# Improving Conditional Coverage via Orthogonal Quantile Regression

arXiv: https://arxiv.org/abs/2106.00394 · HTML: https://mlsanity.com/paper/2106.00394/

## Key facts

- Cited by 48 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 55 references that are in the map.
- First posted on arXiv 1 Jun 2021.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Uncertainty Sets for Image Classifiers using Conformal Prediction](https://mlsanity.com/paper/2009.14193/) (2020)
- [Distribution-Free, Risk-Controlling Prediction Sets](https://mlsanity.com/paper/2101.02703/) (2021)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [Empirical Risk Minimization under Fairness Constraints](https://mlsanity.com/paper/1802.08626/) (2018)
- [Distributional conformal prediction](https://mlsanity.com/paper/1909.07889/) (2019)
- [Nested conformal prediction and quantile out-of-bag ensemble methods](https://mlsanity.com/paper/1910.10562/) (2019)
- [Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty Quantification](https://mlsanity.com/paper/2011.09588/) (2020)
- [A comparison of some conformal quantile regression methods](https://mlsanity.com/paper/1909.05433/) (2019)
- [CD-split and HPD-split: efficient conformal regions in high dimensions](https://mlsanity.com/paper/2007.12778/) (2020)
- [Flexible distribution-free conditional predictive bands using density estimators](https://mlsanity.com/paper/1910.05575/) (2019)
- [Adaptive, Distribution-Free Prediction Intervals for Deep Networks](https://mlsanity.com/paper/1905.10634/) (2019)
- [With Malice Towards None: Assessing Uncertainty via Equalized Coverage](https://mlsanity.com/paper/1908.05428/) (2019)
- [Achieving Equalized Odds by Resampling Sensitive Attributes](https://mlsanity.com/paper/2006.04292/) (2020)
- [Robust Learning with the Hilbert-Schmidt Independence Criterion](https://mlsanity.com/paper/1910.00270/) (2019)
- [The Randomized Dependence Coefficient](https://mlsanity.com/paper/1304.7717/) (2013)
- [Distribution-Free Conditional Median Inference](https://mlsanity.com/paper/2102.07967/) (2021)

## Built on by (most cited first, 30 of 48)

- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Image-to-Image Regression with Distribution-Free Uncertainty Quantification and Applications in Imaging](https://mlsanity.com/paper/2202.05265/) (2022)
- [Online conformal prediction with decaying step sizes](https://mlsanity.com/paper/2402.01139/) (2024)
- [Conformal Prediction for Natural Language Processing: A Survey](https://mlsanity.com/paper/2405.01976/) (2024)
- [Calibrated Multiple-Output Quantile Regression with Representation Learning](https://mlsanity.com/paper/2110.00816/) (2021)
- [Sample-Efficient Safety Assurances using Conformal Prediction](https://mlsanity.com/paper/2109.14082/) (2021)
- [High-Dimensional Prediction for Sequential Decision Making](https://mlsanity.com/paper/2310.17651/) (2023)
- [Achieving Risk Control in Online Learning Settings](https://mlsanity.com/paper/2205.09095/) (2022)
- [Length Optimization in Conformal Prediction](https://mlsanity.com/paper/2406.18814/) (2024)
- [A Large-Scale Study of Probabilistic Calibration in Neural Network Regression](https://mlsanity.com/paper/2306.02738/) (2023)
- [Efficient and Differentiable Conformal Prediction with General Function Classes](https://mlsanity.com/paper/2202.11091/) (2022)
- [Conformal Prediction with Learned Features](https://mlsanity.com/paper/2404.17487/) (2024)
- [Split Localized Conformal Prediction](https://mlsanity.com/paper/2206.13092/) (2022)
- [Integrating Uncertainty Awareness into Conformalized Quantile Regression](https://mlsanity.com/paper/2306.08693/) (2023)
- [Equal Opportunity of Coverage in Fair Regression](https://mlsanity.com/paper/2311.02243/) (2023)
- [Wasserstein-regularized Conformal Prediction under General Distribution Shift](https://mlsanity.com/paper/2501.13430/) (2025)
- [Calibration in Machine Learning Uncertainty Quantification: beyond consistency to target adaptivity](https://mlsanity.com/paper/2309.06240/) (2023)
- [Towards Instance-Wise Calibration: Local Amortized Diagnostics and Reshaping of Conditional Densities (LADaR)](https://mlsanity.com/paper/2205.14568/) (2022)
- [Epistemic Uncertainty in Conformal Scores: A Unified Approach](https://mlsanity.com/paper/2502.06995/) (2025)
- [Conformalized Unconditional Quantile Regression](https://mlsanity.com/paper/2304.01426/) (2023)
- [Adaptive Conformal Regression with Jackknife+ Rescaled Scores](https://mlsanity.com/paper/2305.19901/) (2023)
- [Relaxed Quantile Regression: Prediction Intervals for Asymmetric Noise](https://mlsanity.com/paper/2406.03258/) (2024)
- [Conformalized Interval Arithmetic with Symmetric Calibration](https://mlsanity.com/paper/2408.10939/) (2024)
- [Conditional Coverage Diagnostics for Conformal Prediction](https://mlsanity.com/paper/2512.11779/) (2025)
- [Conformalized Multimodal Uncertainty Regression and Reasoning](https://mlsanity.com/paper/2309.11018/) (2023)
- [Validation of uncertainty quantification metrics: a primer based on the consistency and adaptivity concepts](https://mlsanity.com/paper/2303.07170/) (2023)
- [Uncertainty Quantification on Graph Learning: A Survey](https://mlsanity.com/paper/2404.14642/) (2024)
- [Likelihood Annealing: Fast Calibrated Uncertainty for Regression](https://mlsanity.com/paper/2302.11012/) (2023)
- [Distribution-Free Model-Agnostic Regression Calibration via Nonparametric Methods](https://mlsanity.com/paper/2305.12283/) (2023)
- [Conformal Prediction in Hierarchical Classification with Constrained Representation Complexity](https://mlsanity.com/paper/2501.19038/) (2025)

## Abstract (arXiv)

We develop a method to generate prediction intervals that have a user-specified coverage level across all regions of feature-space, a property called conditional coverage. A typical approach to this task is to estimate the conditional quantiles with quantile regression -- it is well-known that this leads to correct coverage in the large-sample limit, although it may not be accurate in finite samples. We find in experiments that traditional quantile regression can have poor conditional coverage. To remedy this, we modify the loss function to promote independence between the size of the intervals and the indicator of a miscoverage event. For the true conditional quantiles, these two quantities are independent (orthogonal), so the modified loss function continues to be valid. Moreover, we empirically show that the modified loss function leads to improved conditional coverage, as evaluated by several metrics. We also introduce two new metrics that check conditional coverage by looking at the strength of the dependence between the interval size and the indicator of miscoverage.