Adapt the Score or Localize the Calibration? A Finite-Sample Comparison for Conditional Conformal Prediction

Given a fixed nonconformity score $S$ and target coverage $1-\alpha$, should extra labels transform the score before global calibration, or calibrate it locally? Under score regularity, we bound both routes' errors in conditional coverage and oracle-relative length, with high probability over calibration, given the training data. Subtracting an estimated conditional score quantile $q_S^{1-\alpha }(\cdot )$ before global calibration gives a three-term error bound, valid simultaneously wherever the score assumptions hold: the pointwise quantile-estimation error, its average over covariates, and a calibration fluctuation. We prove that this average term is needed: an error confined to one region causes over-coverage elsewhere, even with population calibration. For randomly localized conformal prediction (RLCP), we bound localization bias by the target quantile's variation alone; the conditional score distribution need not be continuous in the covariate. These bounds hold over the realized centre's neighbourhood and account for the smaller effective calibration sample. We apply the global bounds to local-polynomial and deep-network quantile estimators. Neural estimators can exploit additive or compositional structure, whereas our RLCP bound depends on neighbourhood data mass in the full covariate space. The bounds favour quantile learning when the fitting data can capture this structure; local calibration remains useful when the fit is poor but nearby data are plentiful. Simulations and eight real datasets illustrate the global route's gains; low-dimensional simulations show when localization is preferable.