  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.
