# On Volume Minimization in Conformal Regression

arXiv: https://arxiv.org/abs/2502.09985 · HTML: https://mlsanity.com/paper/2502.09985/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2025.

## Built on by (most cited first, 9 of 9)

- [Conformal Risk Control for Semantic Uncertainty Quantification in Computed Tomography](https://mlsanity.com/paper/2503.00136/) (2025)
- [Singleton-Optimized Conformal Prediction](https://mlsanity.com/paper/2509.24095/) (2025)
- [Non-Asymptotic Analysis of Efficiency in Conformalized Regression](https://mlsanity.com/paper/2510.07093/) (2025)
- [Smoothing-Based Conformal Prediction for Balancing Efficiency and Interpretability](https://mlsanity.com/paper/2509.22529/) (2025)
- [Geometry of tail allocation in conformal prediction intervals](https://mlsanity.com/paper/2604.25202/) (2026)
- [Robust Conditional Conformal Prediction via Branched Normalizing Flow](https://mlsanity.com/paper/2605.01868/) (2026)
- [Super-Level-Set Regression: Conditional Quantiles via Volume Minimization](https://mlsanity.com/paper/2605.06210/) (2026)
- [PICPIs: Prediction-Interval-Conditional Prediction Intervals](https://mlsanity.com/paper/2609.25388/) (2026)
- [Valid and Efficient Split Conformal Regression for Time Series](https://mlsanity.com/paper/2609.33866/) (2026)

## Abstract (arXiv)

We study the question of volume optimality in split conformal regression, a topic still poorly understood in comparison to coverage control. Using the fact that the calibration step can be seen as an empirical volume minimization problem, we first derive a finite-sample upper-bound on the excess volume loss of the interval returned by the classical split method. This important quantity measures the difference in length between the interval obtained with the split method and the shortest oracle prediction interval. Then, we introduce EffOrt, a methodology that modifies the learning step so that the base prediction function is selected in order to minimize the length of the returned intervals. In particular, our theoretical analysis of the excess volume loss of the prediction sets produced by EffOrt reveals the links between the learning and calibration steps, and notably the impact of the choice of the function class of the base predictor. We also introduce Ad-EffOrt, an extension of the previous method, which produces intervals whose size adapts to the value of the covariate. Finally, we evaluate the empirical performance and the robustness of our methodologies.