On Learning, Lower Bounds and (un)Keeping Promises

Abstract

We extend the line of research initiated by Fortnow and Klivans [6] that studies the relationship between efficient learning algorithms and circuit lower bounds. In [6], it was shown that if a Boolean circuit class \(\mathcal{C}\) has an efficient deterministic exact learning algorithm, (i.e. an algorithm that uses membership and equivalence queries) then \(\mathsf{EXP}^{\mathsf{NP}} \not \subseteq \mathsf{P/poly}[\mathcal{C}]\). Recently, in [14] EXP NP was replaced by DTIME(n ω(1)). Yet for the models of randomized exact learning or Valiant’s PAC learning, the best result so far is a lower bound against BPEXP (the exponential-time analogue of BPP). In this paper, we derive stronger lower bounds as well as some other consequences from randomized exact learning and PAC learning algorithms, answering an open question posed in [6] and [14]. In particular, we show that