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Title: Distributionally robust formulation of the graphical Lasso Authors:  Sang-Yun Oh - University of California, Santa Barbara (United States) [presenting]
Alexander Petersen - Brigham Young University (United States)
Pedro Cisneros-Velarde - University of California Santa Barbara (United States)
Chau Tran - University of California Santa Barbara (United States)
Abstract: Building on a recent framework for distributionally robust optimization, the inverse covariance matrix estimation is considered for multivariate data. We provide a novel notion of a Wasserstein ambiguity set specifically tailored to this estimation problem. A Special case includes penalized likelihood estimator 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. 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. We also establish a theoretical connection between the confidence level of graphical model selection via the DRO formulation and the asymptotic family-wise error rate of estimating false edges.