Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator

Abstract

Note. The best result in each experiment is highlighted in bold.The optimal solutions of many decision problems such as the Markowitz portfolio allocation and the linear discriminant analysis depend on the inverse covariance matrix of a Gaussian random vector. In “Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator,” Nguyen, Kuhn, and Mohajerin Esfahani propose a distributionally robust inverse covariance estimator, obtained by robustifying the Gaussian maximum likelihood problem with a Wasserstein ambiguity set. In the absence of any prior structural information, the estimation problem has an analytical solution that is naturally interpreted as a nonlinear shrinkage estimator. Besides being invertible and well conditioned, the new shrinkage estimator is rotation equivariant and preserves the order of the eigenvalues of the sample covariance matrix. If there are sparsity constraints, which are typically encountered in Gaussian graphical models, the estimation problem can be solved using a sequential quadratic approximation algorithm.

Adjusted Wasserstein Distributionally Robust Estimator in Statistical Learning

“…We review the existing work related to the proposed adjusted WDRO estimator. WDRO is broadly utilized to solve parameter estimation problems (Shafieezadeh Abadeh et al, 2015;Aolaritei et al, 2022;Nguyen et al, 2022). Multiple algorithms have been developed (Luo and Mehrotra, 2019;Li et al, 2019;Blanchet et al, 2022c) and can be applied to compute the parameter estimators in the WDRO framework.…”

“…WDRO can be applied in statistical learning (Chen and Paschalidis, 2018;Nguyen et al, 2022). In general, the statistical learning model can be written as…”

Equilibrium transport with time-inconsistent costs: An application to matching problems in the job market

“…Many decision-making tasks are based on a nominal probability measure, which may not be accurate. Wasserstein distances were used in distributionally robust optimization by considering alternative measures within a certain Wasserstein distance from the nominal distribution and selecting decision rules that perform well even in the worst-case scenario (Blanchet and Murthy, 2019;Blanchet et al, 2021a;Gao and Kleywegt, 2022;Mohajerin Esfahani and Kuhn, 2018;Nguyen et al, 2022;Kuhn et al, 2019;Blanchet et al, 2021b).…”

Family‐wise error rate control in Gaussian graphical model selection via distributionally robust optimization

“…Recently, Cisneros‐Velarde et al (2020) have formulated the precision matrix estimation problem using the distributionally robust optimization (DRO) framework (Blanchet & Si, 2019; Nguyen et al, 2018). The authors establish the correspondence between the radius of the ambiguity set in the DRO framework—which measures the uncertainity around the empirical measure (see more below)—and the regularization parameter of the graphical lasso estimator.…”