Exact Learning Algorithms, Betting Games, and Circuit Lower Bounds

Abstract

This article extends and improves the work of Fortnow and Klivans [2009], who showed that if a circuit class C has an efficient learning algorithm in Angluin’s model of exact learning via equivalence and membership queries [Angluin 1988], then we have the lower bound EXP NP not C. We use entirely different techniques involving betting games [Buhrman et al. 2001] to remove the NP oracle and improve the lower bound to EXP not C. This shows that it is even more difficult to design a learning algorithm for C than the results of Fortnow and Klivans [2009] indicated. We also investigate the connection between betting games and natural proofs, and as a corollary the existence of strong pseudorandom generators. Our results also yield further evidence that the class of Boolean circuits has no efficient exact learning algorithm. This is because our separation is strong in that it yields a natural proof [Razborov and Rudich 1997] against the class. From this we conclude that an exact learning algorithm for Boolean circuits would imply that strong pseudorandom generators do not exist, which contradicts widely believed conjectures from cryptography. As a corollary we obtain that if strong pseudorandom generators exist, then there is no exact learning algorithm for Boolean circuits.