Noisy Optimization: Convergence with a Fixed Number of Resamplings

Abstract

It is known that evolution strategies in continuous domains might not converge in the presence of noise [3,14]. It is also known that, under mild assumptions, and using an increasing number of resamplings, one can mitigate the effect of additive noise [4] and recover convergence. We show new sufficient conditions for the convergence of an evolutionary algorithm with constant number of resamplings; in particular, we get fast rates (log-linear convergence) provided that the variance decreases around the optimum slightly faster than in the so-called multiplicative noise model.

Evolution Strategies with Additive Noise

“…Such results can be found in [4,5,21,27]. In some cases, the same behavior can be achieved in the noisy case; typically in the case of variance decreasing faster than in the multiplicative model, and if fitness values are averaged over a constant ad-hoc number of resamplings [9].…”

Analysis of Different Types of Regret in Continuous Noisy Optimization

“…, f (x, w r )) and the fitness value used in the comparisons is the average of these evaluations 1 r r i=1 f (x, w i ). In particular, the variance of the noise is divided by r. Several rules have been studied: constant [13], adaptive (polynomial in the inverse of the step-size), polynomial and exponential [5] number of resamplings. A general (µ, λ)-ES is presented in Algorithm 2.…”