# Efficient Neural Causal Discovery without Acyclicity Constraints

arXiv: https://arxiv.org/abs/2107.10483 · HTML: https://mlsanity.com/paper/2107.10483/

## Key facts

- Cited by 82 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 71 references that are in the map.
- First posted on arXiv 22 Jul 2021.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [GAIN: Missing Data Imputation using Generative Adversarial Nets](https://mlsanity.com/paper/1806.02920/) (2018)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Gradient-Based Neural DAG Learning](https://mlsanity.com/paper/1906.02226/) (2019)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [Towards Causal Representation Learning](https://mlsanity.com/paper/2102.11107/) (2021)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Learning Sparse Nonparametric DAGs](https://mlsanity.com/paper/1909.13189/) (2019)
- [On the Role of Sparsity and DAG Constraints for Learning Linear DAGs](https://mlsanity.com/paper/2006.10201/) (2020)
- [A Meta-Transfer Objective for Learning to Disentangle Causal Mechanisms](https://mlsanity.com/paper/1901.10912/) (2019)
- [Causal Discovery with Reinforcement Learning](https://mlsanity.com/paper/1906.04477/) (2019)
- [Invariant Causal Prediction for Nonlinear Models](https://mlsanity.com/paper/1706.08576/) (2017)
- [Learning Neural Causal Models from Unknown Interventions](https://mlsanity.com/paper/1910.01075/) (2019)
- [Permutation-based Causal Inference Algorithms with Interventions](https://mlsanity.com/paper/1705.10220/) (2017)
- [Characterizing and Learning Equivalence Classes of Causal DAGs under Interventions](https://mlsanity.com/paper/1802.06310/) (2018)
- [Variational Autoencoder with Arbitrary Conditioning](https://mlsanity.com/paper/1806.02382/) (2018)
- [Causal Discovery with General Non-Linear Relationships Using Non-Linear ICA](https://mlsanity.com/paper/1904.09096/) (2019)
- [Constraint-based Causal Discovery from Multiple Interventions over Overlapping Variable Sets](https://mlsanity.com/paper/1403.2150/) (2014)
- [Flow Models for Arbitrary Conditional Likelihoods](https://mlsanity.com/paper/1909.06319/) (2019)
- [Causal Generative Neural Networks](https://mlsanity.com/paper/1711.08936/) (2017)
- [Structural Intervention Distance (SID) for Evaluating Causal Graphs](https://mlsanity.com/paper/1306.1043/) (2013)
- [Learning DAGs without imposing acyclicity](https://mlsanity.com/paper/2006.03005/) (2020)

## Built on by (most cited first, 30 of 82)

- [Weakly supervised causal representation learning](https://mlsanity.com/paper/2203.16437/) (2022)
- [CITRIS: Causal Identifiability from Temporal Intervened Sequences](https://mlsanity.com/paper/2202.03169/) (2022)
- [Deep End-to-end Causal Inference](https://mlsanity.com/paper/2202.02195/) (2022)
- [Learning to Induce Causal Structure](https://mlsanity.com/paper/2204.04875/) (2022)
- [Causal Representation Learning for Instantaneous and Temporal Effects in Interactive Systems](https://mlsanity.com/paper/2206.06169/) (2022)
- [Large-Scale Differentiable Causal Discovery of Factor Graphs](https://mlsanity.com/paper/2206.07824/) (2022)
- [Learning Neural Causal Models with Active Interventions](https://mlsanity.com/paper/2109.02429/) (2021)
- [BISCUIT: Causal Representation Learning from Binary Interactions](https://mlsanity.com/paper/2306.09643/) (2023)
- [CausalTime: Realistically Generated Time-series for Benchmarking of Causal Discovery](https://mlsanity.com/paper/2310.01753/) (2023)
- [Active Bayesian Causal Inference](https://mlsanity.com/paper/2206.02063/) (2022)
- [A Survey on Causal Discovery Methods for I.I.D. and Time Series Data](https://mlsanity.com/paper/2303.15027/) (2023)
- [CUTS: Neural Causal Discovery from Irregular Time-Series Data](https://mlsanity.com/paper/2302.07458/) (2023)
- [Stable Differentiable Causal Discovery](https://mlsanity.com/paper/2311.10263/) (2023)
- [From Identifiable Causal Representations to Controllable Counterfactual Generation: A Survey on Causal Generative Modeling](https://mlsanity.com/paper/2310.11011/) (2023)
- [Sample, estimate, aggregate: A recipe for causal discovery foundation models](https://mlsanity.com/paper/2402.01929/) (2024)
- [Deep Learning of Causal Structures in High Dimensions](https://mlsanity.com/paper/2212.04866/) (2022)
- [Variational Causal Dynamics: Discovering Modular World Models from Interventions](https://mlsanity.com/paper/2206.11131/) (2022)
- [Causal Reasoning in the Presence of Latent Confounders via Neural ADMG Learning](https://mlsanity.com/paper/2303.12703/) (2023)
- [Federated Causal Discovery from Heterogeneous Data](https://mlsanity.com/paper/2402.13241/) (2024)
- [CoLiDE: Concomitant Linear DAG Estimation](https://mlsanity.com/paper/2310.02895/) (2023)
- [Predictive, scalable and interpretable knowledge tracing on structured domains](https://mlsanity.com/paper/2403.13179/) (2024)
- [A Meta-Reinforcement Learning Algorithm for Causal Discovery](https://mlsanity.com/paper/2207.08457/) (2022)
- [Deep Causal Learning: Representation, Discovery and Inference](https://mlsanity.com/paper/2211.03374/) (2022)
- [Demystifying amortized causal discovery with transformers](https://mlsanity.com/paper/2405.16924/) (2024)
- [Can Large Language Models Help Experimental Design for Causal Discovery?](https://mlsanity.com/paper/2503.01139/) (2025)
- [Learning Interactive World Model for Object-Centric Reinforcement Learning](https://mlsanity.com/paper/2511.02225/) (2025)
- [Constraint-Free Structure Learning with Smooth Acyclic Orientations](https://mlsanity.com/paper/2309.08406/) (2023)
- [Score matching through the roof: linear, nonlinear, and latent variables causal discovery](https://mlsanity.com/paper/2407.18755/) (2024)
- [DAG Learning on the Permutahedron](https://mlsanity.com/paper/2301.11898/) (2023)
- [On the Generalization and Adaption Performance of Causal Models](https://mlsanity.com/paper/2206.04620/) (2022)

## Abstract (arXiv)

Learning the structure of a causal graphical model using both observational and interventional data is a fundamental problem in many scientific fields. A promising direction is continuous optimization for score-based methods, which, however, require constrained optimization to enforce acyclicity or lack convergence guarantees. In this paper, we present ENCO, an efficient structure learning method for directed, acyclic causal graphs leveraging observational and interventional data. ENCO formulates the graph search as an optimization of independent edge likelihoods, with the edge orientation being modeled as a separate parameter. Consequently, we can provide convergence guarantees of ENCO under mild conditions without constraining the score function with respect to acyclicity. In experiments, we show that ENCO can efficiently recover graphs with hundreds of nodes, an order of magnitude larger than what was previously possible, while handling deterministic variables and latent confounders.