Beyond Marginal Validity: Finite-Sample Guarantees for Localized Conformal Prediction

2026-08-06

AI Summary

This work addresses the limitation of traditional conformal prediction, which guarantees only marginal coverage and often exhibits poor conditional coverage, leading to calibration bias in specific regions of the covariate space. To overcome this, the authors propose Randomized Localized Conformal Prediction (RLCP), a method that performs local calibration within neighborhoods of test points, thereby enhancing conditional coverage while preserving marginal validity. The paper establishes, for the first time, finite-sample, high-probability uniform guarantees for such localized approaches, simultaneously controlling both conditional coverage error and oracle length error. By leveraging Hölder continuity, kernel density estimation, data-splitting-based score learning, and conformal quantile regression, the authors develop a theoretical framework for local coverage, deriving finite-sample bounds on the conditional coverage gap and length error, and demonstrating that improved score estimation enables performance approaching that of the oracle.

Abstract

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.

Problem

• conformal prediction
• conditional coverage
• finite-sample guarantees
• localized prediction
• oracle efficiency

Innovation

• localized conformal prediction
• finite-sample guarantees
• conditional coverage
• oracle efficiency
• score estimation