Fast and Flexible Probabilistic Model Counting

Achlioptas, Dimitris; Hammoudeh, Zayd; Theodoropoulos, Panos

doi:10.1007/978-3-319-94144-8_10

Cited by 7 publications

(11 citation statements)

References 13 publications

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“…There have been recent developments on the use of short parity constraints to speed up computation in WISH style methods (Zhao et al 2016;Ermon et al 2014;Asteris and Dimakis 2016;Achlioptas and Theodoropoulos 2017;Achlioptas, Hammoudeh, and Theodoropoulos 2018). While these approaches reduce the empirical complexity of answering individual optimization oracle queries, our method reduces the number of queries itself, complementing these approaches.…”

Section: Related Workmentioning

confidence: 98%

Towards Efficient Discrete Integration via Adaptive Quantile Queries

Ding,

Wang,

Sabharwal

et al. 2019

Preprint

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Discrete integration in a high dimensional space of n variables poses fundamental challenges. The WISH algorithm reduces the intractable discrete integration problem into n optimization queries subject to randomized constraints, obtaining a constant approximation guarantee. The optimization queries are expensive, which limits the applicability of WISH. We propose AdaWISH, which is able to obtain the same guarantee, but accesses only a small subset of queries of WISH. For example, when the number of function values is bounded by a constant, AdaWISH issues only O(log n) queries. The key idea is to query adaptively, taking advantage of the shape of the weight function. In general, we prove that AdaWISH has a regret of no more than O(log n) relative to an oracle that issues queries at data-dependent optimal points. Experimentally, AdaWISH gives precise estimates for discrete integration problems, of the same quality as that of WISH and better than several competing approaches, on a variety of probabilistic inference benchmarks, while saving substantially on the number of optimization queries compared to WISH. For example, it saves 81.5% of WISH queries while retaining the quality of results on a suite of UAI inference challenge benchmarks.

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Section: Related Workmentioning

confidence: 98%

Towards Efficient Discrete Integration via Adaptive Quantile Queries

Ding,

Wang,

Sabharwal

et al. 2019

Preprint

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“…One approach limited the counting to any set of variables I for which any assignment leads to at most one solution in V , denoting those as minimal independent supports (Chakraborty, Meel, and Vardi 2014;Ivrii et al 2016). Another approach broke with the independent probability p by using each variable the same number of times across r XOR constraints (Achlioptas and Jiang 2015;Achlioptas, Hammoudeh, and Theodoropoulos 2018). Related work on MILP has only focused on upper bounds based on relaxations (Jain, Kadioglu, and Sellmann 2010).…”

Section: Counting and Probabilistic Inferencementioning

confidence: 99%

Empirical Bounds on Linear Regions of Deep Rectifier Networks

Serra

Ramalingam

2020

AAAI

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We can compare the expressiveness of neural networks that use rectified linear units (ReLUs) by the number of linear regions, which reflect the number of pieces of the piecewise linear functions modeled by such networks. However, enumerating these regions is prohibitive and the known analytical bounds are identical for networks with same dimensions. In this work, we approximate the number of linear regions through empirical bounds based on features of the trained network and probabilistic inference. Our first contribution is a method to sample the activation patterns defined by ReLUs using universal hash functions. This method is based on a Mixed-Integer Linear Programming (MILP) formulation of the network and an algorithm for probabilistic lower bounds of MILP solution sets that we call MIPBound, which is considerably faster than exact counting and reaches values in similar orders of magnitude. Our second contribution is a tighter activation-based bound for the maximum number of linear regions, which is particularly stronger in networks with narrow layers. Combined, these bounds yield a fast proxy for the number of linear regions of a deep neural network.

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“…The framework developed so far has focused on modeling probability distributions where the probability masses are (integer) multiples of 1 2 n . Of course, many examples violate this assumption, e.g, a coin with a 1 3 bias towards heads. To handle such distributions, we adapt our framework to condition on certain coin flip outcomes not occurring.…”

Section: Circuits Coin Flips and Model Countingmentioning

confidence: 99%

“…In the preceding sections, we have developed a modeling framework for probabilistic systems based on feeding unbiased coins into Boolean predicates or sequential circuits. Our encodings require only unweighted model counting algorithms for their analysis, and thus directly benefit from the recent dramatic performance gains in unweighted model counting [7,1,25]. In contrast, many previous works on probabilistic inference using model counting algorithms have built on algorithms for weighted model counting.…”

Section: Relationship To Weighted Model Countingmentioning

confidence: 99%

“…Model counting is a promising alternative algorithmic approach to analyzing probabilistic models and probabilistic inference. The recent rapid improvements in SAT-based model counting, particularly in approximate model counting [12,6,23,7,1,24], raise hope that the Herculean improvements in SAT solving could be leveraged for probabilistic inference. In particular, model counting may enable us to transition away from Monte Carlo methods and BDD-based probabilistic model checking, in the same way that SAT solvers resulted in a transition away from explicit-state/BDD-based model checking [2].…”

Section: Introductionmentioning

confidence: 99%

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A Model Counter's Guide to Probabilistic Systems

Vazquez-Chanlatte,

Rabe,

Seshia

2019

Preprint

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In this paper, we systematize the modeling of probabilistic systems for the purpose of analyzing them with model counting techniques. Starting from unbiased coin flips, we show how to model biased coins, correlated coins, and distributions over finite sets. From there, we continue with modeling sequential systems, such as Markov chains, and revisit the relationship between weighted and unweighted model counting. Thereby, this work provides a conceptual framework for deriving #SAT encodings for probabilistic inference.

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Fast and Flexible Probabilistic Model Counting

Cited by 7 publications

References 13 publications

Towards Efficient Discrete Integration via Adaptive Quantile Queries

Towards Efficient Discrete Integration via Adaptive Quantile Queries

Empirical Bounds on Linear Regions of Deep Rectifier Networks

A Model Counter's Guide to Probabilistic Systems

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