The presence of distribution shifts poses a significant challenge for deploying modern machine learning models in real-world applications. This work focuses on the target shift problem in a regression setting (Zhang et al., 2013; Nguyen et al., 2016). More specifically, the target variable $y$ (also known as the response variable), which is continuous, has different marginal distributions in the training source and testing domain, while the conditional distribution of features $\boldsymbol{x}$ given $y$ remains the same. While most literature focuses on classification tasks with finite target space, the regression problem has an *infinite dimensional* target space, which makes many of the existing methods inapplicable. In this work, we show that the continuous target shift problem can be addressed by estimating the importance weight function from an ill-posed integral equation. We propose a nonparametric regularized approach named *ReTaSA* to solve the ill-posed integral equation and provide theoretical justification for the estimated importance weight function. The effectiveness of the proposed method has been demonstrated with extensive numerical studies on synthetic and real-world datasets.

We study the domain adaptation problem with label shift in this work. Under the label shift context, the marginal distribution of the label varies across the training and testing datasets, while the conditional distribution of features given the label is the same. Traditional label shift adaptation methods either suffer from large estimation errors or require cumbersome post-prediction calibrations. To address these issues, we first propose a moment-matching framework for adapting the label shift based on the geometry of the influence function. Under such a framework, we propose a novel method named $\underline{\mathrm{E}}$fficient $\underline{\mathrm{L}}$abel $\underline{\mathrm{S}}$hift $\underline{\mathrm{A}}$daptation (ELSA), in which the adaptation weights can be estimated by solving linear systems. Theoretically, the ELSA estimator is $\sqrt{n}$-consistent ($n$ is the sample size of the source data) and asymptotically normal. Empirically, we show that ELSA can achieve state-of-the-art estimation performances without post-prediction calibrations, thus, gaining computational efficiency.