这篇论文给 RLCP 补上了有限样本保证，证明局部校准误差有界，还能帮你理解带宽怎么选。
本文针对随机局部化共形预测（RLCP）给出了有限样本保证。在条件分数 CDF 满足 Hölder 正则性及标准密度和核假设下，作者证明了条件覆盖缺口和相对于 oracle 的长度误差的高概率界。该界分解为 O(h^β) 的局部化偏差和随校准规模递减的校准项，明确了带宽的偏差-方差权衡。对于数据分割学习分数，当分数针对枢轴分数时，局部保证可分解为固定分数校准和均匀分数估计误差，表明改进学习能强化局部化保证。
Beyond Marginal Validity: Finite-Sample Guarantees for Localized Conformal Prediction
Conformal prediction endows arbitrary black-box predictors with finite-sample, distribution-free marginal coverage, yet marginal validity can hide severe covariate-specific miscalibration, while exact distribution-free conditional coverage is finite-sample unattainable. Randomly localized conformal prediction (RLCP) mitigates this gap by calibrating near the test point while preserving marginal coverage. Existing theory, however, lacks finite-sample guarantees for the realized localized set that jointly control conditional validity and oracle efficiency. We provide such guarantees. For any fixed score, under Hölder regularity of the conditional score CDF and standard density and kernel assumptions, we prove high-probability bounds, uniform over a realized localization neighbourhood, for the conditional-coverage gap and the length error relative to the oracle. The bounds decompose into an $O(h^β)$ localization bias and a calibration term decreasing with calibration size, clarifying the bandwidth bias-variance tradeoff and when RLCP tracks the oracle. We also analyze data-split learned scores: when the score targets a pivotal score, as in conformalized quantile regression, uniform local guarantees decompose into fixed-score calibration and uniform score-estimation errors, showing that improved learning sharpens localized guarantees.