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.

翻译：本文考虑一类相当广泛且通用的马尔可夫链——伊藤链（Ito链），其形式类似于某类随机微分方程的欧拉-丸山离散化。我们研究的链是理论分析的统一框架，其噪声具有几乎任意的各向同性和状态相关性，而非现有相关文献中常见的正态独立噪声。此外，链的漂移项和扩散系数允许存在不精确性，以覆盖诸如随机梯度朗之万动力学、采样、随机梯度下降或随机梯度提升等广泛应用。我们证明了伊藤链与对应随机微分方程的概率分布之间的$W_{2}$距离上界。这些结果改进或涵盖了现有绝大多数估计，且在某些特定情形下为首次理论分析。