# Constraint-Free Structure Learning with Smooth Acyclic Orientations

arXiv: https://arxiv.org/abs/2309.08406 · HTML: https://mlsanity.com/paper/2309.08406/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 45 references that are in the map.
- First posted on arXiv 15 Sep 2023.

## Builds on (most cited first, 30 of 31)

- [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)
- [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)
- [D'ya like DAGs? A Survey on Structure Learning and Causal Discovery](https://mlsanity.com/paper/2103.02582/) (2021)
- [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)
- [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)
- [Amortized Inference for Causal Structure Learning](https://mlsanity.com/paper/2205.12934/) (2022)
- [Masked Gradient-Based Causal Structure Learning](https://mlsanity.com/paper/1910.08527/) (2019)
- [High-dimensional learning of linear causal networks via inverse covariance estimation](https://mlsanity.com/paper/1311.3492/) (2013)
- [Causal Structure Learning](https://mlsanity.com/paper/1706.09141/) (2017)
- [Efficient Neural Causal Discovery without Acyclicity Constraints](https://mlsanity.com/paper/2107.10483/) (2021)
- [BCD Nets: Scalable Variational Approaches for Bayesian Causal Discovery](https://mlsanity.com/paper/2112.02761/) (2021)
- [SparseMAP: Differentiable Sparse Structured Inference](https://mlsanity.com/paper/1802.04223/) (2018)
- [DAGs with No Curl: An Efficient DAG Structure Learning Approach](https://mlsanity.com/paper/2106.07197/) (2021)
- [Unsuitability of NOTEARS for Causal Graph Discovery](https://mlsanity.com/paper/2104.05441/) (2021)
- [Differentiable DAG Sampling](https://mlsanity.com/paper/2203.08509/) (2022)
- [Large-Scale Differentiable Causal Discovery of Factor Graphs](https://mlsanity.com/paper/2206.07824/) (2022)
- [Scaling structural learning with NO-BEARS to infer causal transcriptome networks](https://mlsanity.com/paper/1911.00081/) (2019)
- [On the Convergence of Continuous Constrained Optimization for Structure Learning](https://mlsanity.com/paper/2011.11150/) (2020)
- [Structure Learning with Continuous Optimization: A Sober Look and Beyond](https://mlsanity.com/paper/2304.02146/) (2023)
- [Ordering-Based Causal Structure Learning in the Presence of Latent Variables](https://mlsanity.com/paper/1910.09014/) (2019)
- [BaCaDI: Bayesian Causal Discovery with Unknown Interventions](https://mlsanity.com/paper/2206.01665/) (2022)
- [On Low Rank Directed Acyclic Graphs and Causal Structure Learning](https://mlsanity.com/paper/2006.05691/) (2020)
- [Truncated Matrix Power Iteration for Differentiable DAG Learning](https://mlsanity.com/paper/2208.14571/) (2022)
- [Efficient Marginalization of Discrete and Structured Latent Variables via Sparsity](https://mlsanity.com/paper/2007.01919/) (2020)
- [Optimizing NOTEARS Objectives via Topological Swaps](https://mlsanity.com/paper/2305.17277/) (2023)
- [DAG Learning on the Permutahedron](https://mlsanity.com/paper/2301.11898/) (2023)

## Built on by (most cited first, 10 of 10)

- [Demystifying amortized causal discovery with transformers](https://mlsanity.com/paper/2405.16924/) (2024)
- [Reinforcement Learning for Causal Discovery without Acyclicity Constraints](https://mlsanity.com/paper/2408.13448/) (2024)
- [Interpretable Hierarchical Concept Reasoning through Attention-Guided Graph Learning](https://mlsanity.com/paper/2506.21102/) (2025)
- [LOCAL: Learning with Orientation Matrix to Infer Causal Structure from Time Series Data](https://mlsanity.com/paper/2410.19464/) (2024)
- [Scalable Variational Causal Discovery Unconstrained by Acyclicity](https://mlsanity.com/paper/2407.04992/) (2024)
- [Causal Discovery via Bayesian Optimization](https://mlsanity.com/paper/2501.14997/) (2025)
- [Towards the Formalization of a Trustworthy AI for Mining Interpretable Models explOiting Sophisticated Algorithms](https://mlsanity.com/paper/2510.20621/) (2025)
- [MARLIN: Multi-Agent Reinforcement Learning for Incremental DAG Discovery](https://mlsanity.com/paper/2603.20295/) (2026)
- [Neural Autoregressive Flows for Markov Boundary Learning](https://mlsanity.com/paper/2603.20791/) (2026)
- [Support Selection Beyond Smooth DAG Exactness: Completion Geometry,Score Margins, and Selective Certificates](https://mlsanity.com/paper/2608.08103/) (2026)

## Abstract (arXiv)

The structure learning problem consists of fitting data generated by a Directed Acyclic Graph (DAG) to correctly reconstruct its arcs. In this context, differentiable approaches constrain or regularize the optimization problem using a continuous relaxation of the acyclicity property. The computational cost of evaluating graph acyclicity is cubic on the number of nodes and significantly affects scalability. In this paper we introduce COSMO, a constraint-free continuous optimization scheme for acyclic structure learning. At the core of our method, we define a differentiable approximation of an orientation matrix parameterized by a single priority vector. Differently from previous work, our parameterization fits a smooth orientation matrix and the resulting acyclic adjacency matrix without evaluating acyclicity at any step. Despite the absence of explicit constraints, we prove that COSMO always converges to an acyclic solution. In addition to being asymptotically faster, our empirical analysis highlights how COSMO performance on graph reconstruction compares favorably with competing structure learning methods.