Convergence and error estimates for time-discrete consensus-based optimization algorithms

We present convergence and error estimates of modified versions of the time-discrete consensus-based optimization (CBO) algorithm proposed in Carrillo et al. (ESAIM: Control Optim Calc Var, 2020) for general non-convex functions. In authors’ recent work (Ha et al. in Math Models Meth Appl Sci 30:2417–2444, 2020), rigorous error analysis of a modified version of the first-order consensus-based optimization algorithm proposed in Carrillo et al. (2020) was studied at the particle level without resorting to the kinetic equation via a mean-field limit. However, the error analysis for the corresponding time- discrete algorithm was not done mainly due to lack of discrete analogue of Itô’s stochastic calculus. In this paper, we provide a simple and elementary convergence and error analysis for a general time-discrete consensus-based optimization algorithm, which includes modifications of the three discrete algorithms in Carrillo et al. (2020), two of which are present in Ha et al. (2020). Our analysis provides numerical stability and convergence conditions for the three algorithms, as well as error estimates to the global minimum.