On the Power of Membership Queries in Agnostic Learning

Abstract

We study the properties of the agnostic learning framework of Haussler [Hau92] and Kearns, Schapire and Sellie [KSS94]. In particular, we address the question: is there any situation in which membership queries are useful in agnostic learning? Our results show that the answer is negative for distribution-independent agnostic learning and positive for agnostic learning with respect to a specific marginal distribution. Namely, we give a simple proof that any concept class learnable agnostically by a distribution-independent algorithm with access to membership queries is also learnable agnostically without membership queries. This resolves an open problem posed by Kearns et al. [KSS94]. For agnostic learning with respect to the uniform distribution over $\{0,1\}^n$ we show a concept class that is learnable with membership queries but computationally hard to learn from random examples alone (assuming that one-way functions exist).