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18
Fixing MaxProduct: Convergent Message Passing Algorithms for MAP LPRelaxations
"... We present a novel message passing algorithm for approximating the MAP problem in graphical models. The algorithm is similar in structure to maxproduct but unlike maxproduct it always converges, and can be proven to find the exact MAP solution in various settings. The algorithm is derived via bloc ..."
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Cited by 162 (16 self)
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We present a novel message passing algorithm for approximating the MAP problem in graphical models. The algorithm is similar in structure to maxproduct but unlike maxproduct it always converges, and can be proven to find the exact MAP solution in various settings. The algorithm is derived via block coordinate descent in a dual of the LP relaxation of MAP, but does not require any tunable parameters such as step size or tree weights. We also describe a generalization of the method to cluster based potentials. The new method is tested on synthetic and realworld problems, and compares favorably with previous approaches. Graphical models are an effective approach for modeling complex objects via local interactions. In such models, a distribution over a set of variables is assumed to factor according to cliques of a graph with potentials assigned to each clique. Finding the assignment with highest probability in these models is key to using them in practice, and is often referred to as the MAP (maximum aposteriori) assignment problem. In the general case the problem is NP hard, with complexity exponential in the treewidth of the underlying graph.
Messagepassing for graphstructured linear programs: Proximal methods and rounding schemes
, 2008
"... 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 “treebased ” linear programming (LP) relaxations for the MAP problem. This paper develops a family of superlinearly convergen ..."
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Cited by 62 (0 self)
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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 “treebased ” linear programming (LP) relaxations for the MAP problem. This paper develops a family of superlinearly 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 doubleloop 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 LPsolving algorithms. We establish convergence guarantees for our algorithms, and illustrate their performance via some simulations. We also develop two classes of graphstructured rounding schemes, randomized and deterministic, for obtaining integral configurations from the LP solutions. Our deterministic rounding schemes use a “reparameterization ” property of our algorithms so that when the LP solution is integral, the MAP solution can be obtained even before the LPsolver converges to the optimum. We also propose a graphstructured 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
NormProduct Belief Propagation: PrimalDual MessagePassing for Approximate Inference
, 2008
"... 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 me ..."
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Cited by 53 (10 self)
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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 sumproduct and maxproduct and treerewaighted (TRW) sum and max product algorithms (TRBP) and introduce a new set of convergent algorithms based on ”convexfreeenergy” and LinearProgramming (LP) relaxation as a zerotemprature of a convexfreeenergy. 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
Convergent message passing algorithms  a unifying view
 In Proc. Twentyeighth Conference on Uncertainty in Artificial Intelligence (UAI ’09
, 2009
"... Messagepassing 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 algorithm ..."
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Cited by 42 (0 self)
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Messagepassing 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 messagepassing algorithms. In this paper, we present a unified view of convergent messagepassing algorithms. We algorithm, treeconsistency 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 nonconvergent sumproduct algorithm for tree based variational bounds, is actually convergent with the right update order for the case where trees are monotonic chains. 1
Convergent messagepassing algorithms for inference over general graphs with convex free energy
 In The 24th Conference on Uncertainty in Artificial Intelligence (UAI
, 2008
"... Inference problems in graphical models can be represented as a constrained optimization of a free energy function. It is known that when the Bethe free energy is used, the fixedpoints of the belief propagation (BP) algorithm correspond to the local minima of the free energy. However BP fails to conv ..."
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Cited by 24 (7 self)
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Inference problems in graphical models can be represented as a constrained optimization of a free energy function. It is known that when the Bethe free energy is used, the fixedpoints of the belief propagation (BP) algorithm correspond to the local minima of the free energy. However BP fails to converge in many cases of interest. Moreover, the Bethe free energy is nonconvex for graphical models with cycles thus introducing great difficulty in deriving efficient algorithms for finding local minima of the free energy for general graphs. In this paper we introduce two efficient BPlike algorithms, one sequential and the other parallel, that are guaranteed to converge to the global minimum, for any graph, over the class of energies known as ”convex free energies”. In addition, we propose an efficient heuristic for setting the parameters of the convex free energy based on the structure of the graph. 1
Efficient MRF energy minimization via adaptive diminishing smoothing
 In UAI
, 2012
"... We consider the linear programming relaxation of an energy minimization problem for Markov Random Fields. The dual objective of this problem can be treated as a concave and unconstrained, but nonsmooth function. The idea of smoothing the objective prior to optimization was recently proposed in a se ..."
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Cited by 21 (6 self)
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We consider the linear programming relaxation of an energy minimization problem for Markov Random Fields. The dual objective of this problem can be treated as a concave and unconstrained, but nonsmooth function. The idea of smoothing the objective prior to optimization was recently proposed in a series of papers. Some of them suggested the idea to decrease the amount of smoothing (so called temperature) while getting closer to the optimum. However, no theoretical substantiation was provided. We propose an adaptive smoothing diminishing algorithm based on the duality gap between relaxed primal and dual objectives and demonstrate the efficiency of our approach with a smoothed version of Sequential TreeReweighted Message Passing (TRWS) algorithm. The strategy is applicable to other algorithms as well, avoids adhoc tuning of the smoothing during iterations, and provably guarantees convergence to the optimum. 1
Convexifying the bethe free energy
 in Conference on Uncertainty in Artifical Intelligence (UAI
, 2009
"... The introduction of loopy belief propagation (LBP) revitalized the application of graphical models in many domains. Many recent works present improvements on the basic LBP algorithm in an attempt to overcome convergence and local optima problems. Notable among these are convexified free energy appro ..."
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Cited by 15 (2 self)
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The introduction of loopy belief propagation (LBP) revitalized the application of graphical models in many domains. Many recent works present improvements on the basic LBP algorithm in an attempt to overcome convergence and local optima problems. Notable among these are convexified free energy approximations that lead to inference procedures with provable convergence and quality properties. However, empirically LBP still outperforms most of its convex variants in a variety of settings, as we also demonstrate here. Motivated by this fact we seek convexified free energies that directly approximate the Bethe free energy. We show that the proposed approximations compare favorably with stateofthe art convex free energy approximations. 1
Convergence analysis of reweighted sumproduct algorithms
 In Int. Conf. Acoustic, Speech and Sig. Proc
, 2007
"... Abstract—Markov random fields are designed to represent structured dependencies among large collections of random variables, and are wellsuited to capture the structure of realworld signals. Many fundamental tasks in signal processing (e.g., smoothing, denoising, segmentation etc.) require efficie ..."
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Cited by 13 (3 self)
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Abstract—Markov random fields are designed to represent structured dependencies among large collections of random variables, and are wellsuited to capture the structure of realworld signals. Many fundamental tasks in signal processing (e.g., smoothing, denoising, segmentation etc.) require efficient methods for computing (approximate) marginal probabilities over subsets of nodes in the graph. The marginalization problem, though solvable in linear time for graphs without cycles, is computationally intractable for general graphs with cycles. This intractability motivates the use of approximate “messagepassing ” algorithms. This paper studies the convergence and stability properties of the family of reweighted sumproduct algorithms, a generalization of the widely used sumproduct or belief propagation algorithm, in which messages are adjusted with graphdependent weights. For pairwise Markov random fields, we derive various conditions that are sufficient to ensure convergence, and also provide bounds on the geometric convergence rates. When specialized to the ordinary sumproduct algorithm, these results provide strengthening of previous analyses. We prove that some of our conditions are necessary and sufficient for subclasses of homogeneous models, but not for general models. The experimental simulations on various classes of graphs validate our theoretical results. Index Terms—Approximate marginalization, belief propagation, convergence analysis, graphical models, Markov random fields, sumproduct algorithm. I.
Tightening fractional covering upper bounds on the partition function for highorder region graphs
 In Proceedings of the 28th conference on Uncertainty in artificial intelligence (UAI12
, 2012
"... In this paper we present a new approach for tightening upper bounds on the partition function. Our upper bounds are based on fractional covering bounds on the entropy function, and result in a concave program to compute these bounds and a convex program to tighten them. To solve these programs effec ..."
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Cited by 9 (4 self)
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In this paper we present a new approach for tightening upper bounds on the partition function. Our upper bounds are based on fractional covering bounds on the entropy function, and result in a concave program to compute these bounds and a convex program to tighten them. To solve these programs effectively for general region graphs we utilize the entropy barrier method, thus decomposing the original programs by their dual programs and solve them with dual block optimization scheme. The entropy barrier method provides an elegant framework to generalize the messagepassing scheme to highorder region graph, as well as to solve the block dual steps in closedform. This is a key for computational relevancy for large problems with thousands of regions. 1
Convergent Decomposition Solvers for Treereweighted Free Energies
"... We investigate minimization of treereweighted free energies for the purpose of obtaining approximate marginal probabilities and upper bounds on the partition function of cyclic graphical models. The solvers we present for this problem work by directly tightening treereweighted upper bounds. As a re ..."
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Cited by 6 (1 self)
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We investigate minimization of treereweighted free energies for the purpose of obtaining approximate marginal probabilities and upper bounds on the partition function of cyclic graphical models. The solvers we present for this problem work by directly tightening treereweighted upper bounds. As a result, they are particularly efficient for treereweighted energies arising from a small number of spanning trees. While this assumption may seem restrictive at first, we show how small sets of trees can be constructed in a principled manner. An appealing property of our algorithms, which results from the problem decomposition, is that they are embarrassingly parallel. In contrast to the original message passing algorithm introduced for this problem, we obtain global convergence guarantees. 1