Mean-field Langevin dynamics (MFLD) minimizes an entropy-regularized
nonlinear convex functional defined over the space of probability
distributions. MFLD has gained attention due to its connection with noisy
gradient descent for mean-field two-layer neural networks. Unlike standard
Langevin dynamics, the nonlinearity of the objective functional induces
particle interactions, necessitating multiple particles to approximate the
dynamics in a finite-particle setting. Recent works (Chen et al., 2022; Suzuki
et al., 2023b) have demonstrated the uniform-in-time propagation of chaos for
MFLD, showing that the gap between the particle system and its mean-field limit
uniformly shrinks over time as the number of particles increases. In this work,
we improve the dependence on logarithmic Sobolev inequality (LSI) constants in
their particle approximation errors, which can exponentially deteriorate with
the regularization coefficient. Specifically, we establish an LSI-constant-free
particle approximation error concerning the objective gap by leveraging the
problem structure in risk minimization. As the application, we demonstrate
improved convergence of MFLD, sampling guarantee for the mean-field stationary
distribution, and uniform-in-time Wasserstein propagation of chaos in terms of
particle complexity.

通过改进粒子近似误差的对数 Sobolev 不等式常数依赖性，我们展示了 MFLD 的收敛性提高、对均场稳态分布的采样保证以及粒子复杂度的统一随时间的 Wasserstein 传播。