April 2024 Transfer learning for functional mean estimation: Phase transition and adaptive algorithms
T. Tony Cai, Dongwoo Kim, Hongming Pu
Author Affiliations +
Ann. Statist. 52(2): 654-678 (April 2024). DOI: 10.1214/24-AOS2362


This paper studies transfer learning for estimating the mean of random functions based on discretely sampled data, where in addition to observations from the target distribution, auxiliary samples from similar but distinct source distributions are available. The paper considers both common and independent designs and establishes the minimax rates of convergence for both designs. The results reveal an interesting phase transition phenomenon under the two designs and demonstrate the benefits of utilizing the source samples in the low sampling frequency regime.

For practical applications, this paper proposes novel data-driven adaptive algorithms that attain the optimal rates of convergence within a logarithmic factor simultaneously over a large collection of parameter spaces. The theoretical findings are complemented by a simulation study that further supports the effectiveness of the proposed algorithms.

Funding Statement

The research of Tony Cai was supported in part by NSF Grant DMS-2015259 and NIH Grant R01-GM129781.


The authors extend their gratitude to the Editor, the Associate Editor and the anonymous referees for their constructive and insightful comments and suggestions, which have significantly enhanced the quality of this paper.


Download Citation

T. Tony Cai. Dongwoo Kim. Hongming Pu. "Transfer learning for functional mean estimation: Phase transition and adaptive algorithms." Ann. Statist. 52 (2) 654 - 678, April 2024. https://doi.org/10.1214/24-AOS2362


Received: 1 May 2023; Revised: 1 October 2023; Published: April 2024
First available in Project Euclid: 9 May 2024

Digital Object Identifier: 10.1214/24-AOS2362

Primary: 62J05
Secondary: 62G20

Keywords: Adaptivity , common design , Functional data analysis , independent design , mean function , minimax rate of convergence , phase transition , transfer learning

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 2 • April 2024
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