# Importance Weighting Correction of Regularized Least-Squares for Target Shift

arXiv: https://arxiv.org/abs/2210.09709 · HTML: https://mlsanity.com/paper/2210.09709/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 23 references that are in the map.
- First posted on arXiv 18 Oct 2022.

## Builds on (most cited first, 7 of 7)

- [Detecting and Correcting for Label Shift with Black Box Predictors](https://mlsanity.com/paper/1802.03916/) (2018)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [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)
- [Optimally tackling covariate shift in RKHS-based nonparametric regression](https://mlsanity.com/paper/2205.02986/) (2022)
- [Minimax optimal approaches to the label shift problem in non-parametric settings](https://mlsanity.com/paper/2003.10443/) (2020)
- [Quantification under prior probability shift: the ratio estimator and its extensions](https://mlsanity.com/paper/1807.03929/) (2018)

## Built on by (most cited first, 1 of 1)

- [Learning under Target Shift: Optimal Density Ratio Estimation and Importance-Weighted Regression](https://mlsanity.com/paper/2609.15785/) (2026)

## Abstract (arXiv)

Importance weighting is a standard tool for correcting distribution shift, but its statistical behavior under target shift -- where the label distribution changes between training and testing while the conditional distribution of inputs given the label remains stable -- remains under-explored. We analyze importance-weighted kernel ridge regression under target shift and show that, because the weights depend only on the output variable, reweighting corrects the train-test mismatch without altering the input-space complexity that governs kernel generalization. Under standard RKHS regularity and capacity conditions and a mild Bernstein-type moment condition on the label weights, we obtain finite-sample guarantees showing that the estimator achieves the same convergence behavior as in the no-shift case, with shift severity affecting only the constants through weight moments. We complement these results with matching minimax lower bounds, establishing rate optimality and quantifying the unavoidable dependence on shift severity. We further study more general weighting schemes and prove that weight misspecification induces an irreducible bias: the estimator concentrates around an induced population regression function that generally differs from the desired test regression function unless the weights are accurate. Finally, we derive consequences for plug-in classification under target shift via standard calibration arguments.