On the Role of Sparsity and DAG Constraints for Learning Linear DAGs

Learning graphical structure based on Directed Acyclic Graphs (DAGs) is a challenging problem, partly owing to the large search space of possible graphs. A recent line of work formulates the structure learning problem as a continuous constrained optimization task using the least squares objective and an algebraic characterization of DAGs. However, the formulation requires a hard DAG constraint and may lead to optimization difficulties. In this paper, we study the asymptotic role of the sparsity and DAG constraints for learning DAG models in the linear Gaussian and non-Gaussian cases, and investigate their usefulness in the finite sample regime. Based on the theoretical results, we formulate a likelihood-based score function, and show that one only has to apply soft sparsity and DAG constraints to learn a DAG equivalent to the ground truth DAG.