UC Santa Barbara

Building on a recent framework for distributionally robust optimization, we consider

estimation of the inverse covariance matrix for multivariate data. We provide a novel

notion of a Wasserstein ambiguity set specifically tailored to this estimation problem,

leading to a tractable class of regularized estimators. Special cases include penalized

likelihood estimators for Gaussian data, specifically the graphical lasso estimator. As a

consequence of this formulation, the radius of the Wasserstein ambiguity set is directly

related to the regularization parameter in the estimation problem. Using this relationship, the level of robustness of the estimation procedure can be shown to correspond to

the level of confidence with which the ambiguity set contains a distribution with the population covariance. Furthermore, a unique feature of our formulation is that the radius

can be expressed in closed-form as a function of the ordinary sample covariance matrix.

Taking advantage of this finding, we develop a simple algorithm to determine a regularization parameter for graphical lasso, using only the bootstrapped sample covariance

matrices, meaning that computationally expensive repeated evaluation of the graphical

lasso algorithm is not necessary. Alternatively, the distributionally robust formulation

can also quantify the robustness of the corresponding estimator if one uses an off-the-shelf

method such as cross-validation. Finally, we numerically study the obtained regularization criterion and analyze the robustness of other automated tuning procedures used in practice.