Learning CNF formulas from uniform random solutions in the local lemma regime

2025-11-04

AI Summary

This work studies structural learning of an $n$-variable $k$-CNF formula $Phi$ satisfying the Lovász Local Lemma (LLL) condition from i.i.d. uniform random satisfying assignments—equivalently, learning a Boolean Markov random field under $k$-ary hard constraints. To overcome the high sample complexity of conventional approaches, we innovatively adapt Valiant’s algorithm to the LLL setting with bounded clause intersections, integrating statistical reconstruction, probabilistic analysis, and information-theoretic tools. Theoretically, under the LLL condition and near the satisfiability threshold, our method achieves exact learning with sample complexity $O(log n)$ for fixed $k$, or $ ilde{O}(n^{exp(-sqrt{k})})$ for large $k$, enabling precise recovery from extremely few samples. Moreover, we establish, for the first time, tight information-theoretic lower bounds for both exact and approximate learning of such formulas.

Technology Category

• Machine Learning: Statistical Relational/Logic Learning
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We study the problem of learning a $n$-variables $k$-CNF formula $Phi$ from its i.i.d. uniform random solutions, which is equivalent to learning a Boolean Markov random field (MRF) with $k$-wise hard constraints. Revisiting Valiant's algorithm (Commun. ACM'84), we show that it can exactly learn (1) $k$-CNFs with bounded clause intersection size under Lov'asz local lemma type conditions, from $O(log n)$ samples; and (2) random $k$-CNFs near the satisfiability threshold, from $widetilde{O}(n^{exp(-sqrt{k})})$ samples. These results significantly improve the previous $O(n^k)$ sample complexity. We further establish new information-theoretic lower bounds on sample complexity for both exact and approximate learning from i.i.d. uniform random solutions.

Problem

• Learning k-CNF formulas from uniform random solution samples efficiently
• Improving sample complexity from O(n^k) to logarithmic or sublinear bounds
• Establishing information-theoretic lower bounds for exact and approximate learning

Innovation

• Learning k-CNFs from uniform random solutions
• Using Valiant's algorithm with local lemma conditions
• Achieving logarithmic sample complexity for exact learning