提出镜像版平均场朗之万动力学，解决约束域优化问题。
Mirror Mean-Field Langevin Dynamics
- 将平均场朗之万动力学扩展至镜像框架，支持凸集约束
- 连续版本线性收敛，离散版本保持时不变混沌传播
- 适用于无限宽两层神经网络等受限优化场景
平均场朗之万动力学（MFLD）在ℝᵈ的Wasserstein空间上最小化一个熵正则化的非线性凸泛函，近年来被用作无限宽两层神经网络梯度下降动力学的模型。然而，许多实际问题涉及约束域，现有平均场算法因全局扩散项无法求解。本文提出镜像平均场朗之万动力学（MMFLD），将MFLD拓展至镜像朗之万框架以处理ℝᵈ中凸子集上的概率测度优化。通过统一的对数索博列夫不等式，获得了连续MMFLD的线性收敛保证，并证明了其时间与粒子离散化版本具有统一的时不变混沌传播性质。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。