# Wasserstein Distributionally Robust Quantile Regression

arXiv: https://arxiv.org/abs/2603.14991 · HTML: https://mlsanity.com/paper/2603.14991/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Mar 2026.

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

- [Quantile Geometry Regularization for Distributional Reinforcement Learning](https://mlsanity.com/paper/2605.08182/) (2026)

## Abstract (arXiv)

We study distributionally robust quantile regression using type-p Wasserstein ambiguity sets. We derive a closed-form expression for the worst-case quantile regression loss under general p-Wasserstein uncertainty. We further give a uniqueness result showing that for p>1, the check loss yields the only class of convex loss functions for which such an additive Wasserstein regularization holds. Our analysis also uncovers qualitative differences between the regimes p=1 and p>1. When p>1, the slope coefficients coincide with those of the regularized formulation, while the intercept undergoes a radius-dependent adjustment; the value p affects only this intercept correction, whereas the choice of transport norm influences both. Finally, we establish finite-sample out-of-sample risk guarantees of order O(N-1/2) under mild moment conditions. Numerical experiments illustrate the theoretical findings and the practical implications of the proposed formulation.