Uniform-in-time propagation of chaos for mean field Langevin dynamics

TL;DR

This work proves that when the mean-field energy $F$ is convex in the functional sense and the system satisfies a uniform logarithmic-Sobolev inequality, the mean-field Langevin dynamics $X_t$ converges exponentially to its unique invariant measure $m_\infty$ in $L^p(m_\infty)$ for all $p$, while the corresponding $N$-particle system remains close to $m_\infty^{\otimes N}$ at a uniform exponential rate in both $W_2$ and relative entropy, up to an $O(1/N)$ bias per particle. By establishing $L^p$-hypercontractivity and reverse hypercontractivity for negative $p$, the authors derive comprehensive $L^p$ convergence across all $p$, including $p<1$, and a detailed entropy-based analysis of both the mean-field and particle systems. They then leverage these results to obtain uniform-in-time propagation of chaos, giving time-uniform bounds on Wasserstein and entropy distances between the finite-particle law and the tensorized mean-field law, together with uniform concentration of the mean-field flow. The framework accommodates neural-network-inspired MFL dynamics and regularized Coulomb-type interactions, offering practical guarantees for long-time finite-particle approximations in high-dimensional settings. Overall, the paper advances the theoretical understanding of time-uniform convergence and finite-particle approximations for gradient-type mean-field Langevin systems with functional convexity assumptions, with potential implications for training dynamics in neural networks and related stochastic interacting particle systems.

Abstract

We study the mean field Langevin dynamics and the associated particle system. By assuming the functional convexity of the energy, we obtain the $L^p$-convergence of the marginal distributions towards the unique invariant measure for the mean field dynamics. Furthermore, we prove the uniform-in-time propagation of chaos in both the $L^2$-Wasserstein metric and relative entropy.

Figures (3)

• Figure 1: Data samples $\{z_k, y_k\}_{k=1}^K$ (schematic).
• Figure 2: Individual (shadowed) and $10$-averaged (bold) losses versus time steps.
• Figure 3: Average losses of last $500$ steps for individual trainings (shadowed) and its $10$-average (bold).

Theorems & Definitions (79)

• Remark 2.1: Well-definedness of $\hat{m}$
• Remark 2.2: Functional inequalities
• Remark 2.3
• Theorem 2.1: Energy dissipation of MFL
• Remark 2.4
• Proposition 2.2: $L^2$-convergence
• Proposition 2.3: Hypercontractivity
• Remark 2.5: Optimality of exponent's growth
• Theorem 2.4: $L^p$-convergence of MFL
• Remark 2.6: Necessity of $L^{1+}$-initial condition