Bayesian inference and kernel methods are well established in machine
learning. The neural network Gaussian process in particular provides a concept
to investigate neural networks in the limit of infinitely wide hidden layers by
using kernel and inference methods. Here we build upon this limit and provide a
field-theoretic formalism which covers the generalization properties of
infinitely wide networks. We systematically compute generalization properties
of linear, non-linear, and deep non-linear networks for kernel matrices with
heterogeneous entries. In contrast to currently employed spectral methods we
derive the generalization properties from the statistical properties of the
input, elucidating the interplay of input dimensionality, size of the training
data set, and variability of the data. We show that data variability leads to a
non-Gaussian action reminiscent of a ($\varphi^3+\varphi^4$)-theory. Using our
formalism on a synthetic task and on MNIST we obtain a homogeneous kernel
matrix approximation for the learning curve as well as corrections due to data
variability which allow the estimation of the generalization properties and
exact results for the bounds of the learning curves in the case of infinitely
many training data points.

基于无限宽度神经网络的高斯过程，并结合内核和推理方法，构建了一个场论的形式体系，研究了无限宽度网络的泛化性质，并从输入数据的统计性质得到了泛化性质的提取。