# Nonparametric Quantile Regression: Non-Crossing Constraints and Conformal Prediction

arXiv: https://arxiv.org/abs/2210.10161 · HTML: https://mlsanity.com/paper/2210.10161/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 24 references that are in the map.
- First posted on arXiv 18 Oct 2022.

## Builds on (most cited first, 7 of 7)

- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Conformal Prediction: a Unified Review of Theory and New Challenges](https://mlsanity.com/paper/2005.07972/) (2020)
- [A comparison of some conformal quantile regression methods](https://mlsanity.com/paper/1909.05433/) (2019)
- [Adaptive, Distribution-Free Prediction Intervals for Deep Networks](https://mlsanity.com/paper/1905.10634/) (2019)
- [Deep Non-Crossing Quantiles through the Partial Derivative](https://mlsanity.com/paper/2201.12848/) (2022)

## Built on by (most cited first, 5 of 5)

- [Conformalized Multimodal Uncertainty Regression and Reasoning](https://mlsanity.com/paper/2309.11018/) (2023)
- [Conditional Quantile Estimation for Uncertain Watch Time in Short-Video Recommendation](https://mlsanity.com/paper/2407.12223/) (2024)
- [Simulation-Based Inference with Quantile Regression](https://mlsanity.com/paper/2401.02413/) (2024)
- [Relative Advantage Debiasing for Watch-Time Prediction in Short-Video Recommendation](https://mlsanity.com/paper/2508.11086/) (2025)
- [Any-Quantile Probabilistic Forecasting of Short-Term Electricity Demand](https://mlsanity.com/paper/2404.17451/) (2024)

## Abstract (arXiv)

We propose a nonparametric quantile regression method using deep neural networks with a rectified linear unit penalty function to avoid quantile crossing. This penalty function is computationally feasible for enforcing non-crossing constraints in multi-dimensional nonparametric quantile regression. We establish non-asymptotic upper bounds for the excess risk of the proposed nonparametric quantile regression function estimators. Our error bounds achieve optimal minimax rate of convergence for the Holder class, and the prefactors of the error bounds depend polynomially on the dimension of the predictor, instead of exponentially. Based on the proposed non-crossing penalized deep quantile regression, we construct conformal prediction intervals that are fully adaptive to heterogeneity. The proposed prediction interval is shown to have good properties in terms of validity and accuracy under reasonable conditions. We also derive non-asymptotic upper bounds for the difference of the lengths between the proposed non-crossing conformal prediction interval and the theoretically oracle prediction interval. Numerical experiments including simulation studies and a real data example are conducted to demonstrate the effectiveness of the proposed method.