Relaxed Sparsest-Permutation Formulation for Causal Discovery at Scale

TL;DR

This paper introduces SCOPE, a scalable method for causal discovery that relaxes sparsest-permutation formulations, enabling efficient structure learning in large datasets with improved runtime.

Contribution

It proposes a support-level relaxation of sparsest-permutation learning and a scalable implementation called SCOPE, achieving accurate causal structure recovery at scale.

Findings

SCOPE matches the accuracy of slower baselines in MEC recovery.

SCOPE significantly reduces runtime compared to existing methods.

SCOPE scales to datasets with up to 10,000 variables.

Abstract

Despite the growing availability of large datasets, causal structure learning remains computationally prohibitive at scale. We revisit sparsest-permutation learning for linear structural equation models and show that exact Cholesky factorization is unnecessary for structure recovery. This observation motivates a support-level relaxation that searches for sparse triangular factors over a precision-support screening graph. The relaxed formulation can be efficiently evaluated via masked zero-fill incomplete Cholesky factorization, enabling scalable comparison of candidate orderings. At the population level, we establish soundness for Markov equivalence class (MEC) recovery under no-cancellation and sparsest Markov representation assumptions, as well as robustness to ordering misspecification. Motivated by these guarantees, we introduce SCOPE, a sparse-Cholesky pipeline that provides a scalable implementation of the relaxed formulation. Experiments on synthetic and real datasets demonstrate that SCOPE matches the MEC recovery accuracy of substantially slower baselines, while achieving significantly reduced runtime and scaling to 10k variables.