Sampling in CMA-ES: Low Numbers of Low Discrepancy Points

TL;DR

This paper demonstrates that using small, fixed sets of low-discrepancy points in CMA-ES can outperform traditional uniform sampling, with as few as 16-128 points sufficing for effective optimization across various dimensions.

Contribution

It shows that low-discrepancy point sets with low cardinality can effectively replace random sampling in CMA-ES, improving performance and reducing the number of required samples.

Findings

128 points approximate full sequence performance up to 40 dimensions

16 points outperform uniform sampling in 2D

Lower discrepancy correlates with better empirical performance

Abstract

The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is one of the most successful examples of a derandomized evolution strategy. However, it still relies on randomly sampling offspring, which can be done via a uniform distribution and subsequently transforming into the required Gaussian. Previous work has shown that replacing this uniform sampling with a low-discrepancy sampler, such as Halton or Sobol sequences, can improve performance over a wide set of problems. We show that iterating through small, fixed sets of low-discrepancy points can still perform better than the default uniform distribution. Moreover, using only 128 points throughout the search is sufficient to closely approximate the empirical performance of using the complete pseudorandom sequence up to dimensionality 40 on the BBOB benchmark. For lower dimensionalities (below 10), we find that using as little as 32 unique low discrepancy points performs similar or better than uniform sampling. In 2D, for which we have highly optimized low discrepancy samples available, we demonstrate that using these points yields the highest empirical performance and requires only 16 samples to improve over uniform sampling. Overall, we establish a clear relation between the $L_2$ discrepancy of the used point set and the empirical performance of the CMA-ES.