Some properties of regularized Kernel methods

In regularized kernel methods, the solution of a learning problem is found by minimizing functionals consisting of the sum of a data and a complexity term. In this paper we investigate some properties of a more general form of the above functionals in which the data term corresponds to the expected risk. First, we prove a quantitative version of the representer theorem holding for both regression and classification, for both differentiable and non-differentiable loss functions, and for arbitrary offset terms. Second, we show that the case in which the offset space is non trivial corresponds to solving a standard problem of regularization in a Reproducing Kernel Hilbert Space in which the penalty term is given by a seminorm. Finally, we discuss the issues of existence and uniqueness of the solution. From the specialization of our analysis to the discrete setting it is immediate to establish a connection between the solution properties of sparsity and coefficient boundedness and some properties of the loss function. For the case of Support Vector Machines for classification, we also obtain a complete characterization of the whole method in terms of the Khun-Tucker conditions with no need to introduce the dual formulation.