Summary: We establish learning rates to the Bayes risk for support vector machines (SVMs) using a regularization sequence \(\lambda_{n} = n^{-\alpha}\), where \(\alpha \in (0,1)\) is arbitrary. Under a noise condition recently proposed by Tsybakov these rates can become faster than \(n^{-1/2}\). In order to deal with the approximation error we present a general concept called the approximation error function which describes how well the infinite sample versions of the considered SVMs approximate the data-generating distribution. In addition we discuss in some detail the relation between the “classical” approximation error and the approximation error function. Finally, for distributions satisfying a geometric noise assumption we establish some learning rates when the used RKHS is a Sobolev space.
For the entire collection see [Zbl 1076.68003].