Consistency Guarantees for Greedy Permutation-Based Causal Inference Algorithms

Directed acyclic graph (DAG) models, are widely used to represent complex causal systems. Since the basic task of learning a DAG model from data is NP-hard, a standard approach is greedy search over the space of DAGs or Markov equivalence classes of DAGs. Since the space of DAGs on $p$ nodes and the associated space of Markov equivalence classes are both much larger than the space of permutations, it is desirable to consider permutation-based greedy searches. We here provide the first consistency guarantees, both uniform and high-dimensional, of a greedy permutation-based search. This search corresponds to a simplex-type algorithm over the edge-graph of a sub-polytope of the permutohedron, called the DAG associahedron. Every vertex in this polytope is associated with a DAG, and hence with a collection of permutations that are consistent with the DAG ordering. A walk is performed on the edges of the polytope maximizing the sparsity of the associated DAGs. We also show based on simulations that this permutation search is competitive with current approaches.