# Active, anytime-valid risk controlling prediction sets

arXiv: https://arxiv.org/abs/2406.10490 · HTML: https://mlsanity.com/paper/2406.10490/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 41 references that are in the map.
- First posted on arXiv 15 Jun 2024.

## Builds on (most cited first, 25 of 25)

- [Adaptive Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2106.00170/) (2021)
- [A Holistic Approach to Undesired Content Detection in the Real World](https://mlsanity.com/paper/2208.03274/) (2022)
- [Distribution-Free, Risk-Controlling Prediction Sets](https://mlsanity.com/paper/2101.02703/) (2021)
- [Prediction-Powered Inference](https://mlsanity.com/paper/2301.09633/) (2023)
- [Conformal Language Modeling](https://mlsanity.com/paper/2306.10193/) (2023)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Conformal Inference for Online Prediction with Arbitrary Distribution Shifts](https://mlsanity.com/paper/2208.08401/) (2022)
- [Black-Box Reductions for Parameter-free Online Learning in Banach Spaces](https://mlsanity.com/paper/1802.06293/) (2018)
- [Image-to-Image Regression with Distribution-Free Uncertainty Quantification and Applications in Imaging](https://mlsanity.com/paper/2202.05265/) (2022)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [There are natural scores: Full comment on Shafer, "Testing by betting: A strategy for statistical and scientific communication"](https://mlsanity.com/paper/2102.05569/) (2021)
- [Conformal Decision Theory: Safe Autonomous Decisions from Imperfect Predictions](https://mlsanity.com/paper/2310.05921/) (2023)
- [Active Statistical Inference](https://mlsanity.com/paper/2403.03208/) (2024)
- [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020)
- [Nonparametric Two-Sample Testing by Betting](https://mlsanity.com/paper/2112.09162/) (2021)
- [Anytime-valid off-policy inference for contextual bandits](https://mlsanity.com/paper/2210.10768/) (2022)
- [Efficient Adaptive Experimental Design for Average Treatment Effect Estimation](https://mlsanity.com/paper/2002.05308/) (2020)
- [Semiparametric Efficient Inference in Adaptive Experiments](https://mlsanity.com/paper/2311.18274/) (2023)
- [Sequential Predictive Two-Sample and Independence Testing](https://mlsanity.com/paper/2305.00143/) (2023)
- [Sequential Kernelized Independence Testing](https://mlsanity.com/paper/2212.07383/) (2022)
- [The numeraire e-variable and reverse information projection](https://mlsanity.com/paper/2402.18810/) (2024)
- [Risk-limiting Financial Audits via Weighted Sampling without Replacement](https://mlsanity.com/paper/2305.06884/) (2023)
- [Conformal Policy Learning for Sensorimotor Control Under Distribution Shifts](https://mlsanity.com/paper/2311.01457/) (2023)
- [Anytime-valid sequential testing for elicitable functionals via supermartingales](https://mlsanity.com/paper/2204.05680/) (2022)

## Built on by (most cited first, 14 of 14)

- [Semi-Supervised Risk Control via Prediction-Powered Inference](https://mlsanity.com/paper/2412.11174/) (2024)
- [Adaptive Prediction-Powered AutoEval with Reliability and Efficiency Guarantees](https://mlsanity.com/paper/2505.18659/) (2025)
- [On Continuous Monitoring of Risk Violations under Unknown Shift](https://mlsanity.com/paper/2506.16416/) (2025)
- [Conformal Selective Acting: Anytime-Valid Risk Control for RLVR-Trained LLMs](https://mlsanity.com/paper/2605.20270/) (2026)
- [Selective Risk Certification for LLM Outputs via Information-Lift Statistics: PAC-Bayes, Robustness, and Skeleton Design](https://mlsanity.com/paper/2509.12527/) (2025)
- [Anytime-Valid Conformal Risk Control](https://mlsanity.com/paper/2602.04364/) (2026)
- [When Can Conformal Risk Control Certify LLM Outputs? Bounds, Impossibility, and Adaptation for Structured Generation](https://mlsanity.com/paper/2606.29054/) (2026)
- [Prediction-Powered Risk Monitoring of Deployed Models for Detecting Harmful Distribution Shifts](https://mlsanity.com/paper/2602.02229/) (2026)
- [Certified Adaptive Refresh: Anytime-Valid Monitoring for Federated Conformal RAG](https://mlsanity.com/paper/2605.29139/) (2026)
- [Pay Only for Disagreement: Certified No-Regression Verdicts for Model Updates with Matching Label-Complexity Bounds](https://mlsanity.com/paper/2609.17560/) (2026)
- [Anytime-Valid LLM Leaderboards via Benchmark-weighted and Block-Factorized e-Processes](https://mlsanity.com/paper/2609.32248/) (2026)
- [When Can Fraud Operations Authorize Automation? A Decision-Support Framework for Fresh Audit Evidence and Review Workload](https://mlsanity.com/paper/2608.08577/) (2026)
- [From Uncertainty to Clinical Risk: Severity-Aware Conformal Planning for Interactive Medical Diagnosis](https://mlsanity.com/paper/2608.27847/) (2026)
- [When Can Old Evaluations Certify a New Model? Label-Efficient Release Decisions under Evaluator Drift](https://mlsanity.com/paper/2609.32267/) (2026)

## Abstract (arXiv)

Rigorously establishing the safety of black-box machine learning models concerning critical risk measures is important for providing guarantees about model behavior. Recently, Bates et. al. (JACM '24) introduced the notion of a risk controlling prediction set (RCPS) for producing prediction sets that are statistically guaranteed low risk from machine learning models. Our method extends this notion to the sequential setting, where we provide guarantees even when the data is collected adaptively, and ensures that the risk guarantee is anytime-valid, i.e., simultaneously holds at all time steps. Further, we propose a framework for constructing RCPSes for active labeling, i.e., allowing one to use a labeling policy that chooses whether to query the true label for each received data point and ensures that the expected proportion of data points whose labels are queried are below a predetermined label budget. We also describe how to use predictors (i.e., the machine learning model for which we provide risk control guarantees) to further improve the utility of our RCPSes by estimating the expected risk conditioned on the covariates. We characterize the optimal choices of label policy and predictor under a fixed label budget and show a regret result that relates the estimation error of the optimal labeling policy and predictor to the wealth process that underlies our RCPSes. Lastly, we present practical ways of formulating label policies and empirically show that our label policies use fewer labels to reach higher utility than naive baseline labeling strategies on both simulations and real data.