2019/01/01 by Pedro Cisneros‐Velarde, Cisneros-Velarde, Pedro, Sang‐Yun Oh +4 · 1 citation
Decision Sciences · Mathematics · #62F35 #62F40 #90C90 #Advanced Statistical Methods and Models #Advanced Statistical Process Monitoring #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Risk and Portfolio Optimization #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1905.08975
openalex publication_date 2019/01/01 · openalex created_date 2022/07/29 · openalex updated_date 2026/08/04
Building on a recent framework for distributionally robust optimization, we\nconsider estimation of the inverse covariance matrix for multivariate data. We\nprovide a novel notion of a Wasserstein ambiguity set specifically tailored to\nthis estimation problem, leading to a tractable class of regularized\nestimators. Special cases include penalized likelihood estimators for Gaussian\ndata, specifically the graphical lasso estimator. As a consequence of this\nformulation, the radius of the Wasserstein ambiguity set is directly related to\nthe regularization parameter in the estimation problem. Using this\nrelationship, the level of robustness of the estimation procedure can be shown\nto correspond to the level of confidence with which the ambiguity set contains\na distribution with the population covariance. Furthermore, a unique feature of\nour formulation is that the radius can be expressed in closed-form as a\nfunction of the ordinary sample covariance matrix. Taking advantage of this\nfinding, we develop a simple algorithm to determine a regularization parameter\nfor graphical lasso, using only the bootstrapped sample covariance matrices,\nmeaning that computationally expensive repeated evaluation of the graphical\nlasso algorithm is not necessary. Alternatively, the distributionally robust\nformulation can also quantify the robustness of the corresponding estimator if\none uses an off-the-shelf method such as cross-validation. Finally, we\nnumerically study the obtained regularization criterion and analyze the\nrobustness of other automated tuning procedures used in practice.\n