# Optimal Non-Asymptotic Edgeworth Expansions for Multivariate Neural Network Outputs

arXiv: https://arxiv.org/abs/2605.24072 · HTML: https://mlsanity.com/paper/2605.24072/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 37 references that are in the map.
- First posted on arXiv 22 May 2026.

## Builds on (most cited first, 20 of 20)

- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [Priors in Bayesian Deep Learning: A Review](https://mlsanity.com/paper/2105.06868/) (2021)
- [Student-t Processes as Alternatives to Gaussian Processes](https://mlsanity.com/paper/1402.4306/) (2014)
- [Random Neural Networks in the Infinite Width Limit as Gaussian Processes](https://mlsanity.com/paper/2107.01562/) (2021)
- [Why bigger is not always better: on finite and infinite neural networks](https://mlsanity.com/paper/1910.08013/) (2019)
- [Predicting the outputs of finite deep neural networks trained with noisy gradients](https://mlsanity.com/paper/2004.01190/) (2020)
- [Exact posterior distributions of wide Bayesian neural networks](https://mlsanity.com/paper/2006.10541/) (2020)
- [Quantitative Gaussian Approximation of Randomly Initialized Deep Neural Networks](https://mlsanity.com/paper/2203.07379/) (2022)
- [Random Fully Connected Neural Networks as Perturbatively Solvable Hierarchies](https://mlsanity.com/paper/2204.01058/) (2022)
- [Finite size corrections for neural network Gaussian processes](https://mlsanity.com/paper/1908.10030/) (2019)
- [The Limitations of Large Width in Neural Networks: A Deep Gaussian Process Perspective](https://mlsanity.com/paper/2106.06529/) (2021)
- [Quantitative CLTs in Deep Neural Networks](https://mlsanity.com/paper/2307.06092/) (2023)
- [Rate of Convergence of Polynomial Networks to Gaussian Processes](https://mlsanity.com/paper/2111.03175/) (2021)
- [Wide Deep Neural Networks with Gaussian Weights are Very Close to Gaussian Processes](https://mlsanity.com/paper/2312.11737/) (2023)
- [The Positivity of the Neural Tangent Kernel](https://mlsanity.com/paper/2404.12928/) (2024)
- [Non-asymptotic approximations of Gaussian neural networks via second-order Poincar'e inequalities](https://mlsanity.com/paper/2304.04010/) (2023)
- [Entropic bounds for conditionally Gaussian vectors and applications to neural networks](https://mlsanity.com/paper/2504.08335/) (2025)
- [Finite-Dimensional Gaussian Approximation for Deep Neural Networks: Universality in Random Weights](https://mlsanity.com/paper/2507.12686/) (2025)
- [Bayesian inference with finitely wide neural networks](https://mlsanity.com/paper/2303.02859/) (2023)

## 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.