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

• 成果类型: Article
• 署名作者: Behdin, Kayhan; Chen, Wenyu; Mazumder, Rahul
• 署名单位: Massachusetts Institute of Technology (MIT)
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2023.0381
• 发表日期: 2026
• 关键词: scalable algorithms; regression; CONVERGENCE; selection; Lasso

摘要

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 x 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 & euro;0-penalized version of the pseudo-likelihood function; most earlier approaches are based on the & euro;1-relaxation. Our estimator can be formulated as a convex mixed-integer program (MIP) that can be difficult to compute beyond p approximate to 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.