Uniform-in-time mean-field limit estimate for the Consensus-Based Optimization

We establish a uniform-in-time estimate for the mean-field convergence of the Consensus- Based Optimization (CBO) algorithm by rescaling the consensus point in the dynamics with a small parameter κ ∈ (0, 1). This uniform-in-time estimate is essential, as CBO convergence relies on a sufficiently large time horizon and is crucial for ensuring stable, reliable long-term convergence, the latter being key to the practical effectiveness of CBO methods.