Stochastic moments dynamics: a flexible finite-dimensional random perturbation of Wasserstein gradient descent

2025/05/12 by Pierre Germain, Pierre Monmarché, Germain, Pierre +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Medicine · #Advanced Neuroimaging Techniques and Applications #Caveolin-1 and cellular processes #FOS: Mathematics #Probability (math.PR) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2505.07448

openalex publication_date 2025/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

For optimizing a non-convex function in finite dimension, a method is to add Brownian noise to a gradient descent, allowing for transitions between basins of attractions of different minimizers. To adapt this for optimization over a space of probability distributions requires a suitable noise. For this purpose, we introduce here a simple stochastic process where a number of moments of the distribution are following a chosen finite-dimensional diffusion process, generalizing some previous studies where the expectation of the measure is subject to a Brownian noise. The process may explode in finite time, for instance when trying to force the variance of a distribution to behave like a Brownian motion. We show, up to the possible explosion time, well-posedness and propagation of chaos for the system of mean-field interacting particles with common noise approximating the process.

Citations

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Related

• Beyond Propagation of Chaos: A Stochastic Algorithm for Mean Field Optimization
• Local Characteristics of Random Motion
• Non-Equilibrium Dynamics and First-Passage Properties of Stochastic Processes: From Brownian Motion to Active Particles
• Langevin dynamics for the probability of finite state Markov processes
• Mean-Field Stochastic PDEs: Well-posedness and Quantitative Dimension-Free Propagation of Chaos