# A discrete version of CMA-ES

## TL;DR

This paper introduces a discrete version of CMA-ES, extending the algorithm to handle multivariate binomial distributions for optimizing discrete variables, which was previously limited to continuous variables.

## Contribution

The authors develop a novel discrete CMA-ES variant using multivariate binomial distributions, capable of modeling higher-order interactions for discrete optimization tasks.

## Key findings

- The discrete CMA-ES models correlations efficiently through variable interactions.
- The distribution can estimate pairwise and higher-order interactions.
- The paper provides a complete algorithm for the discrete CMA-ES.

## 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.