Determination of hyper-parameters for kernel based classification and regression

The optimization of the hyper-parameters of a statistical procedure or machine learning task is a crucial step for obtaining a minimal error. Unfortunately, the optimization of hyper-parameters usually requires many runs of the procedure and hence is very costly. A more detailed knowledge of the dependency of the performance of a procedure on its hyper-parameters can help to speed up this process. In this paper, we investigate the case of kernel-based classifiers and regression estimators which belong to the class of convex risk minimization methods from machine learning. In an empirical investigation, the response surfaces of nonlinear support vector machines and kernel logistic regression are analyzed and the performance of several algorithms for determining hyper-parameters is investigated. The rest of the paper is organized as follows: Section 2 briefly outlines kernel based classification and regression methods. Section 3 gives details on several methods for optimizing the hyper-parameters of statistical procedures. Then, some numerical examples are presented in Section 4. Section 5 contains a discussion. Finally, all figures are given in the appendix.