On Exactly Learning Disjunctions and DNFs Without Equivalence Queries

Abstract

In this paper we address the issue of exactly learning boolean functions. The notion of exact learning introduced by [2] endows a learner with access to oracles that can answer two types of queries: membership queries and equivalence queries, in which however, equivalence queries are unrealistically strong and cannot be really carried out. Thus we investigate exact learning without equivalence queries and provide some positive results of exactly learning disjunctions and DNFs as follows (without equivalence queries).

We present a general result for exactly properly learning disjunctions if probability mass of negative inputs and probabilities that all bits are assigned to 0 and 1 are all positive. Moreover, with at most n membership queries, we can reduce sample and time complexity.

We present a general result for exactly properly learning the class of s-DNFs with random examples, and obtain two concrete results under uniform distributions. First, the class of l-term s-DNFs with $l_1$ $\log 2l$-terms can be exactly learned using $O(2^{s+l_1} s\ln n)$ examples in time linear in $((\frac{2en}{s})^s,2^{s+l_1}s\ln n)$. Second, if assume each literal appears in at most d terms, the class of l-term s-DNFs with $l_1$ $\log 4sd$-terms can be exactly learned using $O(2^{s+l_1}\cdot e^{\frac{l}{sd}} s\ln n)$ examples in time linear in $((\frac{2en}{s})^s,2^{s+l_1}\cdot e^{\frac{l}{sd}}s\ln n)$.