In this work, we consider 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, in our chain the drift and diffusion coefficient can be inexact in order to cover wide range of applications as Stochastic Gradient Langevin Dynamics, sampling, Stochastic Gradient Descent or Stochastic Gradient Boosting. We prove the bound in $W_{2}$-distance between the laws of our Ito chain and corresponding differential equation. These results improve or cover most of the known estimates. And for some particular cases, our analysis is the first.

翻译：本文考虑一类相当通用的马尔可夫链——伊藤链，其形式类似于某类随机微分方程的欧拉-丸山离散化。我们研究的链是一个统一的理论分析框架，采用几乎任意各向同性且依赖状态的噪声，而非大多数相关文献中使用的正态且独立于状态的噪声。此外，该链中的漂移与扩散系数可不精确，以覆盖广泛的应用场景，如随机梯度Langevin动力学、采样、随机梯度下降或随机梯度提升。我们证明了伊藤链与对应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大多数已知估计，且在若干特例中，我们的分析尚属首次。