A Discrete Version of CMA-ES

11 Pages Posted: 9 Jan 2019 Last revised: 13 Feb 2019

See all articles by Eric Benhamou

Eric Benhamou

Université Paris Dauphine; EB AI Advisory; AI For Alpha

Jamal Atif

Université Paris Dauphine

Rida Laraki

Université Paris-Dauphine, PSL Research University

Date Written: December 27, 2018

Abstract

Modern machine learning uses more and more advanced optimization techniques to find optimal hyper parameters. Whenever the objective function is non-convex, non continuous and with potentially multiple local minima, standard gradient descent optimization methods fail. A last resource and very different method is to assume that the optimum(s), not necessarily unique, is/are distributed according to a distribution and iteratively to adapt the distribution according to tested points. These strategies originated in the early 1960s, named Evolution Strategy (ES) have culminated with the CMA-ES (Covariance Matrix Adaptation) ES. It relies on a multi variate normal distribution and is supposed to be state of the art for general optimization program. However, it is far from being optimal for discrete variables.

In this paper, we extend the method to multivariate binomial correlated distributions. For such a distribution, we show that it shares similar features to the multi variate normal: independence and correlation is equivalent and correlation is efficiently modeled by interaction between different variables. We discuss this distribution in the framework of the exponential family. We prove that the model can estimate not only pairwise interactions among the two variables but also is capable of modeling higher order interactions.

This allows creating a version of CMA-ES that can accommodate efficiently discrete variables. We provide the corresponding algorithm and conclude.

Suggested Citation

Benhamou, Eric and Atif, Jamal and Laraki, Rida, A Discrete Version of CMA-ES (December 27, 2018). Available at SSRN: https://ssrn.com/abstract=3307212 or http://dx.doi.org/10.2139/ssrn.3307212

Eric Benhamou (Contact Author)

Université Paris Dauphine ( email )

Place du Maréchal de Tassigny
Paris, Cedex 16 75775
France

EB AI Advisory ( email )

35 Boulevard d'Inkermann
Neuilly sur Seine, 92200
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AI For Alpha ( email )

35 boulevard d'Inkermann
Neuilly sur Seine, 92200
France

Jamal Atif

Université Paris Dauphine ( email )

Place du Maréchal de Tassigny
Paris, Cedex 16 75775
France

Rida Laraki

Université Paris-Dauphine, PSL Research University ( email )

Place du Maréchal de Lattre de Tassigny
Paris, 75016
France

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