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38
SampleSearch: Importance Sampling in Presence of Determinism
, 2009
"... The paper focuses on developing effective importance sampling algorithms for mixed probabilistic and deterministic graphical models. The use of importance sampling in such graphical models is problematic because it generates many useless zero weight samples which are rejected yielding an inefficient ..."
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Cited by 35 (4 self)
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The paper focuses on developing effective importance sampling algorithms for mixed probabilistic and deterministic graphical models. The use of importance sampling in such graphical models is problematic because it generates many useless zero weight samples which are rejected yielding an inefficient sampling process. To address this rejection problem, we propose the SampleSearch scheme that augments sampling with systematic constraintbased backtracking search. We characterize the bias introduced by the combination of search with sampling, and derive a weighting scheme which yields an unbiased estimate of the desired statistics (e.g. probability of evidence). When computing the weights exactly is too complex, we propose an approximation which has a weaker guarantee of asymptotic unbiasedness. We present results of an extensive empirical evaluation demonstrating that SampleSearch outperforms other schemes in presence of significant amount of determinism.
Approximate counting by sampling the backtrackfree search space
 In AAAI
, 2007
"... We present a new estimator for counting the number of solutions of a Boolean satisfiability problem as a part of an importance sampling framework. The estimator uses the recently introduced SampleSearch scheme that is designed to overcome the rejection problem associated with distributions having a ..."
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Cited by 26 (6 self)
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We present a new estimator for counting the number of solutions of a Boolean satisfiability problem as a part of an importance sampling framework. The estimator uses the recently introduced SampleSearch scheme that is designed to overcome the rejection problem associated with distributions having a substantial amount of determinism. We show here that the sampling distribution of SampleSearch can be characterized as the backtrackfree distribution and propose several schemes for its computation. This allows integrating SampleSearch into the importance sampling framework for approximating the number of solutions and also allows using SampleSearch for computing a lower bound measure on the number of solutions. Our empirical evaluation demonstrates the superiority of our new approximate counting schemes against recent competing approaches.
Model Counting
, 2008
"... Propositional model counting or #SAT is the problem of computing the number of models for a given propositional formula, i.e., the number of distinct truth assignments to variables for which the formula evaluates to true. For a propositional formula F, we will use #F to denote the model count of F. ..."
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Cited by 23 (0 self)
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Propositional model counting or #SAT is the problem of computing the number of models for a given propositional formula, i.e., the number of distinct truth assignments to variables for which the formula evaluates to true. For a propositional formula F, we will use #F to denote the model count of F. This problem is also referred to as the solution counting problem for SAT. It generalizes SAT and is the canonical #Pcomplete problem. There has been significant theoretical work trying to characterize the worstcase complexity of counting problems, with some surprising results such as model counting being hard even for some polynomialtime solvable problems like 2SAT. The model counting problem presents fascinating challenges for practitioners and poses several new research questions. Efficient algorithms for this problem will have a significant impact on many application areas that are inherently beyond SAT (‘beyond ’ under standard complexity theoretic assumptions), such as boundedlength adversarial and contingency planning, and probabilistic reasoning. For example, various probabilistic inference problems, such as Bayesian net reasoning, can be effectively translated into model counting problems [cf.
Leveraging belief propagation, backtrack search, and statistics for model counting
"... Abstract. We consider the problem of estimating the model count (number of solutions) of Boolean formulas, and present two techniques that compute estimates of these counts, as well as either lower or upper bounds with different tradeoffs between efficiency, bound quality, and correctness guarantee ..."
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Cited by 20 (6 self)
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Abstract. We consider the problem of estimating the model count (number of solutions) of Boolean formulas, and present two techniques that compute estimates of these counts, as well as either lower or upper bounds with different tradeoffs between efficiency, bound quality, and correctness guarantee. For lower bounds, we use a recent framework for probabilistic correctness guarantees, and exploit message passing techniques for marginal probability estimation, namely, variations of Belief Propagation (BP). Our results suggest that BP provides useful information even on structured loopy formulas. For upper bounds, we perform multiple runs of the MiniSat SAT solver with a minor modification, and obtain statistical bounds on the model count based on the observation that the distribution of a certain quantity of interest is often very close to the normal distribution. Our experiments demonstrate that our model counters based on these two ideas, BPCount and MiniCount, can provide very good bounds in time significantly less than alternative approaches. 1
Markov chains on orbits of permutation groups
 In UAI2012
, 2012
"... We present a novel approach to detecting and utilizing symmetries in probabilistic graphical models with two main contributions. First, we present a scalable approach to computing generating sets of permutation groups representing the symmetries of graphical models. Second, we introduce orbital Mark ..."
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Cited by 17 (4 self)
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We present a novel approach to detecting and utilizing symmetries in probabilistic graphical models with two main contributions. First, we present a scalable approach to computing generating sets of permutation groups representing the symmetries of graphical models. Second, we introduce orbital Markov chains, a novel family of Markov chains leveraging model symmetries to reduce mixing times. We establish an insightful connection between model symmetries and rapid mixing of orbital Markov chains. Thus, we present the first lifted MCMC algorithm for probabilistic graphical models. Both analytical and empirical results demonstrate the effectiveness and efficiency of the approach. 1
Probabilistic Planning via Heuristic Forward Search and Weighted Model Counting
"... We present a new algorithm for probabilistic planning with no observability. Our algorithm, called ProbabilisticFF, extends the heuristic forwardsearch machinery of ConformantFF to problems with probabilistic uncertainty about both the initial state and action effects. Specifically, Probabilistic ..."
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Cited by 17 (0 self)
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We present a new algorithm for probabilistic planning with no observability. Our algorithm, called ProbabilisticFF, extends the heuristic forwardsearch machinery of ConformantFF to problems with probabilistic uncertainty about both the initial state and action effects. Specifically, ProbabilisticFF combines ConformantFF’s techniques with a powerful machinery for weighted model counting in (weighted) CNFs, serving to elegantly define both the search space and the heuristic function. Our evaluation of ProbabilisticFF shows its fine scalability in a range of probabilistic domains, constituting a several orders of magnitude improvement over previous results in this area. We use a problematic case to point out the main open issue to be addressed by further research. 1.
Studies in Lower Bounding Probability of Evidence using the Markov Inequality
"... Computing the probability of evidence even with known error bounds is NPhard. In this paper we address this hard problem by settling on an easier problem. We propose an approximation which provides high confidence lower bounds on probability of evidence but does not have any guarantees in terms of ..."
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Cited by 10 (4 self)
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Computing the probability of evidence even with known error bounds is NPhard. In this paper we address this hard problem by settling on an easier problem. We propose an approximation which provides high confidence lower bounds on probability of evidence but does not have any guarantees in terms of relative or absolute error. Our proposed approximation is a randomized importance sampling scheme that uses the Markov inequality. However, a straightforward application of the Markov inequality may lead to poor lower bounds. We therefore propose several heuristic measures to improve its performance in practice. Empirical evaluation of our scheme with stateoftheart lower bounding schemes reveals the promise of our approach. 1
Approximate modelbased diagnosis using greedy stochastic search
 In Proc. SARA’07
, 2007
"... We propose a StochAstic Fault diagnosis AlgoRIthm, called Safari, which trades off guarantees of computing minimal diagnoses for computational efficiency. We empirically demonstrate, using the 74XXX and ISCAS85 suites of benchmark combinatorial circuits, that Safari achieves several ordersofmagnit ..."
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Cited by 9 (6 self)
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We propose a StochAstic Fault diagnosis AlgoRIthm, called Safari, which trades off guarantees of computing minimal diagnoses for computational efficiency. We empirically demonstrate, using the 74XXX and ISCAS85 suites of benchmark combinatorial circuits, that Safari achieves several ordersofmagnitude speedup over two wellknown deterministic algorithms, CDA ∗ and HA ∗ , for multiplefault diagnoses; further, Safari can compute a range of multiplefault diagnoses that CDA ∗ and HA ∗ cannot. We also prove that Safari is optimal for a range of propositional fault models, such as the widelyused weakfault models (models with ignorance of abnormal behavior). We discuss the optimality of Safari in a class of strongfault circuit models with stuckat failure modes. By modeling the algorithm itself as a Markov chain, we provide exact bounds on the minimality of the diagnosis computed. Safari also displays strong anytime behavior, and will return a diagnosis after any nontrivial inference time. 1.
A Scalable Approximate Model Counter
"... Abstract. Propositional model counting (#SAT), i.e., counting the number of satisfying assignments of a propositional formula, is a problem of significant theoretical and practical interest. Due to the inherent complexity of the problem, approximate model counting, which counts the number of satisfy ..."
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Cited by 9 (4 self)
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Abstract. Propositional model counting (#SAT), i.e., counting the number of satisfying assignments of a propositional formula, is a problem of significant theoretical and practical interest. Due to the inherent complexity of the problem, approximate model counting, which counts the number of satisfying assignments to within given tolerance and confidence level, was proposed as a practical alternative to exact model counting. Yet, approximate model counting has been studied essentially only theoretically. The only reported implementation of approximate model counting, due to Karp and Luby, worked only for DNF formulas. A few existing tools for CNF formulas are bounding model counters; they can handle realistic problem sizes, but fall short of providing counts within given tolerance and confidence, and, thus, are not approximate model counters. We present here a novel algorithm, as well as a reference implementation, that is the first scalable approximate model counter for CNF formulas. The algorithm works by issuing a polynomial number of calls to a SAT solver. Our tool, ApproxMC, scales to formulas with tens of thousands of variables. Careful experimental comparisons show that ApproxMC reports, with high confidence, bounds that are close to the exact count, and also succeeds in reporting bounds with small tolerance and high confidence in cases that are too large for computing exact model counts. 1
Computing the Density of States of Boolean Formulas
"... Abstract. In this paper we consider the problem of computing the density of states of a Boolean formula in CNF, a generalization of both MAXSAT and model counting. Given a Boolean formula F, its density of states counts the number of configurations that violate exactly E clauses, for all values of ..."
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Cited by 4 (2 self)
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Abstract. In this paper we consider the problem of computing the density of states of a Boolean formula in CNF, a generalization of both MAXSAT and model counting. Given a Boolean formula F, its density of states counts the number of configurations that violate exactly E clauses, for all values of E. We propose a novel Markov Chain Monte Carlo algorithm based on flat histogram methods that, despite the hardness of the problem, converges quickly to a very accurate solution. Using this method, we show the first known results on the density of states of several widely used formulas and we provide novel insights about the behavior of random 3SAT formulas around the phase transition. 1