# CoLiDE: Concomitant Linear DAG Estimation

arXiv: https://arxiv.org/abs/2310.02895 · HTML: https://mlsanity.com/paper/2310.02895/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 59 references that are in the map.
- First posted on arXiv 4 Oct 2023.

## Builds on (most cited first, 27 of 27)

- [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)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [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)
- [CAM: Causal additive models, high-dimensional order search and penalized regression](https://mlsanity.com/paper/1310.1533/) (2013)
- [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)
- [High-dimensional learning of linear causal networks via inverse covariance estimation](https://mlsanity.com/paper/1311.3492/) (2013)
- [A survey of Bayesian Network structure learning](https://mlsanity.com/paper/2109.11415/) (2021)
- [Efficient Neural Causal Discovery without Acyclicity Constraints](https://mlsanity.com/paper/2107.10483/) (2021)
- [Permutation-Based Causal Structure Learning with Unknown Intervention Targets](https://mlsanity.com/paper/1910.09007/) (2019)
- [BCD Nets: Scalable Variational Approaches for Bayesian Causal Discovery](https://mlsanity.com/paper/2112.02761/) (2021)
- [DAGs with No Fears: A Closer Look at Continuous Optimization for Learning Bayesian Networks](https://mlsanity.com/paper/2010.09133/) (2020)
- [Differentiable DAG Sampling](https://mlsanity.com/paper/2203.08509/) (2022)
- [Towards Scalable Bayesian Learning of Causal DAGs](https://mlsanity.com/paper/2010.00684/) (2020)
- [Efficient Intervention Design for Causal Discovery with Latents](https://mlsanity.com/paper/2005.11736/) (2020)
- [Optimizing NOTEARS Objectives via Topological Swaps](https://mlsanity.com/paper/2305.17277/) (2023)
- [Bayesian Network Learning via Topological Order](https://mlsanity.com/paper/1701.05654/) (2017)
- [Inexact Block Coordinate Descent Algorithms for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/1905.04211/) (2019)
- [Advances in Learning Bayesian Networks of Bounded Treewidth](https://mlsanity.com/paper/1406.1411/) (2014)
- [Global Optimality in Bivariate Gradient-based DAG Learning](https://mlsanity.com/paper/2306.17378/) (2023)
- [DAG Learning on the Permutahedron](https://mlsanity.com/paper/2301.11898/) (2023)
- [Efficient Smoothed Concomitant Lasso Estimation for High Dimensional Regression](https://mlsanity.com/paper/1606.02702/) (2016)
- [Generalized Concomitant Multi-Task Lasso for sparse multimodal regression](https://mlsanity.com/paper/1705.09778/) (2017)
- [dotears: Scalable, consistent DAG estimation using observational and interventional data](https://mlsanity.com/paper/2305.19215/) (2023)

## Built on by (most cited first, 13 of 13)

- [Non-negative Weighted DAG Structure Learning](https://mlsanity.com/paper/2409.07880/) (2024)
- [Convolutional Learning on Directed Acyclic Graphs](https://mlsanity.com/paper/2405.03056/) (2024)
- [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)
- [Revisiting Differentiable Structure Learning: Inconsistency of ℓ1 Penalty and Beyond](https://mlsanity.com/paper/2410.18396/) (2024)
- [Directed Acyclic Graph Convolutional Networks](https://mlsanity.com/paper/2506.12218/) (2025)
- [Non-negative DAG Learning from Time-Series Data](https://mlsanity.com/paper/2512.07267/) (2025)
- [Topology Identification and Inference over Graphs](https://mlsanity.com/paper/2512.10183/) (2025)
- [Directed Graph Topology Inference via Graph Filter Identification](https://mlsanity.com/paper/2606.27455/) (2026)
- [From topology learning to graph generation: A unifying perspective](https://mlsanity.com/paper/2609.02286/) (2026)
- [Nonnegative DAG Learning via Concomitant Estimation](https://mlsanity.com/paper/2609.34595/) (2026)
- [DAG Structure Learning with Node-Varying Heavy-Tailed Disturbances](https://mlsanity.com/paper/2610.13304/) (2026)

## Abstract (arXiv)

We deal with the combinatorial problem of learning directed acyclic graph (DAG) structure from observational data adhering to a linear structural equation model (SEM). Leveraging advances in differentiable, nonconvex characterizations of acyclicity, recent efforts have advocated a continuous constrained optimization paradigm to efficiently explore the space of DAGs. Most existing methods employ lasso-type score functions to guide this search, which (i) require expensive penalty parameter retuning when the textitunknown SEM noise variances change across problem instances; and (ii) implicitly rely on limiting homoscedasticity assumptions. In this work, we propose a new convex score function for sparsity-aware learning of linear DAGs, which incorporates concomitant estimation of scale and thus effectively decouples the sparsity parameter from the exogenous noise levels. Regularization via a smooth, nonconvex acyclicity penalty term yields CoLiDE (Concomitant Linear DAG Estimation), a regression-based criterion amenable to efficient gradient computation and closed-form estimation of noise variances in heteroscedastic scenarios. Our algorithm outperforms state-of-the-art methods without incurring added complexity, especially when the DAGs are larger and the noise level profile is heterogeneous. We also find CoLiDE exhibits enhanced stability manifested via reduced standard deviations in several domain-specific metrics, underscoring the robustness of our novel linear DAG estimator.