1 Rademacher Chaos Complexities for Learning the Kernel Problem Yiming Ying 1 mathying@gmail.com College of Engineering, Mathematics and Physical Sciences University of Exeter, Harrison Building Exeter, EX4 4QF, UK Colin Campbell C.Campbell@bristol.ac.uk Department of Engineering Mathematics University of Bristol, Queen’s Building Bristol, BS8 1TR, UK Abstract In this paper we develop a novel generalization bound for learning the kernel problem. First, we show that the generalization analysis of the kernel learning problem reduces to investigation of the suprema of the Rademacher chaos pro- cess of order two over candidate kernels, which we refer to as Rademacher chaos complexity. Next, we show how to estimate the empirical Rademacher chaos com- plexity by well-established metric entropy integrals and pseudo-dimension of the set of candidate kernels. Our new methodology mainly depends on the principal theory of U-processes and entropy integrals. Finally, we establish satisfactory ex- cess generalization bounds and misclassification error rates for learning Gaussian kernels and general radial basis kernels. Keywords: Learning the kernel, generalization bound, Rademacher chaos, Rademacher averages, entropy integrals 1 Corresponding author. Tel: +44(0)1392 723591 Fax: +44(0)1392 217965