A Joint Finite-Sample Certificate for Adaptive Selective Conformal Risk Control

Selective predictors answer on confident inputs and abstain elsewhere; deploying one safely needs a single finite-sample certificate that simultaneously upper-bounds the selected risk, lower-bounds the acceptance probability pacc above a floor pmin, and lower-bounds the deployment utility. This certificate must be valid under adaptive threshold selection from a finite grid of m pairs on ncert samples. We give such a certificate for bounded, possibly non-monotone losses by treating the selected risk directly as a ratio rather than through a Hoeffding-style range bound. The construction couples three confidence bounds: a variance-adaptive empirical-Bernstein bound on the ratio risk, a Clopper--Pearson bound on acceptance, and a two-sided closeness bound on utility. Together they lower-bound the certified policy's utility absolutely and to within 2gammau of the best over the certified set, both non-vacuous whenever feasible; a regime-scoped third leg matches an external oracle, informative only where the risk margin gammar < α and vacuous at the headline operating points. Relative to the range-only Hoeffding-ratio construction this sharpens the acceptance-floor dependence from 1/pmin to 1/√pmin, and a closed-form corollary identifies a per-pair regime in which our risk bound dominates a Hoeffding conformal risk control (Hoeffding--CRC) selective bound. Empirically, on ImageNet (three ResNets) and COCO val 2017 panoptic, the certificate opens a +22 pp certified-acceptance frontier over Hoeffding--CRC and is ≈10× tighter than a non-vacuous matched-valid baseline; these gains are regime-scoped, not universal, and absent on ADE20K. The certifier runs in O(ncert m) time.