arXiv subjects

Lucia Celli

Publications and source records attributed to Lucia Celli.

3 recordsLinked to original sources

arXiv2026

Optimal Non-Asymptotic Edgeworth Expansions for Multivariate Neural Network Outputs

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 $m\in\mathbb{N}$. 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.

stat.ML↗

arXiv2026

Wide neural networks with general weights: convergence rate and explicit dependence on the hyper-parameters

Using Stein's method techniques introduced by Chatterjee (2008) and further extended by Kasprzak and Peccati (2022) and by Lachi\`eze-Rey and Peccati (2017), we derive novel quantitative bounds on the convergence in distribution of feed-forward fully connected neural networks (with Lipschitz activation functions) towards Gaussian processes, as the hidden layer width $n$ tends to infinity. We consider networks initialized with independent and identically distributed (i.i.d.) weights possessing sufficiently many finite moments, and i.i.d. Gaussian biases independent of the weights. Specifically, when the network is evaluated at a single input, we obtain convergence rates of order $O(n^{-1/2})$ in both total variation and Wasserstein distances. When evaluated at a general finite collection of inputs, we establish bounds of the same order in terms of the convex distance. All bounds are given in explicit and computable form. As a consequence of our estimates, we also deduce a novel convergence result in the regime where the depth of the neural network increases simultaneously with the width $n$, up to order $O\big((\log_2 n)^{1/3}\big)$. To the best of our knowledge, this is the first CLT in the infinite width/depth limit holding for general (nonlinear) Lipschitz activation functions and non-Gaussian weight distributions. Our analysis yields several results of independent interest, including: (i) an explicit lower bound on the determinant of the limiting covariance matrix and (ii) new advances in Stein's method, both for the one-dimensional Stein's equation associated with the square of a Lipschitz function and for the multivariate Stein's equation associated with the tensor product of a Lipschitz function with itself.

math.PR↗

arXiv2025

Entropic bounds for conditionally Gaussian vectors and applications to neural networks

Using entropic inequalities from information theory, we provide new bounds on the total variation and 2-Wasserstein distances between a conditionally Gaussian law and a Gaussian law with invertible covariance matrix. We apply our results to quantify the speed of convergence to Gaussian of a randomly initialized fully connected neural network and its derivatives - evaluated in a finite number of inputs - when the initialization is Gaussian and the sizes of the inner layers diverge to infinity. Our results require mild assumptions on the activation function, and allow one to recover optimal rates of convergence in a variety of distances, thus improving and extending the findings of Basteri and Trevisan (2023), Favaro et al. (2023), Trevisan (2024) and Apollonio et al. (2024). One of our main tools are the quantitative cumulant estimates established in Hanin (2024). As an illustration, we apply our results to bound the total variation distance between the Bayesian posterior law of the neural network and its derivatives, and the posterior law of the corresponding Gaussian limit: this yields quantitative versions of a posterior CLT by Hron et al. (2022), and extends several estimates by Trevisan (2024) to the total variation metric.

math.PR↗