mlsanityData to 7 Oct 2026

DAGs with NO TEARS: Continuous Optimization for Structure Learning

arXiv 1803.01422 ↗First posted 4 Mar 2018stat.ML cs.AI cs.LGcited by 745

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Abstract (arXiv)

Estimating the structure of directed acyclic graphs (DAGs, also known as Bayesian networks) is a challenging problem since the search space of DAGs is combinatorial and scales superexponentially with the number of nodes. Existing approaches rely on various local heuristics for enforcing the acyclicity constraint. In this paper, we introduce a fundamentally different strategy: We formulate the structure learning problem as a purely continuous optimization problem over real matrices that avoids this combinatorial constraint entirely. This is achieved by a novel characterization of acyclicity that is not only smooth but also exact. The resulting problem can be efficiently solved by standard numerical algorithms, which also makes implementation effortless. The proposed method outperforms existing ones, without imposing any structural assumptions on the graph such as bounded treewidth or in-degree. Code implementing the proposed algorithm is open-source and publicly available at https://github.com/xunzheng/notears.

Key facts. Cited by 745 papers in the map (data through 7 Oct 2026). First posted on arXiv 4 Mar 2018.

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Citing papers per year: 2021: 66; 2022: 98; 2023: 141; 2024: 130; 2025: 140; 2026: 170.

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  1. 2023Causal Reasoning and Large Language Models: Opening a New Frontier for Causalitycited by 348
  2. 2021D'ya like DAGs? A Survey on Structure Learning and Causal Discoverycited by 209
  3. 2021Discrete Graph Structure Learning for Forecasting Multiple Time Seriescited by 169
  4. 2021Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Gamecited by 159
  5. 2022Bayesian Structure Learning with Generative Flow Networkscited by 153
  6. 2022Weakly supervised causal representation learningcited by 140
  7. 2022DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterizationcited by 135
  8. 2022Deep Learning for Time Series Anomaly Detection: A Surveycited by 130
  9. 2022Score matching enables causal discovery of nonlinear additive noise modelscited by 108
  10. 2021DiBS: Differentiable Bayesian Structure Learningcited by 105
  11. 2023Nonparametric Identifiability of Causal Representations from Unknown Interventionscited by 105
  12. 2022Deep End-to-end Causal Inferencecited by 93
38 more
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  2. 2022Amortized Inference for Causal Structure Learningcited by 89
  3. 2021A survey of Bayesian Network structure learningcited by 85
  4. 2023Identifiability Guarantees for Causal Disentanglement from Soft Interventionscited by 84
  5. 2021Efficient Neural Causal Discovery without Acyclicity Constraintscited by 82
  6. 2021BCD Nets: Scalable Variational Approaches for Bayesian Causal Discoverycited by 68
  7. 2022Learning to Induce Causal Structurecited by 68
  8. 2022When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learningcited by 64
  9. 2022Diffusion Models for Causal Discovery via Topological Orderingcited by 64
  10. 2022Greedy Relaxations of the Sparsest Permutation Algorithmcited by 63
  11. 2023GFlowNets for AI-Driven Scientific Discoverycited by 63
  12. 2023A Survey on Causal Discovery: Theory and Practicecited by 63
  13. 2023Causal Discovery with Language Models as Imperfect Expertscited by 62
  14. 2021gCastle: A Python Toolbox for Causal Discoverycited by 60
  15. 2022On the Identifiability and Estimation of Causal Location-Scale Noise Modelscited by 56
  16. 2021Ordering-Based Causal Discovery with Reinforcement Learningcited by 55
  17. 2022Causal Inference-Based Root Cause Analysis for Online Service Systems with Intervention Recognitioncited by 55
  18. 2023Integrating Large Language Model for Improved Causal Discoverycited by 54
  19. 2021DAGs with No Curl: An Efficient DAG Structure Learning Approachcited by 53
  20. 2021Systematic Evaluation of Causal Discovery in Visual Model Based Reinforcement Learningcited by 53
  21. 2024Efficient Causal Graph Discovery Using Large Language Modelscited by 51
  22. 2021MIRACLE: Causally-Aware Imputation via Learning Missing Data Mechanismscited by 50
  23. 2022Causal Representation Learning for Instantaneous and Temporal Effects in Interactive Systemscited by 50
  24. 2023Joint Bayesian Inference of Graphical Structure and Parameters with a Single Generative Flow Networkcited by 50
  25. 2022Interventions, Where and How? Experimental Design for Causal Models at Scalecited by 49
  26. 2022Large-Scale Differentiable Causal Discovery of Factor Graphscited by 47
  27. 2021Unsuitability of NOTEARS for Causal Graph Discoverycited by 46
  28. 2022Differentiable DAG Samplingcited by 46
  29. 2022Causal Structure Learning: a Combinatorial Perspectivecited by 46
  30. 2023Causal Order: The Key to Leveraging Imperfect Experts in Causal Inferencecited by 46
  31. 2021Variational Causal Networks: Approximate Bayesian Inference over Causal Structurescited by 43
  32. 2023Causal Discovery from Temporal Data: An Overview and New Perspectivescited by 43
  33. 2025CausalPFN: Amortized Causal Effect Estimation via In-Context Learningcited by 43
  34. 2022Graph-Augmented Normalizing Flows for Anomaly Detection of Multiple Time Seriescited by 42
  35. 2022Causal de Finetti: On the Identification of Invariant Causal Structure in Exchangeable Datacited by 42
  36. 2023Scalable Causal Discovery with Score Matchingcited by 42
  37. 2023Causal Discovery with Score Matching on Additive Models with Arbitrary Noisecited by 41
  38. 2021Learning Neural Causal Models with Active Interventionscited by 40

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