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. This assumption is often violated in practice, where losses may behave non-monotonically due to competing objectives such as coverage and efficiency. We study CRC under non-monotone loss functions when the tuning parameter is selected from a finite grid, a common scenario in thresholding or discretized decision rules. Revisiting a known counterexample, we show that the validity of CRC without monotonicity depends on the relationship between the calibration sample size and the grid resolution. In particular, risk control can still be achieved when the calibration sample is sufficiently large relative to the grid. We provide a finite-sample guarantee for bounded losses over a grid of size $m$, showing that the excess risk above the target level $α$ is of order $\sqrt{\log(m)/n}$, where $n$ is the calibration sample size. A matching lower bound shows 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 impact of non-monotonicity. Methods that account for finite-sample deviations achieve more stable risk control than approaches based on monotonicity transformations, while maintaining competitive prediction-set sizes.

翻译：[translated abstract in Chinese] 保形风险控制（CRC）提供了一种无分布假设的保证方法，用于在用户指定水平上控制期望损失。现有理论通常假设损失随控制预测集大小的调节参数单调递减。然而，由于覆盖率和效率等竞争目标，损失函数在实际中常呈现非单调行为，导致该假设经常被违反。我们研究了在有限网格上选择调节参数时的非单调损失函数下的CRC，这在阈值化或离散化决策规则中是常见场景。通过重新审视已知反例，我们发现无单调性假设下CRC的有效性取决于校准样本量与网格分辨率之间的关系。具体而言，当校准样本相对网格足够大时，仍可实现风险控制。我们针对大小为$m$的网格上有界损失提供了有限样本保证，表明超出目标水平$α$的过量风险阶数为$\sqrt{\log(m)/n}$，其中$n$为校准样本量。匹配的下界证明该速率是极小化最优的。我们还在附加结构条件（包括Lipschitz连续性和单调性）下推导了改进的保证，并通过重要性加权将分析扩展至分布偏移场景。在合成多标签分类和真实目标检测数据上的数值实验验证了非单调性的实际影响。相较于基于单调性变换的方法，考虑有限样本偏差的方法能在保持竞争性预测集大小的同时，实现更稳定的风险控制。