Online Regularized Learning Algorithm for Functional Data

TL;DR

This paper develops and analyzes an online regularized learning algorithm for functional linear models in reproducing kernel Hilbert spaces, providing convergence rates and addressing saturation issues in unregularized methods.

Contribution

It introduces a regularized online learning algorithm for functional data, achieving improved convergence rates and overcoming saturation boundaries of unregularized algorithms.

Findings

Convergence analysis for excess prediction error and estimation error with different step-sizes.

Regularization uplifts saturation boundary, enabling faster convergence without capacity assumptions.

Constant step-size methods achieve competitive convergence rates.

Abstract

In recent years, functional linear models have attracted growing attention in statistics and machine learning, with the aim of recovering the slope function or its functional predictor. This paper considers online regularized learning algorithm for functional linear models in reproducing kernel Hilbert spaces. Convergence analysis of excess prediction error and estimation error are provided with polynomially decaying step-size and constant step-size, respectively. Fast convergence rates can be derived via a capacity dependent analysis. By introducing an explicit regularization term, we uplift the saturation boundary of unregularized online learning algorithms when the step-size decays polynomially, and establish fast convergence rates of estimation error without capacity assumption. However, it remains an open problem to obtain capacity independent convergence rates for the estimation error of the unregularized online learning algorithm with decaying step-size. It also shows that convergence rates of both prediction error and estimation error with constant step-size are competitive with those in the literature.