  Transfer learning algorithms have been developed in various applicational
contexts while only a few of them offer statistical guarantees in
high-dimensions. Among these work, the differences between the target and
sources, a.k.a. the contrasts, are typically modeled as, or at least close to,
vectors with certain low-dimensional structure (e.g., sparsity), resulting in a
separate debiasing step after a preceding pooling estimation procedure. Under
such intuitive yet powerful framework, additional homogeneity conditions on
Hessian matrices of the population loss functions are often imposed to preserve
the delicate low-dimensional structure of the contrasts during pooling, which
is either unrealistic in practice or easily destroyed by basic data
transformation such as standardization. In this article, under the general
M-estimators framework with decomposable regularizers, we highlight the role of
fine-tuning underneath the conspicuous gain of the debiasing step in transfer
learning. Namely, we find it is possible to enhance estimation accuracy by
fine-tuning a primal estimator sufficiently close to the true target one. Our
theory suggests slightly enlarging the pooling regularization strength when
either the contrast's low-dimensional structure or the homogeneity of Hessian
matrices is violated. Traditional linear regression and generalized low-rank
trace regression in high-dimensions are discussed as two specific examples
under our framework. When the informative source datasets are unknown, a novel
truncated-penalized algorithm is proposed to directly output the primal
estimator by simultaneously selecting the useful sources and its oracle
property is proved. Extensive numerical experiments are conducted to validate
the theoretical assertions. A case study on the air quality regulation in China
by transfer learning is also provided for illustration.
