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Covariate-shift generalization, a typical case in out-of-distribution (OOD)generalization, requires a good performance on the unknown test distribution,which varies from the accessible training distribution in the form of covariateshift. Recently, independence-driven importance weighting algorithms in stablelearning literature have shown empirical effectiveness to deal withcovariate-shift generalization on several learning models, including regressionalgorithms and deep neural networks, while their theoretical analyses aremissing. In this paper, we theoretically prove the effectiveness of suchalgorithms by explaining them as feature selection processes. We first specifya set of variables, named minimal stable variable set, that is the minimal andoptimal set of variables to deal with covariate-shift generalization for commonloss functions, such as the mean squared loss and binary cross-entropy loss.Afterward, we prove that under ideal conditions, independence-driven importanceweighting algorithms could identify the variables in this set. Analysis ofasymptotic properties is also provided. These theories are further validated inseveral synthetic experiments.

A Theoretical Analysis on Independence-driven Importance Weighting for Covariate-shift Generalization