Proper learning algorithm for functions of k terms under smooth distributions

Abstract

In this paper, we introduce a probabilistic distribution, called a smooth distribution, which is a generalization of variants of the uniform distribution such as q-bounded distribution and product distribution. Then, we give an algorithm that, under the smooth distribution, properly learns the class of functions of k terms given as Fk?Tkn={g(f1(v), ?, fk(v))|g?Fk, f1, ?, fk?Tn} in polynomial time for constant k, where Fk is the class of all Boolean functions of k variables and Tn is the class of terms over n variables. Although class Fk?Tkn was shown by Blum and Singh to be learned using DNF as the hypothesis class, it has remained open whether it is properly learnable under a distribution-free setting.