  We develop here a novel transfer learning methodology called Profiled
Transfer Learning (PTL). The method is based on the \textit{approximate-linear}
assumption between the source and target parameters. Compared with the commonly
assumed \textit{vanishing-difference} assumption and \textit{low-rank}
assumption in the literature, the \textit{approximate-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
\textit{approximate-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.
