# Conformal Risk Control under Non-Monotone Losses: Theory and Finite-Sample Guarantees

arXiv: https://arxiv.org/abs/2604.01502 · HTML: https://mlsanity.com/paper/2604.01502/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 31 references that are in the map.
- First posted on arXiv 2 Apr 2026.

## Builds on (most cited first, 18 of 18)

- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Conformal Prediction Under Covariate Shift](https://mlsanity.com/paper/1904.06019/) (2019)
- [Conformal prediction beyond exchangeability](https://mlsanity.com/paper/2202.13415/) (2022)
- [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)
- [Conformal Prediction With Conditional Guarantees](https://mlsanity.com/paper/2305.12616/) (2023)
- [Image-to-Image Regression with Distribution-Free Uncertainty Quantification and Applications in Imaging](https://mlsanity.com/paper/2202.05265/) (2022)
- [Distribution-free uncertainty quantification for classification under label shift](https://mlsanity.com/paper/2103.03323/) (2021)
- [Theoretical Foundations of Conformal Prediction](https://mlsanity.com/paper/2411.11824/) (2024)
- [Consistent Accelerated Inference via Confident Adaptive Transformers](https://mlsanity.com/paper/2104.08803/) (2021)
- [How to Trust Your Diffusion Model: A Convex Optimization Approach to Conformal Risk Control](https://mlsanity.com/paper/2302.03791/) (2023)
- [Achieving Risk Control in Online Learning Settings](https://mlsanity.com/paper/2205.09095/) (2022)
- [Non-Exchangeable Conformal Risk Control](https://mlsanity.com/paper/2310.01262/) (2023)
- [Conformal Prediction Sets with Limited False Positives](https://mlsanity.com/paper/2202.07650/) (2022)
- [Cross-Validation Conformal Risk Control](https://mlsanity.com/paper/2401.11974/) (2024)
- [Conformal Risk Control for Non-Monotonic Losses](https://mlsanity.com/paper/2602.20151/) (2026)

## Built on by (most cited first, 3 of 3)

- [A Joint Finite-Sample Certificate for Adaptive Selective Conformal Risk Control](https://mlsanity.com/paper/2606.08517/) (2026)
- [Risk-Controlled Post-Processing of Decision Policies](https://mlsanity.com/paper/2605.06479/) (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)

## Abstract (arXiv)

Conformal risk control (CRC) provides distribution-free guarantees for controlling the expected loss at a user-specified level. Existing theory typically assumes that the loss decreases monotonically with a tuning parameter that governs the size of the prediction set. However, this assumption is often violated in practice, where losses may behave non-monotonically due to competing objectives such as coverage and efficiency. In this paper, we study CRC under non-monotone loss functions when the tuning parameter is selected from a finite grid, a setting commonly arising in thresholding and discretized decision rules. Revisiting a known counterexample, we show that the validity of CRC without monotonicity depends critically on the relationship between the calibration sample size and the grid resolution. In particular, reliable risk control can still be achieved when the calibration sample is sufficiently large relative to the grid size. We establish a finite-sample guarantee for bounded losses over a grid of size m, showing that the excess risk above the target level α scales on the order of √log(m)/n, where n is the calibration sample size. A matching lower bound demonstrates that this rate is minimax optimal. We also derive refined guarantees under additional structural conditions, including Lipschitz continuity and monotonicity, and extend the analysis to settings with distribution shift via importance weighting. Numerical experiments on synthetic multilabel classification and real object detection data illustrate the practical implications of non-monotonicity. Methods that explicitly account for finite-sample uncertainty achieve more stable risk control than approaches based on monotonicity transformations, while maintaining competitive prediction set sizes.