Limite funzionale per reti neurali profonde gaussiane

We consider fully connected feed-forward deep neural networks where weights and biases are independent and identically distributed according to Gaussian distributions. We look at neural networks as stochastic processes, i.e. infinite-dimensional random elements, on the input real space of finite dimension, and we show that: i) a network defines a Gaussian process on the input space; ii) under suitable assumptions on the activation function, a network with rescaled weights converges weakly to a Gaussian process in the large-width limit. Our results contribute to recent theoretical studies on the interplay between infinitely wide deep neural networks and Gaussian processes. In particular, our large-width functional limit, jointly over networks' layers, extends and completes previous results (Matthews et al., 2018) that are limited to the convergence of the neural network on a finite number of fixed distinct inputs in the input space. As a by-product, we show that the limiting Gaussian process have almost surely locally h-Hölder continuous paths, for h<1. &#8203;