mlsanityData to 13 Oct 2026

Finite-Dimensional Gaussian Approximation for Deep Neural Networks: Universality in Random Weights

arXiv 2507.12686 ↗First posted 16 Jul 2025stat.ML cs.LG math.PRcited by 3

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Abstract (arXiv)

We study the Finite-Dimensional Distributions (FDDs) of deep neural networks with randomly initialized weights that have finite-order moments. Specifically, we establish Gaussian approximation bounds in the Wasserstein-1 norm between the FDDs and their Gaussian limit assuming a Lipschitz activation function and allowing the layer widths to grow to infinity at arbitrary relative rates. In the special case where all widths are proportional to a common scale parameter n and there are L-1 hidden layers, we obtain convergence rates of order n^-(1/6)L-1 + ε, for any ε > 0.

Key facts. Cited by 3 papers in the map (data through 13 Oct 2026). Builds on 19 of its 36 references that are in the map. First posted on arXiv 16 Jul 2025.

Builds on

19 of 36 references.

References · 19

  1. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,610
  2. 2017Deep Neural Networks as Gaussian Processescited by 618
  3. 2018Gaussian Process Behaviour in Wide Deep Neural Networkscited by 316
  4. 2017Training Quantized Nets: A Deeper Understandingcited by 74
  5. 2021Random Neural Networks in the Infinite Width Limit as Gaussian Processescited by 43
  6. 2022Quantitative Gaussian Approximation of Randomly Initialized Deep Neural Networkscited by 24
  7. 2023Quantitative CLTs in Deep Neural Networkscited by 19
  8. 2021Non-asymptotic approximations of neural networks by Gaussian processescited by 18
  9. 2022Deep neural networks with dependent weights: Gaussian Process mixture limit, heavy tails, sparsity and compressibilitycited by 13
  10. 2021Rate of Convergence of Polynomial Networks to Gaussian Processescited by 12
  11. 2023Wide Deep Neural Networks with Gaussian Weights are Very Close to Gaussian Processescited by 11
  12. 2023Non-asymptotic approximations of Gaussian neural networks via second-order Poincar'e inequalitiescited by 10
7 more
  1. 2023Gaussian random field approximation via Stein's method with applications to wide random neural networkscited by 10
  2. 2021Deep Stable neural networks: large-width asymptotics and convergence ratescited by 8
  3. 2023A Quantitative Functional Central Limit Theorem for Shallow Neural Networkscited by 8
  4. 2023Infinitely wide limits for deep Stable neural networks: sub-linear, linear and super-linear activation functionscited by 5
  5. 2025Entropic bounds for conditionally Gaussian vectors and applications to neural networkscited by 5
  6. 2023Wide neural networks: From non-gaussian random fields at initialization to the NTK geometry of trainingcited by 3
  7. 2023Posterior Inference on Shallow Infinitely Wide Bayesian Neural Networks under Weights with Unbounded Variancecited by 3

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Cited by year (2026 to 13 Oct)

Citing papers per year: 2025: 1; 2026: 2.

3 papers.

  1. 2025Posterior Bayesian Neural Networks with Dependent Weightscited by 0
  2. 2026Optimal Non-Asymptotic Edgeworth Expansions for Multivariate Neural Network Outputscited by 0
  3. 2026Geometric Dyson Brownian Motions and the Free Log-Normal Limit for a Non-Square Gaussian Matrix Productcited by 0