Transfer Learning for High-dimensional Linear Regression: Prediction, Estimation, and Minimax Optimality

TL;DR

This paper develops transfer learning methods for high-dimensional linear regression, improving prediction and estimation accuracy by leveraging auxiliary data, with theoretical optimality and practical robustness demonstrated in gene expression studies.

Contribution

It introduces optimal transfer learning estimators and the Trans-Lasso procedure for unknown auxiliary sample informativeness in high-dimensional linear regression.

Findings

Faster convergence rates with auxiliary data when informative

Trans-Lasso is robust to non-informative auxiliary samples

Improved gene expression prediction using transfer learning

Abstract

This paper considers the estimation and prediction of a high-dimensional linear regression in the setting of transfer learning, using samples from the target model as well as auxiliary samples from different but possibly related regression models. When the set of "informative" auxiliary samples is known, an estimator and a predictor are proposed and their optimality is established. The optimal rates of convergence for prediction and estimation are faster than the corresponding rates without using the auxiliary samples. This implies that knowledge from the informative auxiliary samples can be transferred to improve the learning performance of the target problem. In the case that the set of informative auxiliary samples is unknown, we propose a data-driven procedure for transfer learning, called Trans-Lasso, and reveal its robustness to non-informative auxiliary samples and its efficiency in knowledge transfer. The proposed procedures are demonstrated in numerical studies and are applied to a dataset concerning the associations among gene expressions. It is shown that Trans-Lasso leads to improved performance in gene expression prediction in a target tissue by incorporating the data from multiple different tissues as auxiliary samples.