  In this paper, we study consensus-based optimization (CBO), which is a
multi-agent metaheuristic derivative-free optimization method that can globally
minimize nonconvex nonsmooth functions and is amenable to theoretical analysis.
Based on an experimentally supported intuition that, on average, CBO performs a
gradient descent of the squared Euclidean distance to the global minimizer, we
devise a novel technique for proving the convergence to the global minimizer in
mean-field law for a rich class of objective functions. The result unveils
internal mechanisms of CBO that are responsible for the success of the method.
In particular, we prove that CBO performs a convexification of a large class of
optimization problems as the number of optimizing agents goes to infinity.
Furthermore, we improve prior analyses by requiring mild assumptions about the
initialization of the method and by covering objectives that are merely locally
Lipschitz continuous. As a core component of this analysis, we establish a
quantitative nonasymptotic Laplace principle, which may be of independent
interest. From the result of CBO convergence in mean-field law, it becomes
apparent that the hardness of any global optimization problem is necessarily
encoded in the rate of the mean-field approximation, for which we provide a
novel probabilistic quantitative estimate. The combination of these results
allows to obtain probabilistic global convergence guarantees of the numerical
CBO method.
