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We consider the problem of learning high-dimensional Gaussian graphicalmodels. The graphical lasso is one of the most popular methods for estimatingGaussian graphical models. However, it does not achieve the oracle rate ofconvergence. In this paper, we propose the graphical nonconvex optimization foroptimal estimation in Gaussian graphical models, which is then approximated bya sequence of convex programs. Our proposal is computationally tractable andproduces an estimator that achieves the oracle rate of convergence. Thestatistical error introduced by the sequential approximation using the convexprograms are clearly demonstrated via a contraction property. The rate ofconvergence can be further improved using the notion of sparsity pattern. Theproposed methodology is then extended to semiparametric graphical models. Weshow through numerical studies that the proposed estimator outperforms otherpopular methods for estimating Gaussian graphical models.

Graphical Nonconvex Optimization for Optimal Estimation in Gaussian Graphical Models