Gaussian random field approximation via Stein's method with applications to wide random neural networks
This work addresses the problem of approximating random fields in machine learning, particularly for neural networks, offering theoretical guarantees that are incremental but novel in extending beyond one-dimensional index sets.
The authors derived upper bounds on the Wasserstein distance between any continuous random field and a Gaussian field using Stein's method, and applied this to provide the first bounds for Gaussian approximation of wide random neural networks of any depth with Lipschitz activations, expressed in terms of network widths and weight moments.
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.