ON THE MEAN-FIELD LIMIT FOR THE CONSENSUS-BASED OPTIMIZATION

The paper proves tightness of the empirical measures for the CBO particle system, identifies any subsequential limit as a weak solution to the mean-field PDE, and uses uniqueness (via well-posedness of the McKean SDE and a linear duality argument) to conclude the whole sequence converges to a deterministic limit (Theorem 3.3; see the model, PDE, and uniqueness discussions around (1.5)–(1.7), Proposition 3.2, and Appendix Theorem 4.3) . The candidate solution reaches the same conclusion but supplies an alternative well-posedness proof for the McKean SDE via a quantitative Lipschitz bound for the weight map μ ↦ X_α(μ) on moment-bounded sets and a short-time contraction in Wasserstein distance, then proceeds with tightness and generator/martingale arguments consistent with the paper’s approach. One minor gap in the candidate’s write-up is that it does not explicitly assume fourth moments of the initial data, which the paper uses to close uniform moment bounds needed for Aldous’ criterion (Lemma 2.1) . Apart from this fixable omission, both arguments are correct; they differ mainly in how the mean-field well-posedness is established.