Distributional PAC-Learning from Nisan's Natural Proofs
This work addresses a theoretical problem in computational learning theory by connecting circuit complexity and learning algorithms, with implications for cryptography and average-case learning models, though it is incremental in refining existing frameworks.
The paper investigates whether natural proofs for circuit lower bounds imply efficient PAC-learning algorithms over arbitrary distributions, showing that standard assumptions from lattice-based cryptography prevent a full generalization. It introduces a new distributional PAC-learning model and proves that natural proofs from communication complexity arguments, including Nisan's, imply algorithms in this model, leading to new distributional PAC-learning algorithms for depth-2 majority circuits, polytopes, and DNFs over natural distributions.
Carmosino et al. (2016) demonstrated that natural proofs of circuit lower bounds for $Λ$ imply efficient algorithms for learning $Λ$-circuits, but only over \textit{the uniform distribution}, with \textit{membership queries}, and provided $\AC^0[p] \subseteq Λ$. We consider whether this implication can be generalized to $Λ\not\supseteq \AC^0[p]$, 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 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 \textit{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 \textit{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.