Asymptotic bias of inexact Markov Chain Monte Carlo methods in high dimension

TL;DR

This paper analyzes how the asymptotic bias of inexact MCMC methods like ULA and uHMC depends on dimension and step size, providing bounds and insights for high-dimensional target distributions.

Contribution

It establishes bounds on Wasserstein distances for inexact MCMC methods, revealing how bias depends on dimension, step size, and properties of the target distribution.

Findings

Asymptotic bias depends on key quantities related to the target distribution.

Bias scales similarly with dimension and step size for certain models.

Provides bounds that clarify the impact of inexactness in high-dimensional settings.

Abstract

Inexact Markov Chain Monte Carlo methods rely on Markov chains that do not exactly preserve the target distribution. Examples include the unadjusted Langevin algorithm (ULA) and unadjusted Hamiltonian Monte Carlo (uHMC). This paper establishes bounds on Wasserstein distances between the invariant probability measures of inexact MCMC methods and their target distributions with a focus on understanding the precise dependence of this asymptotic bias on both dimension and discretization step size. Assuming Wasserstein bounds on the convergence to equilibrium of either the exact or the approximate dynamics, we show that for both ULA and uHMC, the asymptotic bias depends on key quantities related to the target distribution or the stationary probability measure of the scheme. As a corollary, we conclude that for models with a limited amount of interactions such as mean-field models, finite range graphical models, and perturbations thereof, the asymptotic bias has a similar dependence on the step size and the dimension as for product measures.