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 F k ∘ T k n ={ g ( f 1 ( v ), …, f k ( v )) | g ∈ F k , f 1 , …, f k ∈ T n } in polynomial time for constant k , where F k is the class of all Boolean functions of k variables and T n is the class of terms over n variables. Although class F k ∘ T k n 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.