Sacha Braun

Constructing minimum-volume prediction regions that satisfy conditional coverage is a fundamental challenge in multivariate regression. Standard approaches rely on explicitly estimating the full conditional density and subsequently thresholding it. This two-step plug-in process is notoriously difficult, sensitive to estimation errors, and computationally expensive. One would like to instead optimize the region directly. Formulating a direct solution is challenging, however, because it requires minimizing a volume objective that is coupled with the conditional quantiles of the model’s own estimation error. In this work, we address this challenge. We introduce \emphsuper-level-set regression (SLS), a novel mathematical framework that successfully resolves this implicit coupling, allowing us to directly parameterize and optimize the geometric boundaries of the target conditional level sets. By bypassing full distribution estimation and leveraging flexible volume-preserving frontier functions, our approach natively captures complex, multimodal, and disjoint conditional structures end-to-end. Ultimately, SLS offers a new perspective on multivariate conditional quantile regression, replacing the restrictive assumptions of density-first methods with a direct geometric optimization strategy.

Evaluating conditional coverage remains one of the most persistent challenges in assessing the reliability of predictive systems. Although conformal methods can give guarantees on marginal coverage, no method canguarantee to produce sets with correct conditional coverage, leaving practitioners without a clear way to interpret local deviations. To overcome sample-inefficiency and overfitting issues of existing metrics, we cast conditional coverage estimation as a classification problem. Condi- tional coverage is violated if and only if some classifier can achieve lower risk than the target coverage. Through the choice of a (proper) loss function, the resulting risk difference gives a conservative estimate of natural miscoverage measures such as L1 and L2 distance, and can even separate the effects of over- and under-coverage, as well as handle non-constant target coverages. We call the resulting family of metrics excess risk of the target coverage (ERT). We show experimentally that the use of modern classifiers provides much higher statistical power than simple classifiers underlying established metrics like CovGap. Additionally, we use our metric to benchmark different confor- mal prediction methods. Finally, we release an open-source package for ERT as well as previous conditional coverage metrics. Together, these contributions provide a new lens for understanding, diagnosing, and improving the conditional reliability of predictive systems.

Conformal prediction provides a principled framework for constructing predictive sets with finite- sample validity. While much of the focus has been on univariate response variables, existing mul- tivariate methods either impose rigid geometric assumptions or rely on flexible but computation- ally expensive approaches that do not explicitly optimize prediction set volume. We propose an optimization-driven framework based on a novel loss function that directly learns minimum-volume covering sets while ensuring valid coverage. This formulation naturally induces a new nonconfor- mity score for conformal prediction, which adapts to the residual distribution and covariates. Our approach optimizes over prediction sets defined by arbitrary norm balls—including single and multi- norm formulations. Additionally, by jointly optimizing both the predictive model and predictive uncertainty, we obtain prediction sets that are tight, informative, and computationally efficient, as demonstrated in our experiments on real-world datasets.