Maximum Margin Algorithms with Boolean Kernels

Abstract

Recent work has introduced Boolean kernels with which one can learn over a feature space containing all conjunctions of length up to k (for any 1≤ k ≤ n) over the original n Boolean features in the input space. This motivates the question of whether maximum margin algorithms such as support vector machines can learn Disjunctive Normal Form expressions in the PAC learning model using this kernel. We study this question, as well as a variant in which structural risk minimization (SRM) is performed where the class hierarchy is taken over the length of conjunctions. We show that such maximum margin algorithms do not PAC learn t(n)-term DNF for any t(n) = ω(1), even when used with such a SRM scheme. We also consider PAC learning under the uniform distribution and show that if the kernel uses conjunctions of length $\tilde{\omega}(\sqrt{n})$ then the maximum margin hypothesis will fail on the uniform distribution as well. Our results concretely illustrate that margin based algorithms may overfit when learning simple target functions with natural kernels.