Smooth, Sparse, and Stable: Finite-Time Exact Skeleton Recovery via Smoothed Proximal Gradients

TL;DR

This paper introduces a novel optimization method that guarantees finite-time exact recovery of causal graph structures, overcoming limitations of previous approaches that only asymptotically approximate DAGs.

Contribution

The paper proposes the AHOC constraint and SPG-AHOC algorithm, providing the first finite-time guarantee for exact DAG support recovery in continuous optimization.

Findings

SPG-AHOC recovers the true DAG structure in finite iterations.

Theoretical proof of the Finite-Time Oracle Property under standard assumptions.

Empirical results show state-of-the-art accuracy and support the theory.

Abstract

Continuous optimization has significantly advanced causal discovery, yet existing methods (e.g., NOTEARS) generally guarantee only asymptotic convergence to a stationary point. This often yields dense weighted matrices that require arbitrary post-hoc thresholding to recover a DAG. This gap between continuous optimization and discrete graph structures remains a fundamental challenge. In this paper, we bridge this gap by proposing the Hybrid-Order Acyclicity Constraint (AHOC) and optimizing it via the Smoothed Proximal Gradient (SPG-AHOC). Leveraging the Manifold Identification Property of proximal algorithms, we provide a rigorous theoretical guarantee: the Finite-Time Oracle Property. We prove that under standard identifiability assumptions, SPG-AHOC recovers the exact DAG support (structure) in finite iterations, even when optimizing a smoothed approximation. This result eliminates structural ambiguity, as our algorithm returns graphs with exact zero entries without heuristic truncation. Empirically, SPG-AHOC achieves state-of-the-art accuracy and strongly corroborates the finite-time identification theory.