[Paper Review] Tightening LP Relaxations for MAP using Message Passing

David Sontag, Talya Meltzer+3 more|arXiv (Cornell University)|Jun 13, 2012

Gene Regulatory Network AnalysisBiochemistry, Genetics and Molecular Biology19 references251 citations

TL;DR

This paper proposes a dual decomposition-based message-passing algorithm that iteratively strengthens linear programming (LP) relaxations for MAP inference in graphical models by selecting high-impact clusters. By reusing previous solutions and ensuring monotonic dual improvement, the method efficiently tightens relaxations beyond standard LP, successfully solving challenging problems like protein sidechain placement and stereo matching where standard LP fails.

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.

Motivation & Objective

• To address the limitation of standard LP relaxations in graphical models, which often fail to be tight for real-world problems.
• To overcome the exponential computational cost of higher-order cluster-based LP relaxations by enabling efficient, incremental cluster selection.
• To develop a method that monotonically improves the dual objective while reusing prior solutions to maintain efficiency.
• To achieve provable convergence to the MAP configuration in cases where standard LP relaxations fail.
• To enable practical application of tighter LP relaxations in complex inference tasks such as protein design and stereo matching.

Proposed method

• The method uses dual decomposition to iteratively identify and add clusters that most improve the dual objective function.
• It formulates the cluster selection problem as a subproblem in the dual LP, ensuring monotonic improvement with each addition.
• The algorithm reuses the previous solution to the LP relaxation, enabling fast re-optimization after each cluster addition.
• Message-passing techniques are applied to the dual problem to efficiently compute the necessary updates and identify promising clusters.
• The approach combines cluster-based tightening with dual ascent, maintaining convergence guarantees while scaling to larger problems.
• The method is implemented as a dual message-passing algorithm that dynamically selects clusters based on dual variable updates.

Experimental results

Research questions

• RQ1 Can we systematically strengthen LP relaxations for MAP inference in graphical models beyond the standard formulation?
• RQ2 How can we efficiently select high-impact clusters without incurring exponential computational costs?
• RQ3 Can we maintain dual monotonicity and reuse previous solutions to accelerate convergence in iterative tightening?
• RQ4 Does the proposed method achieve provable MAP inference in problems where standard LP relaxations fail?
• RQ5 Can this approach be practically applied to real-world problems such as protein sidechain placement and stereo matching?

Key findings

• The proposed method successfully finds the MAP configuration in protein sidechain placement problems where standard LP relaxations fail.
• It achieves convergence to the optimal solution in stereo matching tasks, even when the standard LP relaxation is not tight.
• The algorithm demonstrates significant computational efficiency by reusing prior solutions, reducing re-optimization costs across iterations.
• Iterative cluster selection leads to monotonic dual improvement, ensuring convergence to a tighter relaxation.
• The method outperforms standard LP relaxation in terms of solution quality on benchmark problems from computational biology and computer vision.