HellingerIEP: Stable Domain Adaptation under Contaminated Covariate Shift

Importance weighting is a standard approach to domain adaptation under covariate shift, but direct density-ratio estimation can be unstable when the unlabeled target sample is contaminated. Existing robust methods often rely on user-specified trimming or reweighting mechanisms. We study how the divergence geometry of a density-ratio objective affects contamination sensitivity. We introduce HellingerIEP, a nonnegative kernel estimator derived from a variational Hellinger objective. Parameterizing the square root of the density ratio yields a convex optimization problem, and, for a fixed kernel dictionary and regularization level, we establish consistency for the corresponding regularized population solution and asymptotic normality under an interiority condition. Under a common square-root ratio parameterization, the Hellinger, KL, and Pearson target-score contributions contain factors of orders $w^{-1}$, $w^{-1/2}$, and $w^{1/2}$, respectively. Thus, Hellinger attenuates added-point score contributions more strongly in high-ratio regions but has greater score magnitude in low-ratio regions, so no uniform sensitivity ordering follows. This nonuniformity suggests a stability–adaptation trade-off, since a large fitted ratio may represent either contamination or genuine target shift. Experiments on synthetic and benchmark domain adaptation tasks show that HellingerIEP is competitive in clean settings and more stable under the contamination regimes considered.