A modified Consensus-Based Optimization model: consensus formation and uniform-in-time propagation of chaos
Young-Pil Choi, Seungchan Lee, Sihyun Song
TL;DR
This work develops a modified Consensus-Based Optimization (CBO) framework with a regularized Gibbs weight, enabling a fully rigorous, dimension-free analysis of finite-particle, mean-field, and optimization dynamics under relaxed regularity on the objective function $f$. The regularized consensus $m_t^h$ stabilizes the dynamics, yielding exponential contraction to a consensus state and linking this state to the global minimizer via weighted-energy estimates and Laplace's principle. The authors establish large-time consensus for the particle system under a quantitative threshold on the drift, prove uniform-in-time propagation of chaos with explicit, dimension-free rates, and analyze the mean-field McKean–Vlasov equation to show deterministic consensus converging to minimizers as $\alpha\to\infty$. Collectively, the results provide a unified and robust theoretical foundation for consensus-based optimization, enabling global convergence guarantees under substantially relaxed regularity assumptions on $f$ and without the need for cutoffs, rescaling, or boundedness constraints. The framework thus offers rigorous insights into how regularization and Gibbs-weighted consensus steer high-dimensional stochastic search toward global optima with provable stability and scalability.
Abstract
We introduce a modified Consensus-Based Optimization model that admits a fully unified and rigorous analysis of its finite-particle dynamics, the associated McKean--Vlasov equation, and their optimization behavior under a single set of structural framework. The key ingredient is a regularized Gibbs weight that stabilizes the consensus point and avoids degeneracies present in the classical formulation, eliminating the need for cutoffs, rescaling, or boundedness assumptions on the objective function. Our first main result establishes large-time consensus for the particle system: when the drift exceeds an explicit threshold, all particles converge exponentially to a common random limit that concentrates near the global minimizer. Our second result proves uniform-in-time propagation of chaos, providing quantitative and dimension-free convergence of the empirical measure to the McKean--Vlasov dynamics. Finally, we show that the mean-field system reaches deterministic consensus and that its consensus point approaches the global minimizer in the regime of highly concentrated Gibbs weights. Together, these results yield a unified and internally consistent theoretical framework for consensus-based optimization under substantially relaxed regularity assumptions on the objective function.