*Submitted Paper*

**Inserted:** 10 jul 2023

**Last Updated:** 4 jun 2024

**Year:** 2023

**Abstract:**

In this paper we provide explicit upper bounds on
some distances between the (law of the) output of
a *random Gaussian neural network* and (the law of) a random Gaussian vector. Our results concern both *shallow* random Gaussian neural networks with univariate output and fully connected and *deep* random Gaussian neural networks, with a rather general activation function. The upper bounds show how the widths of the layers, the activation function and other architecture parameters affect the Gaussian approximation of the output. Our techniques, relying on Stein's method and integration by parts formulas for the Gaussian law, yield estimates on distances which are indeed integral probability metrics, and include the total variation and the convex distances. These latter metrics are defined by testing against indicator functions of suitable measurable sets, and so allow for accurate estimates of the probability that the output is localized in some region of the space. Such estimates have a significant interest both from a practitioner's and a theorist's perspective.

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