A discrete Consensus-Based Global Optimization Method with Noisy Objective Function

Consensus based optimization is a derivative-free particles-based method for the solution of global optimization problems. Several versions of the method have been proposed in the literature, and different convergence results have been proved. However, all existing results assume the objective function to be evaluated exactly at each iteration of the method. In this work, we extend the convergence analysis of a discrete-time CBO method to the case where only a noisy stochastic estimator of the objective function can be computed at a given point. In particular we prove that under suitable assumptions on the oracle's noise, the expected value of the mean squared distance of the particles from the solution can be made arbitrarily small in a finite number of iterations. Numerical experiments showing the impact of noise are also given.