mlsanityData to 7 Oct 2026

On the Convergence of Continuous Constrained Optimization for Structure Learning

arXiv 2011.11150 ↗First posted 23 Nov 2020cs.LG stat.MLcited by 38

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Abstract (arXiv)

Recently, structure learning of directed acyclic graphs (DAGs) has been formulated as a continuous optimization problem by leveraging an algebraic characterization of acyclicity. The constrained problem is solved using the augmented Lagrangian method (ALM) which is often preferred to the quadratic penalty method (QPM) by virtue of its standard convergence result that does not require the penalty coefficient to go to infinity, hence avoiding ill-conditioning. However, the convergence properties of these methods for structure learning, including whether they are guaranteed to return a DAG solution, remain unclear, which might limit their practical applications. In this work, we examine the convergence of ALM and QPM for structure learning in the linear, nonlinear, and confounded cases. We show that the standard convergence result of ALM does not hold in these settings, and demonstrate empirically that its behavior is akin to that of the QPM which is prone to ill-conditioning. We further establish the convergence guarantee of QPM to a DAG solution, under mild conditions. Lastly, we connect our theoretical results with existing approaches to help resolve the convergence issue, and verify our findings in light of an empirical comparison of them.

Key facts. Cited by 38 papers in the map (data through 7 Oct 2026). First posted on arXiv 23 Nov 2020.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 3; 2022: 4; 2023: 13; 2024: 9; 2025: 6; 2026: 3.

38 papers.

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  1. 2022DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterizationcited by 135
  2. 2021DiBS: Differentiable Bayesian Structure Learningcited by 105
  3. 2023Causal Discovery from Temporal Data: An Overview and New Perspectivescited by 43
  4. 2023Structure Learning with Continuous Optimization: A Sober Look and Beyondcited by 34
  5. 2023Discovering Causal Relations and Equations from Datacited by 33
  6. 2021Towards Federated Bayesian Network Structure Learning with Continuous Optimizationcited by 31
  7. 2023Stable Differentiable Causal Discoverycited by 30
  8. 2022Truncated Matrix Power Iteration for Differentiable DAG Learningcited by 23
  9. 2021FedDAG: Federated DAG Structure Learningcited by 17
  10. 2023Optimizing NOTEARS Objectives via Topological Swapscited by 17
  11. 2024Federated Causal Discovery from Heterogeneous Datacited by 15
  12. 2023MM-DAG: Multi-task DAG Learning for Multi-modal Data -- with Application for Traffic Congestion Analysiscited by 13
26 more
  1. 2023Global Optimality in Bivariate Gradient-based DAG Learningcited by 10
  2. 2023Constraint-Free Structure Learning with Smooth Acyclic Orientationscited by 10
  3. 2023DAG Learning on the Permutahedroncited by 9
  4. 2023Evaluation of Induced Expert Knowledge in Causal Structure Learning by NOTEARScited by 8
  5. 2023Effective Causal Discovery under Identifiable Heteroscedastic Noise Modelcited by 8
  6. 2024Causality for Tabular Data Synthesis: A High-Order Structure Causal Benchmark Frameworkcited by 7
  7. 2024Markov Equivalence and Consistency in Differentiable Structure Learningcited by 6
  8. 2023Boosting Differentiable Causal Discovery via Adaptive Sample Reweightingcited by 5
  9. 2024Optimal Transport for Structure Learning Under Missing Datacited by 5
  10. 2024Causal Representation Learning in Temporal Data via Single-Parent Decodingcited by 5
  11. 2025Analytic DAG Constraints for Differentiable DAG Learningcited by 5
  12. 2022On the Sparse DAG Structure Learning Based on Adaptive Lassocited by 3
  13. 2023Tree Search in DAG Space with Model-based Reinforcement Learning for Causal Discoverycited by 3
  14. 2024psiDAG: Projected Stochastic Approximation Iteration for DAG Structure Learningcited by 3
  15. 2024Interventional Causal Structure Discovery over Graphical Models with Convergence and Optimality Guaranteescited by 1
  16. 2024Revisiting Differentiable Structure Learning: Inconsistency of ℓ1 Penalty and Beyondcited by 1
  17. 2025Toward Temporal Causal Representation Learning with Tensor Decompositioncited by 1
  18. 2025Score-based Greedy Search for Structure Identification of Partially Observed Linear Causal Modelscited by 1
  19. 2025Differentially Private and Federated Structure Learning in Bayesian Networkscited by 1
  20. 2022Differentiable and Transportable Structure Learningcited by 0
  21. 2024Boosting Causal Additive Modelscited by 0
  22. 2025Differentiable Structure Learning and Causal Discovery for General Binary Datacited by 0
  23. 2025Humanoid-inspired Causal Representation Learning for Domain Generalizationcited by 0
  24. 2026Stable Causal Discovery via Directed Acyclic Graph Aggregationcited by 0
  25. 2026MARCEDES: Score-based causal discovery under non-Gaussianity with continuous optimizationcited by 0
  26. 2026Nonnegative DAG Learning via Concomitant Estimationcited by 0