High-Dimensional Poisson DAG Model Learning Using $\ell_1$-Regularized Regression

In this paper we develop a new approach for learning high-dimensional Poisson directed acyclic graphical (DAG) models from only observational data without strong assumptions such as faithfulness and strong-sparsity. A key component of our method is to decouple the ordering estimation or parents search where the problems can be efficiently addressed using the l_1-regularized regression and the mean-variance relationship. We show that the sample size n = Omega( d^{7/3} log^{7} p) is suffices for our polynomial time algorithm to recover the true graph, where p is the number of nodes and d is the maximum indegree. We verify through simulations that our algorithm is statistically consistent in the high-dimensional p > n setting, and performs well compared to state-of-the-art ODS, GES, MMHC algorithms.