# Learning under Target Shift: Optimal Density Ratio Estimation and Importance-Weighted Regression

arXiv: https://arxiv.org/abs/2609.15785 · HTML: https://mlsanity.com/paper/2609.15785/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 45 references that are in the map.
- First posted on arXiv 14 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [Detecting and Correcting for Label Shift with Black Box Predictors](https://mlsanity.com/paper/1802.03916/) (2018)
- [Regularized Learning for Domain Adaptation under Label Shifts](https://mlsanity.com/paper/1903.09734/) (2019)
- [A Unified View of Label Shift Estimation](https://mlsanity.com/paper/2003.07554/) (2020)
- [Telescoping Density-Ratio Estimation](https://mlsanity.com/paper/2006.12204/) (2020)
- [Density Ratio Estimation via Infinitesimal Classification](https://mlsanity.com/paper/2111.11010/) (2021)
- [Optimally tackling covariate shift in RKHS-based nonparametric regression](https://mlsanity.com/paper/2205.02986/) (2022)
- [When is Importance Weighting Correction Needed for Covariate Shift Adaptation?](https://mlsanity.com/paper/2303.04020/) (2023)
- [Estimating Unbounded Density Ratios: Applications in Error Control under Covariate Shift](https://mlsanity.com/paper/2504.01031/) (2025)
- [Spectral Algorithms under Covariate Shift](https://mlsanity.com/paper/2504.12625/) (2025)
- [Spectral Algorithms in Misspecified Regression: Convergence under Covariate Shift](https://mlsanity.com/paper/2509.05106/) (2025)
- [ReTaSA: A Nonparametric Functional Estimation Approach for Addressing Continuous Target Shift](https://mlsanity.com/paper/2401.16410/) (2024)
- [Importance Weighting Correction of Regularized Least-Squares for Target Shift](https://mlsanity.com/paper/2210.09709/) (2022)
- [Unbounded Density Ratio Estimation and Its Application to Covariate Shift Adaptation](https://mlsanity.com/paper/2603.29725/) (2026)

## Abstract (arXiv)

We study density ratio estimation and importance-weighted regression under target shift with continuous outputs. Under target shift, the conditional distribution of the inputs given the outputs remains invariant across the training and test distributions, while the output marginal distribution may change. Although this problem has been extensively studied for discrete outputs, the continuous setting is substantially less understood: the importance weights are determined by an unknown density ratio function, for which existing estimation methods lack explicit finite-sample convergence rates. We propose a spectral regularization method in a reproducing kernel Hilbert space (RKHS) for estimating the continuous density ratio from labeled training samples and unlabeled test inputs. Under a source condition with regularity parameter iota>0, we establish high-probability finite-sample guarantees and show that the estimator achieves the capacity-independent minimax-optimal RKHS-norm rate O(nη-iota/(2iota+2)). We then incorporate the estimated density ratio into importance-weighted regression and characterize the propagation of density-ratio estimation error to the final predictor. When sufficiently many samples are available for density ratio estimation, the resulting regression estimator attains the minimax-optimal rates of standard kernel regression. These results establish a finite-sample theory for continuous density ratio estimation and importance-weighted learning under target shift.