Nonnegative DAG Learning via Concomitant Estimation

Abstract

We study the problem of learning directed acyclic graphs (DAGs) with nonnegative edge weights from observational data. We propose the Nonnegative and Concomitant (NoCo) DAG estimator, which jointly recovers the weighted graph structure and the exogenous noise variances in the linear structural equation model for the observations. Different from prior art, this noise-adaptive formulation blends a smoothed concomitant lasso criterion with a simpler log-determinant acyclicity constraint that exploits nonnegativity and yields a more benign optimization landscape. Specifically, nonnegative weights allow us to impose acyclicity directly on the adjacency matrix without elementwise squaring of its entries, thus avoiding the well-documented degeneracy of the Karush-Kuhn-Tucker conditions. Computationally, we develop a method of multipliers' algorithm that accommodates both homoscedastic and heteroscedastic noise profiles. Within each iteration, we use block successive convex approximation to minimize the augmented Lagrangian, alternating between proximal gradient steps for the adjacency matrix and closed-form updates for the noise scales. Simulated experiments demonstrate NoCo's improved structural and edge-weight recovery relative to competing methods in a variety of settings, highlighting the benefits of exploiting nonnegativity along with noise adaptivity.