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

Youhei Akimoto 1 
1 TAO - Machine Learning and Optimisation
LRI - Laboratoire de Recherche en Informatique, UP11 - Université Paris-Sud - Paris 11, Inria Saclay - Ile de France, CNRS - Centre National de la Recherche Scientifique : UMR8623
Abstract : In this paper we investigate the convergence properties of a variant of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). Our study is based on the recent theoretical foundation that the pure rank-mu update CMA-ES performs the natural gradient descent on the parameter space of Gaussian distributions. We derive a novel variant of the natural gradient method where the parameters of the Gaussian distribution are updated along the natural gradient to improve a newly defined function on the parameter space. We study this algorithm on composites of a monotone function with a convex quadratic function. We prove that our algorithm adapts the covariance matrix so that it becomes proportional to the inverse of the Hessian of the original objective function. We also show the speed of covariance matrix adaptation and the speed of convergence of the parameters. We introduce a stochastic algorithm that approximates the natural gradient with finite samples and present some simulated results to evaluate how precisely the stochastic algorithm approximates the deterministic, ideal one under finite samples and to see how similarly our algorithm and the CMA-ES perform.
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Submitted on : Wednesday, April 18, 2012 - 6:08:21 PM
Last modification on : Tuesday, October 25, 2022 - 4:18:08 PM
Long-term archiving on: : Thursday, July 19, 2012 - 2:31:37 AM


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  • HAL Id : hal-00688909, version 1
  • ARXIV : 1204.4141


Youhei Akimoto. Analysis of a Natural Gradient Algorithm on Monotonic Convex-Quadratic-Composite Functions. Genetic and Evolutionary Computation Conference (GECCO 2012), Jul 2012, Philadelphia, United States. ⟨hal-00688909⟩



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