NOTEARS: Continuous Optimization for Causal Structure Learning
NOTEARS Continuous DAG Structure Learning · Also known as: DAGs with NO TEARS, Continuous Structure Learning, Continuous DAG Optimization, Sürekli DAG Yapı Öğrenimi
NOTEARS (No Tears: Acyclicity Regression Structure) is a causal structure learning algorithm introduced by Zheng, Aragam, Ravikumar, and Xing in 2018 at NeurIPS. It reformulates the combinatorially hard problem of learning a directed acyclic graph (DAG) from observational data as a continuous, smooth optimization problem, enabling the use of standard gradient-based solvers and removing the need for exhaustive combinatorial search over graph space.
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When to use it
Use NOTEARS when you have continuous observational data and seek to discover a causal DAG among variables under a linear structural equation model (SEM) assumption with Gaussian noise. It suits moderate to high-dimensional settings where combinatorial constraint-based methods become intractable. Key assumptions include linearity, causal sufficiency (no hidden confounders), and identifiability conditions. It is not appropriate for discrete, count, or highly non-Gaussian data without extensions. Consider PC algorithm or LiNGAM as alternatives depending on distributional assumptions.
Strengths & limitations
- Reformulates intractable combinatorial DAG search as smooth continuous optimization, enabling scalability to hundreds of variables.
- Provides a theoretically exact and differentiable acyclicity characterization h(W), compatible with automatic differentiation frameworks.
- Achieves state-of-the-art structure recovery accuracy on linear Gaussian benchmarks at the time of publication.
- Simple algorithmic pipeline (L-BFGS + augmented Lagrangian) with well-understood convergence properties.
- Assumes a linear structural equation model; performance degrades under nonlinear causal mechanisms without extensions such as DAG-GNN or NOTEARS-MLP.
- Causal sufficiency is required — latent confounders can produce spurious edges that the algorithm cannot detect.
- The augmented Lagrangian approach may converge to local optima on non-convex landscapes, particularly with dense graphs.
- Thresholding the weight matrix to recover the DAG introduces a tuning step that can affect the precision-recall trade-off.
Frequently asked
Does NOTEARS guarantee recovery of the true DAG?
Under the linear Gaussian SEM with equal noise variances and causal sufficiency, NOTEARS is consistent and recovers the Markov equivalence class of the true DAG as sample size grows. Unique edge orientation requires additional identifiability conditions, such as equal noise variances or non-Gaussian noise as leveraged by LiNGAM.
How does NOTEARS differ from constraint-based methods like the PC algorithm?
The PC algorithm operates via conditional independence tests and is model-agnostic but relies on sequential hypothesis testing with accumulating Type I error. NOTEARS instead optimizes a global score function subject to a smooth acyclicity constraint, making it more amenable to high-dimensional settings and gradient-based computation, but requiring explicit distributional assumptions.
Can NOTEARS handle nonlinear causal relationships?
The original NOTEARS is restricted to linear SEMs. Extensions such as NOTEARS-MLP replace the linear weight matrix with neural network modules, enabling nonlinear structure learning while preserving the continuous acyclicity constraint. These nonlinear variants are substantially more computationally demanding and require larger datasets for reliable estimation.
Sources
- Zheng, X., Aragam, B., Ravikumar, P., & Xing, E. P. (2018). DAGs with NO TEARS: Continuous optimization for structure learning. Advances in Neural Information Processing Systems, 31. link ↗
How to cite this page
ScholarGate. (2026, June 2). NOTEARS Continuous DAG Structure Learning. ScholarGate. https://scholargate.app/en/causal-inference/notears
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