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Colorful Pinball: Density-Weighted Quantile Regression for Conditional Guarantee of Conformal Prediction

Qianyi Chen, Bo Li

TL;DR

This work tackles the challenge of achieving reliable conditional coverage in finite-sample conformal prediction. It introduces Colorful Pinball Conformal Prediction (CPCP), which directly minimizes the mean squared conditional error by weighting the pinball loss with the conditional density at the target quantile, estimated via a three-headed quantile network. The authors derive a sharp, Taylor-based surrogate for MSCE, establish exact non-asymptotic excess-risk guarantees, and provide a robust algorithm with stability mechanisms. Empirical results on eight large-scale, real-world datasets show consistent improvements in conditional coverage (MSCE and WSC) over strong baselines, validating the practical impact of density-weighted quantile regression in conformal UQ.

Abstract

While conformal prediction provides robust marginal coverage guarantees, achieving reliable conditional coverage for specific inputs remains challenging. Although exact distribution-free conditional coverage is impossible with finite samples, recent work has focused on improving the conditional coverage of standard conformal procedures. Distinct from approaches that target relaxed notions of conditional coverage, we directly minimize the mean squared error of conditional coverage by refining the quantile regression components that underpin many conformal methods. Leveraging a Taylor expansion, we derive a sharp surrogate objective for quantile regression: a density-weighted pinball loss, where the weights are given by the conditional density of the conformity score evaluated at the true quantile. We propose a three-headed quantile network that estimates these weights via finite differences using auxiliary quantile levels at \(1-α\pm δ\), subsequently fine-tuning the central quantile by optimizing the weighted loss. We provide a theoretical analysis with exact non-asymptotic guarantees characterizing the resulting excess risk. Extensive experiments on diverse high-dimensional real-world datasets demonstrate remarkable improvements in conditional coverage performance.

Colorful Pinball: Density-Weighted Quantile Regression for Conditional Guarantee of Conformal Prediction

TL;DR

This work tackles the challenge of achieving reliable conditional coverage in finite-sample conformal prediction. It introduces Colorful Pinball Conformal Prediction (CPCP), which directly minimizes the mean squared conditional error by weighting the pinball loss with the conditional density at the target quantile, estimated via a three-headed quantile network. The authors derive a sharp, Taylor-based surrogate for MSCE, establish exact non-asymptotic excess-risk guarantees, and provide a robust algorithm with stability mechanisms. Empirical results on eight large-scale, real-world datasets show consistent improvements in conditional coverage (MSCE and WSC) over strong baselines, validating the practical impact of density-weighted quantile regression in conformal UQ.

Abstract

While conformal prediction provides robust marginal coverage guarantees, achieving reliable conditional coverage for specific inputs remains challenging. Although exact distribution-free conditional coverage is impossible with finite samples, recent work has focused on improving the conditional coverage of standard conformal procedures. Distinct from approaches that target relaxed notions of conditional coverage, we directly minimize the mean squared error of conditional coverage by refining the quantile regression components that underpin many conformal methods. Leveraging a Taylor expansion, we derive a sharp surrogate objective for quantile regression: a density-weighted pinball loss, where the weights are given by the conditional density of the conformity score evaluated at the true quantile. We propose a three-headed quantile network that estimates these weights via finite differences using auxiliary quantile levels at , subsequently fine-tuning the central quantile by optimizing the weighted loss. We provide a theoretical analysis with exact non-asymptotic guarantees characterizing the resulting excess risk. Extensive experiments on diverse high-dimensional real-world datasets demonstrate remarkable improvements in conditional coverage performance.
Paper Structure (64 sections, 5 theorems, 102 equations, 1 figure, 7 tables, 2 algorithms)

This paper contains 64 sections, 5 theorems, 102 equations, 1 figure, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. Split Conformal Prediction.
  5. Rectified Conformal Prediction (RCP).
  6. Conditional coverage.
  7. Quantile regression.
  8. Approximation to MSCE
  9. Colorful Pinball Conformal Prediction
  10. Algorithm details
  11. Towards better finite-sample stability
  12. Theoretical Analysis
  13. Experiments
  14. Data.
  15. Baselines.
  16. ...and 49 more sections

Key Result

Proposition 3.1

Assume the conditional CDF of scores is $L_F$-Lipschitz, under mild regularity conditions, holds. Here, $\mathcal{L}_x(\cdot) = \mathbb{E}_Y[\rho_{\tau} (\cdot,s(x,Y))]$ denotes the expected pinball loss (with expectation taken over $Y\mid X$).

Figures (1)

  • Figure 1: Illustration of the Colorful Pinball Conformal Prediction framework. We estimate density-based weights via auxiliary quantiles, fine-tune the central quantile with density-weighted pinball loss, and apply conformalization with rectified conformity scores.

Theorems & Definitions (13)

  • Proposition 3.1
  • Remark 3.3
  • Proposition 3.4
  • Example 3.1
  • Remark 3.5
  • Theorem 5.2
  • Remark B.4
  • Remark B.5
  • Lemma B.6: Norm equivalence for Hölder classes
  • proof
  • ...and 3 more