We consider the transfer learning problem in the high dimensional linear regression setting, where the feature dimension is larger than the sample size. To learn transferable information, which may vary across features or the source samples, we propose an adaptive transfer learning method that can detect and aggregate the feature-wise (F-AdaTrans) or sample-wise (S-AdaTrans) transferable structures. We achieve this by employing a fused-penalty, coupled with weights that can adapt according to the transferable structure. To choose the weight, we propose a theoretically informed, data-driven procedure, enabling F-AdaTrans to selectively fuse the transferable signals with the target while filtering out non-transferable signals, and S-AdaTrans to obtain the optimal combination of information transferred from each source sample. We show that, with appropriately chosen weights, F-AdaTrans achieves a convergence rate close to that of an oracle estimator with a known transferable structure, and S-AdaTrans recovers existing near-minimax optimal rates as a special case. The effectiveness of the proposed method is validated using both simulation and real data, demonstrating favorable performance compared to the existing methods.

翻译：本文研究高维线性回归背景下的迁移学习问题，其中特征维度大于样本容量。为学习可能随特征或源样本变化的可迁移信息，我们提出一种自适应迁移学习方法，能够检测并聚合特征维度（F-AdaTrans）或样本维度（S-AdaTrans）的可迁移结构。该方法通过采用融合惩罚项实现，其权重可根据可迁移结构自适应调整。为选择权重，我们提出一种理论指导的数据驱动流程：使F-AdaTrans能够选择性融合可迁移信号至目标域并过滤非迁移信号，使S-AdaTrans能够获得各源样本信息的最优迁移组合。我们证明，在适当选择权重的情况下，F-AdaTrans的收敛速率接近已知可迁移结构的Oracle估计器，而S-AdaTrans作为特例可恢复现有的近极小化最优速率。通过仿真与真实数据验证，所提方法相比现有方法展现出更优性能。