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In this paper we investigate the convergence properties of a variant of theCovariance Matrix Adaptation Evolution Strategy (CMA-ES). Our study is based onthe recent theoretical foundation that the pure rank-mu update CMA-ES performsthe natural gradient descent on the parameter space of Gaussian distributions.We derive a novel variant of the natural gradient method where the parametersof the Gaussian distribution are updated along the natural gradient to improvea newly defined function on the parameter space. We study this algorithm oncomposites of a monotone function with a convex quadratic function. We provethat our algorithm adapts the covariance matrix so that it becomes proportionalto the inverse of the Hessian of the original objective function. We also showthe speed of covariance matrix adaptation and the speed of convergence of theparameters. We introduce a stochastic algorithm that approximates the naturalgradient with finite samples and present some simulated results to evaluate howprecisely the stochastic algorithm approximates the deterministic, ideal oneunder finite samples and to see how similarly our algorithm and the CMA-ESperform.

Analysis of a Natural Gradient Algorithm on Monotonic Convex-Quadratic-Composite Functions