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

TL;DR

This paper extends the analysis of a consensus-based global optimization method to scenarios with noisy objective function evaluations, demonstrating convergence properties and providing numerical insights into noise effects.

Contribution

It introduces a convergence analysis for a discrete CBO method when only noisy estimators of the objective are available, broadening its applicability.

Findings

Expected mean squared distance can be made arbitrarily small with enough iterations.

Convergence holds under certain assumptions on the noise.

Numerical experiments illustrate the impact of noise on the method.

Abstract

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.