Distributional PAC-Learning from Nisan's Natural Proofs

(Abridged) Carmosino et al. (2016) demonstrated that natural proofs of circuit lower bounds for \Lambda imply efficient algorithms for learning \Lambda-circuits, but only over the uniform distribution, with membership queries, and provided \AC^0[p] \subseteq \Lambda. We consider whether this implication can be generalized to \Lambda \not\supseteq \AC^0[p], and to learning algorithms in Valiant's PAC model, which use only random examples and learn over arbitrary example distributions. We give results of both positive and negative flavor. On the negative side, we observe that if, for every circuit class \Lambda, the implication from natural proofs for \Lambda to learning \Lambda-circuits in Valiant's PAC model holds, then there is a polynomial time solution to O(n^{1.5})-uSVP (unique Shortest Vector Problem), and polynomial time quantum solutions to O(n^{1.5})-SVP (Shortest Vector Problem) and O(n^{1.5})-SIVP (Shortest Independent Vector Problem). This indicates that whether natural proofs for \Lambda imply efficient learning algorithms for \Lambda in Valiant's PAC model may depend on \Lambda. On the positive side, our main result is that specific natural proofs arising from a type of communication complexity argument (e.g., Nisan (1993), for depth-2 majority circuits) imply PAC-learning algorithms in a new distributional variant of Valiant's 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.