Early stopping and polynomial smoothing in regression with reproducing kernels
This work addresses the challenge of determining optimal stopping points in kernel regression for practitioners, though it is incremental as it builds on existing techniques like the minimum discrepancy principle.
The paper tackles the problem of early stopping for iterative learning algorithms in RKHS-based nonparametric regression, proposing a data-driven rule without a validation set that is proven to be minimax-optimal across kernel spaces and shows comparable performance to cross-validation in simulations.
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.