  We consider the problem of discovering $K$ related Gaussian directed acyclic
graphs (DAGs), where the involved graph structures share a consistent causal
order and sparse unions of supports. Under the multi-task learning setting, we
propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning
$K$ linear structural equation models. We theoretically show that the joint
estimator, by leveraging data across related tasks, can achieve a better sample
complexity for recovering the causal order (or topological order) than separate
estimations. Moreover, the joint estimator is able to recover non-identifiable
DAGs, by estimating them together with some identifiable DAGs. Lastly, our
analysis also shows the consistency of union support recovery of the
structures. To allow practical implementation, we design a continuous
optimization problem whose optimizer is the same as the joint estimator and can
be approximated efficiently by an iterative algorithm. We validate the
theoretical analysis and the effectiveness of the joint estimator in
experiments.
