  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. In this
paper, 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.
