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Information Geometry of Gaussian Distributions in View of Stochastic Optimization

Luigi Malagò 1, 2 Giovanni Pistone 3
2 TAO - Machine Learning and Optimisation
CNRS - Centre National de la Recherche Scientifique : UMR8623, Inria Saclay - Ile de France, UP11 - Université Paris-Sud - Paris 11, LRI - Laboratoire de Recherche en Informatique
Abstract : We study the optimization of a continuous function by its stochastic relaxation, i.e., the optimization of the expected value of the function itself with respect to a density in a statistical model. We focus on gradient descent techniques applied to models from the exponential family and in particular on the multivariate Gaussian distribution. From the theory of the exponential family, we reparametrize the Gaussian distribution using natural and expectation parameters, and we derive formulas for natural gradients in both parameterizations. We discuss some advantages of the natural parameterization for the identification of sub-models in the Gaussian distribution based on conditional independence assumptions among variables. Gaussian distributions are widely used in stochastic optimization and in particular in model-based Evolutionary Computation, as in Estimation of Distribution Algorithms and Evolutionary Strategies. By studying natural gradient flows over Gaussian distributions our analysis and results directly apply to the study of CMA-ES and NES algorithms
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https://hal.inria.fr/hal-01108986
Contributor : Luigi Malagò <>
Submitted on : Friday, January 23, 2015 - 6:13:15 PM
Last modification on : Friday, April 30, 2021 - 9:54:29 AM

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  • HAL Id : hal-01108986, version 1

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Luigi Malagò, Giovanni Pistone. Information Geometry of Gaussian Distributions in View of Stochastic Optimization. Foundations of Genetic Algorithms XIII, Jun He, Thomas Jansen, Gabriela Ochoa and Christine Zarges, Jan 2015, Aberystwyth, United Kingdom. pp.150-162. ⟨hal-01108986⟩

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