The mean-field Langevin dynamics (MFLD) minimizes an entropy-regularized nonlinear convex functional on the Wasserstein space over $\mathbb{R}^d$, and has gained attention recently as a model for the gradient descent dynamics of interacting particle systems such as infinite-width two-layer neural networks. However, many problems of interest have constrained domains, which are not solved by existing mean-field algorithms due to the global diffusion term. We study the optimization of probability measures constrained to a convex subset of $\mathbb{R}^d$ by proposing the \emph{mirror mean-field Langevin dynamics} (MMFLD), an extension of MFLD to the mirror Langevin framework. We obtain linear convergence guarantees for the continuous MMFLD via a uniform log-Sobolev inequality, and uniform-in-time propagation of chaos results for its time- and particle-discretized counterpart.

翻译：均场朗之万动力学（MFLD）在 $\mathbb{R}^d$ 上的Wasserstein空间中最小化熵正则化的非线性凸泛函，近期作为交互粒子系统（如无限宽双层神经网络）梯度下降动力学的模型而受到关注。然而，许多感兴趣的问题具有约束域，由于全局扩散项，现有均场算法无法解决此类问题。我们通过提出\emph{镜像均场朗之万动力学}（MMFLD），即MFLD扩展到镜像朗之万框架，研究约束在 $\mathbb{R}^d$ 凸子集上的概率测度优化问题。利用一致对数Sobolev不等式，我们获得了连续MMFLD的线性收敛保证，并证明了其时间和粒子离散化版本的一致时间传播混沌结果。