Adaptive Kernel Ridge Regression with Linear Structure: Sharp Oracle Inequalities and Minimax Optimality

TL;DR

This paper introduces an enhanced kernel ridge regression method that explicitly models linear components, achieving optimal prediction risk and better bias-variance trade-offs without extra tuning.

Contribution

It proposes a modified KRR with an explicit linear component, maintaining computational efficiency and achieving minimax optimality in diverse settings.

Findings

The method achieves sharp oracle inequalities.

It adaptively captures both linear and nonlinear structures.

Simulation studies confirm theoretical advantages.

Abstract

Kernel ridge regression (KRR) is a widely used nonparametric method due to its strong theoretical guarantees and computational convenience. However, standard KRR does not distinguish between linear and nonlinear components in the signal, instead applying a single functional regularization to the entire function. This may lead to unnecessary shrinkage of linear structure and consequently suboptimal prediction performance. In this paper, we propose a modified regression procedure that augments KRR with an explicit linear component. The proposed method has the same computational complexity as standard KRR and introduces no additional tuning parameters. Theoretically, we establish a sharp oracle inequality for the proposed estimator and show that it adaptively captures both linear and nonlinear structure, achieving minimax optimal prediction risk under general kernels. Compared with standard KRR, the proposed method improves both the bias and approximation error at the expense of only an additional parametric variance term, which is negligible in low- and moderate-dimensional settings. In high-dimensional regimes, incorporating ridge regularization for the linear component yields a procedure that performs uniformly no worse than KRR. Extensive simulation studies support the theoretical findings.