In this work, we consider the problem of building distribution-free prediction intervals with finite-sample conditional coverage guarantees. Conformal prediction (CP) is an increasingly popular framework for building prediction intervals with distribution-free guarantees, but these guarantees only ensure marginal coverage: the probability of coverage is averaged over a random draw of both the training and test data, meaning that there might be substantial undercoverage within certain subpopulations. Instead, ideally, we would want to have local coverage guarantees that hold for each possible value of the test point's features. While the impossibility of achieving pointwise local coverage is well established in the literature, many variants of conformal prediction algorithm show favorable local coverage properties empirically. Relaxing the definition of local coverage can allow for a theoretical understanding of this empirical phenomenon. We aim to bridge this gap between theoretical validation and empirical performance by proving achievable and interpretable guarantees for a relaxed notion of local coverage. Building on the localized CP method of Guan (2023) and the weighted CP framework of Tibshirani et al. (2019), we propose a new method, randomly-localized conformal prediction (RLCP), which returns prediction intervals that are not only marginally valid but also achieve a relaxed local coverage guarantee and guarantees under covariate shift. Through a series of simulations and real data experiments, we validate these coverage guarantees of RLCP while comparing it with the other local conformal prediction methods.

翻译：本研究关注构建具有有限样本条件覆盖保证的无分布预测区间问题。共形预测（CP）作为构建具备无分布保证的预测区间框架日益流行，但其保证仅确保边缘覆盖：覆盖概率通过对训练数据和测试数据的随机抽取进行平均，这意味着在某些子群体中可能存在显著覆盖不足。理想情况下，我们期望获得适用于测试点特征每种可能取值的局部覆盖保证。尽管文献已充分证明实现逐点局部覆盖的不可能性，许多共形预测算法变体在实证中展现出良好的局部覆盖特性。放宽局部覆盖的定义可以为理解这一实证现象提供理论依据。本文通过为放宽的局部覆盖概念证明可达成且可解释的保证，旨在弥合理论验证与实证性能之间的差距。基于Guan（2023）的局部化CP方法与Tibshirani等人（2019）的加权CP框架，我们提出了一种新方法——随机局部化共形预测（RLCP），该方法返回的预测区间不仅具有边缘有效性，同时能实现放宽的局部覆盖保证及协变量偏移下的保证。通过系列模拟实验与真实数据实验，我们在对比其他局部共形预测方法的同时，验证了RLCP的覆盖保证。