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

2026-08-06 · 15:59 UTC ·Anton Conrad, Rustam Isaev, Denis Belomestny... ·1 min read

Brief

Randomly localized conformal prediction (RLCP) receives finite-sample, high-probability uniform guarantees for the realized localized set: under Hölder regularity of the conditional score CDF (exponent β) and standard density/kernel assumptions the authors bound the conditional-coverage gap and length error. Bounds split into an O(h^β) localization bias and a calibration term shrinking with calibration size; data-split learned scores admit further decomposition into calibration and uniform score-estimation errors, clarifying when RLCP attains oracle-like performance.

Why it matters

The paper proves high-probability finite-sample bounds, uniform over the realized localization neighborhood, for RLCP's conditional-coverage gap and length error relative to the oracle under any fixed score; assumptions include Hölder regularity of the conditional score CDF (exponent β) and standard density/kernel conditions.

Key details

The derived error bounds decompose into an O(h^β) localization bias plus a calibration term that decreases with calibration size, making the bandwidth bias–variance tradeoff explicit and characterizing when RLCP tracks the oracle; for data-split learned scores (e.g., conformalized quantile regression) guarantees further split into fixed-score calibration and uniform score-estimation errors, so better score learning sharpens localized guarantees.

Metadata: Anton Conrad et al., arXiv:2608.06206v1 (published 2026-08-06), 68 pages, 8 figures, 2 tables (PDF available).

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.

Comment: 68 pages, 8 figures, 2 tables