Greedy Causal Discovery Is Geometric

Finding a directed acyclic graph (DAG) that best encodes the conditional independence statements observable from data is a central question within causality. Algorithms that greedily transform one candidate DAG into another given a fixed set of moves have been particularly successful, for example, the greedy equivalence search, greedy interventional equivalence search, and max-min hill climbing algorithms. In 2010, Studený, Hemmecke, and Lindner introduced the characteristic imset (CIM) polytope, , whose vertices correspond to Markov equivalence classes, as a way of transforming causal discovery into a linear optimization problem. We show that the moves of the aforementioned algorithms are included within classes of edges of and that restrictions placed on the skeleton of the candidate DAGs correspond to faces of . Thus, we observe that greedy equivalence search, greedy interventional equivalence search, and max-min hill climbing all have geometric realizations as greedy edge-walks along . Furthermore, the identified edges of strictly generalize the moves of these algorithms. Exploiting this generalization, we introduce a greedy simplex-type algorithm called greedy CIM, and a hybrid variant, skeletal greedy CIM, that outperforms current competitors among hybrid and constraint-based algorithms.