Partial Optimality of Dual Decomposition for MAP Inference in Pairwise MRFs

TL;DR

This paper investigates the relationship between dual decomposition and LP relaxation in MAP inference for binary pairwise MRFs, revealing that solutions from both methods share partial optimality even when LP relaxation is not tight.

Contribution

It demonstrates that dual decomposition solutions partially agree with LP relaxation solutions in non-tight cases, extending understanding of their relationship in MAP inference.

Findings

Dual decomposition solutions partially match LP relaxation solutions.

Both methods share partial optimality in binary pairwise MRFs.

Results apply even when LP relaxation is not tight.

Abstract

Markov random fields (MRFs) are a powerful tool for modelling statistical dependencies for a set of random variables using a graphical representation. An important computational problem related to MRFs, called maximum a posteriori (MAP) inference, is finding a joint variable assignment with the maximal probability. It is well known that the two popular optimisation techniques for this task, linear programming (LP) relaxation and dual decomposition (DD), have a strong connection both providing an optimal solution to the MAP problem when a corresponding LP relaxation is tight. However, less is known about their relationship in the opposite and more realistic case. In this paper, we explain how the fully integral assignments obtained via DD partially agree with the optimal fractional assignments via LP relaxation when the latter is not tight. In particular, for binary pairwise MRFs the corresponding result suggests that both methods share the partial optimality property of their solutions.