QWO: Speeding Up Permutation-Based Causal Discovery in LiGAMs

Causal discovery is essential for understanding relationships among variables of interest in many scientific domains. In this paper, we focus on permutation-based methods for learning causal graphs in Linear Gaussian Acyclic Models (LiGAMs), where the permutation encodes a causal ordering of the variables. Existing methods in this setting are not scalable due to their high computational complexity. These methods are comprised of two main components: (i) constructing a specific DAG, $\mathcal{G}^\pi$, for a given permutation $\pi$, which represents the best structure that can be learned from the available data while adhering to $\pi$, and (ii) searching over the space of permutations (i.e., causal orders) to minimize the number of edges in $\mathcal{G}^\pi$. We introduce QWO, a novel approach that significantly enhances the efficiency of computing $\mathcal{G}^\pi$ for a given permutation $\pi$. QWO has a speed-up of $O(n^2)$ ($n$ is the number of variables) compared to the state-of-the-art BIC-based method, making it highly scalable. We show that our method is theoretically sound and can be integrated into existing search strategies such as GRASP and hill-climbing-based methods to improve their performance.