Enhancing Exploration in Global Optimization by Noise Injection in the Probability Measures Space

McKean--Vlasov (MKV) systems provide a unifying framework for modern particle-based methods in global optimization. While individual particles evolve stochastically, the associated probability law follows a deterministic dynamics in the mean-field limit, which can hinder exploration in multimodal landscapes. To address this limitation, we introduce a general plug-in for MKV-based methods that enhances exploration by injecting noise directly into the probability law dynamics, referred to as ρ-noise. We leverage an existing ρ-noise coming from conditional MKV theory, namely Stochastic Moment Dynamics (SMD), and propose a geometrically motivated variant, Geometric Conmmon Noise (GCN), that exploits tangent space structure. Our analysis further shows that SMD can be interpreted as a limiting case of GCN. Through extensive experiments on multimodal benchmarks, we demonstrate that these noise-injection strategies consistently improve both exploration and convergence across various dynamics, including Langevin methods, Consensus-Based Optimization, and Stein Boltzmann Sampling, yielding a flexible and effective framework for global optimization.