Many existing two-phase kernel-based hypothesis transfer learning algorithms employ the same kernel regularization across phases and rely on the known smoothness of functions to obtain optimality. Therefore, they fail to adapt to the varying and unknown smoothness between the target/source and their offset in practice. In this paper, we address these problems by proposing Smoothness Adaptive Transfer Learning (SATL), a two-phase kernel ridge regression(KRR)-based algorithm. We first prove that employing the misspecified fixed bandwidth Gaussian kernel in target-only KRR learning can achieve minimax optimality and derive an adaptive procedure to the unknown Sobolev smoothness. Leveraging these results, SATL employs Gaussian kernels in both phases so that the estimators can adapt to the unknown smoothness of the target/source and their offset function. We derive the minimax lower bound of the learning problem in excess risk and show that SATL enjoys a matching upper bound up to a logarithmic factor. The minimax convergence rate sheds light on the factors influencing transfer dynamics and demonstrates the superiority of SATL compared to non-transfer learning settings. While our main objective is a theoretical analysis, we also conduct several experiments to confirm our results.

翻译：许多现有的两阶段基于核的假设迁移学习算法在各阶段使用相同的核正则化，并依赖函数的已知光滑性来达到最优性。因此，在实际应用中，它们无法适应目标/源函数及其偏移之间变化且未知的光滑性。本文提出平滑自适应迁移学习（SATL），一种基于两阶段核岭回归（KRR）的算法，以解决这些问题。我们首先证明，在仅使用目标数据的KRR学习中，采用错误指定的固定带宽高斯核可以实现极小极大最优性，并推导出适应未知Sobolev光滑性的自适应过程。利用这些结果，SATL在两个阶段均使用高斯核，从而使估计量能够自适应目标/源函数及其偏移函数的未知光滑性。我们推导了学习问题超额风险的极小极大下界，并证明SATL在达到该下界时仅差一个对数因子。极小极大收敛速度揭示了影响迁移动态的因素，并展示了SATL相对于非迁移学习设置的优越性。尽管我们的主要目标是理论分析，但我们也进行了若干实验来验证所得结果。