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Title: Distributionally robust formulation of the graphical lasso Authors:  Sang-Yun Oh - University of California, Santa Barbara (United States) [presenting]
Abstract: Building on a recent framework for distributionally robust optimization, estimation of the inverse covariance matrix is considered 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 corresponds to the level of confidence with which the ambiguity set contains a distribution with the population covariance. Furthermore, 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 the graphical lasso, using only the bootstrapped sample covariance matrices, avoiding repeated evaluation of the graphical lasso algorithm during regularization parameter tuning, for example, with cross-validation. Finally, we numerically study the obtained regularization criterion and analyze the robustness of other automated tuning procedures used in practice.