Early stopping and polynomial smoothing in regression with reproducing kernels

TL;DR

This paper introduces a data-driven early stopping rule for kernel-based regression algorithms that is theoretically optimal and performs well in practice without requiring a validation set.

Contribution

It proposes a novel early stopping rule based on the minimum discrepancy principle for RKHS regression, with proven minimax optimality across various kernel spaces.

Findings

The rule is minimax-optimal over multiple kernel classes.

Simulation results show competitive performance with cross-validation.

The method requires only the RKHS assumption on the regression function.

Abstract

In this paper, we study the problem of early stopping for iterative learning algorithms in a reproducing kernel Hilbert space (RKHS) in the nonparametric regression framework. In particular, we work with the gradient descent and (iterative) kernel ridge regression algorithms. We present a data-driven rule to perform early stopping without a validation set that is based on the so-called minimum discrepancy principle. This method enjoys only one assumption on the regression function: it belongs to a reproducing kernel Hilbert space (RKHS). The proposed rule is proved to be minimax-optimal over different types of kernel spaces, including finite-rank and Sobolev smoothness classes. The proof is derived from the fixed-point analysis of the localized Rademacher complexities, which is a standard technique for obtaining optimal rates in the nonparametric regression literature. In addition to that, we present simulation results on artificial datasets that show the comparable performance of the designed rule with respect to other stopping rules such as the one determined by V-fold cross-validation.