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On Volume Minimization in Conformal Regression

Batiste Le Bars, Pierre Humbert

TL;DR

This work analyzes volume minimization within split conformal regression, framing the problem as a minimum-volume set under a coverage constraint. It shows that calibrating the split step yields an empirical solution to this objective and derives a finite-sample bound on the excess interval length. To achieve true volume efficiency, the paper introduces EffOrt, which learns the base predictor by minimizing the empirical $(1-\\alpha)$-QAE, followed by a calibration step, and extends to Ad-EffOrt for covariate-adaptive interval sizes. The empirical results across synthetic and real data demonstrate that EffOrt and Ad-EffOrt produce valid prediction sets with notably reduced lengths and lower variability compared to standard split CP and related adaptive methods. Overall, the approach highlights the critical role of the learning step in achieving efficient, distribution-free prediction sets with practical applicability to regression tasks.

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.

On Volume Minimization in Conformal Regression

TL;DR

This work analyzes volume minimization within split conformal regression, framing the problem as a minimum-volume set under a coverage constraint. It shows that calibrating the split step yields an empirical solution to this objective and derives a finite-sample bound on the excess interval length. To achieve true volume efficiency, the paper introduces EffOrt, which learns the base predictor by minimizing the empirical -QAE, followed by a calibration step, and extends to Ad-EffOrt for covariate-adaptive interval sizes. The empirical results across synthetic and real data demonstrate that EffOrt and Ad-EffOrt produce valid prediction sets with notably reduced lengths and lower variability compared to standard split CP and related adaptive methods. Overall, the approach highlights the critical role of the learning step in achieving efficient, distribution-free prediction sets with practical applicability to regression tasks.

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.

Paper Structure

This paper contains 35 sections, 8 theorems, 59 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Background
  4. Preliminaries on Split Conformal Prediction
  5. Problem statement
  6. On closed-form expressions for \ref{['eq:formal-opt']}.
  7. Related work
  8. Minimum Volume Sets and Density Level Sets estimation.
  9. Efficient Conformal Prediction.
  10. Restriction to intervals with constant size
  11. Base predictor $f\in\mathcal{F}$ is given: optimality of the conformal step
  12. Base predictor $f\in\mathcal{F}$ is not given: sub-optimality of the least-square regressor
  13. EffOrt: EFFiciency-ORienTed split conformal regression
  14. Optimization of the empirical $(1-\alpha)$-QAE
  15. Theoretical analysis
  16. ...and 20 more sections

Key Result

Proposition 1

Let $\hat{t} = \widehat{Q}((1-\alpha)\frac{n_c+1}{n_c};\{S_i\}_{i=1}^{n_c})$ and $C^{1-\alpha}_{f,\hat{t}}$ the corresponding set. If the points in ${\cal D}^{cal}$ are i.i.d., and if $(n_c+1)(1-\alpha)$ is not an integer, then with probability greater than $1-\delta$ we have:

Figures (7)

  • Figure 1: Boxplots of the $50$ empirical expected lengths obtained by evaluating EffOrt in Section \ref{['sec:xpEffort']} (top) and Ad-EffOrt in Section \ref{['sec:xpADEffort']} (bottom). The white circle corresponds to the mean.
  • Figure 2: Synthetic data: Boxplots of the $50$ empirical coverages obtained by evaluating EffOrt (see Section \ref{['sec:xpEffort']}). The white circle corresponds to the mean.
  • Figure 3: Synthetic data: Boxplots of the $50$ empirical expected lengths (top) and coverages (bottom) obtained by evaluating EffOrt (see Section \ref{['sec:xpEffort']}). The white circle corresponds to the mean.
  • Figure 4: Synthetic data: Boxplots of the $50$ empirical coverages obtained by evaluating Ad-EffOrt (see Section \ref{['sec:xpADEffort']}). The white circle corresponds to the mean.
  • Figure 5: Synthetic data: Example of sets returned by Ad-EffOrt (left), LW-CP (middle), and CQR (right).
  • ...and 2 more figures

Theorems & Definitions (21)

  • Example 1
  • Proposition 1
  • Corollary 1
  • proof
  • Remark 1
  • Proposition 2
  • Theorem 1
  • proof : Proof sketch - Details in Appendix \ref{['app:proof-main']}
  • Remark 2
  • Lemma 1
  • ...and 11 more