# Deep Huber quantile regression networks

arXiv: https://arxiv.org/abs/2306.10306 · HTML: https://mlsanity.com/paper/2306.10306/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 109 references that are in the map.
- First posted on arXiv 17 Jun 2023.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Don't Decay the Learning Rate, Increase the Batch Size](https://mlsanity.com/paper/1711.00489/) (2017)
- [A review of predictive uncertainty estimation with machine learning](https://mlsanity.com/paper/2209.08307/) (2022)
- [Of Quantiles and Expectiles: Consistent Scoring Functions, Choquet Representations, and Forecast Rankings](https://mlsanity.com/paper/1503.08195/) (2015)
- [A Survey on Epistemic (Model) Uncertainty in Supervised Learning: Recent Advances and Applications](https://mlsanity.com/paper/2111.01968/) (2021)
- [Boosting algorithms in energy research: A systematic review](https://mlsanity.com/paper/2004.07049/) (2020)
- [A review of machine learning concepts and methods for addressing challenges in probabilistic hydrological post-processing and forecasting](https://mlsanity.com/paper/2206.08998/) (2022)
- [Quantile Regression Neural Networks: A Bayesian Approach](https://mlsanity.com/paper/2009.13591/) (2020)
- [Hydrological post-processing for predicting extreme quantiles](https://mlsanity.com/paper/2202.13166/) (2022)
- [Quantification of predictive uncertainty in hydrological modelling by harnessing the wisdom of the crowd: A large-sample experiment at monthly timescale](https://mlsanity.com/paper/1909.00247/) (2019)

## Built on by (most cited first, 3 of 3)

- [A Robust Quantile Huber Loss With Interpretable Parameter Adjustment In Distributional Reinforcement Learning](https://mlsanity.com/paper/2401.02325/) (2024)
- [Boosted Enhanced Quantile Regression Neural Networks with Spatiotemporal Permutation Entropy for Complex System Prognostics](https://mlsanity.com/paper/2507.14194/) (2025)
- [Variable transformations in consistent loss functions](https://mlsanity.com/paper/2502.16542/) (2025)

## Abstract (arXiv)

Typical machine learning regression applications aim to report the mean or the median of the predictive probability distribution, via training with a squared or an absolute error scoring function. The importance of issuing predictions of more functionals of the predictive probability distribution (quantiles and expectiles) has been recognized as a means to quantify the uncertainty of the prediction. In deep learning (DL) applications, that is possible through quantile and expectile regression neural networks (QRNN and ERNN respectively). Here we introduce deep Huber quantile regression networks (DHQRN) that nest QRNN and ERNN as edge cases. DHQRN can predict Huber quantiles, which are more general functionals in the sense that they nest quantiles and expectiles as limiting cases. The main idea is to train a DL algorithm with the Huber quantile scoring function, which is consistent for the Huber quantile functional. As a proof of concept, DHQRN are applied to predict house prices in Melbourne, Australia and Boston, United States (US). In this context, predictive performances of three DL architectures are discussed along with evidential interpretation of results from two economic case studies. Additional simulation experiments and applications to real-world case studies using open datasets demonstrate a satisfactory absolute performance of DHQRN.