# Nonnegative DAG Learning via Concomitant Estimation

arXiv: https://arxiv.org/abs/2609.34595 · HTML: https://mlsanity.com/paper/2609.34595/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 30 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [D'ya like DAGs? A Survey on Structure Learning and Causal Discovery](https://mlsanity.com/paper/2103.02582/) (2021)
- [On the Role of Sparsity and DAG Constraints for Learning Linear DAGs](https://mlsanity.com/paper/2006.10201/) (2020)
- [Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Game](https://mlsanity.com/paper/2102.13647/) (2021)
- [DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization](https://mlsanity.com/paper/2209.08037/) (2022)
- [D-VAE: A Variational Autoencoder for Directed Acyclic Graphs](https://mlsanity.com/paper/1904.11088/) (2019)
- [High-dimensional learning of linear causal networks via inverse covariance estimation](https://mlsanity.com/paper/1311.3492/) (2013)
- [DAGs with No Fears: A Closer Look at Continuous Optimization for Learning Bayesian Networks](https://mlsanity.com/paper/2010.09133/) (2020)
- [Causal Structure Learning: a Combinatorial Perspective](https://mlsanity.com/paper/2206.01152/) (2022)
- [On the Convergence of Continuous Constrained Optimization for Structure Learning](https://mlsanity.com/paper/2011.11150/) (2020)
- [Structure Learning with Continuous Optimization: A Sober Look and Beyond](https://mlsanity.com/paper/2304.02146/) (2023)
- [CoLiDE: Concomitant Linear DAG Estimation](https://mlsanity.com/paper/2310.02895/) (2023)
- [Inexact Block Coordinate Descent Algorithms for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/1905.04211/) (2019)
- [Learning DAGs from Data with Few Root Causes](https://mlsanity.com/paper/2305.15936/) (2023)
- [Non-negative Weighted DAG Structure Learning](https://mlsanity.com/paper/2409.07880/) (2024)
- [Efficient Smoothed Concomitant Lasso Estimation for High Dimensional Regression](https://mlsanity.com/paper/1606.02702/) (2016)
- [BUILD with Precision: Bottom-Up Inference of Linear DAGs](https://mlsanity.com/paper/2512.16111/) (2025)
- [Exploiting Non-Negativity in DAG Structure Learning](https://mlsanity.com/paper/2605.19947/) (2026)
- [Concomitant DAG Learning: On the Roles of Noise Adaptivity, Sparsity, and Non-negativity](https://mlsanity.com/paper/2605.23537/) (2026)

## Abstract (arXiv)

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.