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Scalable Algorithms for Approximate DNF Model Counting

Paul Burkhardt, David G. Harris, Kevin T Schmitt

TL;DR

A new Monte Carlo approach with an adaptive stopping rule and short-circuit formula evaluation is developed that achieves Probably Approximately Correct (PAC) learning bounds and is asymptotically more efficient than the previous methods.

Abstract

Model counting of Disjunctive Normal Form (DNF) formulas is a critical problem in applications such as probabilistic inference and network reliability. For example, it is often used for query evaluation in probabilistic databases. Due to the computational intractability of exact DNF counting, there has been a line of research into a variety of approximation algorithms. These include Monte Carlo approaches such as the classical algorithms of Karp, Luby, and Madras (1989), as well as methods based on hashing (Soos et al. 2023), and heuristic approximations based on Neural Nets (Abboud, Ceylan, and Lukasiewicz 2020). We develop a new Monte Carlo approach with an adaptive stopping rule and short-circuit formula evaluation. We prove it achieves Probably Approximately Correct (PAC) learning bounds and is asymptotically more efficient than the previous methods. We also show experimentally that it out-performs prior algorithms by orders of magnitude, and can scale to much larger problems with millions of variables.

Scalable Algorithms for Approximate DNF Model Counting

TL;DR

A new Monte Carlo approach with an adaptive stopping rule and short-circuit formula evaluation is developed that achieves Probably Approximately Correct (PAC) learning bounds and is asymptotically more efficient than the previous methods.

Abstract

Model counting of Disjunctive Normal Form (DNF) formulas is a critical problem in applications such as probabilistic inference and network reliability. For example, it is often used for query evaluation in probabilistic databases. Due to the computational intractability of exact DNF counting, there has been a line of research into a variety of approximation algorithms. These include Monte Carlo approaches such as the classical algorithms of Karp, Luby, and Madras (1989), as well as methods based on hashing (Soos et al. 2023), and heuristic approximations based on Neural Nets (Abboud, Ceylan, and Lukasiewicz 2020). We develop a new Monte Carlo approach with an adaptive stopping rule and short-circuit formula evaluation. We prove it achieves Probably Approximately Correct (PAC) learning bounds and is asymptotically more efficient than the previous methods. We also show experimentally that it out-performs prior algorithms by orders of magnitude, and can scale to much larger problems with millions of variables.
Paper Structure (17 sections, 9 theorems, 28 equations, 8 figures, 2 tables, 2 algorithms)

This paper contains 17 sections, 9 theorems, 28 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Lazy Monte Carlo Sampling
  4. Our Algorithm
  5. Analysis of the Algorithm
  6. Data structures and implementation notes
  7. Empirical evaluation and discussion
  8. Scaling DNF size
  9. Scaling the PAC bounds
  10. Comparison with Neural#DNF and previous benchmarks
  11. Discussion on algorithm performance
  12. Random sampling.
  13. Clause selection heuristic.
  14. Memory access pattern.
  15. Variable ordering.
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\beta$ be an arbitrary non-zero constant and $\varepsilon,\delta\in(0,3/4)$, and all the values $\rho(v)$ are in the range $[b, 1-b]$ for some constant $b > 0$. Then the expected work of our Algorithm 2 is and the expected randomness complexity is

Figures (8)

  • Figure 1: Average work as problem size increases, $\varepsilon = \delta = 0.05$. Light gray lines indicate an ideal scaling from $m=2^{10}$.
  • Figure 2: $(\varepsilon,\delta)$-Scalability. $n=m=2^{12}$
  • Figure 3: Accuracy Tests. $\delta = 0.05, \varepsilon \in [0.005,0.1]$
  • Figure 4: work comparison of the competing methods. $n=\,$15,000, $m=\,$11,250, $\varepsilon=0.1$, and $\delta=0.05$.
  • Figure 5: Mean running times for the two regimes on the six different scenarios of uniform width in {3,5,8,13,21,34}.
  • ...and 3 more figures

Theorems & Definitions (18)

  • Theorem 1
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • proof : Proof of Theorem \ref{['main-analysis-theorem']}
  • Theorem 6
  • Proposition 7
  • ...and 8 more