Ito Diffusion Approximation of Universal Ito Chains for Sampling, Optimization and Boosting

TL;DR

This paper introduces a broad class of Ito chains that generalize discretizations of SDEs, providing theoretical bounds on their approximation accuracy for various stochastic algorithms.

Contribution

It develops a unified framework for analyzing Ito chains with inexact coefficients and arbitrary noise, extending existing results and covering new cases in stochastic sampling and optimization.

Findings

Proves bounds in $W_{2}$-distance for Ito chains approximating SDEs.

Extends analysis to inexact drift and diffusion coefficients.

Provides first theoretical results for some specific cases.

Abstract

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.