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Transfer learning for scalar-on-function regression via control variates

Yuping Yang, Zhiyang Zhou

TL;DR

This paper adapts the control-variates transfer-learning (CVS) framework to scalar-on-function regression (SoFR), enabling privacy-preserving information transfer using dataset-specific summaries rather than pooling subject-level data. It formalizes the problem across a target and multiple sources, introduces two CVS-based estimators (CVS and penalized CVS, pCVS), and links them to existing transfer-learning strategies such as offset-TL (O-TL) and aggregation-based OTl. The authors derive convergence rates that explicitly account for smoothing and measurement error in discretely observed functional data and depend on covariance-function similarity across datasets, validated by simulations and a stock-return application. The work demonstrates that CVS-based methods can achieve competitive estimation and prediction while preserving data privacy, and outlines practical extensions and avenues for improving variance estimation. Overall, the framework broadens TL in functional data analysis with rigorous theory and practical privacy-preserving benefits.

Abstract

Transfer learning (TL) has emerged as a powerful tool for improving estimation and prediction performance by leveraging information from related datasets. In this paper, we repurpose the control-variates (CVS) method for TL in the context of scalar-on-function regression. Our proposed framework relies exclusively on dataset-specific summary statistics, avoiding the need to pool subject-level data and thus remaining applicable in privacy-restricted or decentralized settings. We establish theoretical connections among several existing TL strategies and derive convergence rates for our CVS-based proposals. These rates explicitly account for the typically overlooked smoothing error and reveal how the similarity among covariance functions across datasets influences convergence behavior. Numerical studies support the theoretical findings and demonstrate that the proposed methods achieve competitive estimation and prediction performance compared with existing alternatives.

Transfer learning for scalar-on-function regression via control variates

TL;DR

This paper adapts the control-variates transfer-learning (CVS) framework to scalar-on-function regression (SoFR), enabling privacy-preserving information transfer using dataset-specific summaries rather than pooling subject-level data. It formalizes the problem across a target and multiple sources, introduces two CVS-based estimators (CVS and penalized CVS, pCVS), and links them to existing transfer-learning strategies such as offset-TL (O-TL) and aggregation-based OTl. The authors derive convergence rates that explicitly account for smoothing and measurement error in discretely observed functional data and depend on covariance-function similarity across datasets, validated by simulations and a stock-return application. The work demonstrates that CVS-based methods can achieve competitive estimation and prediction while preserving data privacy, and outlines practical extensions and avenues for improving variance estimation. Overall, the framework broadens TL in functional data analysis with rigorous theory and practical privacy-preserving benefits.

Abstract

Transfer learning (TL) has emerged as a powerful tool for improving estimation and prediction performance by leveraging information from related datasets. In this paper, we repurpose the control-variates (CVS) method for TL in the context of scalar-on-function regression. Our proposed framework relies exclusively on dataset-specific summary statistics, avoiding the need to pool subject-level data and thus remaining applicable in privacy-restricted or decentralized settings. We establish theoretical connections among several existing TL strategies and derive convergence rates for our CVS-based proposals. These rates explicitly account for the typically overlooked smoothing error and reveal how the similarity among covariance functions across datasets influences convergence behavior. Numerical studies support the theoretical findings and demonstrate that the proposed methods achieve competitive estimation and prediction performance compared with existing alternatives.
Paper Structure (18 sections, 10 theorems, 128 equations, 2 figures, 4 algorithms)

This paper contains 18 sections, 10 theorems, 128 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Methodology
  4. Offset transfer learning
  5. Control-variates method
  6. Another look at the control-covariates method
  7. Theoretical results
  8. Numerical illustration
  9. Simulation study
  10. Application to the prediction of stock returns
  11. Conclusion and discussion
  12. Technical details
  13. Deriving Eq. \ref{['eq:U_star']}
  14. Approximating expectations and variances conditional on contaminated observations
  15. Inverse matrices in Eqs. \ref{['eq:l_c_delta']} and \ref{['eq:pl_c_delta']}
  16. ...and 3 more sections

Key Result

Proposition 1

Under conditions cond:trajectories, cond:rho_J, and cond:xi, and

Figures (2)

  • Figure 1: Boxplots of values of REE (left panel) and RPE (right panel) for O-TL, AO-TL, CVS, and pCVS based on 100 simulative datasets. In each subfigure, boxplots are generated for different values of $\eta$.
  • Figure 2: Boxplots of RPE values for O-TL, AO-TL, CVS, and pCVS based on 100 replications. Each subfigure corresponds to the prediction of MRs in a given month using MCRs from the preceding month. For each month pair, we cycle through the 11 sectors: basic industries (BI), capital goods (CG), consumer durable (CD), consumer non-durable (CND), consumer services (CS), energy (E), finance (Fin), health care (HC), public utility (PU), technology (Tech), and transportation (Trans). Values of RPE are computed by treating each sector in turn as the target and the remaining sectors as sources.

Theorems & Definitions (19)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 1: MarshallOlkinArnold2011, pp. 340--341
  • Lemma 2: MerikoskiKumar2004, Theorems 1 and 7
  • Lemma 3
  • Lemma 4: LiHsing2007, Theorem 4 and its proof
  • Lemma 5
  • ...and 9 more