[Paper Review] Lagrangian Relaxation for MAP Estimation in Graphical Models

Jason K. Johnson, Dmitry Malioutov+1 more|ArXiv.org|Sep 28, 2007

Bayesian Modeling and Causal Inference15 references73 citations

TL;DR

This paper introduces a general Lagrangian relaxation framework for MAP estimation in discrete and Gaussian graphical models by reformulating intractable problems on a more tractable augmented graph with constraints, then relaxing these via Lagrange multipliers to yield a convex dual problem. The method achieves optimal MAP estimates when the duality gap vanishes, and introduces multiscale relaxations with summary variables that accelerate convergence and reduce duality gaps.

ABSTRACT

We develop a general framework for MAP estimation in discrete and Gaussian graphical models using Lagrangian relaxation techniques. The key idea is to reformulate an intractable estimation problem as one defined on a more tractable graph, but subject to additional constraints. Relaxing these constraints gives a tractable dual problem, one defined by a thin graph, which is then optimized by an iterative procedure. When this iterative optimization leads to a consistent estimate, one which also satisfies the constraints, then it corresponds to an optimal MAP estimate of the original model. Otherwise there is a ``duality gap'', and we obtain a bound on the optimal solution. Thus, our approach combines convex optimization with dynamic programming techniques applicable for thin graphs. The popular tree-reweighted max-product (TRMP) method may be seen as solving a particular class of such relaxations, where the intractable graph is relaxed to a set of spanning trees. We also consider relaxations to a set of small induced subgraphs, thin subgraphs (e.g. loops), and a connected tree obtained by ``unwinding'' cycles. In addition, we propose a new class of multiscale relaxations that introduce ``summary'' variables. The potential benefits of such generalizations include: reducing or eliminating the ``duality gap'' in hard problems, reducing the number or Lagrange multipliers in the dual problem, and accelerating convergence of the iterative optimization procedure.

Motivation & Objective

• Address the intractability of exact MAP estimation in large-scale graphical models with complex dependencies.
• Overcome the exponential complexity of junction tree methods by introducing a convex relaxation framework.
• Develop a unified approach that generalizes existing methods like tree-reweighted max-product (TRMP) through Lagrangian relaxation.
• Reduce or eliminate the duality gap in hard inference problems via structured augmentations, including multiscale relaxations with summary variables.
• Enable efficient optimization via iterative dual ascent with marginal and max-marginal matching, applicable to both discrete and Gaussian models.

Proposed method

• Reformulate the original intractable graphical model as a constrained optimization problem on an augmented graph with replicated or structured subgraphs.
• Apply Lagrangian relaxation to the constraints, transforming the problem into a convex dual optimization problem over Lagrange multipliers.
• Use block coordinate descent to iteratively minimize the dual function by updating multipliers for replica and cross-scale constraints.
• Enforce consistency via marginal and max-marginal matching in discrete models, and moment-matching in Gaussian models using precision matrices and mean parameters.
• Introduce multiscale relaxations by adding summary variables that coarsen the model, with constraints linking fine-scale and coarse-scale variables.
• Derive closed-form update rules for Lagrange multipliers in Gaussian models using inverse covariance and mean parameters, ensuring moment-matching across scales.

Experimental results

Research questions

• RQ1 Can Lagrangian relaxation be systematically applied to MAP estimation in both discrete and Gaussian graphical models to achieve tractable inference?
• RQ2 How can the duality gap in MAP estimation be reduced or eliminated through strategic model augmentation?
• RQ3 What is the role of multiscale relaxations with summary variables in improving convergence speed and reducing duality gaps?
• RQ4 In what cases does the relaxed dual solution yield the exact MAP estimate, and how can this be detected?
• RQ5 How do different graph structures (e.g., spanning trees, loops, unwound cycles) compare in terms of duality gap and convergence behavior?

Key findings

• The proposed framework generalizes the tree-reweighted max-product (TRMP) method as a special case of Lagrangian relaxation over spanning trees.
• When the duality gap is zero, the iterative dual optimization yields an exact MAP estimate that satisfies all original constraints.
• The occurrence of ties in max-marginals signals a non-zero duality gap and indicates the need for further model augmentation.
• In Gaussian models, the method yields a tight bound and exact MAP estimate whenever the relaxation is well-posed, with valid upper bounds on variances.
• Multiscale relaxations significantly accelerate convergence—demonstrated in a 1024-node 1D membrane model where multiscale LR outperformed both block Gauss-Seidel and single-scale LR.
• Closed-form updates for Lagrange multipliers in Gaussian models ensure moment-matching between fine-scale and coarse-scale variables, maintaining model consistency.