  Covariate shift occurs prevalently in practice, where the input distributions
of the source and target data are substantially different. Despite its
practical importance in various learning problems, most of the existing methods
only focus on some specific learning tasks and are not well validated
theoretically and numerically. To tackle this problem, we propose a unified
analysis of general nonparametric methods in a reproducing kernel Hilbert space
(RKHS) under covariate shift. Our theoretical results are established for a
general loss belonging to a rich loss function family, which includes many
commonly used methods as special cases, such as mean regression, quantile
regression, likelihood-based classification, and margin-based classification.
Two types of covariate shift problems are the focus of this paper and the sharp
convergence rates are established for a general loss function to provide a
unified theoretical analysis, which concurs with the optimal results in
literature where the squared loss is used. Extensive numerical studies on
synthetic and real examples confirm our theoretical findings and further
illustrate the effectiveness of our proposed method.
