A probability approximation framework: Markov process approach

TL;DR

This paper introduces a probability approximation framework using Markov processes, applying it to analyze errors in stochastic gradient descent, SDE discretization, and normal approximation, with bounds in Wasserstein-1 distance.

Contribution

It develops a novel Markov process-based approach for probability approximation, extending classical principles to new stochastic process applications.

Findings

Error bounds for SGD approximated by SDEs in Wasserstein-1 distance

Error bounds for Euler-Maruyama discretization of Ornstein-Uhlenbeck processes

Multivariate normal approximation error bounds in Wasserstein-1 distance

Abstract

We view the classical Lindeberg principle in a Markov process setting to establish a probability approximation framework by the associated It\^{o}'s formula and Markov operator. As applications, we study the error bounds of the following three approximations: approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by multiplicative Brownian motion, Euler-Maruyama (EM) discretization for multi-dimensional Ornstein-Uhlenbeck stable process, and multivariate normal approximation. All these error bounds are in Wasserstein-1 distance.