Numerical Integration Method for Training Neural Network
We propose a new numerical integration method for training a shallow neural network by using the ridgelet transform with a fast convergence guarantee. Given a training dataset, the ridgelet transform can provide the parameters of the neural network that attains the global optimum of the training problem. In other words, we can obtain the global minimizer of the training problem by numerically computing the ridgelet transform, instead of by numerically optimizing the so-called backpropagation training problem. We employed the kernel quadrature for the basis of the numerical integration, because it is known to converge faster, i.e. $O(1/p)$ with the hidden unit number $p$, than other random methods, i.e. $O(1/\sqrt{p})$, such as Monte Carlo integration methods. Originally, the kernel quadrature has been developed for the purpose of computing posterior means, where the measure is assumed to be a probability measure, and the final product is a single number. On the other hand, our problem is the computation of an integral transform, where the measure is generally a signed measure, and the final product is a function. In addition, the performance of kernel quadrature is sensitive to the selection of its kernel. In this paper, we develop a generalized kernel quadrature method with a fast convergence guarantee in a function norm that is applicable to signed measures, and propose a natural choice of kernels.