The Boolean Fourier representation has been widely used in learning theory, particularly for learning Disjunctive Normal Form (DNF) under uniform and product distributions. Extending these results to non-product distributions has remained a longstanding open problem. We address this challenge by introducing a generalized Fourier representation that enables learning under a broad class of non-product distributions. Our approach represents any distribution $D$ as a Bayesian network (BN) and derives a corresponding Fourier expansion. We show that standard Fourier-based learning techniques using membership queries to identify heavy coefficients can be adapted to this generalized representation with minor modifications. We prove that the $L_1$ spectral norm of conjunctions remains bounded under this expansion for difference-bounded tree BNs, significantly generalizing the known result for uniform distributions; matching lower bounds demonstrate the necessity of these constraints. Using these results, we establish the learnability of DNF and the agnostic learnability of decision trees under such distributions. Finally, we present an algorithm for learning difference-bounded tree BN distributions, extending our results to settings where the distribution is unknown.

翻译：暂无翻译