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

arXiv: https://arxiv.org/abs/2606.08517 · HTML: https://mlsanity.com/paper/2606.08517/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 32 references that are in the map.
- First posted on arXiv 7 Jun 2026.

## Builds on (most cited first, 28 of 28)

- [SegFormer: Simple and Efficient Design for Semantic Segmentation with Transformers](https://mlsanity.com/paper/2105.15203/) (2021)
- [Masked-attention Mask Transformer for Universal Image Segmentation](https://mlsanity.com/paper/2112.01527/) (2021)
- [Do ImageNet Classifiers Generalize to ImageNet?](https://mlsanity.com/paper/1902.10811/) (2019)
- [Selective Classification for Deep Neural Networks](https://mlsanity.com/paper/1705.08500/) (2017)
- [SelectiveNet: A Deep Neural Network with an Integrated Reject Option](https://mlsanity.com/paper/1901.09192/) (2019)
- [Conformal Risk Control](https://mlsanity.com/paper/2208.02814/) (2022)
- [Distribution-Free, Risk-Controlling Prediction Sets](https://mlsanity.com/paper/2101.02703/) (2021)
- [Learn then Test: Calibrating Predictive Algorithms to Achieve Risk Control](https://mlsanity.com/paper/2110.01052/) (2021)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Non-Exchangeable Conformal Risk Control](https://mlsanity.com/paper/2310.01262/) (2023)
- [Efficiently Controlling Multiple Risks with Pareto Testing](https://mlsanity.com/paper/2210.07913/) (2022)
- [Automatically Adaptive Conformal Risk Control](https://mlsanity.com/paper/2406.17819/) (2024)
- [Post-selection inference for e-value based confidence intervals](https://mlsanity.com/paper/2203.12572/) (2022)
- [Confidence on the Focal: Conformal Prediction with Selection-Conditional Coverage](https://mlsanity.com/paper/2403.03868/) (2024)
- [Semi-Supervised Risk Control via Prediction-Powered Inference](https://mlsanity.com/paper/2412.11174/) (2024)
- [Conformal Selective Prediction with General Risk Control](https://mlsanity.com/paper/2603.24704/) (2026)
- [Selective conformal inference with false coverage-statement rate control](https://mlsanity.com/paper/2301.00584/) (2023)
- [Selective Conformal Risk Control](https://mlsanity.com/paper/2512.12844/) (2025)
- [Cross-Validation Conformal Risk Control](https://mlsanity.com/paper/2401.11974/) (2024)
- [Conformal Risk Control for Ordinal Classification](https://mlsanity.com/paper/2405.00417/) (2024)
- [Conformal Risk Control for Non-Monotonic Losses](https://mlsanity.com/paper/2602.20151/) (2026)
- [Two-stage Risk Control with Application to Ranked Retrieval](https://mlsanity.com/paper/2404.17769/) (2024)
- [Anytime-Valid Conformal Risk Control](https://mlsanity.com/paper/2602.04364/) (2026)
- [Generalization and Informativeness of Weighted Conformal Risk Control Under Covariate Shift](https://mlsanity.com/paper/2501.11413/) (2025)
- [Online Conformal Abstention Under Adversarial Bandit Feedback](https://mlsanity.com/paper/2506.14067/) (2025)
- [Conformal Risk Control under Non-Monotone Losses: Theory and Finite-Sample Guarantees](https://mlsanity.com/paper/2604.01502/) (2026)
- [Multiply Robust Conformal Risk Control with Coarsened Data](https://mlsanity.com/paper/2508.15489/) (2025)

## Built on by (most cited first, 5 of 5)

- [Calibrate What You SHIP: Post-Selection Risk Control for Verifier-Guided Text-to-Image Generation](https://mlsanity.com/paper/2608.21748/) (2026)
- [Certify or Refuse: A Cross-Model Map for Selective Risk Control with Coverage Floors under Covariate Shift](https://mlsanity.com/paper/2608.10893/) (2026)
- [Conditional Validity for Adaptive Modality Acquisition: When the Policy Chooses Its Own Calibration Group](https://mlsanity.com/paper/2608.15520/) (2026)
- [SafeRestore: Detector-Relative Risk Certificates for Selective Industrial Image Restoration](https://mlsanity.com/paper/2609.03475/) (2026)
- [Neyman-Pearson Classification with Indecisions: Simultaneous Control of Selective Type I and Type II Errors](https://mlsanity.com/paper/2610.04233/) (2026)

## Abstract (arXiv)

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.