Bandwidth Selection for Gaussian Kernel Ridge Regression via Jacobian Control

TL;DR

This paper introduces a fast, Jacobian-based heuristic for selecting the Gaussian kernel bandwidth in kernel ridge regression, offering comparable performance to traditional methods but with significantly reduced computational cost.

Contribution

A novel, closed-form bandwidth selection method based on Jacobian regularization, reducing computational complexity in Gaussian kernel ridge regression.

Findings

Method achieves similar accuracy to cross-validation and marginal likelihood methods.

Significantly faster, up to six orders of magnitude in computational speed.

Effective on both real and synthetic datasets.

Abstract

Most machine learning methods require tuning of hyper-parameters. For kernel ridge regression with the Gaussian kernel, the hyper-parameter is the bandwidth. The bandwidth specifies the length scale of the kernel and has to be carefully selected to obtain a model with good generalization. The default methods for bandwidth selection, cross-validation and marginal likelihood maximization, often yield good results, albeit at high computational costs. Inspired by Jacobian regularization, we formulate an approximate expression for how the derivatives of the functions inferred by kernel ridge regression with the Gaussian kernel depend on the kernel bandwidth. We use this expression to propose a closed-form, computationally feather-light, bandwidth selection heuristic, based on controlling the Jacobian. In addition, the Jacobian expression illuminates how the bandwidth selection is a trade-off between the smoothness of the inferred function and the conditioning of the training data kernel matrix. We show on real and synthetic data that compared to cross-validation and marginal likelihood maximization, our method is on pair in terms of model performance, but up to six orders of magnitude faster.