Proper learning algorithm for functions of κ terms under smooth distributions

Abstract

Algorithms for learning feasibly Boolean functions from examples are explored. A class of functions we deal with is written as F1 oF2k = {g(f1(v),...fk(v)) g ∈ F1, f1...,fk ∈ F2} for classes F1 and F2 given by somewhat "simple" description. Letting Γ = {0,1}, we denote by F1 and F2 a class of functions from Γk to Γ and that of functions from Γn to Γ, respectively. For exa.mple, let FOr consist of an OR function of k variables, and let Fn be the class of all monomials of n variables. In the distribution free setting, it is known that FORo Fnk, denoted usually k-term DNF, is not learnable unless P≠NP In this paper, we first introduce a probabilistic distribution, called a smooth distribution, which is a generalization of both q-bounded distribution and product distribution, and define the learnability under this distribution. Then, we give an algorithm that properly learns FkoTnk under smooth distribution in polynomial time for constant k, where Fk is the class of all Boolean functions of k variables. The class FkoTnk is called the functions of k terms and although it was shown by Blum and Singh to be learned using DNF as a hypothesis class, it remains open whether it is properly learnable under distribution free setting.