elimination (exact sum) 10
11 Variable Elimination (exact sum) Cost: exponential in the tree-width 11
12 Variable Elimination (exact sum / max) Interpretation as message-passing on trees Algorithm similar for max (dynamic programming)
13 Variable Elimination (exact mixed) Interpretation as message-passing on trees Algorithm similar for max (dynamic programming) Mixed-inference is harder! Elimination orders are restricted: max sum Example from D. Koller and N. Friedman (2009) 13
14 Variable Elimination (exact max /mixed) Interpretation as message-passing on trees Algorithm similar for max (dynamic programming) Mixed-inference is harder! Elimination orders are restricted: max sum Example from D. Koller and N. Friedman (2009) 14
15 Variational approaches Replace elimination with optimization over distributions (maximum: q = 1(x*) ) : set of joint distributions over x Equivalently n terms of, the marginal polytope 15
16 Variational approaches Replace elimination with optimization over distributions (maximum: q = 1(x*) ) : set of joint distributions over x Equivalently n terms of, the marginal polytope (maximum: q = p) 16
17 Variational approaches Replace elimination with optimization over distributions (maximum: q = 1(x*) ) : set of joint distributions over x Equivalently n terms of, the marginal polytope (maximum: q = p) Proof: (Kullback Leibler divergence)
18 Variational approximations Replace q 2 P and H(q) with simpler approximations 18
19 Variational approximations Replace q 2 P and H(q) with simpler approximations Algorithms & their properties: Method distributions entropy value Max: Sum: Linear programming Mean field Belief propagation Tree-reweighted n/a exact 19
20 Variational methods for marginal MAP max (B) sum (A) max part sum part Apply the same approach to each part: 20
21 Variational methods for marginal MAP dual where (Truncate the entropies of the max nodes) General framework for approximate algorithms Truncated Bethe approximation Truncated TRW approximation Truncated mean field approximation H(x) = H(x B ) + H(x A x B ) 21
22 Truncated free energy approximations dual where (Truncate the entropies of the max nodes) Truncated Bethe approximation Truncated tree-reweighted approximation 22
23 A-B trees In sum or max-inference, trees are tractable subproblems In mixed inference, they may not be A-B trees Extend the notion to mixed inference Graph structure that remains a tree during elimination Type 1 Example from D. Koller and N. Friedman (2009) Type 2 23
24 Designing message passing algorithms Can write as generic weighted objective Derive messages (minor generalization of TRW) Take limit as some weights! 0 (minor generalization of Weiss et al. 2007) max (B) sum (A) Can opt to take limit directly on message update equations 24
25 Mixed product message passing A! A [ B B! B B! A Sum- product Max- product Match max and sum max (B) sum (A) 25
26 Mixed product message passing Satisfies a reparameterization property, where i 2 A i, j 2 B! B i 2 B, j 2 A Sum- product Max- product Match max and sum Can use this to show local optimality properties similar to max-product 26
27 Variational methods for marginal MAP dual where (Truncate the entropies of the max nodes) Double-loop algorithms (CCCP & similar): Example: Truncated Bethe approximation Solve summation problem: 27
28 Variational methods for marginal MAP dual where (Truncate the entropies of the max nodes) Double-loop algorithms (CCCP & similar): Example: Truncated Bethe approximation Solve summation problem: Remove excess entropy: 28
29 Variational methods for marginal MAP dual where (Truncate the entropies of the max nodes) Double-loop algorithms (CCCP & similar): Example: Truncated Bethe approximation Solve summation problem: Remove excess entropy: Iterate: (a bit like annealing makes the function sharper )
30 Experiments: trees
31 Experiments: cycles Attractive Mixed
32 Conclusions Consider mixed inference tasks (marginal MAP) Derive a variational framework Develop analogues of Bethe, TRW, etc. Approximations and bounds Develop algorithms Message passing & double-loop methods Directions Extend to more general mixed problems Algorithmic improvements 32
33 Conclusions Thanks! Consider mixed inference tasks (marginal MAP) Derive a variational framework Develop analogues of Bethe, TRW, etc. Approximations and bounds Develop algorithms Message passing & double-loop methods Directions Extend to more general mixed problems Algorithmic improvements 33
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