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

TL;DR

This paper presents new algorithms and bounds for learning $k$-CNF formulas from uniform random solutions, improving sample complexity and extending understanding in the local lemma regime.

Contribution

It extends Valiant's algorithm to learn $k$-CNFs under local lemma conditions and near the satisfiability threshold, with significantly reduced sample complexity.

Findings

Exact learning of $k$-CNFs with bounded clause intersection from $O( ext{log} n)$ samples

Learning random $k$-CNFs near the satisfiability threshold from $ ilde{O}(n^{ ext{exp}(- ext{sqrt}(k))})$ samples

New information-theoretic lower bounds on sample complexity for learning from uniform solutions

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.