# Clouds of points optimization in convex polygons with evolutionary algorithms based on Wasserstein barycenters

Babacar SOW (EMSE, LIMOS), Rodolphe LE RICHE (CNRS LIMOS)

Julien PELAMATTI(EDF R&D), Merlin KELLER (EDF R&D), Sanaa ZANNANE (EDF R&D)

**Universités:** Ecole Nationale Supérieure des Mines de Saint-Etienne(EMSE), Université Clermont Auvergne(UCA)

**Laboratoire:** LIMOS

**Email of contact:**

babacar.sow@emse.fr

anr®

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## Bayesian optimization: MMD as kernel and WBGEA for the optimization of the EI

* Let $H$ be a Reproducing Kernel Hilbert Space with a characteristic kernel $k_H$ such as Matérn $5/2$. The characteristic nature guarantees the injectivity of the embedding map [1]: $P_X \mapsto \mu_X = \int P_X(x)k_H(x, \cdot)dx$. $\theta_1$ and $\theta_2$ are 2 hyper-parameters scaling the dimensions between two points $\mathbf{x}$ and $\mathbf{x}'$ through $|x_1 - x'_1|/\theta_1$ and $|x_2 - x'_2|/\theta_2$.

* The correlation kernel $K(X, X') = \sigma \exp(-\frac{\|\mu_X - \mu_{X'}\|^2_H}{2\theta^2})$ is symmetric and semi-definite positive. We denote this kernel by MMD for Maximum Mean Discrepancy and use it to define a Gaussian process over clouds of points.

* The acquisition function is the expected improvement (EI) with the following formula: $EI(X) = (y^{\text{min}} - \mu(X))\Phi(\frac{(y^{\text{min}} - \mu(X))}{\sqrt{\Sigma(X,X)}}) + \sqrt{\Sigma(X,X)}\Phi(\frac{(y^{\text{min}} - \mu(X))}{\sqrt{\Sigma(X,X)}})$ where $\mu(X)$ and $\Sigma(X, X)$ denote respectively the mean and the variance of the prediction. $\Phi$ and $\phi$ are respectively the cumulative distribution and density function of the standard normal law. The EI is defined over the same search space as the functions to optimize. It is optimized with WBGEA.

* We fix the budget of evaluations to 100. A random initial set of 50 clouds is chosen. The hyper-parameters of the kernel are updated every 5 iterations by maximization of the likelihood. The acquisition criterion is optimized at each iteration. We present below the percentage of the maximum value attained by Bayesian optimization (BO) with respect to that attained by WBGEA (denoted by Percentage\_BO\_WBGEA):

<table><thead><tr><th>Functions</th><th>F<sub>0</sub></th><th>F<sub>0</sub><sub>pen</sub></th><th>F<sub>inert</sub></th><th>F<sub>minDist</sub></th></tr></thead><tbody><tr><td>Percentage\_BO\_WBGEA</td><td>96.05%</td><td>90.12%</td><td>69.35%</td><td>65.80%</td></tr></tbody></table>

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<figure><img src="image_6.png" /><figcaption>**Figure 2:** Average over 10 (+/- std. deviation) of the evolutions of the maximum of the functions over the BO iterations.</figcaption></figure>

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The first one has a contracting property while the second increases systematically the spread of clouds.

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