  We propose an improved evolution strategy (ES) using a novel nonlocal
gradient operator for high-dimensional black-box optimization. Standard ES
methods with $d$-dimensional Gaussian smoothing suffer from the curse of
dimensionality due to the high variance of Monte Carlo (MC) based gradient
estimators. To control the variance, Gaussian smoothing is usually limited in a
small region, so existing ES methods lack nonlocal exploration ability required
for escaping from local minima. We develop a nonlocal gradient operator with
directional Gaussian smoothing (DGS) to address this challenge. The DGS
conducts 1D nonlocal explorations along $d$ orthogonal directions in
$\mathbb{R}^d$, each of which defines a nonlocal directional derivative as a 1D
integral. We then use Gauss-Hermite quadrature, instead of MC sampling, to
estimate the $d$ 1D integrals to ensure high accuracy (i.e., small variance).
Our method enables effective nonlocal exploration to facilitate the global
search in high-dimensional optimization. We demonstrate the superior
performance of our method in three sets of examples, including benchmark
functions for global optimization, and real-world science and engineering
applications.
