  Human learners have the natural ability to use knowledge gained in one
setting for learning in a different but related setting. This ability to
transfer knowledge from one task to another is essential for effective
learning. In this paper, we study transfer learning in the context of
nonparametric classification based on observations from different distributions
under the posterior drift model, which is a general framework and arises in
many practical problems.
  We first establish the minimax rate of convergence and construct a
rate-optimal two-sample weighted $K$-NN classifier. The results characterize
precisely the contribution of the observations from the source distribution to
the classification task under the target distribution. A data-driven adaptive
classifier is then proposed and is shown to simultaneously attain within a
logarithmic factor of the optimal rate over a large collection of parameter
spaces. Simulation studies and real data applications are carried out where the
numerical results further illustrate the theoretical analysis. Extensions to
the case of multiple source distributions are also considered.
