Early stopping and non-parametric regression: An optimal data-dependent stopping rule

TL;DR

This paper introduces a data-dependent early stopping rule for non-parametric regression in RKHS, achieving minimax-optimal rates without cross-validation, and demonstrates its effectiveness through theoretical analysis and simulations.

Contribution

It proposes a novel, data-dependent early stopping rule for gradient descent in non-parametric regression that does not require hold-out data and attains optimal convergence rates.

Findings

The stopping rule achieves minimax-optimal rates for various kernel classes.

Simulation results show the stopping rule outperforms other methods.

The approach is closely connected to kernel ridge regression solutions.

Abstract

The strategy of early stopping is a regularization technique based on choosing a stopping time for an iterative algorithm. Focusing on non-parametric regression in a reproducing kernel Hilbert space, we analyze the early stopping strategy for a form of gradient-descent applied to the least-squares loss function. We propose a data-dependent stopping rule that does not involve hold-out or cross-validation data, and we prove upper bounds on the squared error of the resulting function estimate, measured in either the $L^2(P)$ and $L^2(P_n)$ norm. These upper bounds lead to minimax-optimal rates for various kernel classes, including Sobolev smoothness classes and other forms of reproducing kernel Hilbert spaces. We show through simulation that our stopping rule compares favorably to two other stopping rules, one based on hold-out data and the other based on Stein's unbiased risk estimate. We also establish a tight connection between our early stopping strategy and the solution path of a kernel ridge regression estimator.