# Quantile regression with deep ReLU Networks: Estimators and minimax rates

arXiv: https://arxiv.org/abs/2010.08236 · HTML: https://mlsanity.com/paper/2010.08236/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2020.

## Built on by (most cited first, 24 of 24)

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- [Distributional Off-Policy Evaluation with Deep Quantile Process Regression](https://mlsanity.com/paper/2604.18143/) (2026)
- [ConquerNet: Convolution-Smoothed Quantile ReLU Neural Networks with Minimax Guarantees](https://mlsanity.com/paper/2605.06265/) (2026)
- [On Pairwise Quantile Regression - Statistical Guarantees and Applications](https://mlsanity.com/paper/2607.04431/) (2026)
- [Adaptive deep nonparametric regression from dependent data under covariate shift](https://mlsanity.com/paper/2607.20309/) (2026)
- [Conditional Diffusion for Nonparametric Instrumental Variable Quantile Regression](https://mlsanity.com/paper/2608.08204/) (2026)
- [A Distribution Mapping Approach to Counterfactually Fair Reinforcement Learning](https://mlsanity.com/paper/2608.08743/) (2026)
- [Transfer Learning in Nonparametric Regression with Deep ReLU Networks](https://mlsanity.com/paper/2608.20255/) (2026)

## Abstract (arXiv)

Quantile regression is the task of estimating a specified percentile response, such as the median, from a collection of known covariates. We study quantile regression with rectified linear unit (ReLU) neural networks as the chosen model class. We derive an upper bound on the expected mean squared error of a ReLU network used to estimate any quantile conditional on a set of covariates. This upper bound only depends on the best possible approximation error, the number of layers in the network, and the number of nodes per layer. We further show upper bounds that are tight for two large classes of functions: compositions of H"older functions and members of a Besov space. These tight bounds imply ReLU networks with quantile regression achieve minimax rates for broad collections of function types. Unlike existing work, the theoretical results hold under minimal assumptions and apply to general error distributions, including heavy-tailed distributions. Empirical simulations on a suite of synthetic response functions demonstrate the theoretical results translate to practical implementations of ReLU networks. Overall, the theoretical and empirical results provide insight into the strong performance of ReLU neural networks for quantile regression across a broad range of function classes and error distributions. All code for this paper is publicly available at https://github.com/tansey/quantile-regression.