Convergence Analysis of Kernel Conjugate Gradient for Functional Linear Regression

TL;DR

This paper analyzes the convergence behavior of a kernel conjugate gradient algorithm for functional linear regression, establishing rates that match known minimax bounds and highlighting the role of regularity and eigenvalue decay.

Contribution

It provides a detailed convergence analysis of the kernel conjugate gradient method in functional linear models, incorporating early stopping as regularization.

Findings

Convergence rates depend on the regularity of the slope function.

Rates match the minimax optimal bounds from existing literature.

Eigenvalue decay influences the convergence speed.

Abstract

In this paper, we discuss the convergence analysis of the conjugate gradient-based algorithm for the functional linear model in the reproducing kernel Hilbert space framework, utilizing early stopping results in regularization against over-fitting. We establish the convergence rates depending on the regularity condition of the slope function and the decay rate of the eigenvalues of the operator composition of covariance and kernel operator. Our convergence rates match the minimax rate available from the literature.