Wasserstein Distributionally Robust Quantile Regression

TL;DR

This paper develops a distributionally robust quantile regression framework using Wasserstein ambiguity sets, providing explicit formulas, theoretical insights, and risk guarantees, with practical demonstrations.

Contribution

It introduces a closed-form worst-case loss expression for Wasserstein-based quantile regression and characterizes the unique convex loss function for $p>1$, revealing qualitative differences between $p=1$ and $p>1$ regimes.

Findings

Explicit worst-case loss formula derived

Unique convex loss function identified for $p>1$

Finite-sample risk guarantees established

Abstract

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.