Learning kernel-based halfspaces with the zero-one loss

We describe and analyze a new algorithm for agnostically learning kernel-based halfspaces with respect to the zero-one loss function. Unlike most previous formulations which rely on surrogate convex loss functions (e.g. hinge-loss in SVM and log-loss in logistic regression), we provide finite time/sample guarantees with respect to the more natural zero-one loss function. The proposed algorithm can learn kernel-based halfspaces in worst-case time poly (exp (L log(L/ε))), for any distribution, where L is a Lipschitz constant (which can be thought of as the reciprocal of the margin), and the learned classifier is worse than the optimal halfspace by at most ε. We also prove a hardness result, showing that under a certain cryptographic assumption, no algorithm can learn kernel-based halfspaces in time polynomial in L.