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Bucket and minibucket Schemes for M Best Solutions over Graphical Models
"... The paper focuses on finding the m best solutions of a combinatorial optimization problem defined over a graphical model (e.g., the m most probable explanations for a Bayesian network). We describe elimmopt, a new bucket elimination algorithm for solving the mbest task, provide efficient implement ..."
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Cited by 7 (3 self)
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The paper focuses on finding the m best solutions of a combinatorial optimization problem defined over a graphical model (e.g., the m most probable explanations for a Bayesian network). We describe elimmopt, a new bucket elimination algorithm for solving the mbest task, provide efficient implementation of its defining combination and marginalization operators, analyze its worstcase performance, and compare it with that of recent related algorithms. An extension to the minibucket framework, yielding a collection of bounds for each of the mbest solutions is discussed and empirically evaluated. We also formulate the mbest task as a regular reasoning task over general graphical models defined axiomatically, which makes all other inference algorithms applicable. 1
Enabling Local Computation for Partially Ordered Preferences
"... Abstract. Many computational problems linked to uncertainty and preference management can be expressed in terms of computing the marginal(s) of a combination of a collection of valuation functions. Shenoy and Shafer showed how such a computation can be performed using a local computation scheme. A m ..."
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Cited by 4 (1 self)
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Abstract. Many computational problems linked to uncertainty and preference management can be expressed in terms of computing the marginal(s) of a combination of a collection of valuation functions. Shenoy and Shafer showed how such a computation can be performed using a local computation scheme. A major strength of this work is that it is based on an algebraic description: what is proved is the correctness of the local computation algorithm under a few axioms on the algebraic structure. The instantiations of the framework in practice make use of totally ordered scales. The present paper focuses on the use of partially ordered scales and examines how such scales can be cast in the ShaferShenoy framework and thus benefit from local computation algorithms. It also provides several examples of such scales, thus showing that each of the algebraic structures explored here is of interest.
A Distributed Algorithm for Optimising over Pure Strategy Nash Equilibria
, 2010
"... We develop an efficient algorithm for computing pure strategy Nash equilibria that satisfy various criteria (such as the utilitarian or Nash–Bernoulli social welfare functions) in games with sparse interaction structure. Our algorithm, called Valued Nash Propagation (VNP), integrates the optimisatio ..."
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Cited by 3 (0 self)
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We develop an efficient algorithm for computing pure strategy Nash equilibria that satisfy various criteria (such as the utilitarian or Nash–Bernoulli social welfare functions) in games with sparse interaction structure. Our algorithm, called Valued Nash Propagation (VNP), integrates the optimisation problem of maximising a criterion with the constraint satisfaction problem of finding a game’s equilibria to construct a criterion that defines a c–semiring. Given a suitably compact game structure, this criterion can be efficiently optimised using message–passing. To this end, we first show that VNP is complete in games whose interaction structure forms a hypertree. Then, we go on to provide theoretic and empirical results justifying its use on games with arbitrary structure; in particular, we show that it computes the optimum>82 % of the time and otherwise selects an equilibrium that is always within 2 % of the optimum on average.
Local Computation Schemes with Partially Ordered Preferences
"... Abstract. Many computational problems linked to uncertainty and preference management can be expressed in terms of computing the marginal(s) of a combination of a collection of valuation functions. Shenoy and Shafer showed how such a computation can be performed using a local computation scheme. A m ..."
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Cited by 2 (1 self)
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Abstract. Many computational problems linked to uncertainty and preference management can be expressed in terms of computing the marginal(s) of a combination of a collection of valuation functions. Shenoy and Shafer showed how such a computation can be performed using a local computation scheme. A major strength of this work is that it is based on an algebraic description: what is proved is the correctness of the local computation algorithm under a few axioms on the algebraic structure. The instantiations of the framework in practice make use of totally ordered scales. The present paper focuses on the use of partially ordered scales and examines how such scales can be cast in the ShaferShenoy framework and thus benefit from local computation algorithms. It also provides many examples of such scales, thus showing that each of the algebraic structures explored here is of interest.
Inference Schemes for M Best Solutions for Soft CSPs
"... Abstract. The paper present a formalization of the mbest task within the unifying framework of semirings. As a consequence, known inference algorithms are defined and their correctness and completeness for the mbest task are immediately implied. We also describe and analyze a Bucket Elimination al ..."
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Cited by 1 (0 self)
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Abstract. The paper present a formalization of the mbest task within the unifying framework of semirings. As a consequence, known inference algorithms are defined and their correctness and completeness for the mbest task are immediately implied. We also describe and analyze a Bucket Elimination algorithm for solving the mbest task, elimmopt, presented in an earlier workshop 1 and introduce an extension to the minibucket framework, yielding a collection of bounds for each of the mbest solutions. Some empirical demonstration of the algorithms and their potential for approximations are provided. 1
SemiringInduced Propositional Logic: Definition and Basic Algorithms
"... In this paper we introduce an extension of propositional logic that allows clauses to be weighted with values from a generic semiring. The main interest of this extension is that different instantiations of the semiring model different interesting computational problems such as finding a model, coun ..."
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In this paper we introduce an extension of propositional logic that allows clauses to be weighted with values from a generic semiring. The main interest of this extension is that different instantiations of the semiring model different interesting computational problems such as finding a model, counting the number of models, finding the best model with respect to an objective function, finding the best model with respect to several independent objective functions, or finding the set of paretooptimal models with respect to several objective functions. Then we show that this framework unifies several solving techniques and, even more importantly, rephrases them from an algorithmic language to a logical language. As a result, several solving techniques can be trivially and elegantly transferred from one computational problem to another. As an example, we extend the basic DPLL algorithm to our framework producing an algorithm that we call SDPLL. Then we enhance the basic SDPLL in order to incorporate the three features that are common in all modern SAT solvers: backjumping, learning and restarts. As a result, we obtain an extremely simple algorithm that captures, unifies and extends in a welldefined logical language several techniques that are valid for arbitrary semirings.
Updating Credal Networks is Approximable in Polynomial Time
"... Credal networks relax the precise probability requirement of Bayesian networks, enabling a richer representation of uncertainty in the form of closed convex sets of probability measures. The increase in expressiveness comes at the expense of higher computational costs. In this paper, we present a ne ..."
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Credal networks relax the precise probability requirement of Bayesian networks, enabling a richer representation of uncertainty in the form of closed convex sets of probability measures. The increase in expressiveness comes at the expense of higher computational costs. In this paper, we present a new variable elimination algorithm for exactly computing posterior inferences in extensively specified credal networks, which is empirically shown to outperform a stateoftheart algorithm. The algorithm is then turned into a provably good approximation scheme, that is, a procedure that for any input is guaranteed to return a solution not worse than the optimum by a given factor. Remarkably, we show that when the networks have bounded treewidth and bounded number of states per variable the approximation algorithm runs in time polynomial in the input size and in the inverse of the error factor, thus being the first known fully polynomialtime approximation scheme for inference in credal networks.