MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies (1206.5286v1)

Abstract: Finding the most probable assignment (MAP) in a general graphical model is known to be NP hard but good approximations have been attained with max-product belief propagation (BP) and its variants. In particular, it is known that using BP on a single-cycle graph or tree reweighted BP on an arbitrary graph will give the MAP solution if the beliefs have no ties. In this paper we extend the setting under which BP can be used to provably extract the MAP. We define Convex BP as BP algorithms based on a convex free energy approximation and show that this class includes ordinary BP with single-cycle, tree reweighted BP and many other BP variants. We show that when there are no ties, fixed-points of convex max-product BP will provably give the MAP solution. We also show that convex sum-product BP at sufficiently small temperatures can be used to solve linear programs that arise from relaxing the MAP problem. Finally, we derive a novel condition that allows us to derive the MAP solution even if some of the convex BP beliefs have ties. In experiments, we show that our theorems allow us to find the MAP in many real-world instances of graphical models where exact inference using junction-tree is impossible.

• The paper introduces convex belief propagation (BP), a novel framework linking various BP variants and linear programming relaxations to reliably solve maximum a posteriori (MAP) estimation in graphical models, even for NP-hard cases.
• This approach provides theoretical guarantees for obtaining the MAP estimate under specific conditions, including methods to handle situations with tied beliefs.
• Practical experiments demonstrate the method's robustness and ability to find global optima in over 90% of instances for challenging large-scale problems like protein side-chain prediction.

MAP Estimation, Linear Programming, and Belief Propagation with Convex Free Energies

The paper by Weiss et al. explores the use of belief propagation (BP), specifically convex BP, to solve the maximum a posteriori (MAP) estimation in graphical models. It addresses the limitations encountered in traditional BP approaches and introduces methods for utilizing convex free energies to reliably extract the MAP assignment. This work extends the applicability of belief propagation to solve NP-hard problems found in various fields such as image understanding, error-correcting codes, and protein folding by leveraging linear programming (LP) relaxations.

Weiss et al. propose a novel framework, referred to as convex BP, that encompasses ordinary BP on single-cycle graphs, tree reweighted BP, and several other variants. These algorithms are based on convex approximations of free energy and provide theoretical guarantees for obtaining the MAP estimate under specific conditions. Importantly, when fixed points of convex max-product BP are free of ties, the MAP solution can be assured. Furthermore, this paper establishes conditions under which even tied beliefs can be used to extract the MAP, adding robustness to the method.

The implications of the research are significant. Theorems derived in this paper enable the identification of MAP solutions in many real-world graphical models where traditional algorithms such as junction-tree inference become infeasible due to the computational complexity involved. The practical benefits extend to large-scale models, as demonstrated in the experiments conducted with real-world datasets from computational biology and error-correcting codes. This work provides strategies for resolving complex inference tasks by blending principles of belief propagation with linear optimization techniques.

Key numerical results from the experiments indicate that convex BP formulas can reliably converge even in challenging scenarios where the LP relaxation is partially fractional. Notably, in tasks like protein side-chain prediction, the proposed methods allow the extraction of the global optimum in over 90% of instances, highlighting the robustness and practical utility of the approach. Other techniques, such as ordinary BP, while capable of achieving correct results, lack the theoretical assurances provided by convex BP.

This paper opens avenues for further exploration in the field of inference algorithms. The strong link established between LP relaxations and a broader class of BP variants, including ordinary BP, suggests potential advancements in deriving correctness proofs for BP variants across more complex problems. Additionally, the practice of finding MAP beyond LP relaxation may lead to similar results in diverse applications, pushing the boundaries on what can be achieved with MAP estimation in graphical models.

In conclusion, the work of Weiss et al. underscores the power and flexibility of belief propagation algorithms when equipped with convex free energy approximations, offering promising solutions to computationally challenging MAP estimation tasks. Future research may further leverage these results to tackle even larger and more complex graphical models, providing practical benefits across multiple domains in artificial intelligence.