PSA-CMA-ES: CMA-ES with population size adaptation

The population size, i.e., the number of candidate solutions generated at each iteration, is the most critical strategy parameter in the covariance matrix adaptation evolution strategy, CMA-ES, which is one of the state-of-the-art search algorithms for black-box continuous optimization. The population size is required to be larger than its default value when the objective function is well-structured multimodal and/or noisy, while we want to keep it as small as possible for optimization speed. However, the strategy parameter tuning based on trial and error is, in general, prohibitively expensive in black-box optimization scenario. This paper proposes a novel strategy to adapt the population size for CMA-ES. The population size is adapted based on the estimated accuracy of the update of the normal distribution parameters. The CMA-ES with the proposed population size adaptation mechanism, PSA-CMA-ES, is tested both on noiseless and noisy benchmark functions, and compared with existing strategies. The results revealed that the PSA-CMA-ES works well on well-structured multimodal and/or noisy functions, but causes inefficient increase of the population size on unimodal functions. Furthermore, it is shown that the PSA-CMA-ES can tackle noise and multimodality at the same time.