Distributional PAC-Learning from Nisan's Natural Proofs

Carmosino et al. (2016) demonstrated that natural proofs of circuit lower bounds for Lambda imply efficient algorithms for learning Lambda-circuits, but only over textitthe uniform distribution, with textitmembership queries, and provided AC0[p] subseteq Lambda. We consider whether this implication can be generalized to Lambda notsupseteq 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 Lambda, there is an implication from natural proofs for Lambda to PAC-learning for Lambda, 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 the 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 textitdistributional 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 textitboosting-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.