mlsanityData to 14 Oct 2026

Sample Complexity Bounds for Score-Matching: Causal Discovery and Generative Modeling

arXiv 2310.18123 ↗First posted 27 Oct 2023cs.LG stat.MLcited by 16

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Abstract (arXiv)

This paper provides statistical sample complexity bounds for score-matching and its applications in causal discovery. We demonstrate that accurate estimation of the score function is achievable by training a standard deep ReLU neural network using stochastic gradient descent. We establish bounds on the error rate of recovering causal relationships using the score-matching-based causal discovery method of Rolland et al. [2022], assuming a sufficiently good estimation of the score function. Finally, we analyze the upper bound of score-matching estimation within the score-based generative modeling, which has been applied for causal discovery but is also of independent interest within the domain of generative models.

Key facts. Cited by 16 papers in the map (data through 14 Oct 2026). Builds on 34 of its 50 references that are in the map. First posted on arXiv 27 Oct 2023.

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34 of 50 references.

References · 34

  1. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  2. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,255
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  4. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,354
  5. 2018A Convergence Theory for Deep Learning via Over-Parameterizationcited by 787
  6. 2018Gradient Descent Provably Optimizes Over-parameterized Neural Networkscited by 659
  7. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modelingcited by 602
  8. 2018Gradient Descent Finds Global Minima of Deep Neural Networkscited by 574
  9. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptionscited by 445
  10. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 384
  11. 2013Causal Discovery with Continuous Additive Noise Modelscited by 255
  12. 2022Convergence of denoising diffusion models under the manifold hypothesiscited by 242
22 more
  1. 2019Gradient-Based Neural DAG Learningcited by 227
  2. 2020Proving the Lottery Ticket Hypothesis: Pruning is All You Needcited by 201
  3. 2022Convergence of score-based generative modeling for general data distributionscited by 181
  4. 2022Convergence for score-based generative modeling with polynomial complexitycited by 177
  5. 2019Towards moderate overparameterization: global convergence guarantees for training shallow neural networkscited by 176
  6. 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Datacited by 174
  7. 2020Generative Modeling with Denoising Auto-Encoders and Langevin Samplingcited by 119
  8. 2022Score matching enables causal discovery of nonlinear additive noise modelscited by 111
  9. 2020Tight Bounds on the Smallest Eigenvalue of the Neural Tangent Kernel for Deep ReLU Networkscited by 87
  10. 2022Causal Machine Learning for Healthcare and Precision Medicinecited by 68
  11. 2022Diffusion Models for Causal Discovery via Topological Orderingcited by 66
  12. 2022Statistical Efficiency of Score Matching: The View from Isoperimetrycited by 64
  13. 2021Ordering-Based Causal Discovery with Reinforcement Learningcited by 55
  14. 2017Learning linear structural equation models in polynomial time and sample complexitycited by 51
  15. 2022Denoising Likelihood Score Matching for Conditional Score-based Data Generationcited by 46
  16. 2023Scalable Causal Discovery with Score Matchingcited by 45
  17. 2023Causal Discovery with Score Matching on Additive Models with Arbitrary Noisecited by 43
  18. 2020Blindness of score-based methods to isolated components and mixing proportionscited by 34
  19. 2020Distribution Approximation and Statistical Estimation Guarantees of Generative Adversarial Networkscited by 23
  20. 2022Robustness in deep learning: The good (width), the bad (depth), and the ugly (initialization)cited by 18
  21. 2017Bayesian model comparison with the Hyv"arinen score: computation and consistencycited by 10
  22. 2014Justifying Information-Geometric Causal Inferencecited by 6

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 6; 2025: 7; 2026: 3.

16 papers.

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  1. 2024Score matching through the roof: linear, nonlinear, and latent variables causal discoverycited by 10
  2. 2024Identifiability Guarantees for Causal Disentanglement from Purely Observational Datacited by 7
  3. 2025Optimal and Diffusion Transports in Machine Learningcited by 6
  4. 2024Hybrid Top-Down Global Causal Discovery with Local Search for Linear and Nonlinear Additive Noise Modelscited by 5
  5. 2024Differentially Private Kernel Density Estimationcited by 3
  6. 2025A Non-Asymptotic Convergent Analysis for Scored-Based Graph Generative Model via a System of Stochastic Differential Equationscited by 3
  7. 2025Causal Structure and Representation Learning with Biomedical Applicationscited by 3
  8. 2025Regularization can make diffusion models more efficientcited by 2
  9. 2025Understanding Generalization in Diffusion Distillation via Probability Flow Distancecited by 2
  10. 2024Causal Order Discovery based on Monotonic SCMscited by 1
  11. 2024Shallow Diffuse: Robust and Invisible Watermarking through Low-Dimensional Subspaces in Diffusion Modelscited by 1
  12. 2025When Additive Noise Meets Unobserved Mediators: Bivariate Denoising Diffusion for Causal Discoverycited by 1
4 more
  1. 2026Ordering-based Causal Discovery via Generalized Score Matchingcited by 1
  2. 2025Linear Causal Discovery with Interventional Constraintscited by 0
  3. 2026Bridging Dynamics Gaps via Diffusion Schr"odinger Bridge for Cross-Domain Reinforcement Learningcited by 0
  4. 2026Intervention-Based Time Series Causal Discovery via Simulator-Generated Interventional Distributionscited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2025Score-informed Neural Operator for Enhancing Ordering-based Causal Discovery
  2. 2025Algorithm- and Data-Dependent Generalization Bounds for Diffusion Models
  3. 2026Non-asymptotic Convergence of Stochastic Gradient Descent in Score-based Generative Models
  4. 2025Improved Sample Complexity For Diffusion Model Training Without Empirical Risk Minimizer Access
  5. 2024Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization
  6. 2026Sample complexity bounds for categorical Markov random fields via Discrete Diffusions
  7. 2024An analysis of the noise schedule for score-based generative models
  8. 2024Supervised Score-Based Modeling by Gradient Boosting