发表机构
EPITA（法国高等计算机与电子技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对随机局部共形预测（RLCP）缺乏同时控制条件有效性和先验效率的有限样本保证的问题，本文在特定假设下推导了相关高概率界，阐明了带宽偏差-方差权衡，还分析了数据拆分学习得分的情况，为RLCP提供了理论支撑。
AI 中文摘要
共形预测为任意黑箱预测器赋予了有限样本、与分布无关的边缘覆盖率，但边缘有效性可能掩盖严重的协变量特定校准误差，而精确的与分布无关的条件覆盖率在有限样本下是无法实现的。随机局部共形预测（RLCP）通过在测试点附近进行校准同时保持边缘覆盖率来缩小这一差距。然而，现有理论缺乏对已实现的局部集合的有限样本保证，该保证需同时控制条件有效性和先验效率。我们提供了此类保证：对于任意固定得分，在条件得分累积分布函数（CDF）的Hölder正则性以及标准密度和核假设下，我们证明了在已实现的局部邻域上一致的高概率界，该界涉及条件覆盖率差距和相对于先验的长度误差。这些界分解为O(h^β)的局部化偏差和随校准大小减小的校准项，阐明了带宽的偏差-方差权衡以及RLCP何时跟踪先验。我们还分析了数据拆分学习得分：当得分以共形分位数回归中那样的关键得为目标时，一致局部保证分解为固定得分校准和一致得分估计误差，表明改进的学习可增强局部化保证。
英文摘要
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.
Comments68 pages, 8 figures, 2 tables
