Proper learning of k-term DNF formulas from satisfying assignments

Abstract

In certain applications there may be only positive examples available to learn concepts of a class of interest. Furthermore, learning has to be done properly, i.e. the hypothesis space has to coincide with the concept class, and without false positives, i.e. the hypothesis always has to be a subset of the real concept (one-sided error). For the well studied class of k-term DNF formulas it has been known that learning is difficult. Unless RP = NP, it is not feasible to learn k-term DNF formulas properly in a distribution-free sense even if both positive and negative examples are available and even if false positives are allowed.This paper constructs an efficient algorithm that, for fixed but arbitrary k and q, if examples are drawn from q-bounded distributions, it properly learns the class of k-term DNFs without false positives from positive examples alone with arbitrarily small relative error.