mlsanityData to 7 Oct 2026

DAGs with No Fears: A Closer Look at Continuous Optimization for Learning Bayesian Networks

arXiv 2010.09133 ↗First posted 18 Oct 2020cs.LG stat.MLcited by 66

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Abstract (arXiv)

This paper re-examines a continuous optimization framework dubbed NOTEARS for learning Bayesian networks. We first generalize existing algebraic characterizations of acyclicity to a class of matrix polynomials. Next, focusing on a one-parameter-per-edge setting, it is shown that the Karush-Kuhn-Tucker (KKT) optimality conditions for the NOTEARS formulation cannot be satisfied except in a trivial case, which explains a behavior of the associated algorithm. We then derive the KKT conditions for an equivalent reformulation, show that they are indeed necessary, and relate them to explicit constraints that certain edges be absent from the graph. If the score function is convex, these KKT conditions are also sufficient for local minimality despite the non-convexity of the constraint. Informed by the KKT conditions, a local search post-processing algorithm is proposed and shown to substantially and universally improve the structural Hamming distance of all tested algorithms, typically by a factor of 2 or more. Some combinations with local search are both more accurate and more efficient than the original NOTEARS.

Key facts. Cited by 66 papers in the map (data through 7 Oct 2026). First posted on arXiv 18 Oct 2020.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 9; 2022: 9; 2023: 13; 2024: 14; 2025: 11; 2026: 10.

66 papers.

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  1. 2021D'ya like DAGs? A Survey on Structure Learning and Causal Discoverycited by 209
  2. 2021Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Gamecited by 159
  3. 2022DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterizationcited by 135
  4. 2021DiBS: Differentiable Bayesian Structure Learningcited by 105
  5. 2021BCD Nets: Scalable Variational Approaches for Bayesian Causal Discoverycited by 68
  6. 2021DAGs with No Curl: An Efficient DAG Structure Learning Approachcited by 53
  7. 2023A Survey on Causal Discovery Methods for I.I.D. and Time Series Datacited by 35
  8. 2023Structure Learning with Continuous Optimization: A Sober Look and Beyondcited by 34
  9. 2021Towards Federated Bayesian Network Structure Learning with Continuous Optimizationcited by 31
  10. 2023Stable Differentiable Causal Discoverycited by 30
  11. 2022Truncated Matrix Power Iteration for Differentiable DAG Learningcited by 23
  12. 2021FedDAG: Federated DAG Structure Learningcited by 17
38 more
  1. 2023Optimizing NOTEARS Objectives via Topological Swapscited by 17
  2. 2024Federated Causal Discovery from Heterogeneous Datacited by 15
  3. 2023CoLiDE: Concomitant Linear DAG Estimationcited by 12
  4. 2022GFlowCausal: Generative Flow Networks for Causal Discoverycited by 11
  5. 2023Global Optimality in Bivariate Gradient-based DAG Learningcited by 10
  6. 2023Causality-based CTR Prediction using Graph Neural Networkscited by 9
  7. 2023Evaluation of Induced Expert Knowledge in Causal Structure Learning by NOTEARScited by 8
  8. 2024Non-negative Weighted DAG Structure Learningcited by 7
  9. 2024Dagma-DCE: Interpretable, Non-Parametric Differentiable Causal Discoverycited by 6
  10. 2024Markov Equivalence and Consistency in Differentiable Structure Learningcited by 6
  11. 2022Learning Latent Structural Causal Modelscited by 5
  12. 2023Boosting Differentiable Causal Discovery via Adaptive Sample Reweightingcited by 5
  13. 2024Optimal Transport for Structure Learning Under Missing Datacited by 5
  14. 2024Scalable and Flexible Causal Discovery with an Efficient Test for Adjacencycited by 5
  15. 2024On the Identifiability of Sparse ICA without Assuming Non-Gaussianitycited by 5
  16. 2024Continuous Bayesian Model Selection for Multivariate Causal Discoverycited by 5
  17. 2025Analytic DAG Constraints for Differentiable DAG Learningcited by 5
  18. 2022Tearing Apart NOTEARS: Controlling the Graph Prediction via Variance Manipulationcited by 4
  19. 2023Discovering Dynamic Causal Space for DAG Structure Learningcited by 4
  20. 2025The Robustness of Differentiable Causal Discovery in Misspecified Scenarioscited by 4
  21. 2025Differentiable Constraint-Based Causal Discoverycited by 4
  22. 2025BUILD with Precision: Bottom-Up Inference of Linear DAGscited by 4
  23. 2022Differentiable Invariant Causal Discoverycited by 3
  24. 2022Invariant Structure Learning for Better Generalization and Causal Explainabilitycited by 3
  25. 2022On the Sparse DAG Structure Learning Based on Adaptive Lassocited by 3
  26. 2024Reinforcement Learning for Causal Discovery without Acyclicity Constraintscited by 3
  27. 2026Exploiting Non-Negativity in DAG Structure Learningcited by 3
  28. 2021Scalable Causal Structure Learning: Scoping Review of Traditional and Deep Learning Algorithms and New Opportunities in Biomedicinecited by 2
  29. 2025Robust Causal Discovery under Imperfect Structural Constraintscited by 2
  30. 2026Concomitant DAG Learning: On the Roles of Noise Adaptivity, Sparsity, and Non-negativitycited by 2
  31. 2023Causal disentanglement of multimodal datacited by 1
  32. 2023Towards Human-like Perception: Learning Structural Causal Model in Heterogeneous Graphcited by 1
  33. 2024OCDB: Revisiting Causal Discovery with a Comprehensive Benchmark and Evaluation Frameworkcited by 1
  34. 2024Interventional Causal Structure Discovery over Graphical Models with Convergence and Optimality Guaranteescited by 1
  35. 2024Revisiting Differentiable Structure Learning: Inconsistency of ℓ1 Penalty and Beyondcited by 1
  36. 2024CGLearn: Consistent Gradient-Based Learning for Out-of-Distribution Generalizationcited by 1
  37. 2024Fast Causal Discovery by Approximate Kernel-based Generalized Score Functions with Linear Computational Complexitycited by 1
  38. 2025Your Assumed DAG is Wrong and Here's How To Deal With Itcited by 1

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