Evolution Strategies with Additive Noise: A Convergence Rate Lower Bound

We consider the problem of optimizing functions corrupted with additive noise. It is known that Evolutionary Algorithms can reach a Simple Regret O(1/√n) within logarithmic factors, when n is the number of function evaluations. Here, Simple Regret at evaluation $n$ is the difference between the evaluation of the function at the current recommendation point of the algorithm and at the real optimum. We show mathematically that this bound is tight, for any family of functions that includes sphere functions, at least for a wide set of Evolution Strategies without large mutations.