# Regularization Strategies for Quantile Regression

arXiv: https://arxiv.org/abs/2102.05135 · HTML: https://mlsanity.com/paper/2102.05135/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 40 references that are in the map.
- First posted on arXiv 9 Feb 2021.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Optimization with Non-Differentiable Constraints with Applications to Fairness, Recall, Churn, and Other Goals](https://mlsanity.com/paper/1809.04198/) (2018)
- [Satisfying Real-world Goals with Dataset Constraints](https://mlsanity.com/paper/1606.07558/) (2016)
- [Deep Lattice Networks and Partial Monotonic Functions](https://mlsanity.com/paper/1709.06680/) (2017)
- [Pairwise Fairness for Ranking and Regression](https://mlsanity.com/paper/1906.05330/) (2019)
- [Two-Player Games for Efficient Non-Convex Constrained Optimization](https://mlsanity.com/paper/1804.06500/) (2018)
- [Monotonic Calibrated Interpolated Look-Up Tables](https://mlsanity.com/paper/1505.06378/) (2015)
- [Neural Likelihoods via Cumulative Distribution Functions](https://mlsanity.com/paper/1811.00974/) (2018)
- [Fair quantile regression](https://mlsanity.com/paper/1907.08646/) (2019)
- [Parsimonious Quantile Regression of Financial Asset Tail Dynamics via Sequential Learning](https://mlsanity.com/paper/2010.08263/) (2020)

## Built on by (most cited first, 1 of 1)

- [Air Traffic Controller Task Demand via Graph Neural Networks: An Interpretable Approach to Airspace Complexity](https://mlsanity.com/paper/2507.13423/) (2025)

## Abstract (arXiv)

We investigate different methods for regularizing quantile regression when predicting either a subset of quantiles or the full inverse CDF. We show that minimizing an expected pinball loss over a continuous distribution of quantiles is a good regularizer even when only predicting a specific quantile. For predicting multiple quantiles, we propose achieving the classic goal of non-crossing quantiles by using deep lattice networks that treat the quantile as a monotonic input feature, and we discuss why monotonicity on other features is an apt regularizer for quantile regression. We show that lattice models enable regularizing the predicted distribution to a location-scale family. Lastly, we propose applying rate constraints to improve the calibration of the quantile predictions on specific subsets of interest and improve fairness metrics. We demonstrate our contributions on simulations, benchmark datasets, and real quantile regression problems.