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In this paper, we consider fully connected feed-forward deep neural networkswhere weights and biases are independent and identically distributed accordingto Gaussian distributions. Extending previous results (Matthews et al.,2018a;b; Yang, 2019) we adopt a function-space perspective, i.e. we look atneural networks as infinite-dimensional random elements on the input space$\mathbb{R}^I$. Under suitable assumptions on the activation function we showthat: i) a network defines a continuous Gaussian process on the input space$\mathbb{R}^I$; ii) a network with re-scaled weights converges weakly to acontinuous Gaussian process in the large-width limit; iii) the limitingGaussian process has almost surely locally $\gamma$-H\"older continuous paths,for $0 < \gamma <1$. Our results contribute to recent theoretical studies onthe interplay between infinitely wide deep neural networks and Gaussianprocesses by establishing weak convergence in function-space with respect to astronger metric.

Large-width functional asymptotics for deep Gaussian neural networks