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Fast Conformal Prediction using Conditional Interquantile Intervals

Naixin Guo, Rui Luo, Zhixin Zhou

TL;DR

This work tackles the problem of constructing finite-sample conformal prediction intervals for regression that remain effective under skewed conditional distributions. It introduces Conformal Interquantile Regression (CIR), which derives prediction sets from conditional interquantile intervals via a calibration procedure, and its enhancement CIR+ that leverages interval widths to tighten predictions. The authors prove marginal coverage guarantees for CIR (and CIR+) and show conditional coverage is achieved asymptotically under standard assumptions, while avoiding histogram-based discretization to reduce computation. Through extensive synthetic and real-data experiments, CIR and CIR+ achieve competitive predictive accuracy with substantially lower computational cost than Conformal Histogram Regression (CHR), providing a practical, scalable approach to distribution-aware uncertainty quantification. The methods balance conservatism and efficiency: CIR yields robust intervals, while CIR+ delivers narrower intervals with minimal additional computation, with extensions proposed for time-series data.

Abstract

We introduce Conformal Interquantile Regression (CIR), a conformal regression method that efficiently constructs near-minimal prediction intervals with guaranteed coverage. CIR leverages black-box machine learning models to estimate outcome distributions through interquantile ranges, transforming these estimates into compact prediction intervals while achieving approximate conditional coverage. We further propose CIR+ (Conditional Interquantile Regression with More Comparison), which enhances CIR by incorporating a width-based selection rule for interquantile intervals. This refinement yields narrower prediction intervals while maintaining comparable coverage, though at the cost of slightly increased computational time. Both methods address key limitations of existing distributional conformal prediction approaches: they handle skewed distributions more effectively than Conformalized Quantile Regression, and they achieve substantially higher computational efficiency than Conformal Histogram Regression by eliminating the need for histogram construction. Extensive experiments on synthetic and real-world datasets demonstrate that our methods optimally balance predictive accuracy and computational efficiency compared to existing approaches.

Fast Conformal Prediction using Conditional Interquantile Intervals

TL;DR

This work tackles the problem of constructing finite-sample conformal prediction intervals for regression that remain effective under skewed conditional distributions. It introduces Conformal Interquantile Regression (CIR), which derives prediction sets from conditional interquantile intervals via a calibration procedure, and its enhancement CIR+ that leverages interval widths to tighten predictions. The authors prove marginal coverage guarantees for CIR (and CIR+) and show conditional coverage is achieved asymptotically under standard assumptions, while avoiding histogram-based discretization to reduce computation. Through extensive synthetic and real-data experiments, CIR and CIR+ achieve competitive predictive accuracy with substantially lower computational cost than Conformal Histogram Regression (CHR), providing a practical, scalable approach to distribution-aware uncertainty quantification. The methods balance conservatism and efficiency: CIR yields robust intervals, while CIR+ delivers narrower intervals with minimal additional computation, with extensions proposed for time-series data.

Abstract

We introduce Conformal Interquantile Regression (CIR), a conformal regression method that efficiently constructs near-minimal prediction intervals with guaranteed coverage. CIR leverages black-box machine learning models to estimate outcome distributions through interquantile ranges, transforming these estimates into compact prediction intervals while achieving approximate conditional coverage. We further propose CIR+ (Conditional Interquantile Regression with More Comparison), which enhances CIR by incorporating a width-based selection rule for interquantile intervals. This refinement yields narrower prediction intervals while maintaining comparable coverage, though at the cost of slightly increased computational time. Both methods address key limitations of existing distributional conformal prediction approaches: they handle skewed distributions more effectively than Conformalized Quantile Regression, and they achieve substantially higher computational efficiency than Conformal Histogram Regression by eliminating the need for histogram construction. Extensive experiments on synthetic and real-world datasets demonstrate that our methods optimally balance predictive accuracy and computational efficiency compared to existing approaches.
Paper Structure (19 sections, 3 theorems, 51 equations, 5 figures, 3 tables, 2 algorithms)

This paper contains 19 sections, 3 theorems, 51 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Proposed Method
  4. Problem Setup
  5. Conditional Interquantile Regression
  6. Conditional Interquantile Regression with More Comparison (CIR+)
  7. Theoretical Results
  8. Remark:
  9. Experiments
  10. Synthetic Data Analysis
  11. Real Data Analysis
  12. Conclusion
  13. Additional Theoretical Results
  14. Proof of Theorem\ref{['thm:cir2']}
  15. Empirical Results
  16. ...and 4 more sections

Key Result

Theorem 4.1

(Finite-sample) If $(X_i,Y_i)$, for $i \in \mathcal{I}_\text{cal}\cup\mathcal{I}_\text{test}$, are exchangeable, then for $i \in \mathcal{I}_\text{test}$, the output of Alg. alg:CIR satisfies:

Figures (5)

  • Figure 1: Comparison of CIR, CIR+, CQR romano2019conformalized, and CHR sesia2021conformal in a single-variable example. All methods use the same deep quantile model and guarantee 90% marginal coverage. Left: Prediction bands as a function of $X$. Empirical marginal and estimated conditional coverage for all methods is 0.9 (except for CQR conditional coverage: 0.7). Average lengths: CIR (3.0), CIR+ (2.7), CHR (3.0), CQR (5.2). Right: Interquantile intervals at $X \approx 0.4$, using 50 quantiles with truncated high-density regions. Narrower intervals indicate higher densities. CIR and CIR+ select narrower intervals, with CIR+ choosing one fewer due to its scaled length being less than $\widehat{s}-s$.
  • Figure 2: Performance of our method (CIR) compared to that of naive uncalibrated prediction intervals based on the same deep neural network regression model. Note that the top part of this plot shows marginal coverage.
  • Figure 3: Performance comparison of our methods (CIR and CIR+) and benchmarks on synthetic data based on neural networks. Upper: Effects of sample size. The dashed lines and curves correspond to an omniscient oracle. The vertical error bars span two standard errors from the mean. Lower: Effects of distribution skewness of the conditional distribution of the response, with a sample size of 5000. The maximum skewness (near 3) corresponds to data in the upper figure.
  • Figure 4: Performance comparison of our methods (CIR and CIR+) and benchmarks on synthetic data based on random forest. Upper: Effects of sample size. The dashed lines and curves correspond to an omniscient oracle. The vertical error bars span two standard errors from the mean. Lower: Effects of distribution skewness of the conditional distribution of the response, with a sample size of 5000. The maximum skewness (near 3) corresponds to data in the upper figure.
  • Figure 5: Performance of our methods compared to that of naive uncalibrated prediction intervals based on the random forest regression model. Note that the top part of this plot shows marginal coverage.

Theorems & Definitions (5)

  • Theorem 4.1
  • proof
  • Theorem 4.2
  • Theorem A.1
  • proof