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 _k^k_n={g(f₁(v), …, f_k(v)) | g_k, f₁, …, f_kn} in polynomial time for constant k, where _k is the class of all Boolean functions of k variables and _n is the class of terms over n variables. Although class _k^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.