# Distributionally Robust Formulation and Model Selection for the Graphical Lasso

arXiv: https://arxiv.org/abs/1905.08975 · HTML: https://mlsanity.com/paper/1905.08975/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 May 2019.

## Built on by (most cited first, 7 of 7)

- [A Finite Sample Complexity Bound for Distributionally Robust Q-learning](https://mlsanity.com/paper/2302.13203/) (2023)
- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [Testing Group Fairness via Optimal Transport Projections](https://mlsanity.com/paper/2106.01070/) (2021)
- [Robust Graph Learning Under Wasserstein Uncertainty](https://mlsanity.com/paper/2105.04210/) (2021)
- [Distributionally Robust Skeleton Learning of Discrete Bayesian Networks](https://mlsanity.com/paper/2311.06117/) (2023)
- [Relaxed Sparsest-Permutation Formulation for Causal Discovery at Scale](https://mlsanity.com/paper/2605.05568/) (2026)
- [A Unified Perspective on Conformal Prediction and Wasserstein Distributionally Robust Optimization for Uncertainty Quantification](https://mlsanity.com/paper/2608.29789/) (2026)

## Abstract (arXiv)

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.