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.

Quantitative CLTs in Deep Neural Networks

We study the distribution of a fully connected neural network with random Gaussian weights and biases in which the hidden layer widths are proportional to a large constant $n$. Under mild assumptions on the non-linearity, we obtain quantitative bounds on normal approximations valid at large but finite $n$ and any fixed network depth. Our theorems show both for the finite-dimensional distributions and the entire process, that the distance between a random fully connected network (and its derivatives) to the corresponding infinite width Gaussian process scales like $n^{-\gamma}$ for $\gamma>0$, with the exponent depending on the metric used to measure discrepancy. Our bounds are strictly stronger in terms of their dependence on network width than any previously available in the literature; in the one-dimensional case, we also prove that they are optimal, i.e., we establish matching lower bounds.