Analysis of information geometric optimization with isotropic gaussian distribution under finite samples

In this article, we theoretically investigate the convergence properties of the information geometric optimization (IGO) algorithm given the family of isotropic Gaussian distributions on the sphere function. Differently from previous studies, where the exact natural gradient is taken, i.e., the infinite samples are assumed, we consider the case that the natural gradient is estimated from finite samples. We derive the rates of the expected decrease of the squared distance to the optimum and the variance parameter as functions of the learning rates, dimension, and sample size. From the rates of decrease deduces that the rates of decreases of the squared distance to the optimum and the variance parameter must agree for geometric convergence of the algorithm. In other words, the ratio between the squared distance to the optimum and the variance must be stable, which is observed empirically but is not derived in the previous theoretical studies. We further derive the condition on the learning rates that the rates of decreases agree and derive the stable value of the ratio. We confirm in simulation that the derived rates of decreases and the stable value of the ratio well approximate the behavior of the IGO algorithm.