Super-Level-Set Regression: Conditional Quantiles via Volume Minimization
For practitioners needing reliable prediction regions in multivariate regression, SLS offers a direct optimization alternative to two-step density-based methods, which are often difficult and computationally expensive.
This paper introduces super-level-set regression (SLS), a framework that directly optimizes minimum-volume prediction regions with conditional coverage, bypassing the need for full density estimation. SLS captures complex, multimodal conditional structures end-to-end via geometric optimization.
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 super-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.