Estimation and inference in sparse multivariate regression and conditional Gaussian graphical models under an unbalanced distributed setting

This paper proposes a distributed estimation and inferential framework for sparse multivariate regression and conditional Gaussian graphical models under the unbalanced splitting setting. This type of data splitting arises when the datasets from different sources cannot be aggregated on one single machine or when the available machines are of different powers. In this paper, the number of covariates, responses and machines grow with the sample size, while sparsity is imposed. Debiased estimators of the coefficient matrix and of the precision matrix are proposed on every single machine and theoretical guarantees are provided. Moreover, new aggregated estimators that pool information across the machines using a pseudo log-likelihood function are proposed. It is shown that they enjoy efficiency and asymptotic normality as the number of machines grows with the sample size. The performance of these estimators is investigated via a simulation study and a real data example. It is shown empirically that the performances of these estimators are close to those of the non-distributed estimators which use the entire dataset.