Conrad et al. prove finite-sample guarantees for localized conformal prediction

Conformal prediction has long promised marginal coverage — valid prediction intervals for any black-box model without distributional assumptions. But marginal validity can mask severe miscalibration in specific covariate regions, and exact conditional coverage is impossible in finite samples. Randomly Localized Conformal Prediction (RLCP) addresses this by calibrating near each test point, yet until now lacked finite-sample theoretical guarantees for the realized localized set. In this 68-page paper, Anton Conrad, Rustam Isaev, Denis Belomestny, Eric Moulines, and Sergey Samsonov supply exactly those guarantees, bridging a critical gap between practice and theory.

The authors prove high-probability bounds, uniform over the realized localization neighborhood, for both the conditional-coverage gap and the length error relative to the oracle. Under Hölder regularity of the conditional score CDF plus standard density and kernel assumptions, the bounds decompose into an O(h^β) localization bias and a calibration term that decreases with calibration size. This decomposition reveals the bandwidth bias-variance tradeoff and shows precisely when RLCP tracks the oracle. The paper also analyzes data-split learned scores — as in conformalized quantile regression — showing uniform local guarantees factor into fixed-score calibration and score-estimation errors, meaning better learned scores directly sharpen localized validity.