On Multiplicative Noise Models for Stochastic Search

In this paper we investigate multiplicative noise models in the context of continuous optimization. We illustrate how some intrinsic properties of the noise model imply the failure of reasonable search algorithms for locating the optimum of the noiseless part of the objective function. Those findings are rigorously investigated on the (1 + 1)-ES for the minimization of the noisy sphere function. Assuming a lower bound on the support of the noise distribution, we prove that the (1 + 1)-ES diverges when the lower bound allows to sample negative fitness with positive probability and converges in the opposite case. We provide a discussion on the practical applications and non applications of those outcomes and explain the differences with previous results obtained in the limit of infinite search-space dimensionality.