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.

翻译：均值场朗之万动力学（MFLD）最小化定义在概率分布空间上的熵正则化非线性凸泛函。由于MFLD与均值场双层神经网络的带噪梯度下降之间的联系，它受到了广泛关注。与标准朗之万动力学不同，目标泛函的非线性引入了粒子间的相互作用，这要求在有限粒子设置下需要多个粒子来逼近动力学过程。近期研究（Chen et al., 2022; Suzuki et al., 2023b）证明了MFLD具有时间均匀的混沌传播特性，表明随着粒子数增加，粒子系统与其均值场极限之间的差距随时间均匀缩小。在本工作中，我们改进了其粒子逼近误差中对数索博列夫不等式（LSI）常数的依赖关系，该常数可能随正则化系数指数级恶化。具体而言，我们通过利用风险最小化问题中的结构，建立了关于目标间隙的、与LSI常数无关的粒子逼近误差。作为应用，我们展示了MFLD收敛性的改进、均值场平稳分布的采样保证，以及在粒子复杂度意义上时间均匀的Wasserstein混沌传播。