Wasserstein Distributionally Robust Quantile Regression
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915865703743488 |
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| author | Zhang, Chunxu Mao, Tiantian Wang, Ruodu |
| author_facet | Zhang, Chunxu Mao, Tiantian Wang, Ruodu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14991 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wasserstein Distributionally Robust Quantile Regression Zhang, Chunxu Mao, Tiantian Wang, Ruodu Statistics Theory Optimization and Control 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. |
| title | Wasserstein Distributionally Robust Quantile Regression |
| topic | Statistics Theory Optimization and Control |
| url | https://arxiv.org/abs/2603.14991 |