  This paper studies kernel ridge regression in high dimensions under covariate
shifts and analyzes the role of importance re-weighting. We first derive the
asymptotic expansion of high dimensional kernels under covariate shifts. By a
bias-variance decomposition, we theoretically demonstrate that the re-weighting
strategy allows for decreasing the variance. For bias, we analyze the
regularization of the arbitrary or well-chosen scale, showing that the bias can
behave very differently under different regularization scales. In our analysis,
the bias and variance can be characterized by the spectral decay of a
data-dependent regularized kernel: the original kernel matrix associated with
an additional re-weighting matrix, and thus the re-weighting strategy can be
regarded as a data-dependent regularization for better understanding. Besides,
our analysis provides asymptotic expansion of kernel functions/vectors under
covariate shift, which has its own interest.
