Lp- and Risk Consistency of Localized SVMs

Abstract

Kernel-based regularized risk minimizers, also called support vector machines (SVMs), are known to possess many desirable properties but suffer from their super-linear computational requirements when dealing with large data sets. This problem can be tackled by using localized SVMs instead, which also offer the additional advantage of being able to apply different hyperparameters to different regions of the input space. In this paper, localized SVMs are analyzed with regards to their consistency. It is proven that they inherit $L_p$- as well as risk consistency from global SVMs under very weak conditions and even if the regions underlying the localized SVMs are allowed to change as the size of the training data set increases.