Variational Algorithms for Marginal MAP Qiang Liu Alexander Ihler

Department of Computer Science, University of California, Irvine

AbstractMarginal MAP tasks seek an optimal configuration of the marginal distribution over a subset of variables. Marginal MAP can be computationally much harder than more common inference tasks.

We show• a general variational framework for marginal MAP problems• analogues to Bethe, tree-reweighted, & mean-field approximations• novel upper bounds via the tree-reweighted free energy• “mixed” message passing and CCCP-based solvers• conditions for global or local optimality of the solutions• close connections to EM and variational EM approaches

Variational Form

Graphical ModelsGraphical models:• Factors & exponential family form

• Factors are associated with cliques of a graph G=(V,E)

Tasks: max (B) sum (A)

Harder

Mixed inference problems can be hard even in trees, since

• A-B trees extend notion of efficient structure to mixed inference• Ensure graph structure remains a tree during inference• Two example sub-types:

sum

max

sum

max

Example from D. Koller and N. Friedman (2009)

Mixed-Inference (marginal MAP, MAP)

Sum-Inference (partition function, probability of evidence)

Max-Inference (MAP, MPE)

Variational Algorithms

Sum- product

Max- product

Match max and sum

max (B) sum (A)A ! A [ B

B ! B

B ! A

Mixed-product message passing• start with “standard” weighted message passing• Generalize zero-temperature limit results of Weiss et al. (2007)• Apply limit directly to messages ( for Bethe, for TRW)

• Match updates interpretable as a “local” marginal MAP problem• Mixed marginals satisfy a reparameterization property• Fixed points are locally optimal (similar to max-product results)• Convergence can be an issue

Double-loop algorithms• Decompose H into two parts H=H+ - H- & iteratively linearize H-

• CCCP algorithm: take H+, H- to be convex• Can also take H+ to be the Bethe approximation (non-convex)• Iteratively solve sum-product and apply truncation correction

“Type 1” “Type 2”

Connections to EM• Restrict to the mean-field like product subspace

• Coordinate-wise updates = in the primal:

• Reformulate inference as a distributional optimization problem

• Define and

Sum-Inference

Max-Inference

Mixed-Inference

Sum-inference: Mixed-inference:

(with equality when q=p) (with equality when q = p(A|B) 1(B=B*) or similar)

This work

Variational ApproximationsBethe approximation (exact on A-B tree)• “Truncated” free energy

Tree-reweighted approximation (convex comb. of A-B trees)• Dual in terms of edge appearances

ExperimentsChain graphs• GA is a tree• TRW1: type-1 only• TRW2: ½ type-1, ½ type 2• Bethe: most accurate• EM: stuck quickly (2-3 iter.)

Grid graphs• Attractive or mixed potentials • GA has cycles• Similar trends

Attractive Mixed

% correct solutions Energy relative error