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

• URL: http://arxiv.org/abs/2606.08517v1
• Date: Sun, 07 Jun 2026 08:55:44 GMT
• Title: A Joint Finite-Sample Certificate for Adaptive Selective Conformal Risk Control
• Abstract summary: We give a certificate for bounded, possibly non-monotone losses by treating the selected risk directly as a ratio.<n>Relative to the range-only Hoeffding-ratio construction this sharpens the acceptance-floor dependence from $1/pmin$ to $1/sqrtpmin$.<n>The certificate opens a $+22$ pp certified-acceptance frontier over Hoeffding--CRC.
• Score: 0.9668407688201359
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $2\gammau$ of the best over the \emph{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 $1/\sqrt{\pmin}$, 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 ${\approx}10{\times}$ 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.

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