|Stochastic Systems Group|
Multi-Scale Lagrangian Relaxation for Estimation in Discrete and Gaussian Graphical Models
Jason K. Johnson
We develop a general framework for MAP estimation in discrete and Gaussian graphical models using Lagrangian relaxation techniques. The basic idea is to reformulate an intractable estimation problem as one defined on a more tractable graph, but subject to additional complicating constraints. Relaxing these constraints gives a tractable dual problem, one defined by a collection of tractable graphs (e.g., small subgraphs or thin graphs such as trees). The popular tree-reweighted max-product (TRMP) method may be seen as an instance of this general approach. We develop a distributed optimization procedure for minimizing the dual function arising in these approaches. For a certain class of Gaussian models, where the objective decomposes as a sum of convex objectives on small subsets of variables, we show that the optimal MAP estimate is recovered and we obtain an upper-bound on the variance of each variable. In discrete models, the optimal estimate is recovered only if the method leads to a consistent estimate, one which also satisfies the relaxed constraints. Otherwise, there is a "duality gap", and we consider methods to augment the formulation so as to reduce and possibly eliminate the duality gap.
In addition, we propose a new class of multi-scale relaxations that introduce auxiliary variables and associated constraints. The benefits of the multi-scale approach include accelerated convergence of our iterative method and reduction of the duality gap in discrete problems.
Joint work with Dmitry Malioutov and Alan Willsky.
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