Log-log Convergence for Noisy Optimization

Weconsider noisy optimization problems,without the assumption of variance vanishing in the neighborhood of the optimum. We show mathematically that simple rules with exponential number of resamplings lead to a log-log convergence rate. In particular, in this case the log of the distance to the optimum is linear on the log of the number of resamplings. As well as with number of resamplings polynomial in the inverse step-size. We show empirically that this convergence rate is obtained also with polynomial number of resamplings. In this polynomial resampling setting, using classical evolution strategies and an ad hoc choice of the number of resamplings, we seemingly get the same rate as those obtained with specific Estimation of Distribution Algorithms designed for noisy setting. We also experiment non-adaptive polynomial resamplings. Compared to the state of the art, our results provide (i) proofs of log-log convergence for evolution strategies (which were not covered by existing results) in the case of objective functions with quadratic expectations and constant noise, (ii) log-log rates also for objective functions with expectation E[f(x)] = ||x − x∗||p, where x∗ represents the optimum (iii) experiments with different parameterizations than those considered in the proof. These results propose some simple revaluation schemes. This paper extends [1].