An O(nlog log n) Learning Algorithm for DNF under the Uniform Distribution

摘要

We show that a DNF with terms of size at most d can be approximated by a function at most dO(d log 1/ϵ), nonzero Fourier coefficients such that the expected error squared, with respect to the uniform distribution, is at most ϵ. This property is used to derive a learning algorithm for DNF, under the uniform distribution. The learning algorithm uses queries and learns, with respect to the uniform distribution, a DNF with terms of size at most d in time polynomial in n and dO(d log 1/ϵ). The interesting implications are for the case when epsilon is constant. In this case our algorithm learns a DNF with a polynomial number of terms in time nO(log log n), and a DNF with terms of size at most O(log n/log log n) in polynomial time.