Reproducing kernel Hilbert spaces in learning theory

We analyze the regularized least square algorithm in learning theory with Reproducing Kernel Hilbert Spaces (RKHS). Explicit convergence rates for the regression and binary classification problems are obtained in particular for the polynomial and Gaussian kernels on the n-dimensional sphere and the hypercube. There are two major ingredients in our approach: (i) a law of large numbers for Hilbert space-valued random variables; (ii) Mercer's theorem and the spectrum of the integral operator associated with the given reproducing kernel. Our work also illustrates RKHS as a unifying framework for solving many problems encountered in computational learning theory.