Title: Acquiring DNF via Generalized Fourier Expansions

Abstract: While the Boolean Fourier representation is a staple in learning theory—especially for learning Disjunctive Normal Form (DNF) formulas under uniform and product distributions—extending these successes to non-product distributions has long stood as an unresolved challenge. This paper tackles that issue by proposing a generalized Fourier representation capable of facilitating learning across a wide spectrum of non-product distributions. Specifically, we model any distribution $D$ as a Bayesian network (BN) and construct a corresponding Fourier expansion for it. We demonstrate that conventional Fourier-based learning methods, which utilize membership queries to pinpoint significant coefficients, can be effectively adapted to this new representation with only slight adjustments. Our analysis confirms that the $L_1$ spectral norm of conjunctions stays bounded within this expansion when applied to difference-bounded tree BNs, thereby substantially broadening previous findings that were limited to uniform distributions; furthermore, matching lower bounds underscore the essential nature of these specific constraints. Leveraging these theoretical insights, we prove that DNF is learnable and that decision trees are agnostically learnable under such distributions. Additionally, we introduce an algorithm designed to learn difference-bounded tree BN distributions, thereby expanding the applicability of our results to scenarios where the underlying distribution is not known a priori.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC