This paper studies transfer learning for estimating the mean of random functions based on discretely sampled data, where, in addition to observations from the target distribution, auxiliary samples from similar but distinct source distributions are available. The paper considers both common and independent designs and establishes the minimax rates of convergence for both designs. The results reveal an interesting phase transition phenomenon under the two designs and demonstrate the benefits of utilizing the source samples in the low sampling frequency regime. For practical applications, this paper proposes novel data-driven adaptive algorithms that attain the optimal rates of convergence within a logarithmic factor simultaneously over a large collection of parameter spaces. The theoretical findings are complemented by a simulation study that further supports the effectiveness of the proposed algorithms

翻译：本文研究基于离散采样数据对随机函数均值估计的迁移学习问题，其中除了目标分布的观测数据外，还可获得来自相似但不同源分布的辅助样本。本文同时考虑共同设计与独立设计两种采样方案，并建立了两种方案下的极小化收敛速度。研究结果揭示了两种设计方案下有趣的相变现象，并证明了在低采样频率场景下利用源样本的优势。针对实际应用，本文提出了新颖的数据驱动自适应算法，该算法能在大量参数空间上同时达到对数因子内的最优收敛速度。仿真实验进一步验证了所提算法的有效性，与理论结果相互印证。