# Distributionally robust linear regression through the lens of adversarial training

arXiv: https://arxiv.org/abs/2609.39449 · HTML: https://mlsanity.com/paper/2609.39449/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 42 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 18 of 18)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [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)
- [The Many Faces of Adversarial Risk](https://mlsanity.com/paper/2201.08956/) (2022)
- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [Regularization properties of adversarially-trained linear regression](https://mlsanity.com/paper/2310.10807/) (2023)
- [A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2205.00362/) (2022)
- [Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization](https://mlsanity.com/paper/2303.03900/) (2023)
- [Wasserstein Distributionally Robust Estimation in High Dimensions: Performance Analysis and Optimal Hyperparameter Tuning](https://mlsanity.com/paper/2206.13269/) (2022)
- [Tikhonov Regularization is Optimal Transport Robust under Martingale Constraints](https://mlsanity.com/paper/2210.01413/) (2022)
- [On Generalization and Regularization via Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2212.05716/) (2022)
- [Disciplined Saddle Programming](https://mlsanity.com/paper/2301.13427/) (2023)
- [Efficient Optimization Algorithms for Linear Adversarial Training](https://mlsanity.com/paper/2410.12677/) (2024)
- [Universal generalization guarantees for Wasserstein distributionally robust models](https://mlsanity.com/paper/2402.11981/) (2024)
- [The out-of-sample prediction error of the square-root-LASSO and related estimators](https://mlsanity.com/paper/2211.07608/) (2022)
- [Kernel Learning with Adversarial Features: Numerical Efficiency and Adaptive Regularization](https://mlsanity.com/paper/2510.20883/) (2025)

## Abstract (arXiv)

Distributionally robust optimization (DRO) studies parameter estimation under uncertainty in the underlying probability distribution and has emerged as a principled framework for analyzing robustness and generalization. In particular, Wasserstein DRO, with distributional uncertainty induced by the Wasserstein distance, generalizes several popular regularizers. This paper studies Wasserstein DRO linear regression, unifying square-root Lasso and adversarial linear regression as important special cases. We prove that many properties of these two special cases carry over to this general method. In particular, we show (i) deterministic and non-asymptotic in-sample error bounds O(n-1/2) in general and O(n-1) under design matrix and sparsity conditions; (ii) insensitivity to the noise level, also known as the pivotal property; and (iii) solution equivalences for small and large ambiguity sets. The key proof step is to recast the method into a quadratic form, mimicking adversarial linear regression. We also show that the method can be solved efficiently, and we validate our findings through numerical simulations.