Efficient Permutation Discovery in Causal DAGs

TL;DR

This paper introduces an efficient algorithm for discovering permutations in causal DAGs that induce sparse graphs, outperforming existing methods in runtime and applicability to dense graphs, based on DAG-specific structure.

Contribution

The paper presents a novel, efficient permutation discovery algorithm for causal DAGs that leverages DAG-specific structure, improving over prior NP-hard approaches.

Findings

Runs in polynomial time for Gaussian distributions with depth w

Outperforms algorithms for sparse elimination orderings

Achieves near-perfect results on dense graphs

Abstract

The problem of learning a directed acyclic graph (DAG) up to Markov equivalence is equivalent to the problem of finding a permutation of the variables that induces the sparsest graph. Without additional assumptions, this task is known to be NP-hard. Building on the minimum degree algorithm for sparse Cholesky decomposition, but utilizing DAG-specific problem structure, we introduce an efficient algorithm for finding such sparse permutations. We show that on jointly Gaussian distributions, our method with depth $w$ runs in $O(p^{w+3})$ time. We compare our method with $w = 1$ to algorithms for finding sparse elimination orderings of undirected graphs, and show that taking advantage of DAG-specific problem structure leads to a significant improvement in the discovered permutation. We also compare our algorithm to provably consistent causal structure learning algorithms, such as the PC algorithm, GES, and GSP, and show that our method achieves comparable performance with a shorter runtime. Thus, our method can be used on its own for causal structure discovery. Finally, we show that there exist dense graphs on which our method achieves almost perfect performance, so that unlike most existing causal structure learning algorithms, the situations in which our algorithm achieves both good performance and good runtime are not limited to sparse graphs.