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. 1

This paper introduces dual decomposition as a framework for deriving inference algorithms for NLP problems. The approach relies on standard dynamic-programming algorithms as oracle solvers for sub-problems, together with a simple method for forcing agreement between the different oracles. The approach provably solves a linear programming (LP) relaxation of the global inference problem. It leads to algorithms that are simple, in that they use existing decoding algorithms; efficient, in that they avoid exact algorithms for the full model; and often exact, in that empirically they often recover the correct solution in spite of using an LP relaxation. We give experimental results on two problems: 1) the combination of two lexicalized parsing models; and 2) the combination of a lexicalized parsing model and a trigram part-of-speech tagger. 1

A number of linear programming relaxations have been proposed for finding most likely settings of the variables (MAP) in large probabilistic models. The relaxations are often succinctly expressed in the dual and reduce to different types of reparameterizations of the original model. The dual objectives are typically solved by performing local block coordinate descent steps. In this work, we show how to perform block coordinate descent on spanning trees of the graphical model. We also show how all of the earlier dual algorithms are related to each other, giving transformations from one type of reparameterization to another while maintaining monotonicity relative to a common objective function. Finally, we quantify when the MAP solution can and cannot be decoded directly from the dual LP relaxation. 1

The problem of computing a maximum a posteriori (MAP) configuration is a central computational challenge associated with Markov random fields. A line of work has focused on “tree-based ” linear programming (LP) relaxations for the MAP problem. This paper develops a family of super-linearly convergent algorithms for solving these LPs, based on proximal minimization schemes using Bregman divergences. As with standard messagepassing on graphs, the algorithms are distributed and exploit the underlying graphical structure, and so scale well to large problems. Our algorithms have a double-loop character, with the outer loop corresponding to the proximal sequence, and an inner loop of cyclic Bregman divergences used to compute each proximal update. Different choices of the Bregman divergence lead to conceptually related but distinct LP-solving algorithms. We establish convergence guarantees for our algorithms, and illustrate their performance via some simulations. We also develop two classes of graph-structured rounding schemes, randomized and deterministic, for obtaining integral configurations from the LP solutions. Our deterministic rounding schemes use a “re-parameterization ” property of our algorithms so that when the LP solution is integral, the MAP solution can be obtained even before the LP-solver converges to the optimum. We also propose a graph-structured randomized rounding scheme that applies to iterative LP solving algorithms in general. We analyze the performance of our rounding schemes, giving bounds on the number of iterations required, when the LP is integral, for the rounding schemes to obtain the MAP solution. These bounds are expressed in terms of the strength of the potential functions, and the energy gap, which measures how well the integral MAP solution is separated from other integral configurations. We also report simulations comparing these rounding schemes. 1

Markov random field (MRF, CRF) models are popular in computer vision. However, in order to be computationally tractable they are limited to incorporate only local interactions and cannot model global properties, such as connectedness, which is a potentially useful high-level prior for object segmentation. In this work, we overcome this limitation by deriving a potential function that enforces the output labeling to be connected and that can naturally be used in the framework of recent MAP-MRF LP relaxations. Using techniques from polyhedral combinatorics, we show that a provably tight approximation to the MAP solution of the resulting MRF can still be found efficiently by solving a sequence of max-flow problems. The efficiency of the inference procedure also allows us to learn the parameters of a MRF with global connectivity potentials by means of a cutting plane algorithm. We experimentally evaluate our algorithm on both synthetic data and on the challenging segmentation task of the PASCAL VOC 2008 data set. We show that in both cases the addition of a connectedness prior significantly reduces the segmentation error. 1.

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Message-passing algorithms have emerged as powerful techniques for approximate inference in graphical models. When these algorithms converge, they can be shown to find local (or sometimes even global) optima of variational formulations to the inference problem. But many of the most popular algorithms are not guaranteed to converge. This has lead to recent interest in convergent message-passing algorithms. In this paper, we present a unified view of convergent message-passing algorithms. We algorithm, tree-consistency bound optimization (TCBO) that is provably convergent in both its sum and max product forms. We then show that many of the existing convergent algorithms are instances of our TCBO algorithm, and obtain novel convergent algorithms “for free ” by exchanging maximizations and summations in existing algorithms. In particular, we show that Wainwright’s non-convergent sum-product algorithm for tree based variational bounds, is actually convergent with the right update order for the case where trees are monotonic chains. 1

We propose a new class of consistency constraints for Linear Programming (LP) relaxations for finding the most probable (MAP) configuration in graphical models. Usual cluster-based LP relaxations enforce joint consistency of the beliefs of a cluster of variables, with computational cost increasing exponentially with the size of the clusters. By partitioning the state space of a cluster and enforcing consistency only across partitions, we obtain a class of constraints which, although less tight, are computationally feasible for large clusters. We show how to solve the cluster selection and partitioning problem monotonically in the dual LP, using the current beliefs to guide these choices. We obtain a dual message-passing algorithm and apply it to protein design problems where the variables have large state spaces and the usual cluster-based relaxations are very costly. 1

Abstract—Inference problems in graphical models can be represented as a constrained optimization of a free energy function. In this paper we treat both forms of probabilistic inference, estimating marginal probabilities of the joint distribution and finding the most probable assignment, through a unified messagepassing algorithm architecture. In particular we generalize the Belief Propagation (BP) algorithms of sum-product and maxproduct and tree-rewaighted (TRW) sum and max product algorithms (TRBP) and introduce a new set of convergent algorithms based on ”convex-free-energy ” and Linear-Programming (LP) relaxation as a zero-temprature of a convex-free-energy. The main idea of this work arises from taking a general perspective on the existing BP and TRBP algorithms while observing that they all are reductions from the basic optimization formula of f + ∑ i hi

Abstract—Interior-point methods are state-of-the-art algorithms for solving linear programming (LP) problems with polynomial complexity. Specifically, the Karmarkar algorithm typically solves LP problems in time O(n 3.5), where n is the number of unknown variables. Karmarkar’s celebrated algorithm is known to be an instance of the log-barrier method using the Newton iteration. The main computational overhead of this method is in inverting the Hessian matrix of the Newton iteration. In this contribution, we propose the application of the Gaussian belief propagation (GaBP) algorithm as part of an efficient and distributed LP solver that exploits the sparse and symmetric structure of the Hessian matrix and avoids the need for direct matrix inversion. This approach shifts the computation from realm of linear algebra to that of probabilistic inference on graphical models, thus applying GaBP as an efficient inference engine. Our construction is general and can be used for any interior-point algorithm which uses the Newton method, including non-linear program solvers. I.