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Convergent Message-Passing Algorithms for Inference over General Graphs with Convex Free Energies  [PDF]
Tamir Hazan,Amnon Shashua
Computer Science , 2012,
Abstract: 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 non-convex 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 BP-like 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.
Belief Propagation for Linear Programming  [PDF]
Andrew Gelfand,Jinwoo Shin,Michael Chertkov
Computer Science , 2013,
Abstract: Belief Propagation (BP) is a popular, distributed heuristic for performing MAP computations in Graphical Models. BP can be interpreted, from a variational perspective, as minimizing the Bethe Free Energy (BFE). BP can also be used to solve a special class of Linear Programming (LP) problems. For this class of problems, MAP inference can be stated as an integer LP with an LP relaxation that coincides with minimization of the BFE at ``zero temperature". We generalize these prior results and establish a tight characterization of the LP problems that can be formulated as an equivalent LP relaxation of MAP inference. Moreover, we suggest an efficient, iterative annealing BP algorithm for solving this broader class of LP problems. We demonstrate the algorithm's performance on a set of weighted matching problems by using it as a cutting plane method to solve a sequence of LPs tightened by adding ``blossom'' inequalities.
LPQP for MAP: Putting LP Solvers to Better Use  [PDF]
Patrick Pletscher,Sharon Wulff
Computer Science , 2012,
Abstract: MAP inference for general energy functions remains a challenging problem. While most efforts are channeled towards improving the linear programming (LP) based relaxation, this work is motivated by the quadratic programming (QP) relaxation. We propose a novel MAP relaxation that penalizes the Kullback-Leibler divergence between the LP pairwise auxiliary variables, and QP equivalent terms given by the product of the unaries. We develop two efficient algorithms based on variants of this relaxation. The algorithms minimize the non-convex objective using belief propagation and dual decomposition as building blocks. Experiments on synthetic and real-world data show that the solutions returned by our algorithms substantially improve over the LP relaxation.
A Gaussian Belief Propagation Solver for Large Scale Support Vector Machines  [PDF]
Danny Bickson,Elad Yom-Tov,Danny Dolev
Mathematics , 2008,
Abstract: Support vector machines (SVMs) are an extremely successful type of classification and regression algorithms. Building an SVM entails solving a constrained convex quadratic programming problem, which is quadratic in the number of training samples. We introduce an efficient parallel implementation of an support vector regression solver, based on the Gaussian Belief Propagation algorithm (GaBP). In this paper, we demonstrate that methods from the complex system domain could be utilized for performing efficient distributed computation. We compare the proposed algorithm to previously proposed distributed and single-node SVM solvers. Our comparison shows that the proposed algorithm is just as accurate as these solvers, while being significantly faster, especially for large datasets. We demonstrate scalability of the proposed algorithm to up to 1,024 computing nodes and hundreds of thousands of data points using an IBM Blue Gene supercomputer. As far as we know, our work is the largest parallel implementation of belief propagation ever done, demonstrating the applicability of this algorithm for large scale distributed computing systems.
Polynomial Linear Programming with Gaussian Belief Propagation  [PDF]
Danny Bickson,Yoav Tock,Ori Shental,Danny Dolev
Mathematics , 2008, DOI: 10.1109/ALLERTON.2008.4797652
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.
Norm-Product Belief Propagation: Primal-Dual Message-Passing for Approximate Inference  [PDF]
Tamir Hazan,Amnon Shashua
Mathematics , 2009,
Abstract: 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 message-passing algorithm architecture. We generalize the Belief Propagation (BP) algorithms of sum-product and max-product 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 $f + \sum_i h_i$ where the function $f$ is an extended-valued, strictly convex but non-smooth and the functions $h_i$ are extended-valued functions (not necessarily convex). We use tools from convex duality to present the "primal-dual ascent" algorithm which is an extension of the Bregman successive projection scheme and is designed to handle optimization of the general type $f + \sum_i h_i$. Mapping the fractional-free-energy variational principle to this framework introduces the "norm-product" message-passing. Special cases include sum-product and max-product (BP algorithms) and the TRBP algorithms. When the fractional-free-energy is set to be convex (convex-free-energy) the norm-product is globally convergent for estimating of marginal probabilities and for approximating the LP-relaxation. We also introduce another branch of the norm-product, the "convex-max-product". The convex-max-product is convergent (unlike max-product) and aims at solving the LP-relaxation.
Stability in -convex programming
Ebrahim A. Youness
International Journal of Mathematics and Mathematical Sciences , 2001, DOI: 10.1155/s0161171201006317
Abstract: We define and analyze two kinds of stability in E-convex programming problem in which the feasible domain is affected by an operator E. The first kind of this stability is that the set of all operators E that make an optimal set stable while the other kind is that the set of all operators E that make certain side of the feasible domain still active.
Algorithms for Convex Quadratic Programming  [PDF]
Philipp Hungerl?nder
Mathematics , 2014,
Abstract: The main contribution of this thesis is the development of a new algorithm for solving convex quadratic programs. It consists in combining the method of multipliers with an infeasible active-set method. Our approach is iterative. In each step we calculate an augmented Lagrange function. Then we minimize this function using an infeasible active-set method that was already successfully applied to similar problems. After this we update the Lagrange multiplier for the equality constraints. Finally we try to solve convex quadratic program directly, again with the infeasible active-set method, starting from the optimal solution of the actual Lagrange function. Computational experience with our method indicates that typically only few (most of the time only one) outer iterations (multiplier-updates) and also only few (most of the time less than ten) inner iterations (minimization of the Lagrange function) are required to reach the optimal solution. The diploma thesis is organized as follows. We close this chapter with some nota- tion used throughout. In Chapter 2 we show the equivalence of different QP problem formulations and present some important so-called direct methods for solving equality-constrained QPs. We cover the most important aspects for practically successful interior point methods for linear and convex quadratic programming in Chapter 3. Chapter 4 deals with ingredients for practically efficient feasible active set methods. Finally Chapter 5 provides a close description of our Lagrangian infeasible active set method and further gives a convergence analysis of the subalgorithms involved.
Belief Revision in the GOAL Agent Programming Language  [PDF]
Johannes Svante Spurkeland,Andreas Schmidt Jensen,J?rgen Villadsen
ISRN Artificial Intelligence , 2013, DOI: 10.1155/2013/632319
Abstract: Agents in a multiagent system may in many cases find themselves in situations where inconsistencies arise. In order to properly deal with these, a good belief revision procedure is required. This paper illustrates the usefulness of such a procedure: a certain belief revision algorithm is considered in order to deal with inconsistencies and, particularly, the issue of inconsistencies, and belief revision is examined in relation to the GOAL agent programming language. 1. Introduction When designing artificial intelligence, it is desirable to mimic the human way of reasoning as closely as possible to obtain a realistic intelligence albeit still artificial. This includes the ability to not only construct a plan for solving a given problem but also to be able to adapt the plan or discard it in favor of a new. In these situations the environment in which the agents act should be considered as dynamic and complicated as the world it is representing. This will lead to situations where an agent’s beliefs may be inconsistent and need to be revised. Therefore, an important issue in the subject of modern artificial intelligence is that of belief revision. This paper presents an algorithm for belief revision proposed in [1] and shows some examples of situations where belief revision is desired in order to avoid inconsistencies in an agent’s knowledge base. The agent programming language GOAL will be introduced and belief revision will be discussed in this context. Finally, the belief revision algorithm used in this paper will be compared to other approaches dealing with inconsistency. 2. Motivation In many situations, assumptions are made in order to optimize and simplify an artificial intelligent system. This often leads to solutions which are elegant and planning can be done without too many complications. However, such systems tend to be more difficult to realize in the real world—simply because the assumptions made are too restrictive to model the complex real world. The first thing to notice when modeling intelligence is that human thoughts are themselves inconsistent as considered in [2]. It also considers an example of an expert system from [3], where the classical logical representation of the experts’ statements leads to inconsistency when attempting to reason with it. From this, one can realize how experts of a field not necessarily agree with one another and in order to properly reason with their statements inconsistencies need to be taken into account. This is an example where it is not possible to uniquely define the cause and effect in the real world.
Max-Product Belief Propagation for Linear Programming: Applications to Combinatorial Optimization  [PDF]
Sejun Park,Jinwoo Shin
Computer Science , 2014,
Abstract: Max-product belief propagation (BP) is a popular message-passing algorithm for computing a maximum-a-posteriori (MAP) assignment in a joint distribution represented by a graphical model (GM). It has been shown that BP can solve a few classes of Linear Programming (LP) formulations to combinatorial optimization problems including maximum weight matching and shortest path, i.e., BP can be a distributed solver for certain LPs. However, those LPs and corresponding BP analysis are very sensitive to underlying problem setups, and it has been not clear what extent these results can be generalized to. In this paper, we obtain a generic criteria that BP converges to the optimal solution of given LP, and show that it is satisfied in LP formulations associated to many classical combinatorial optimization problems including maximum weight perfect matching, shortest path, traveling salesman, cycle packing and vertex cover. More importantly, our criteria can guide the BP design to compute fractional LP solutions, while most prior results focus on integral ones. Our results provide new tools on BP analysis and new directions on efficient solvers for large-scale LPs.
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