L2 Regularization for Learning Kernels

TL;DR

This paper investigates L2 regularization for learning kernels in regression, providing theoretical analysis, an efficient algorithm, and experimental results showing its advantages over L1 regularization, especially with many kernels.

Contribution

It introduces an L2 regularization framework for kernel learning, derives the solution form, and offers theoretical and empirical evidence of its effectiveness.

Findings

L2 regularization improves performance with many kernels

Theoretical stability bounds are established for orthogonal kernels

L1 regularization can degrade performance in large-scale cases

Abstract

The choice of the kernel is critical to the success of many learning algorithms but it is typically left to the user. Instead, the training data can be used to learn the kernel by selecting it out of a given family, such as that of non-negative linear combinations of p base kernels, constrained by a trace or L1 regularization. This paper studies the problem of learning kernels with the same family of kernels but with an L2 regularization instead, and for regression problems. We analyze the problem of learning kernels with ridge regression. We derive the form of the solution of the optimization problem and give an efficient iterative algorithm for computing that solution. We present a novel theoretical analysis of the problem based on stability and give learning bounds for orthogonal kernels that contain only an additive term O(pp/m) when compared to the standard kernel ridge regression stability bound. We also report the results of experiments indicating that L1 regularization can lead to modest improvements for a small number of kernels, but to performance degradations in larger-scale cases. In contrast, L2 regularization never degrades performance and in fact achieves significant improvements with a large number of kernels.