A new similarity measure for covariate shift with applications to nonparametric regression

TL;DR

This paper introduces a novel measure of distribution mismatch for covariate shift in nonparametric regression, providing sharper convergence rates and detailed insights into the estimation process under distributional changes.

Contribution

It proposes a new, more precise similarity measure for covariate shift, improving theoretical understanding and convergence rate analysis in nonparametric regression.

Findings

New measure leads to sharper convergence rates

Characterizes minimax estimation under covariate shift

Illustrates differences with existing transfer exponent concept

Abstract

We study covariate shift in the context of nonparametric regression. We introduce a new measure of distribution mismatch between the source and target distributions that is based on 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\"older 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.