On Volume Minimization in Conformal Regression

TL;DR

This paper investigates volume optimality in split conformal regression, deriving bounds on interval length and proposing methods to minimize interval size while adapting to covariates, with theoretical and empirical validation.

Contribution

It introduces EffOrt and Ad-EffOrt, novel methods that optimize interval volume in conformal regression and analyze their theoretical properties and practical performance.

Findings

EffOrt reduces interval length compared to classical methods.

Theoretical bounds link learning choices to interval volume.

Ad-EffOrt adapts interval size based on covariates.

Abstract

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.