Sparse Gaussian Graphical Models with Discrete Optimization: Computational and Statistical Perspectives

We consider the problem of learning a sparse graph underlying an undirected Gaussian graphical model, which is a key problem in statistical machine learning. Given n samples from a multivariate Gaussian distribution with p variables, the goal is to estimate the $p\times p$ inverse covariance matrix (aka precision matrix), assuming it is sparse (i.e., has a few nonzero entries). We propose GraphL0BnB, a new estimator based on an ${\mathcal{\ell }}_0$-penalized version of the pseudo-likelihood function; most earlier approaches are based on the ${\mathcal{\ell }}_1$-relaxation. Our estimator can be formulated as a convex mixed-integer program (MIP) that can be difficult to compute beyond $p\approx 100$ using off-the-shelf commercial solvers. To solve the MIP, we propose a custom nonlinear branch-and-bound (BnB) framework that solves node relaxations with tailored first-order methods. As a key component of our BnB framework, we propose large-scale solvers for obtaining good primal solutions that are of independent interest. We derive novel statistical guarantees (estimation and variable selection) for our estimator and discuss how our approach improves upon existing estimators. Our numerical experiments on real and synthetic data sets suggest that our BnB framework offers significant advantages over off-the-shelf commercial solvers, and our approach has favorable performance (both in terms of runtime and statistical performance) compared with the state-of-the-art approaches for learning sparse graphical models.

Funding: This research is supported in part by grants from the Office of Naval Research [N000142212665 and N000142112841].

Supplemental Material: The online appendices are available at https://doi.org/10.1287/opre.2023.0381.