Natural gradient Variational Bayes without matrix inversion

This paper presents an approach for efficiently approximating the inverse of Fisher information, a key component in variational Bayes inference. A notable aspect of our approach is the avoidance of analytically computing the Fisher information matrix and its explicit inversion. Instead, we introduce an iterative procedure for generating a sequence of matrices that converge to the inverse of Fisher information. The natural gradient variational Bayes algorithm without matrix inversion is provably convergent and achieves a convergence rate of order O(log s/s), with s the number of iterations. We also obtain a central limit theorem for the iterates. Our algorithm exhibits versatility, making it applicable across a diverse array of variational Bayes domains, including Gaussian approximation and normalizing flow Variational Bayes. We offer a range of numerical examples to demonstrate the efficiency and reliability of the proposed variational Bayes method.