This work considers a rather general and broad class of Markov chains, Ito
chains that look like Euler-Maryama discretization of some Stochastic
Differential Equation. The chain we study is a unified framework for
theoretical analysis. It comes with almost arbitrary isotropic and
state-dependent noise instead of normal and state-independent one, as in most
related papers. Moreover, our chain's drift and diffusion coefficient can be
inexact to cover a wide range of applications such as Stochastic Gradient
Langevin Dynamics, sampling, Stochastic Gradient Descent, or Stochastic
Gradient Boosting. We prove an upper bound for $W_{2}$-distance between laws of
the Ito chain and the corresponding Stochastic Differential Equation. These
results improve or cover most of the known estimates. Moreover, for some
particular cases, our analysis is the first.

这项工作考虑了一类非常普遍而广泛的 Markov 链，即类似于某些随机微分方程的 Euler-Maryama 离散化的 Ito 链。我们研究的链是一个统一的理论分析框架，与大多数相关论文中的正常和状态独立的噪声不同，它具有几乎任意的各向同性和状态相关的噪声。此外，我们的链的漂移和扩散系数可以是不精确的，以涵盖各种应用，如随机梯度 Langevin 动力学、采样、随机梯度下降或随机梯度提升。我们证明了 Ito 链和相应的随机微分方程之间 $W_2$ 距离的上界。这些结果改进或涵盖了大多数已知的估计。此外，对于某些特殊情况，我们的分析是首次的。