  In transfer learning, we wish to make inference about a target population
when we have access to data both from the distribution itself, and from a
different but related source distribution. We introduce a flexible framework
for transfer learning in the context of binary classification, allowing for
covariate-dependent relationships between the source and target distributions
that are not required to preserve the Bayes decision boundary. Our main
contributions are to derive the minimax optimal rates of convergence (up to
poly-logarithmic factors) in this problem, and show that the optimal rate can
be achieved by an algorithm that adapts to key aspects of the unknown transfer
relationship, as well as the smoothness and tail parameters of our
distributional classes. This optimal rate turns out to have several regimes,
depending on the interplay between the relative sample sizes and the strength
of the transfer relationship, and our algorithm achieves optimality by careful,
decision tree-based calibration of local nearest-neighbour procedures.
