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 $\nabla^2 V$.

翻译：我们提出了一种分析随机微分方程(SDE)中点离散化的新方法，该方法在马尔可夫链蒙特卡洛(MCMC)方法中常用于从目标测度$\pi \propto \exp(-V)$中采样。借鉴Malliavin演算技术，我们计算了$L^2([0, T); \mathbb{R}^d)$上过程的Radon-Nikodym导数估计，这些过程可能预期布朗运动，即它们可能在不同时间适应于滤波。将此应用于各种流行的中点离散化方案，我们改进了文献中采样方法的正则性与交叉正则性结果。在$\nabla^2 V$满足对数凹性与强光滑性假设下，我们还获得了$\widetilde{O}(\frac{\kappa^{5/4} d^{1/4}}{\varepsilon^{1/2}})$的查询复杂度上界，用于获得$\mathsf{KL}$散度意义上$\varepsilon^2$精度的样本。