We consider the problem of estimation and prediction of a high-dimensional linear regression in the setting of transfer learning, using samples from the target model as well as samples from some different but possibly related regression models. If some auxiliary samples are known to be ``informative'', we show that the minimax optimal rates for prediction and estimation are faster than methods that do not use the auxiliary data. That is, some knowledge from the informative auxiliary data can been transferred to improve the learning performance of the target problem. Without knowing which auxiliary samples are informative, we propose a data-driven method for transfer learning (Trans-Lasso) and show that under proper conditions its performance is comparable to the oracle case where the set of informative samples are known.
Our proposed approaches are demonstrated in various numerical studies and are applied to a dataset concerning the associations among gene expressions in a target tissue with samples from multiple other tissues as auxiliary data. We show that Trans-Lasso leads to improved performance in gene expression prediction when certain relevant tissues are used.