Large-width functional asymptotics for deep Gaussian neural networks

Daniele Bracale · Stefano Favaro · Sandra Fortini · Stefano Peluchetti


Keywords: [ stochastic process ] [ gaussian process ] [ infinitely wide neural network ] [ deep learning theory ]

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Tue 4 May 1 a.m. PDT — 3 a.m. PDT

Abstract: In this paper, we consider fully connected feed-forward deep neural networks where weights and biases are independent and identically distributed according to Gaussian distributions. Extending previous results (Matthews et al., 2018a;b;Yang, 2019) we adopt a function-space perspective, i.e. we look at neural networks as infinite-dimensional random elements on the input space $\mathbb{R}^I$. Under suitable assumptions on the activation function we show that: 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 a continuous Gaussian process in the large-width limit; iii) the limiting Gaussian process has almost surely locally $\gamma$-Hölder continuous paths, for $0 < \gamma <1$. Our results contribute to recent theoretical studies on the interplay between infinitely wide deep neural networks and Gaussian processes by establishing weak convergence in function-space with respect to a stronger metric.

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