Finite-Sample Optimal Conformal Prediction Sets under Covariate Shift
Abstract
We study conformal prediction under covariate shift with a known covariate density ratio, seeking prediction sets with minimum expected length under the target distribution. For every calibration sample size, we prove that the rule combining the inverse conditional-density score with randomized weighted conformal calibration minimizes this expected length among all rules valid over the full weighted-exchangeability class. We approximate the oracle using quantile and kernel histograms, including forest-weighted estimators, and select estimators and tuning parameters by resampling to estimate calibrated target length. On a bounded response domain, we bound excess length linearly in L1 density error at fixed calibration size, without requiring a uniform positive lower bound on the true density. We also bound selection-pool and resampling errors. Under explicit growth conditions, the selected procedure approaches the same oracle while the calibration size remains fixed. Experiments on 36 known-density configurations and four public regression datasets compare coverage, prediction-set length, and runtime with existing methods. In targeted low-coverage and strong-shift comparisons, inverse-density selection yields shorter mean sets in most cases and lower mean runtime than the evaluated weighted baselines, with some differences in achieved coverage.
Related
- Backward Conformal Prediction under Covariate Shift
- Sharp training-conditional coverage for conformal prediction under covariate shift
- Sketched Calibration for Conformal Prediction under Covariate Shift
- Sample-Efficient Generative Conformal Prediction
- Minimum Volume Optimal Transport for Multivariate Conformal Prediction
BibTeX
@misc{li2026finitesample,
title = {{Finite-Sample Optimal Conformal Prediction Sets under Covariate Shift}},
author = {Zhongyang Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37866},
eprint = {2610.37866},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37866},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite-Sample Optimal Conformal Prediction Sets under Covariate Shift
AU - Zhongyang Li
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37866
UR - https://arxiv.org/abs/2610.37866
AB - We study conformal prediction under covariate shift with a known covariate density ratio, seeking prediction sets with minimum expected length under the target distribution. For every calibration sample size, we prove that the rule combining the inverse conditional-density score with randomized weighted conformal calibration minimizes this expected length among all rules valid over the full weighted-exchangeability class. We approximate the oracle using quantile and kernel histograms, including forest-weighted estimators, and select estimators and tuning parameters by resampling to estimate calibrated target length. On a bounded response domain, we bound excess length linearly in L^1 density error at fixed calibration size, without requiring a uniform positive lower bound on the true density. We also bound selection-pool and resampling errors. Under explicit growth conditions, the selected procedure approaches the same oracle while the calibration size remains fixed. Experiments on 36 known-density configurations and four public regression datasets compare coverage, prediction-set length, and runtime with existing methods. In targeted low-coverage and strong-shift comparisons, inverse-density selection yields shorter mean sets in most cases and lower mean runtime than the evaluated weighted baselines, with some differences in achieved coverage.
ER -