Gaussian random field approximation via Stein's method with applications to wide random neural networks

• URL: http://arxiv.org/abs/2306.16308v2
• Date: Wed, 1 May 2024 01:39:18 GMT
• Title: Gaussian random field approximation via Stein's method with applications to wide random neural networks
• Authors: Krishnakumar Balasubramanian, Larry Goldstein, Nathan Ross, Adil Salim,
• Abstract summary: We develop a novel Gaussian smoothing technique that allows us to transfer a bound in a smoother metric to the $W_$ distance. We obtain the first bounds on the Gaussian random field approximation of wide random neural networks. Our bounds are explicitly expressed in terms of the widths of the network and moments of the random weights.
• Score: 20.554836643156726
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We derive upper bounds on the Wasserstein distance ($W_1$), with respect to $\sup$-norm, between any continuous $\mathbb{R}^d$ valued random field indexed by the $n$-sphere and the Gaussian, based on Stein's method. We develop a novel Gaussian smoothing technique that allows us to transfer a bound in a smoother metric to the $W_1$ distance. The smoothing is based on covariance functions constructed using powers of Laplacian operators, designed so that the associated Gaussian process has a tractable Cameron-Martin or Reproducing Kernel Hilbert Space. This feature enables us to move beyond one dimensional interval-based index sets that were previously considered in the literature. Specializing our general result, we obtain the first bounds on the Gaussian random field approximation of wide random neural networks of any depth and Lipschitz activation functions at the random field level. Our bounds are explicitly expressed in terms of the widths of the network and moments of the random weights. We also obtain tighter bounds when the activation function has three bounded derivatives.

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