Proper learning algorithm for functions of κ terms under smooth distributions

Algorithms for learning feasibly Boolean functions from examples are explored. A class of functions we deal with is written as F₁ oF₂^k = {g(f₁(v),...f_k(v)) g ∈ F₁, f₁...,f_k ∈ 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 Γ^k to Γ and that of functions from Γⁿ to Γ, respectively. For exa.mple, let F_Or consist of an OR function of k variables, and let F_n be the class of all monomials of n variables. In the distribution free setting, it is known that F_ORo F_n^k, 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 F_koT_n^k under smooth distribution in polynomial time for constant k, where F_k is the class of all Boolean functions of k variables. The class F_koT_n^k 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.