# A Quantitative Functional Central Limit Theorem for Shallow Neural Networks

arXiv: https://arxiv.org/abs/2306.16932 · HTML: https://mlsanity.com/paper/2306.16932/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 30 references that are in the map.
- First posted on arXiv 29 Jun 2023.

## Builds on (most cited first, 13 of 13)

- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [The Principles of Deep Learning Theory](https://mlsanity.com/paper/2106.10165/) (2021)
- [Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity](https://mlsanity.com/paper/1602.05897/) (2016)
- [Deep Equals Shallow for ReLU Networks in Kernel Regimes](https://mlsanity.com/paper/2009.14397/) (2020)
- [Non-Gaussian processes and neural networks at finite widths](https://mlsanity.com/paper/1910.00019/) (2019)
- [Random Neural Networks in the Infinite Width Limit as Gaussian Processes](https://mlsanity.com/paper/2107.01562/) (2021)
- [Time/Accuracy Tradeoffs for Learning a ReLU with respect to Gaussian Marginals](https://mlsanity.com/paper/1911.01462/) (2019)
- [Quantitative Gaussian Approximation of Randomly Initialized Deep Neural Networks](https://mlsanity.com/paper/2203.07379/) (2022)
- [Kolmogorov Width Decay and Poor Approximators in Machine Learning: Shallow Neural Networks, Random Feature Models and Neural Tangent Kernels](https://mlsanity.com/paper/2005.10807/) (2020)
- [Non-asymptotic approximations of neural networks by Gaussian processes](https://mlsanity.com/paper/2102.08668/) (2021)
- [Rate of Convergence of Polynomial Networks to Gaussian Processes](https://mlsanity.com/paper/2111.03175/) (2021)
- [Non-asymptotic approximations of Gaussian neural networks via second-order Poincar'e inequalities](https://mlsanity.com/paper/2304.04010/) (2023)

## Built on by (most cited first, 8 of 8)

- [Quantitative CLTs in Deep Neural Networks](https://mlsanity.com/paper/2307.06092/) (2023)
- [Wide Deep Neural Networks with Gaussian Weights are Very Close to Gaussian Processes](https://mlsanity.com/paper/2312.11737/) (2023)
- [Finite-Dimensional Gaussian Approximation for Deep Neural Networks: Universality in Random Weights](https://mlsanity.com/paper/2507.12686/) (2025)
- [Spectral complexity of deep neural networks](https://mlsanity.com/paper/2405.09541/) (2024)
- [Finite Neural Networks as Mixtures of Gaussian Processes: From Provable Error Bounds to Prior Selection](https://mlsanity.com/paper/2407.18707/) (2024)
- [Fractal and Regular Geometry of Deep Neural Networks](https://mlsanity.com/paper/2504.06250/) (2025)
- [Quantitative Gaussian-Process limits of Tensor Programs](https://mlsanity.com/paper/2607.06290/) (2026)
- [Critical Points of Random Neural Networks](https://mlsanity.com/paper/2505.17000/) (2025)

## Abstract (arXiv)

We prove a Quantitative Functional Central Limit Theorem for one-hidden-layer neural networks with generic activation function. The rates of convergence that we establish depend heavily on the smoothness of the activation function, and they range from logarithmic in non-differentiable cases such as the Relu to √n for very regular activations. Our main tools are functional versions of the Stein-Malliavin approach; in particular, we exploit heavily a quantitative functional central limit theorem which has been recently established by Bourguin and Campese (2020).