Algorithms from Natural Lower Bounds

Circuit analysis algorithms such as learning, SAT, minimum circuit size, and compression imply circuit lower bounds. We show a generic implication in the opposite direction: natural properties (in the sense of Razborov and Rudich) imply randomized learning and compression algorithms. This is the first such implication outside of the derandomization setting. As an application, we use known natural lower bounds for AC[p] circuits (due to Razborov and Smolensky) to get the first quasi-polynomial time algorithm for learning AC[p] functions, in the PAC model over the uniform distribution, with membership queries. ∗This work was partially supported by the Simons Foundation and NSF grants #CNS-1523467 and CCF-121351 (M. Carmosino, R. Impagliazzo) and by NSERC Discovery grants (V. Kabanets, A. Kolokolova). This work was done in part while all authors were visiting Simons Institute for the Theory of Computing. †Department of Computer Science, University of California San Diego, La Jolla, CA; [email protected] ‡Department of Computer Science, University of California San Diego, La Jolla, CA; [email protected] §School of Computing Science, Simon Fraser University, Burnaby, BC, Canada; [email protected] ¶Department of Computer Science, Memorial University of Newfoundland, St. John’s, NL, Canada; [email protected]