Conformal Risk Control

The core innovation of CRC is its ability to control any loss function that is monotone and bounded. Unlike standard conformal prediction, which only guarantees marginal coverage (1 - α), CRC can guarantee that the expected value of a loss function L(Y, S(X)) is below a user-specified threshold. The loss function must satisfy a monotonicity property: larger prediction sets must not increase the loss. This allows practitioners to optimize for domain-specific costs, such as guaranteeing that the false negative rate is below 5% or that the prediction set size is minimized subject to a constraint on the false discovery rate.

CRC produces prediction sets that are calibrated to a specific risk tolerance, not just a coverage probability. The algorithm learns a threshold λ from a calibration set that minimizes the set size while ensuring the empirical risk is controlled. The resulting prediction sets are adaptive: they are larger when the model is uncertain and smaller when it is confident, but the size is driven by the chosen loss function. For example, in medical diagnosis, a CRC set might be empty for healthy patients and large for ambiguous cases, all while guaranteeing that the false negative rate across the population remains below a predefined bound.

Like its conformal prediction parent, CRC provides distribution-free, finite-sample guarantees. The risk control guarantee holds exactly for any sample size, not just asymptotically, provided the calibration and test data are exchangeable. This is achieved through a simple calibration procedure: a threshold λ is selected as the quantile of a modified empirical risk distribution on the calibration set. The proof relies on the same exchangeability argument as split conformal prediction, making CRC a robust tool for high-stakes applications where asymptotic approximations are unacceptable.

A canonical application of CRC is guaranteeing a bound on the false negative rate (FNR). In multi-class classification, a standard conformal set guarantees that the true class is included with probability 1 - α, but it does not directly control the rate at which specific classes are missed. CRC can construct sets that guarantee the FNR for a critical class is below a threshold. This is achieved by defining a loss function that penalizes the exclusion of the true label when it belongs to the protected class, enabling rigorous safety guarantees for applications like disease screening or fraud detection.

CRC is model-agnostic and task-agnostic. It can wrap any pre-trained model—from image classifiers to large language models—and any prediction task where a monotone loss function can be defined. For LLMs, CRC can guarantee that a set of generated answers contains a correct response with high probability. For object detection, it can control the number of missed objects. The only requirement is the ability to define a nested family of prediction sets parameterized by a scalar λ, where larger λ produces larger sets, and a loss function that is non-increasing in λ.

CRC uses a learn-then-test framework that separates model training from risk calibration. First, a base model is trained on a proper training set to produce a scoring function. Then, a held-out calibration set is used to learn the optimal threshold λ that controls the risk. This avoids the computational cost of retraining and ensures that the risk guarantee is not compromised by overfitting. The calibration step involves solving a simple one-dimensional optimization problem over λ, making CRC computationally lightweight and easy to integrate into existing ML pipelines.