Optimal Scaling of Langevin Proposals with Generalized Acceptance Rules

2026-09-15

AI Summary

本文研究了使用广义接受规则的Langevin提案的最佳缩放问题，针对高维目标提出了不同接受规则下的最优接受概率。

Abstract

Langevin-based Markov chain Monte Carlo (MCMC) algorithms use gradient information to improve sampling, particularly in high dimensions. Classical optimal scaling theory for these algorithms has largely focused on the Metropolis-Hastings (MH) acceptance rule. However, there has been a recent surge in acceptance rules beyond MH for applications spanning differential privacy, stochastic MCMC, diffusion models, and molecular dynamics. We develop optimal scaling results for Langevin proposals employed with generalized acceptance rules belonging to a suitable class. For high-dimensional targets, we recover the usual $O(d^{-1/3})$ scaling, while different acceptance rules lead to different optimal acceptance probabilities.

Problem

• Langevin Proposals
• Generalized Acceptance Rules
• Optimal Scaling
• High-dimensional Targets
• Acceptance Probability

Innovation

• Optimal Scaling
• Generalized Acceptance Rules
• Langevin Proposals
• High-dimensional Targets