Marginal Likelihoods for Distributed Parameter Estimation of Gaussian Graphical Models

We consider distributed estimation of the inverse covariance matrix, also called the concentration or precision matrix, in Gaussian graphical models. Traditional centralized estimation often requires global inference of the covariance matrix, which can be computationally intensive in large dimensions. Approximate inference based on message-passing algorithms, on the other hand, can lead to unstable and biased estimation in loopy graphical models. Here, we propose a general framework for distributed estimation based on a maximum marginal likelihood (MML) approach. This approach computes local parameter estimates by maximizing marginal likelihoods defined with respect to data collected from local neighborhoods. Due to the non-convexity of the MML problem, we introduce and solve a convex relaxation. The local estimates are then combined into a global estimate without the need for iterative message-passing between neighborhoods. The proposed algorithm is naturally parallelizable and computationally efficient, thereby making it suitable for high-dimensional problems. In the classical regime where the number of variables
$p$
is fixed and the number of samples
$T$
increases to infinity, the proposed estimator is shown to be asymptotically consistent and to improve monotonically as the local neighborhood size increases. In the high-dimensional scaling regime where both
$p$
and
$T$
increase to infinity, the convergence rate to the true parameters is derived and is seen to be comparable to centralized maximum-likelihood estimation. Extensive numerical experiments demonstrate the improved performance of the two-hop version of the proposed estimator, which suffices to almost close the gap to the centralized maximum likelihood estimator at a reduced computational cost.