On the Inconsistency of Kernel Ridgeless Regression in Fixed Dimensions

TL;DR

This paper demonstrates that kernel ridge regression with translation-invariant kernels does not exhibit benign overfitting in fixed dimensions, as the estimator fails to converge to the true function regardless of sample size or bandwidth choice.

Contribution

It provides the first fixed design analysis showing the lack of benign overfitting for translation-invariant kernels in fixed dimensions, with explicit error decompositions.

Findings

Kernel methods with translation-invariant kernels do not achieve benign overfitting.

The generalization error does not vanish with increasing sample size.

Bandwidth selection does not lead to convergence to the true function.

Abstract

``Benign overfitting'', the ability of certain algorithms to interpolate noisy training data and yet perform well out-of-sample, has been a topic of considerable recent interest. We show, using a fixed design setup, that an important class of predictors, kernel machines with translation-invariant kernels, does not exhibit benign overfitting in fixed dimensions. In particular, the estimated predictor does not converge to the ground truth with increasing sample size, for any non-zero regression function and any (even adaptive) bandwidth selection. To prove these results, we give exact expressions for the generalization error, and its decomposition in terms of an approximation error and an estimation error that elicits a trade-off based on the selection of the kernel bandwidth. Our results apply to commonly used translation-invariant kernels such as Gaussian, Laplace, and Cauchy.