标题： Langevin中点方法的分析使用前瞻性Girsanov定理 显示英文标题

标题： Analysis of Langevin midpoint methods using an anticipative Girsanov theorem

摘要： 我们引入了一种新方法，用于分析随机微分方程（SDE）的中点离散化，这些离散化在从目标测度$\pi \propto \exp(-V)$进行采样的马尔可夫链蒙特卡洛（MCMC）方法中被频繁使用。 借鉴马里亚文微积分的技术，我们计算了在$L^2([0, T); \mathbb{R}^d)$上可能预知布朗运动的过程的Radon-Nikodym导数的估计值，即它们可能不同时适应滤波。 将这些应用到各种流行的中点离散化方法上，我们能够改进文献中关于采样方法的正则性和交叉正则性结果。 在对$\nabla^2 V$做出对数凹性和强光滑性假设的情况下，我们还得到了在$\mathsf{KL}$散度下获得$\varepsilon^2$精度样本的查询复杂度界为$\widetilde{O}(\frac{\kappa^{5/4} d^{1/4}}{\varepsilon^{1/2}})$。 显示英文摘要

摘要： 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 $\nabla^2 V$.