Convergence rates of Kernel Conjugate Gradient for random design regression

TL;DR

This paper establishes convergence rates for kernel conjugate gradient regression with early stopping, showing near-optimal performance depending on the target function's regularity and data complexity.

Contribution

It provides the first statistical convergence rates for kernel conjugate gradient regression with early stopping, matching minimax bounds under various conditions.

Findings

Convergence rates depend on target function regularity and data complexity.

Rates match known minimax lower bounds for prediction and Hilbert norms.

Additional unlabeled data improve convergence when the true function is outside the RKHS.

Abstract

We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the $\mathcal{L}^2$ (prediction) norm as well as for the stronger Hilbert norm, if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available.