CMA-ES with Adaptive Reevaluation for Multiplicative Noise

The covariance matrix adaptation evolution strategy (CMA-ES) is a powerful optimization method for continuous black-box optimization problems. Several noise-handling methods have been proposed to bring out the optimization performance of the CMA-ES on noisy objective functions. The adaptations of the population size and the learning rate are two major approaches that perform well under additive Gaussian noise. The reevaluation technique is another technique that evaluates each solution multiple times. In this paper, we discuss the difference between those methods from the perspective of stochastic relaxation that considers the maximization of the expected utility function. We derive that the set of maximizers of the noise-independent utility, which is used in the reevaluation technique, certainly contains the optimal solution, while the noise-dependent utility, which is used in the population size and leaning rate adaptations, does not satisfy it under multiplicative noise. Based on the discussion, we develop the reevaluation adaptation CMA-ES (RA-CMA-ES), which computes two update directions using half of the evaluations and adapts the number of reevaluations based on the estimated correlation of those two update directions. The numerical simulation shows that the RA-CMA-ES outperforms the comparative method under multiplicative noise, maintaining competitive performance under additive noise.