An O(n^{log log n}) Learning Algorithm for DNF under the Uniform Distribution

We show that a DNF with terms of size at most d can be approximated by a function at most d^{O(d log 1/ε{lunate})}, nonzero Fourier coefficients such that the expected error squared, with respect to the uniform distribution, is at most ε{lunate}. 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 d^{O(d log 1/ε{lunate})}. 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 n^{O(log log n)}, and a DNF with terms of size at most O(log n/log log n) in polynomial time.