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Optimal Non-Asymptotic Edgeworth Expansions for Multivariate Neural Network Outputs

arXiv 2605.24072 ↗First posted 22 May 2026stat.ML cs.LG math.PRcited by 0

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Abstract (arXiv)

Finite-width fully connected neural networks with Gaussian-initialized weights deviate from their infinite-width Gaussian limit, exhibiting non-vanishing higher-order cumulants. We approximate these deviations, for a neural network evaluated in a finite number of inputs, using multidimensional Edgeworth expansions of arbitrary order 4m-1, with m∈mathbbN. Assuming that the corresponding Gaussian limit has an invertible covariance matrix and that the activation function is polynomially bounded, we establish a bound of order n-m on the total variation distance between the law of the true network output and its Edgeworth approximation, with matching lower bounds. As an application, we quantify the error in Bayesian posterior distributions when the prior is replaced by its Edgeworth expansion. Our results are more general and also apply to sequences of conditionally Gaussian vectors converging to a Gaussian vector with invertible covariance.

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References · 20

  1. 2017Deep Neural Networks as Gaussian Processescited by 622
  2. 2018Gaussian Process Behaviour in Wide Deep Neural Networkscited by 317
  3. 2021Priors in Bayesian Deep Learning: A Reviewcited by 101
  4. 2014Student-t Processes as Alternatives to Gaussian Processescited by 45
  5. 2021Random Neural Networks in the Infinite Width Limit as Gaussian Processescited by 44
  6. 2019Why bigger is not always better: on finite and infinite neural networkscited by 41
  7. 2020Predicting the outputs of finite deep neural networks trained with noisy gradientscited by 31
  8. 2020Exact posterior distributions of wide Bayesian neural networkscited by 30
  9. 2022Quantitative Gaussian Approximation of Randomly Initialized Deep Neural Networkscited by 25
  10. 2022Random Fully Connected Neural Networks as Perturbatively Solvable Hierarchiescited by 24
  11. 2019Finite size corrections for neural network Gaussian processescited by 22
  12. 2021The Limitations of Large Width in Neural Networks: A Deep Gaussian Process Perspectivecited by 21
8 more
  1. 2023Quantitative CLTs in Deep Neural Networkscited by 20
  2. 2021Rate of Convergence of Polynomial Networks to Gaussian Processescited by 13
  3. 2023Wide Deep Neural Networks with Gaussian Weights are Very Close to Gaussian Processescited by 13
  4. 2024The Positivity of the Neural Tangent Kernelcited by 13
  5. 2023Non-asymptotic approximations of Gaussian neural networks via second-order Poincar'e inequalitiescited by 11
  6. 2025Entropic bounds for conditionally Gaussian vectors and applications to neural networkscited by 7
  7. 2025Finite-Dimensional Gaussian Approximation for Deep Neural Networks: Universality in Random Weightscited by 5
  8. 2023Bayesian inference with finitely wide neural networkscited by 1

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