Proper Learning Algorithm for Functions of k Terms under Smooth Distributions

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 ○ script T sign^k_n = {g(f₁(v), ..., f_k(v))\g∈ ℱ_k, f₁, ..., f_k ∈ script T sign_n} in polynomial time for constant k, where ℱ_k is the class of all Boolean functions of k variables and sript T sign_n is the class of terms over n variables. Although class ℱ_k ○ script T sign^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.