Analysis of Langevin midpoint methods using an anticipative Girsanov theorem

TL;DR

This paper introduces a novel analytical approach for midpoint discretizations of SDEs used in MCMC, leveraging anticipative Girsanov theorem and Malliavin calculus to improve regularity results and establish a new query complexity bound.

Contribution

It develops an anticipative Girsanov theorem-based method to analyze midpoint SDE discretizations, enhancing understanding of sampling algorithms and providing a new complexity bound.

Findings

Improved regularity and cross-regularity results for sampling methods.

Derived a query complexity bound of O(/4 d^{1/4}/\u03b5^{1/2}) for -accurate sampling.

Applicable under log-concavity and smoothness assumptions.

Abstract

We introduce a new method for analyzing midpoint discretizations of stochastic differential equations (SDEs), which are frequently used in Markov chain Monte Carlo (MCMC) methods for sampling from a target measure $\pi \propto \exp(-V)$. Borrowing techniques from Malliavin calculus, we compute estimates for the Radon-Nikodym derivative for processes on $L^2([0, T); \mathbb{R}^d)$ which may anticipate the Brownian motion, in the sense that they may not be adapted to the filtration at the same time. Applying these to various popular midpoint discretizations, we are able to improve the regularity and cross-regularity results in the literature on sampling methods. We also obtain a query complexity bound of $\widetilde{O}(\frac{\kappa^{5/4} d^{1/4}}{\varepsilon^{1/2}})$ for obtaining a $\varepsilon^2$-accurate sample in $\mathsf{KL}$ divergence, under log-concavity and strong smoothness assumptions for $\nabla^2 V$.