We study covariate shift in the context of nonparametric regression. We introduce a new measure of distribution mismatch between the source and target distributions using the integrated ratio of probabilities of balls at a given radius. We use the scaling of this measure with respect to the radius to characterize the minimax rate of estimation over a family of H{รถ}lder continuous functions under covariate shift. In comparison to the recently proposed notion of transfer exponent, this measure leads to a sharper rate of convergence and is more fine-grained. We accompany our theory with concrete instances of covariate shift that illustrate this sharp difference.
Free, publicly-accessible full text available July 1, 2023
