Do natural proofs imply efficient learning algorithms? Carmosino et al. (2016) demonstrated that natural proofs of circuit lower bounds for Λ imply efficient algorithms for learning Λ-circuits, but only over the uniform distribution, with membership queries, and provided AC⁰[p] ⊆ Λ. We consider whether this implication can be generalized to Λ ⊉ AC⁰[p], and to learning algorithms which use only random examples and learn over arbitrary example distributions (Valiant’s PAC-learning model).

We first observe that, if, for any circuit class Λ, there is an implication from natural proofs for Λ to PAC-learning for Λ, then standard assumptions from lattice-based cryptography do not hold. In particular, we observe that depth-2 majority circuits are a (conditional) counter example to this fully general implication, since Nisan (1993) gave a natural proof, but Klivans and Sherstov (2009) showed hardness of PAC-Learning under lattice-based assumptions. We thus ask: what learning algorithms can we reasonably expect to follow from Nisan’s natural proofs?

Our main result is that all natural proofs arising from a type of communication complexity argument, including Nisan’s, imply PAC-learning algorithms in a new distributional variant (i.e., an "average-case" relaxation) of Valiant’s PAC model. Our distributional PAC model is stronger than the average-case prediction model of Blum et al. (1993) and the heuristic PAC model of Nanashima (2021), and has several important properties which make it of independent interest, such as being boosting-friendly. The main applications of our result are new distributional PAC-learning algorithms for depth-2 majority circuits, polytopes and DNFs over natural target distributions, as well as the nonexistence of encoded-input weak PRFs that can be evaluated by depth-2 majority circuits.