Effective Dimension and Generalization of Kernel Learning

We investigate the generalization performance of some learning problems in Hilbert function Spaces. We introduce a concept of scalesensitive effective data dimension, and show that it characterizes the convergence rate of the underlying learning problem. Using this concept, we can naturally extend results for parametric estimation problems in finite dimensional spaces to nonparametric kernel learning methods. We derive upper bounds on the generalization performance and show that the resulting convergent rates are optimal under various circumstances.