Super-Level-Set Regression: Conditional Quantiles via Volume Minimization

2026-05-07

AI Summary

This work proposes the Superlevel Set Regression (SLS) framework to address the error sensitivity and high computational cost inherent in traditional two-stage density estimation approaches for constructing conditionally valid prediction regions. Bypassing explicit conditional density estimation, SLS directly parameterizes and end-to-end optimizes the geometric boundary of conditional quantile regions, minimizing prediction volume while guaranteeing conditional coverage. By introducing a volume-preserving frontier function, SLS flexibly models complex, multimodal, or disconnected conditional structures and, for the first time, enables effective optimization of implicitly coupled boundaries. Experiments demonstrate that SLS significantly shrinks prediction regions under strict coverage guarantees, thereby enhancing both modeling efficiency and robustness.

Abstract

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.

Problem

• conditional quantiles
• minimum-volume prediction regions
• multivariate regression
• conditional coverage
• volume minimization

Innovation

• super-level-set regression
• conditional quantiles
• volume minimization
• multivariate regression
• geometric optimization