The mean-field Langevin dynamics (MFLD) is a nonlinear generalization of the
Langevin dynamics that incorporates a distribution-dependent drift, and it
naturally arises from the optimization of two-layer neural networks via (noisy)
gradient descent. Recent works have shown that MFLD globally minimizes an
entropy-regularized convex functional in the space of measures. However, all
prior analyses assumed the infinite-particle or continuous-time limit, and
cannot handle stochastic gradient updates. We provide an general framework to
prove a uniform-in-time propagation of chaos for MFLD that takes into account
the errors due to finite-particle approximation, time-discretization, and
stochastic gradient approximation. To demonstrate the wide applicability of
this framework, we establish quantitative convergence rate guarantees to the
regularized global optimal solution under (i) a wide range of learning problems
such as neural network in the mean-field regime and MMD minimization, and (ii)
different gradient estimators including SGD and SVRG. Despite the generality of
our results, we achieve an improved convergence rate in both the SGD and SVRG
settings when specialized to the standard Langevin dynamics.

本文提出了一个新的框架来证明具有有限粒子逼近，时间离散化和随机梯度逼近误差的 MFLD 的混沌传播具有时间一致性，并在学习问题和不同梯度估计器的广泛范围内建立了量化的收敛速率保证，包括 SGD 和 SVRG 算法。

均场 Langevin 动力学的收敛性：时间和空间离散化、随机梯度和方差缩减

Convergence of mean-field Langevin dynamics: Time and space  discretization, stochastic gradient, and variance reduction

We propose a new randomized coordinate descent method for a convex
optimization template with broad applications. Our analysis relies on a novel
combination of four ideas applied to the primal-dual gap function: smoothing,
acceleration, homotopy, and coordinate descent with non-uniform sampling. As a
result, our method features the first convergence rate guarantees among the
coordinate descent methods, that are the best-known under a variety of common
structure assumptions on the template. We provide numerical evidence to support
the theoretical results with a comparison to state-of-the-art algorithms.

提出一种新的随机坐标下降方法，应用广泛，采用四种新颖的思想，包括平滑、加速、同伦、坐标下降和非均匀采样，具有迭代收敛保证，数值实验表明其优于现有算法。