Wasserstein Distributionally Robust Estimation in High Dimensions: Performance Analysis and Optimal Hyperparameter Tuning

Distributionally robust optimization (DRO) has become a powerful framework for estimation under uncertainty, offering strong out-of-sample performance and principled regularization. In this paper, we propose a DRO-based method for linear regression and address a central question: how to optimally choose the robustness radius, which controls the trade-off between robustness and accuracy. Focusing on high-dimensional settings where the dimension and the number of samples are both large and comparable in size, we employ tools from high-dimensional asymptotic statistics to precisely characterize the estimation error of the resulting estimator. Remarkably, this error can be recovered by solving a simple convex-concave optimization problem involving only four scalar variables. This characterization enables efficient selection of the radius that minimizes the estimation error. In doing so, it achieves the same effect as cross-validation, but at a fraction of the computational cost. Numerical experiments confirm that our theoretical predictions closely match empirical performance and that the optimal radius selected through our method aligns with that chosen by cross-validation, highlighting both the accuracy and the practical benefits of our approach.