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

Pierre Germain, Pierre Monmarché

TL;DR

This work introduces stochastic moment dynamics (SMD), a finite-dimensional diffusion on a selected set of distribution moments, to perturb Wasserstein gradient flows and promote basin-hopping between minimizers in nonconvex probability landscapes. Built within the conditional McKean–Vlasov framework with a common noise, the approach yields a particle-system representation whose moments follow an SDE, and the authors prove well-posedness up to a potential explosion time, alongside a propagation-of-chaos result in the mean-field limit. They develop a robust two-pronged strategy to handle explosions: a Lyapunov-based non-explosion criterion and a globally regularized SMD that preserves exploration while guaranteeing non-explosive evolution. The paper further provides explicit moment-control examples, numerical illustrations of explosive vs non-explosive behavior, and demonstrates SMD as an effective exploration mechanism in a double-well MV setting, with practical guidance on choosing observables and regularization. Overall, the work advances optimization over probability measures by coupling moment-driven stochastic perturbations with Wasserstein gradient dynamics, enabling controlled exploration and potential transitions between multiple minimizers in complex landscapes.

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.

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

TL;DR

This work introduces stochastic moment dynamics (SMD), a finite-dimensional diffusion on a selected set of distribution moments, to perturb Wasserstein gradient flows and promote basin-hopping between minimizers in nonconvex probability landscapes. Built within the conditional McKean–Vlasov framework with a common noise, the approach yields a particle-system representation whose moments follow an SDE, and the authors prove well-posedness up to a potential explosion time, alongside a propagation-of-chaos result in the mean-field limit. They develop a robust two-pronged strategy to handle explosions: a Lyapunov-based non-explosion criterion and a globally regularized SMD that preserves exploration while guaranteeing non-explosive evolution. The paper further provides explicit moment-control examples, numerical illustrations of explosive vs non-explosive behavior, and demonstrates SMD as an effective exploration mechanism in a double-well MV setting, with practical guidance on choosing observables and regularization. Overall, the work advances optimization over probability measures by coupling moment-driven stochastic perturbations with Wasserstein gradient dynamics, enabling controlled exploration and potential transitions between multiple minimizers in complex landscapes.

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.
Paper Structure (28 sections, 20 theorems, 159 equations, 4 figures)

This paper contains 28 sections, 20 theorems, 159 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Derivation of the process
  4. Overview of the work
  5. General notations
  6. On conditional McKean-Vlasov equations
  7. Probabilistic framewok
  8. Study of the Lipschitz case
  9. Definition of the stochastic moments dynamics
  10. Proofs of Section \ref{['sec:def-process']}
  11. Proof of Proposition \ref{['prop:stochasticmomentswelldefined']}
  12. Proof of Proposition \ref{['prop:PoC']}
  13. Controlling Explosion: Lyapunov condition and Regularized Dynamics
  14. Non-explosion Criterion
  15. Regularized process
  16. ...and 13 more sections

Key Result

Proposition 2

(Existence of a regular flow of conditional laws) Let $(X_t)_{0\leq t\leq T}$ be a $\mathbb{F}$ adapted process, with continuous paths in $\mathbb R^d$, such that it exists $\alpha\geq1$ such that $\mathbb{E}(\sup_{0\leq t\leq T}|X_t|^\alpha)< \infty$. Then it exists a random process $(\mu_t)_{0\leq

Figures (4)

  • Figure 1: Explosive/Non explosive behavior of the SMD depending on the parameter $\delta$ (top: $\delta=1$; bottom: $\delta=3$).
  • Figure 2: Explosive/Non explosive behavior of the regularized SMD with $f(x)=x^2$ depending on the parameter $\eta$ (top: $\eta=0$; bottom: $\eta=1$)
  • Figure 3: Explosive/Non explosive behavior of the regularized SMD with $f(x)=\tanh(x)$ depending on the parameter $\eta$ (top: $\eta=0$; bottom: $\eta=0.5$)
  • Figure 4: Evolution of the "noisy" McKean Vlasov dynamic depending on the intensity of the noise $\gamma$

Theorems & Definitions (42)

  • Remark 1
  • Definition 1
  • Proposition 2
  • Remark 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • ...and 32 more