# On the Sparse DAG Structure Learning Based on Adaptive Lasso

arXiv: https://arxiv.org/abs/2209.02946 · HTML: https://mlsanity.com/paper/2209.02946/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 58 references that are in the map.
- First posted on arXiv 7 Sep 2022.

## Builds on (most cited first, 27 of 27)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Causality for Machine Learning](https://mlsanity.com/paper/1911.10500/) (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)
- [Learning Sparse Nonparametric DAGs](https://mlsanity.com/paper/1909.13189/) (2019)
- [DYNOTEARS: Structure Learning from Time-Series Data](https://mlsanity.com/paper/2002.00498/) (2020)
- [On the Role of Sparsity and DAG Constraints for Learning Linear DAGs](https://mlsanity.com/paper/2006.10201/) (2020)
- [Learning Bayesian Networks: The Combination of Knowledge and Statistical Data](https://mlsanity.com/paper/1302.6815/) (2013)
- [Causal Discovery with Reinforcement Learning](https://mlsanity.com/paper/1906.04477/) (2019)
- [Deep End-to-end Causal Inference](https://mlsanity.com/paper/2202.02195/) (2022)
- [Masked Gradient-Based Causal Structure Learning](https://mlsanity.com/paper/1910.08527/) (2019)
- [DAGs with No Fears: A Closer Look at Continuous Optimization for Learning Bayesian Networks](https://mlsanity.com/paper/2010.09133/) (2020)
- [Learning Gaussian Networks](https://mlsanity.com/paper/1302.6808/) (2013)
- [A Tutorial on Learning With Bayesian Networks](https://mlsanity.com/paper/2002.00269/) (2020)
- [Differentiable Causal Discovery Under Unmeasured Confounding](https://mlsanity.com/paper/2010.06978/) (2020)
- [Structural Agnostic Modeling: Adversarial Learning of Causal Graphs](https://mlsanity.com/paper/1803.04929/) (2018)
- [On the Convergence of Continuous Constrained Optimization for Structure Learning](https://mlsanity.com/paper/2011.11150/) (2020)
- [Towards Federated Bayesian Network Structure Learning with Continuous Optimization](https://mlsanity.com/paper/2110.09356/) (2021)
- [Causality Learning: A New Perspective for Interpretable Machine Learning](https://mlsanity.com/paper/2006.16789/) (2020)
- [Truncated Matrix Power Iteration for Differentiable DAG Learning](https://mlsanity.com/paper/2208.14571/) (2022)
- [MissDAG: Causal Discovery in the Presence of Missing Data with Continuous Additive Noise Models](https://mlsanity.com/paper/2205.13869/) (2022)
- [Reliable Causal Discovery with Improved Exact Search and Weaker Assumptions](https://mlsanity.com/paper/2201.05666/) (2022)
- [FedDAG: Federated DAG Structure Learning](https://mlsanity.com/paper/2112.03555/) (2021)
- [Targeted VAE: Variational and Targeted Learning for Causal Inference](https://mlsanity.com/paper/2009.13472/) (2020)
- [Proximal Quasi-Newton for Computationally Intensive L1-regularized M-estimators](https://mlsanity.com/paper/1406.7321/) (2014)
- [The restricted consistency property of leave-nv-out cross-validation for high-dimensional variable selection](https://mlsanity.com/paper/1308.5390/) (2013)

## Built on by (most cited first, 3 of 3)

- [Structure Learning with Continuous Optimization: A Sober Look and Beyond](https://mlsanity.com/paper/2304.02146/) (2023)
- [Constraint-Free Structure Learning with Smooth Acyclic Orientations](https://mlsanity.com/paper/2309.08406/) (2023)
- [GPO-VAE: Modeling Explainable Gene Perturbation Responses utilizing GRN-Aligned Parameter Optimization](https://mlsanity.com/paper/2501.18973/) (2025)

## Abstract (arXiv)

Learning the underlying Bayesian Networks (BNs), represented by directed acyclic graphs (DAGs), of the concerned events from purely-observational data is a crucial part of evidential reasoning. This task remains challenging due to the large and discrete search space. A recent flurry of developments followed NOTEARS[1] recast this combinatorial problem into a continuous optimization problem by leveraging an algebraic equality characterization of acyclicity. However, the continuous optimization methods suffer from obtaining non-spare graphs after the numerical optimization, which leads to the inflexibility to rule out the potentially cycle-inducing edges or false discovery edges with small values. To address this issue, in this paper, we develop a completely data-driven DAG structure learning method without a predefined value to post-threshold small values. We name our method NOTEARS with adaptive Lasso (NOTEARS-AL), which is achieved by applying the adaptive penalty method to ensure the sparsity of the estimated DAG. Moreover, we show that NOTEARS-AL also inherits the oracle properties under some specific conditions. Extensive experiments on both synthetic and a real-world dataset demonstrate that our method consistently outperforms NOTEARS.