TriOpt: A Scalable Algorithm for Linear Causal Discovery

TL;DR

TriOpt is a scalable linear causal discovery algorithm that combines ordering-based and continuous optimization methods, enabling fast and accurate DAG learning in high-dimensional data.

Contribution

It introduces a novel two-stage approach that decomposes the problem, achieving significant scalability improvements without sacrificing accuracy.

Findings

TriOpt achieves orders-of-magnitude speedups over existing methods.

It maintains comparable or better accuracy in high-dimensional datasets.

Theoretically guarantees exact recovery under the true ordering.

Abstract

Learning causal relations from observational data is challenging because the graph search space grows super-exponentially with the number of variables. Ordering-based methods reduce this space by first identifying the topological ordering, whereas continuous optimization methods explore most likely regions of the space by casting DAG learning as a differentiable objective with an acyclicity constraint. Despite their conceptual appeal, both paradigms face significant scalability limitations in high-dimensional settings, restricting their practical applicability. In this work, we introduce a new formulation for linear causal discovery that tightly integrates these two paradigms to achieve substantial gains in scalability without sacrificing accuracy. Our approach, TriOpt, decomposes the problem into two efficient stages. First, it recovers the topological ordering by exploiting the Sherman-Morrison rank-1 downdate together with the additive structure of linear kernels, enabling fast and scalable ordering estimation. Second, given this ordering, we reformulate structure learning as a convex continuous optimization problem that entirely avoids the need for enforcing costly acyclicity constraints. We theoretically show that, under the true ordering, TriOpt exactly recovers the underlying linear DAG. Empirically, across synthetic, semi-synthetic, and real-world datasets, TriOpt achieves orders-of-magnitude speedups over state-of-the-art linear causal discovery methods in high-dimensional regimes, while maintaining comparable or superior accuracy.