  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.
