Analysis of Langevin midpoint methods using an anticipative Girsanov theorem

2025-07-17

AI Summary

This work investigates the theoretical performance of the Langevin midpoint discretization for MCMC sampling, focusing on Radon–Nikodym derivative estimation under non-adapted (i.e., anticipative) stochastic processes—where standard adapted filtering assumptions fail and classical Girsanov theory is inapplicable. To address this, we introduce the first synthesis of anticipative Girsanov theory with Malliavin calculus, establishing a novel framework for analyzing regularity and cross-regularity of the midpoint scheme. Under log-concavity of the target distribution and second-order smoothness of its gradient, we derive a query complexity bound of $widetilde{O}(kappa^{5/4} d^{1/4} / varepsilon^{1/2})$ to achieve $varepsilon^2$-accuracy in KL divergence, significantly improving prior bounds. The key contribution lies in overcoming the adaptivity constraint, providing the first rigorous theoretical analysis tool for SDE discretizations driven by non-Markovian noise.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning Theory
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Humans versus LLMs for data annotation and labeling

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 $πpropto exp(-V)$. 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 $widetilde{O}(frac{κ^{5/4} d^{1/4}}{varepsilon^{1/2}})$ for obtaining a $varepsilon^2$-accurate sample in $mathsf{KL}$ divergence, under log-concavity and strong smoothness assumptions for $ abla^2 V$.

Problem

• Analyzing midpoint discretizations of SDEs for MCMC sampling
• Improving regularity results for sampling methods using Malliavin calculus
• Deriving query complexity bounds for KL divergence accuracy

Innovation

• Uses anticipative Girsanov theorem for SDE analysis
• Applies Malliavin calculus to estimate Radon-Nikodym derivatives
• Improves regularity results for midpoint discretizations