  Wasserstein distributionally robust optimization has recently emerged as a
powerful framework for robust estimation, enjoying good out-of-sample
performance guarantees, well-understood regularization effects, and
computationally tractable reformulations. In such framework, the estimator is
obtained by minimizing the worst-case expected loss over all probability
distributions which are close, in a Wasserstein sense, to the empirical
distribution. In this paper, we propose a Wasserstein distributionally robust
estimation framework to estimate an unknown parameter from noisy linear
measurements, and we focus on the task of analyzing the squared error
performance of such estimators. Our study is carried out in the modern
high-dimensional proportional regime, where both the ambient dimension and the
number of samples go to infinity at a proportional rate which encodes the
under/over-parametrization of the problem. Under an isotropic Gaussian features
assumption, we show that the squared error can be recovered as the solution of
a convex-concave optimization problem which, surprinsingly, involves at most
four scalar variables. Importantly, the precise quantification of the squared
error allows to accurately and efficiently compare different ambiguity radii
and to understand the effect of the under/over-parametrization on the
estimation error. We conclude the paper with a list of exciting research
directions enabled by our results.
