  In supervised learning, training and test datasets are often sampled from
distinct distributions. Domain adaptation techniques are thus required.
Covariate shift adaptation yields good generalization performance when domains
differ only by the marginal distribution of features. Covariate shift
adaptation is usually implemented using importance weighting, which may fail,
according to common wisdom, due to small effective sample sizes (ESS). Previous
research argues this scenario is more common in high-dimensional settings.
However, how effective sample size, dimensionality, and model
performance/generalization are formally related in supervised learning,
considering the context of covariate shift adaptation, is still somewhat
obscure in the literature. Thus, a main challenge is presenting a unified
theory connecting those points. Hence, in this paper, we focus on building a
unified view connecting the ESS, data dimensionality, and generalization in the
context of covariate shift adaptation. Moreover, we also demonstrate how
dimensionality reduction or feature selection can increase the ESS, and argue
that our results support dimensionality reduction before covariate shift
adaptation as a good practice.
