# Wasserstein Distributionally Robust Nonparametric Regression

arXiv: https://arxiv.org/abs/2505.07967 · HTML: https://mlsanity.com/paper/2505.07967/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 49 references that are in the map.
- First posted on arXiv 12 May 2025.

## Builds on (most cited first, 22 of 22)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Rademacher Complexity for Adversarially Robust Generalization](https://mlsanity.com/paper/1810.11914/) (2018)
- [Regularization via Mass Transportation](https://mlsanity.com/paper/1710.10016/) (2017)
- [Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality](https://mlsanity.com/paper/2009.04382/) (2020)
- [Adversarial Learning Guarantees for Linear Hypotheses and Neural Networks](https://mlsanity.com/paper/2004.13617/) (2020)
- [Generalized Resilience and Robust Statistics](https://mlsanity.com/paper/1909.08755/) (2019)
- [Theoretical Analysis of Adversarial Learning: A Minimax Approach](https://mlsanity.com/paper/1811.05232/) (2018)
- [How do noise tails impact on deep ReLU networks?](https://mlsanity.com/paper/2203.10418/) (2022)
- [Confidence Regions in Wasserstein Distributionally Robust Estimation](https://mlsanity.com/paper/1906.01614/) (2019)
- [Wasserstein Generative Learning of Conditional Distribution](https://mlsanity.com/paper/2112.10039/) (2021)
- [Approximation bounds for norm constrained neural networks with applications to regression and GANs](https://mlsanity.com/paper/2201.09418/) (2022)
- [Principled learning method for Wasserstein distributionally robust optimization with local perturbations](https://mlsanity.com/paper/2006.03333/) (2020)
- [A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2205.00362/) (2022)
- [Sensitivity analysis of Wasserstein distributionally robust optimization problems](https://mlsanity.com/paper/2006.12022/) (2020)
- [Robust W-GAN-Based Estimation Under Wasserstein Contamination](https://mlsanity.com/paper/2101.07969/) (2021)
- [Exact Generalization Guarantees for (Regularized) Wasserstein Distributionally Robust Models](https://mlsanity.com/paper/2305.17076/) (2023)
- [Universal generalization guarantees for Wasserstein distributionally robust models](https://mlsanity.com/paper/2402.11981/) (2024)

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

- [Wasserstein Regression as a Variational Approximation of Probabilistic Trajectories through the Bernstein Basis](https://mlsanity.com/paper/2510.26607/) (2025)
- [Concave Certificates: Geometric Framework for Distributionally Robust Risk and Complexity Analysis](https://mlsanity.com/paper/2601.01311/) (2026)
- [RepFlow: Representation Enhanced Flow Matching for Causal Effect Estimation](https://mlsanity.com/paper/2605.05890/) (2026)

## Abstract (arXiv)

Wasserstein distributionally robust optimization (WDRO) strengthens statistical learning under model uncertainty by minimizing the local worst-case risk within a prescribed ambiguity set. Although WDRO has been extensively studied in parametric settings, its theoretical properties in nonparametric frameworks remain underexplored. This paper investigates WDRO for nonparametric regression. We first establish a structural distinction based on the order k of the Wasserstein distance, showing that k=1 induces Lipschitz-type regularization, whereas k > 1 corresponds to gradient-norm regularization. To address model misspecification, we analyze the excess local worst-case risk, deriving non-asymptotic error bounds for estimators constructed using norm-constrained feedforward neural networks. This analysis is supported by new covering number and approximation bounds that simultaneously control both the function and its gradient. The proposed estimator achieves a convergence rate of n-2β/(d+2β) up to logarithmic factors, where β depends on the target's smoothness and network parameters. This rate is shown to be minimax optimal under conditions commonly satisfied in high-dimensional settings. Moreover, these bounds on the excess local worst-case risk imply guarantees on the excess natural risk, ensuring robustness against any distribution within the ambiguity set. We show the framework's generality across regression and classification problems. Simulation studies and an application to the MNIST dataset further illustrate the estimator's robustness.