Infinite Kernel Learning: Generalization Bounds and Algorithms

Abstract

Kernel learning is a fundamental problem both in recent research and application of kernel methods. Existing kernel learning methods commonly use some measures of generalization errors to learn the optimal kernel in a convex (or conic) combination of prescribed basic kernels. However, the generalization bounds derived by these measures usually have slow convergence rates, and the basic kernels are finite and should be specified in advance. In this paper, we propose a new kernel learning method based on a novel measure of generalization error, called principal eigenvalue proportion (PEP), which can learn the optimal kernel with sharp generalization bounds over the convex hull of a possibly infinite set of basic kernels. We first derive sharp generalization bounds based on the PEP measure. Then we design two kernel learning algorithms for finite kernels and infinite kernels respectively, in which the derived sharp generalization bounds are exploited to guarantee faster convergence rates, moreover, basic kernels can be learned automatically for infinite kernel learning instead of being prescribed in advance. Theoretical analysis and empirical results demonstrate that the proposed kernel learning method outperforms the state-of-the-art kernel learning methods.