Exploiting Non-Negativity in DAG Structure Learning

TL;DR

This paper introduces a novel approach for learning DAGs with non-negative edge weights, simplifying the acyclicity constraint and improving optimization and accuracy over existing methods.

Contribution

It exploits non-negativity to characterize acyclicity more simply, developing a regularized learning method with favorable optimization landscape properties.

Findings

The true DAG is the unique global minimizer in the population regime.

The proposed method outperforms state-of-the-art alternatives on synthetic and real data.

The landscape has no spurious interior stationary points, aiding optimization.

Abstract

This work addresses the problem of learning directed acyclic graphs (DAGs) from nodal observations generated by a linear structural equation model. DAG learning is a central task in signal processing, machine learning, and causal inference, but it remains challenging because acyclicity is a global combinatorial property. Continuous acyclicity constraints have led to important algorithmic advances by replacing the discrete DAG constraint with smooth equality constraints. 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 characterization of acyclicity. Building on this characterization, we formulate a regularized non-negative DAG learning problem and develop an algorithm based on the method of multipliers. We further analyze the benign optimization landscape induced by non-negativity. In the population regime, we show that the true DAG is the unique global minimizer of the proposed augmented-Lagrangian formulation; moreover, the landscape contains no spurious interior stationary points, and the true DAG is the only acyclic KKT point. Numerical experiments on synthetic and real-world data show that the proposed method improves over state-of-the-art continuous DAG-learning alternatives.