# Consistency Guarantees for Greedy Permutation-Based Causal Inference Algorithms

## TL;DR

This paper introduces the first consistency guarantees for a permutation-based greedy search algorithm for learning DAG models, which is more scalable and competitive with existing methods.

## Contribution

It provides the first theoretical guarantees for a permutation-based greedy DAG learning algorithm, connecting it to a simplex-like polytope structure.

## Key findings

- Permutation search is consistent both uniformly and in high dimensions.
- The method is competitive with existing DAG learning algorithms.
- The approach leverages a polytope structure called the DAG associahedron.

## Abstract

Directed acyclic graphical models, or DAG models, are widely used to represent complex causal systems. Since the basic task of learning such a model from data is NP-hard, a standard approach is greedy search over the space of directed acyclic graphs or Markov equivalence classes of directed acyclic graphs. As the space of directed acyclic graphs 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. Here, we provide the first consistency guarantees, both uniform and high-dimensional, of a greedy permutation-based search. This search corresponds to a simplex-like algorithm operating over the edge-graph of a sub-polytope of the permutohedron, called a DAG associahedron. Every vertex in this polytope is associated with a directed acyclic graph, and hence with a collection of permutations that are consistent with the directed acyclic graph ordering. A walk is performed on the edges of the polytope maximizing the sparsity of the associated directed acyclic graphs. We show via simulated and real data that this permutation search is competitive with current approaches.