Lagrangian Relaxation for MAP Estimation in Graphical Models

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.Comment: 10 pages, presented at 45th Allerton conference on communication, control and computing, to appear in proceeding

MAP inference via Block-Coordinate Frank-Wolfe Algorithm

We present a new proximal bundle method for Maximum-A-Posteriori (MAP) inference in structured energy minimization problems. The method optimizes a Lagrangean relaxation of the original energy minimization problem using a multi plane block-coordinate Frank-Wolfe method that takes advantage of the specific structure of the Lagrangean decomposition. We show empirically that our method outperforms state-of-the-art Lagrangean decomposition based algorithms on some challenging Markov Random Field, multi-label discrete tomography and graph matching problems

Getting Feasible Variable Estimates From Infeasible Ones: MRF Local Polytope Study

This paper proposes a method for construction of approximate feasible primal solutions from dual ones for large-scale optimization problems possessing certain separability properties. Whereas infeasible primal estimates can typically be produced from (sub-)gradients of the dual function, it is often not easy to project them to the primal feasible set, since the projection itself has a complexity comparable to the complexity of the initial problem. We propose an alternative efficient method to obtain feasibility and show that its properties influencing the convergence to the optimum are similar to the properties of the Euclidean projection. We apply our method to the local polytope relaxation of inference problems for Markov Random Fields and demonstrate its superiority over existing methods.Comment: 20 page, 4 figure

Blending Learning and Inference in Structured Prediction

In this paper we derive an efficient algorithm to learn the parameters of structured predictors in general graphical models. This algorithm blends the learning and inference tasks, which results in a significant speedup over traditional approaches, such as conditional random fields and structured support vector machines. For this purpose we utilize the structures of the predictors to describe a low dimensional structured prediction task which encourages local consistencies within the different structures while learning the parameters of the model. Convexity of the learning task provides the means to enforce the consistencies between the different parts. The inference-learning blending algorithm that we propose is guaranteed to converge to the optimum of the low dimensional primal and dual programs. Unlike many of the existing approaches, the inference-learning blending allows us to learn efficiently high-order graphical models, over regions of any size, and very large number of parameters. We demonstrate the effectiveness of our approach, while presenting state-of-the-art results in stereo estimation, semantic segmentation, shape reconstruction, and indoor scene understanding

The power of linear programming for general-valued CSPs

Let $D$, called the domain, be a fixed finite set and let $\Gamma$, called the valued constraint language, be a fixed set of functions of the form $f:D^m\to\mathbb{Q}\cup\{\infty\}$, where different functions might have different arity $m$. We study the valued constraint satisfaction problem parametrised by $\Gamma$, denoted by VCSP$(\Gamma)$. These are minimisation problems given by $n$ variables and the objective function given by a sum of functions from $\Gamma$, each depending on a subset of the $n$ variables. Finite-valued constraint languages contain functions that take on only rational values and not infinite values. Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation (BLP). For a valued constraint language $\Gamma$, BLP is a decision procedure for $\Gamma$ if and only if $\Gamma$ admits a symmetric fractional polymorphism of every arity. For a finite-valued constraint language $\Gamma$, BLP is a decision procedure if and only if $\Gamma$ admits a symmetric fractional polymorphism of some arity, or equivalently, if $\Gamma$ admits a symmetric fractional polymorphism of arity 2. Using these results, we obtain tractability of several novel classes of problems, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) $k$-submodular on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: A full version of a FOCS'12 paper by the last two authors (arXiv:1204.1079) and an ICALP'13 paper by the first author (arXiv:1207.7213) to appear in SIAM Journal on Computing (SICOMP