Exploiting Non-Negativity in DAG Structure Learning

Abstract

This work addresses the problem of learning directed acyclic graphs (DAGs) from nodal observations adhering to a linear structural equation model. DAG structure learning is a central task in signal processing, machine learning, and causal inference, but it remains challenging because acyclicity is a global combinatorial property and the search space grows superexponentially in the number of nodes. Continuous acyclicity characterizations have led to important algorithmic advances by replacing the discrete DAG restriction with smooth equality constraints on the candidate adjacency matrix. However, existing formulations still involve difficult non-convex optimization landscapes and may suffer from degenerate first-order optimality conditions. Here, we restrict attention to DAGs with non-negative edge weights and exploit this additional structure to obtain a simpler acyclicity characterization devoid of Hadamard products. We propose a new non-negative constrained DAG estimator and develop a tractable, locally convergent algorithm based on the method of multipliers. We further analyze the benign optimization landscape induced by the non-negativity constraints. In the large-sample regime, we show that the true DAG is the unique global minimizer of the population augmented Lagrangian. Moreover, the landscape contains no spurious interior stationary points, and the true DAG is the only acyclic Karush--Kuhn--Tucker (KKT) point. Critically, this allows us to guarantee local convergence to the true DAG given infinite samples. Comprehensive numerical experiments show that the proposed structure identification method improves over state-of-the-art continuous DAG-learning baselines.