Tightening LP Relaxations for MAP using Message Passing (1206.3288v1)

Abstract: Linear Programming (LP) relaxations have become powerful tools for finding the most probable (MAP) configuration in graphical models. These relaxations can be solved efficiently using message-passing algorithms such as belief propagation and, when the relaxation is tight, provably find the MAP configuration. The standard LP relaxation is not tight enough in many real-world problems, however, and this has lead to the use of higher order cluster-based LP relaxations. The computational cost increases exponentially with the size of the clusters and limits the number and type of clusters we can use. We propose to solve the cluster selection problem monotonically in the dual LP, iteratively selecting clusters with guaranteed improvement, and quickly re-solving with the added clusters by reusing the existing solution. Our dual message-passing algorithm finds the MAP configuration in protein sidechain placement, protein design, and stereo problems, in cases where the standard LP relaxation fails.

• The paper proposes a dual coordinate descent-based message-passing algorithm that iteratively tightens LP relaxations for accurate MAP assignments.
• The method outperforms traditional approaches by guaranteeing improvements through strategic cluster selection, validated on protein design and stereo vision tasks.
• The work bridges theory and practice in graphical model optimization, offering actionable insights for solving NP-hard, large-scale inference problems.

Tightening LP Relaxations for MAP using Message Passing

The paper "Tightening LP Relaxations for MAP using Message Passing" investigates improving LP relaxations for finding the maximum a posteriori (MAP) assignment in graphical models. Linear programming relaxations serve as a powerful methodology for approximating MAP problems, yet they frequently encounter limitations due to insufficiently tight approximations that fail to produce an integer solution, especially with standard LP relaxations. This paper proposes an approach to iteratively enhance these relaxations by adding clusters based on a dual coordinate descent framework, thereby effectively tightening the relaxation in cases where the standard approach fails to yield appropriate solutions.

The motivation for the work lies in the discrepancy between the simplicity of solving LP relaxations using message-passing algorithms and the vastly imperfect outcomes these relaxations provide by default in various real-world problems. The dual message-passing algorithm introduced is designed explicitly to address complexity arising from the marginal polytope, which is typically characterized by an exponential number of constraints. By incrementally selecting clusters that promise guaranteed improvements in the dual objective, the algorithm strategically navigates around computational inefficiencies.

The paper's contribution rests heavily upon the proposed dual coordinate descent-based mechanism for selecting cluster additions in an LP relaxation. Unlike traditional primal-based algorithms, which struggle with convergence guarantees and warm-start mechanisms post-adjustment, the dual approach here delivers a monotonically decreasing bound upon the objective and thrives on direct improvement assessments. Notably, the research reports success in the protein design problem context, highlighting cases where traditional methods such as TRBP could not find MAP solutions.

Experimentally, the approach is validated across benchmark problems including protein side-chain placement and stereo vision challenges. For protein design, among 97 models attempted, all but one were resolved to find exact MAP configurations, representing a significant advancement over existing methods. The result is noteworthy due to the intractability of achieving these solutions through conventional LP solvers attributed to complex graph structures with high treewidth demands. Such scalable resolution demonstrates the algorithm's robustness in handling large-scale, NP-hard problems.

The implications of this research are significant both practically and theoretically. Practically, it offers a potent tool for solving complex MAP inference issues in domains where problems can be modeled through graphical frameworks. Theoretically, it emphasizes the merit of dual decomposition in optimization, setting a precedent for addressing similar problems with expansive state spaces without exponential cost inflation.

Future developments might explore extending this approach to non-binary graphical models and multimodal inference problems. Additionally, integrating more advanced message-passing strategies could yield further improvements in both efficiency and accuracy in broader applications, such as in industry-scale protein design, computer vision, and beyond.

This paper, thus, makes key contributions to the field of graphical models and optimization by providing a keen insight into the utilization of dualities within linear programming frameworks to achieve better approximations of solutions where traditional means fall short. It neatly bridges the gap between theoretical potential and practical application, laying down groundwork with commendable effectiveness for real-world problem-solving using LP relaxations.