Posterior drift refers to changes in the relationship between responses and covariates while the distributions of the covariates remain unchanged. In this work, we explore functional linear regression under posterior drift with transfer learning. Specifically, we investigate when and how auxiliary data can be leveraged to improve the estimation accuracy of the slope function in the target model when posterior drift occurs. We employ the approximated least square method together with a lasso penalty to construct an estimator that transfers beneficial knowledge from source data. Theoretical analysis indicates that our method avoids negative transfer under posterior drift, even when the contrast between slope functions is quite large. Specifically, the estimator is shown to perform at least as well as the classical estimator using only target data, and it enhances the learning of the target model when the source and target models are sufficiently similar. Furthermore, to address scenarios where covariate distributions may change, we propose an adaptive algorithm using aggregation techniques. This algorithm is robust against non-informative source samples and effectively prevents negative transfer. Simulation and real data examples are provided to demonstrate the effectiveness of the proposed algorithm.

翻译：后验漂移指响应变量与协变量间的关系发生变化，而协变量的分布保持不变。本研究探讨后验漂移条件下结合迁移学习的函数线性回归问题。具体而言，我们研究当后验漂移发生时，如何以及何时可以利用辅助数据提高目标模型中斜率函数的估计精度。我们采用近似最小二乘法结合lasso惩罚项构建估计量，以从源数据中迁移有益知识。理论分析表明，即使在斜率函数间差异较大的情况下，我们的方法也能避免后验漂移导致的负迁移。具体来说，该估计量的性能至少不劣于仅使用目标数据的经典估计量，且当源模型与目标模型足够相似时能提升目标模型的学习效果。此外，针对协变量分布可能发生变化的情形，我们提出一种基于聚合技术的自适应算法。该算法对非信息性源样本具有鲁棒性，并能有效防止负迁移。我们通过仿真和实际数据案例验证了所提算法的有效性。