Profiled Transfer Learning for High Dimensional Linear Model

We develop here a novel transfer learning methodology called Profiled Transfer Learning (PTL). The method is based on the textitapproximate-linear assumption between the source and target parameters. Compared with the commonly assumed textitvanishing-difference assumption and textitlow-rank assumption in the literature, the textitapproximate-linear assumption is more flexible and less stringent. Specifically, the PTL estimator is constructed by two major steps. Firstly, we regress the response on the transferred feature, leading to the profiled responses. Subsequently, we learn the regression relationship between profiled responses and the covariates on the target data. The final estimator is then assembled based on the textitapproximate-linear relationship. To theoretically support the PTL estimator, we derive the non-asymptotic upper bound and minimax lower bound. We find that the PTL estimator is minimax optimal under appropriate regularity conditions. Extensive simulation studies are presented to demonstrate the finite sample performance of the new method. A real data example about sentence prediction is also presented with very encouraging results.