Transfer learning for functional linear regression via control variates

TL;DR

This paper introduces a novel transfer learning approach for scalar-on-function regression using control variates, which relies on summary statistics and is suitable for privacy-sensitive or decentralized data settings.

Contribution

It establishes a theoretical link between offset transfer learning and control variates, and derives convergence rates considering discretization errors in functional data.

Findings

Proposed CVS-based estimators perform competitively in estimation and prediction.

Theoretical connection between O-TL and CVS-based TL is established.

Convergence rates account for discretization smoothing errors.

Abstract

Transfer learning (TL) has emerged as a powerful tool for improving estimation and prediction performance by leveraging information from related datasets, with the offset TL (O-TL) being a prevailing implementation. In this paper, we adapt the control-variates (CVS) method for TL and develop CVS-based estimators for scalar-on-function regression, one of the most fundamental models in functional data analysis. These estimators rely exclusively on dataset-specific summary statistics, thereby avoiding the pooling of subject-level data and remaining applicable in privacy-restricted or decentralized settings. We establish, for the first time, a theoretical connection between O-TL and CVS-based TL, showing that these two seemingly distinct TL strategies adjust local estimators in fundamentally similar ways. We further derive convergence rates that explicitly account for the unavoidable but typically overlooked smoothing error arising from discretely observed functional predictors, and clarify how similarity among covariance functions across datasets governs the performance of TL. Numerical studies support the theoretical findings and demonstrate that the proposed methods achieve competitive estimation and prediction performance compared with existing alternatives.