On the convergence of the $(1+1)$-ES in noisy spherical environments

In this paper, we analyze the convergence of a $(1+1)$-ES on unimodal functions perturbed by noise. We investigate two models for the noise: (1) a model where the noise is scaled proportionally to the step-size and (2) a model where the noise is scaled proportionally to the (non-noisy part of the) fitness function. Those models were previously studied in the literature using gaussian noise and asymptotic estimations when the dimension of the search space grows to infinity. Similar results for both of them were obtained. For lower bounded noise (that does not include gaussian noise), we show that those models exhibit different behaviors: premature convergence occurs with probability one for the first model whenever the noise takes strictly negative values, whereas linear convergence always occurs for the second model. Moreover we exhibit for the second model a case where the convergence rate is equal to zero for any choice of the step-size.